Essentials Crystallography By EDWARD HENRY KRAUS, Ph.D. \\ Junior Professor of Mineralogy in the University of Michigan THE UNIVERSITY OF WITH 427 FIGURES GEORGE WAHR, Publisher Ann Arbor, Mich. 1906 COPYRIGHT, 1906 by EDWARD H. KRAUS EIL!8 PUBLISHING COMPANY, PRINTEI BATTIE CRfctK. MICH. PREFACE. This text is intended for beginners and aims to present the essential features of geometrical crystallography from a standpoint icm ERRATA. Page 26, last line, for 78 31 ' 44" read 70 31' 44". , ra:ma:aOO~\ . ra:ma:OOa1 Page 35, line 8, for + I I read + I -j- Page 37, line 6 from the bottom, for effected read affected. 4O2 462 Page 42, last line, for A = -f - / read A. = + - /. 2 4 Page 45, line 3, for planes read plane. Page 47, line 14, for triangle read triangles. roi--xKx-i.v,.ft.i. r --v*~ ngures nave oeen muwn have been taken from various sources. I am indebted to Mr. W. F. Hunt, Instructor in Mineralogy, also to Messrs. I. D. Scott and C. W. Cook, Assistants in Mineralogy in the University of Michigan, for aid in the reading of the proof. EDWARD H. KRAUS. Mineralogical Laboratory, University of Michigan, Ann Arbor, Mich., June, 1906. 196447 EIL!8 PUBLISHING COMPANY, PRINTERS BATTIE CRttK. MICH. PREFACE. This text is intended for beginners and aims to present the essential features of geometrical crystallography from a standpoint which combines the ideas of symmetry with those of holohedrism, hemihedrism, etc. All forms possible in the thirty-two classes of symmetry are discussed even though representatives of several of the classes have not as yet been observed among minerals or artificial salts. In each system, however, the important classes are indicated so that the book may readily be used for abridged courses. No attempt has been made to discuss the measurement and projection of crystals inasmuch as they involve a somewhat compre- hensive knowledge of forms and, hence, cannot be treated adequately in an introductory course. For a similar reason very little reference has been made to the various structural theories of crystals. How- ever, the bibliography of the more important works, which have been published in English and German during the past thirty years, will prove useful to those who may desire to pursue the study of crystal- lography further than the scope of this text permits. Free use has been made of the various standard texts on crystal- lography, but especial acknowledgments are due von Groth, Linck, Bruhns, Tschermak, and E. S. Dana. A very large majority of the figures have been drawn especially for this text, the others, however, have been taken from various sources. I am indebted to Mr. W. F. Hunt, Instructor in Mineralogy, also to Messrs. I. D. Scott and C. W. Cook, Assistants in Mineralogy in the University of Michigan, for aid in the reading of the proof. EDWARD H. KRAUS. Mineralogical Laboratory, University of Michigan, Ann Arbor, Mich., June, 1906. 196447 [in] TABLE OF CONTENTS. INTRODUCTION. PAGE Crystals and Crystalline Structure 1 Amorphous Structure 1 Some Properties of Crystallized and Amorphous Substances 2 Crystallography 2 Constancy of Interfacial Angles 2 Crystal Habit 4 Crystallographic Axes 4 Crystal Systems 5 Parameters and Parametral Ratio 5 Fundamental Forms 6 Combinations 7 Axial Ratio 7 Rationality of Coefficients 9 Symbols 10 Symbols of Naumann and Dana 11 Miller's System 12 Elements of Symmetry 12 Planes of Symmetry 12 Axes of Symmetry 13 Center of Symmetry 14 Angular Position of Faces 14 Classes of Symmetry 14 Holohedrism 15 Hemihedrism 15 Tetartohedrism , 15 Ogdohedrism 15 Correlated Forms 15 Hemimorphism 16 CUBIC SYSTEM. Crystallographic Axes 17 Classes of Symmetry 17 Hexoctahedral Class (1) 17 Hextetrahedral Class (2) 26 Dyakisdodecahedral Class (3) 32 Pentagonal Icositetrahedral Class (4) 36 Tetrahedral Pentagonal Dodecahedral Class (5) 38 HEXAGONAL SYSTEM. Crystallographic Axes 43 Classes of Symmetry 44 [V] VI TABLE OF CONTENTS. Dihexagonal Bipyramidal Class (6) 44 Dihexagonal Pyramidal Class (7) 53 Hexagonal Hemihedrisms 55 Ditrigonal Bipyramidal Class (8) 56 Ditrigonal Scalenohedral Class (9) 60 Hexagonal Bipyramidal Class (10) 6"> Hexagonal Trapezohedral Class (11) 68 Ditrigonal Pyramidal Class (12) 70 Hexagonal Pyramidal Class (13) 74 Hexagonal Tetartohedrisms 76 Trigonal Bipyramidal Class (14) ... 76 Trigonal Trapezohedral Class (15) 81 Trigonal Rhombohedral Class (16) 85 Trigonal Pyramidal Class (17) 89 TETRAGONAL SYSTEM. Crystallographic Axes 92 Classes of Symmetry 92 Ditetragonal Bipyramidal Class (18) 93 Ditetragonal Pyramidal Class (19) 99 Tetragonal Hemihedrisms 101 Tetragonal Scalenohedral Class (20) 102 Tetragonal Bipyramidal Class (21) 105 Tetragonal Trapezohedral Class (22) 108 Tetragonal Pyramidal Class (23) 110 Tetragonal Bisphenoidal Class (24) Ill ORTHORHOMBIC SYSTEM. Crystallographic Axes 115 Classes of Symmetry 115 Orthorhombic Bipyramidal Class (25) 115 Orthorhombic Pyramidal Class (26) 121 Orthorhombic Bisphenoidal Class (27) 123 MONOCUNIC SYSTEM. Crystallographic Axes 126 Classes of Symmetry 126 Prismatic Class (28) . . 126 Domatic Class (29) 132 Sphenoidal Class (30) 135 TRICLINIC SYSTEM. Crystallographic Axes 138 Classes of Symmetry 138 Pinacoidal Class (31) 138 Asymmetric Class (32) 142 TABLE OF CONTENTS. VII COMPOUND CRYSTALS. Parallel Grouping 144 Twin Crystals 144 Common Twinning Laws Cubic System 145 Hexagonal System 146 Tetragonal System 147 Orthorhombic System 147 Monoclinic System 147 Triclinic System 148 Repeated Twinning 148 Mimicry 149 TABULAR CLASSIFICATION OF THE THIRTY-TWO CLASSES OF CRYSTALS. Cubic System 150 Hexagonal System 151 Tetragonal System 153 Orthorhombic System 154 Monoclinic System 155 Triclinic System K6 Index .... . 157 BIBLIOGRAPHY. In the following list will be found some of the more recent works which treat crystallography more or less in detail. Bauer, M., Lehrbuch der Mineralogie, 2d Ed., Stuttgart, 1904. Bauerman, H., Text Book of Systematic Mineralogy, London, 1881. Baumhauer, H., Das Reich der Krystalle, Freiburg in Br. , 1884. Die neuere Entwickelung der Krystallographie, Leipzig, 1905. Brezina, A., Methodik der Krystallbestimmung, Vienna, 1884. Bruhns, W., Elemente der Krystallographie, Liepzig and Vienna, 1902. Brush-Penfield, Manual of Determinative Mineralogy, New York, 1898. Dana, E. S., Text Book of Mineralogy, New Edition, New York, 1898. Fock, A., Introduction to Chemical Crystallography, translated by W. J. Pope, Oxford, 1895. Goldschmidt, V., Krystallographische Winkeltabellen, Berlin, 1897. Index der Krystallformen der Mineralien, 3 vols., Berlin, 1886-1891. Anwendung der Linearprojection zum Berechnen der Krystalle, Berlin, 1887. Groth, P., Physikalische Krystallographie, 4th Ed., Leipzig, 1905. Einleitung in die Chemische Krystallographie, Leip- zig, 1904. English Translation by H. Marshall, New York, 1906. Heinrich, F. , Lehrbuch der Krystallberechnung, Stuttgart, 1886. Klein, C., Einleitung in die Krystallberechnung, Stuttgart, 1876. Klockmann, F., Lehrbuch der Mineralogie, 3d Ed., Stuttgart, 1903. Lewis, W. J., Crystallography, Cambridge, 1899. Liebisch, T. , Geometrische Krystallographie, Leipzig, 1881. Physikalische Krystallographie, Leipzig, 1891. Grundriss der Physikalischen Krystallographie, Leip- zig, 1896. Linck, G., Grundriss der Krystallographie, Jena, 1896. [IX] X BIBLIOGRAPHY. Miers, H. A., Mineralogy, London and New York, 1902. Moses and Parsons, Mineralogy, Crystallography and Blowpipe Analysis, Revised Edition, New York, 1904. Moses, A. J., Characters of Crystals, New York, 1899. Naumann-Zirkel, Elemente der Mineralogie, 1 4th Ed., Leipzig, 1901. Neis, A., Allgemeine Krystallbeschriebung auf Grund einer verein- fachten Methode der Krystallzeichnung, Stutt- gart, 1895. Rose, G., Elemente der Krystallographie, 3d Ed., Revised by A. Sadebeck, Berlin, 1873. Sadebeck, A., Angewandte Krystallographie, Berlin, 1876. Schonflies, A., Krystallsysteme and Krystallstructur, Leipzig, 1891. Sohncke, L., Entwickelung einer Theorie der Krystallstructur, Leip- zig, 1879. Story-Maskelyne, N., Crystallography, Oxford, 1895. Tschermak, G., Lehrbuch der Mineralogie, 6th Ed., Vienna, 1905. Viola, C., Grundzuge der Kristallographie, Leipzig, 1904. Voigt, W., Die Fundamentalen Physikalischen Eigenschaften del Krystalle, Leipzig, 1898. Von Kobell, F. , Lehrbuch der Mineralogie, 6th Ed., Leipzig, 1899. Websky, M., Anwendung der Linearprojection zum Berechnen del Krystalle, Berlin, 1887. Williams, G. H., Elements of Crystallography, 3d Ed., New York 1890. Wiilfing, E. A. , Tabellarische Uebersicht der Einfachen Formen del 32 Krystallographischen Symmetriegruppen, Stuttgart, 1895. Also, S. L. Penfield, Stereographic Projection and its Possibilities from a Graphic Standpoint. American Journa of Science, 1901, XI, 1-24 and 115-144. On the Solution of Problems in Crystallography b) Means of Graphical Methods, Ibid., 1902, XIV, 249-284. On Crystal Drawing, Ibid., 1905, XIX, 39-75. UNIVERSITY * Li FOR N^ INTRODUCTION.^ Crystals and Crystalline Structure, Nearly all homogeneous substances possessing a definite chemical composition, when solidify- ing from either a solution, state of fusion or vapor, attempt to crys- tallize, that is, to assume certain characteristic forms. If the process of solidification is slow enough and uninterrupted, regular forms, bounded by plane surfaces, usually result. These regular polyhedral forms are termed crystals. If, however, the solidification is so rapid that the substance cannot assume well defined forms, bounded by plane surfaces, the solid mass, which results, is said to be crystalline. Crystalline substances consist of aggregations of crystals, which have been hindered in their development. Their outline and orientation are, hence, irrregular. Figure I shows a well developed crystal of \ Fig- 1. Fig. 2. calcite (CaCO 3 ), while figure 2 1 ) represents a cross-section through a crystalline aggregate of the same substance. Here, no definite outline is to be recognized on the component parts of the mass. Amorphous Structure. Those substances, however, which do not attempt to crystallize, when solidifying, are termed amorphous. i) After Weinschenk. [1] 2 INTRODUCTION. They are without form. Opal (SiO 2 . x H 2 O), for example, never occurs in welj. .defined crystals or in crystalline masses. ': .Some -"Pro'pBrties of Crystallized and Amorphous Substances. There; are; 'tnany interesting features to be noted in the study of crystallized and amorphous substances. The attempt at crystalliza- tion and the assuming of regular polyhedral forms, which is charac- teristic of bodies belonging to the first class, are only some of the expressions of the nature of such substances. The various physical properties, hardness, elasticity, cohesion, transmission of heat and light, usually vary in such crystallized substances with the direction according to fixed laws. This is not the case with amorphous sub- stances. Hence, we may define a crystal as a solid body, which is bounded by natural -plane surfaces and whose form is dependent upon its physical and chemical properties. Crystallography. This science treats of the various properties of crystals and crystallized bodies. It may be subdivided as follows: 1. Geometrical Crystallography. 2. Physical Crystallography, j. Chemical Crystallography. Geometrical crystallography, as the term implies, describes the various forms occurring upon crystals. The relationships existing between the crystal form and the physical and chemical properties of crystals are the subjects of discussion of the second and third sub- divisions of this science, respectively. Only the essentials of the first subdivision geometrical crystallography will be treated in this text. Constancy of Interfacial Angles. As indicated on page i, crystals may, in general, result from solidification from a solution, state of fusion, or vapor. Let us suppose that some ammonium alum, (NH 4 ) 2 A1 2 (SO 4 ) 4 .24H 2 O, has been dissolved in water and the solution allowed to evaporate slowly. As the alum begins to crystal- lize, it will be noticed that the crystals are, for the most part, bounded by eight plane surfaces. If these surfaces are all of the same size, that is, have been equally developed, the crystals will possess an outline as represented by figure 3. Such a form is termed an octahedron. The octahedron is bounded by eight equilateral triangles. The angles between any two adjoining surfaces or faces, as they are often INTRODUCTION. 3 called, is the same, namely, 109 28^'. On most of the crystals, Fig. 3. Fig 4. Fig. 5. Fig. 6. Fig. 7. Fig. 8. however, it will be seen that the various faces have been developed unequally, giving rise to the forms illustrated by figures 4 and 5. Similar cross-sections through these forms are shown in figures 6, 7, and 8, and it is readily seen that, although the size of the faces and, hence, the resulting shapes have been materially changed, the angle between the adjoining faces has re- F - 9 mained the same, namely, 109 28^'. Such forms of the octahedron are said to be misshapen or distorted. Dist or t ion is quite common on all crystals regardless of their chemical composition. It was the Danish physician and natural Fig. 11. Fig. 12. INTRODUCTION. scientist, Nicolas Steno, who in 1669 first showed that the angles between similar faces on crystals of quartz remain constant regard- less of their development. Figures 9 and 10 represent two crystals of quartz with similar cross-sections (figures 11 and 12). Further observations, however, showed that this applies not only to quartz but to all crystallized substances and, hence, we may state the law as follows: Measured at the same temperature, similar angles on crystals of the same substance remain constant regardless of the size or shape of the crystal. Crystal Habit. The various shapes of crystals, resulting from the unequal development of their faces, are oftentimes called their habits. Figures 3, 4, and 5 show some of the habits assumed by alum crystals. In figure 3, the eight faces are about equally developed and this may be termed the octahedral habit. The tab- ular habit, figure 4, is due to the predominance of two parallel faces. Figure 5 shows four parallel faces predominating, and the resulting form is the prismatic habit. Crystallographic Axes. Inasmuch as the crystal form of any substance is dependent upon its physical and chemical properties, page 2, it necessarily follows that an almost infinite variety of forms is possible. In order, however, to study these forms and define the position of the faces occurring on them advantageously, straight lines of definite lengths are assumed to pass through the ideal center of each crystal. These lines are the crystallographic axes. Their intersection forms the axial cross. Figure 13 shows the octahedron referred to its three crystal axes. In this case the axes are of equal lengths and termed a axes. The extremities of the axes are differenti- ated by the use of the plus and minus signs, as shown in figure 13. If the axes are of unequal lengths, the one extending from front to rear is termed the a axis, the one from right to left the b y Fig. 13. +6.' -c Fig. 14. INTRODUCTION. 5 while the vertical axis is called the c axis. This is illustrated by figure 14. The axes are always referred to in the following order, viz: a, b, c. Crystal Systems. Although a great variety of crystal forms is possible, it has been shown in many ways that all forms may be classified into six large groups, called crystal systems. In the group- ing of crystal forms into systems, we are aided by the crystallographic axes. The systems may be differentiated by means of the axes as follows : 1. Cubic System. Three axes, all of equal lengths, intersect at right angles. The axes are designated by the letters, #, a, a. 2. Hexagonal System. Four axes, three of which are equal and in a horizontal plane intersecting at angles of 60. These three axes are often termed the lateral or secondary axes, and are desig- nated by a, a, a. Perpendicular to the plane of the lateral axes is the vertical axis, which may be longer or shorter than the a axes. This fourth axis is called the ^principal or c axis. 3. Tetragonal System. Three axes, two of which are equal, horizontal, and perpendicular to each other. The vertical, c, axis is at right angles to and either longer or shorter than the horizontal or lateral, a, axes. 4. Orthorhombic System, Three axes of unequal lengths intersect at right angles. These axes are designated by a, b, c, as shown in figure 14. 5. Monoclinic System. Three axes, all unequal, two of which (#', c) intersect at an oblique angle, the third axis (b) being perpendicular to these two. 6. Triclinic System, Three axes (a, b, c) are all unequal and intersect at oblique angles. Parameters and Parametral Ratio, In order to determine the position of a face on a crystal, it must be referred to the crystal- lographic axes. Figure I 5 shows an axial cross of the orthorhombic system. The axes, a, b, c, are, therefore, unequal and perpendic- ular to each other. The plane ABC cuts the three axes at the points A, B, and C, hence, at the distance OA a, OB=b, OC=c, from the center, O. These distances, OA, OB, and OC, are known as 6 INTRODUCTION. the parameters and the ratio, OA : OB : OC, as the parametral ratio of the plane ABC. This ratio may, however, be abbreviated to a : b : c. Fig. 15. Fig. 16. There are, however, seven other planes possible about this axial cross which possess parameters of the same lengths as those of the plane ABC, figure 16. The simplified ratios of these planes are: a -b c a b -c a -b -c -a b c -a -b c -a b -c -a -b -c These eight planes, which are similarly located with respect to the crystallographic axes, constitute a crystal form, and may be represented by the general ratio (a : b : c). The number of faces in a crystal form depends, moreover, not only upon the intercepts or parameters but also upon the elements of symmetry possessed by the crystal, see page 12. Those forms, which enclose space, are called closed forms. Figure 16 is such a form. Those, however, which do not enclose space on all sides, as Fig. 17. shown in figure 17, are termed open forms. Fundamental Form. In figure 18, the enclosed form possesses the general ratio, a : b : c. The face ABM, however, has the INTRODUCTION. parametral ratio, oA : oB : oM, where oA = , oB = , and oM = 3oC 3<7. Hence, this ratio may be written a \ b \ $c. But, as in the previous case, this ratio repre- sents a form consisting of eight faces as shown in the figure. That form, the parameters of which are selected as the unit lengths of the crystallo- graphic axes, is known as the unit or fundamental form. In figure 1 8 the inner pyramid is a so-called unit, whereas the other one is a modified pyramid. Combinations. Several differ- ent forms may occur simultaneously upon a crystal, giving rise to a com- bination. Figure 19 shows a com- bination of two pyramids observed on sulphur; p = a : b : c (unit) and s = a : b : Y^c (modified). Figure 20 shows the two forms, o = a:a:a, and h = a: oca : ooa, seepage 19. Axial Ratio. If the intercepts of a unit form cutting all three axes be expressed in figures, the intercept along the b axis being considered as unity, we obtain the axial ratio. In figure 19, which represents a crystal of sulphur, the axial ratio is: a : b : c= .8131 : I : 1.9034-" Every crystallized substance has its own axial ratio. This is illustrated by the ratios of three minerals crystallizing in the ortho- rhombic system. Fig. 19. Fig. 18. Fig. 20. 8 INTRODUCTION. Aragonite, CaCO 3 , a : b : c = .6228 : i : .7204 Anglesite, PbSO 4 , a : b : c = .7852 : I : 1.2894 Topaz, A1 2 (F.OH) 2 SiO 4 , a : b : ^ = .5281 : I : .9442 In the hexagonal and tetragonal systems, since the horizontal axes are all equal, i. c., a = b, see page 5, the axial ratio is reduced to a : c; a now being unity. Thus, the axial ratio of zircon (ZrSiO 4 ) which is tetragonal, may be expressed as follows: a : c= i : .6404; that of quartz (SiO 2 ), hexagonal, by a : c = I : 1.0999. Obviously, in the cubic system, page 5, where all three axes are equal, this is unnecessary. However, in the monoclinic and triclinic systems, where either one or more axes intersect obliquely, it is not only necessary to give the axial ratio but also to indicate the values of the angles between the crystallographic axes. For example, gypsum (CaSO 4 . 2H 2 O) crystallizes in the monoclinic system and has the following axial ratio: a' : b : c .6896 : I : .4133 and the inclination of the a' axis to the c is 98 58'. This angle is known as , figure 2 1 . -b- -c Fig. 21, fb Fig. 22. In the triclinic system, since all axes are inclined to each other, it is also necessary to know the value of the three angles, which are located as shown in figure 22, viz: b/\c=a, a A c = /?, a A b = : y. The axial ratio and the angles showing the inclination of the axes are termed the elements of crystallization. The triclinic mineral albite (NaAlSi 3 O 8 ) possesses the following elements of crystallization: a:5:t= .6330 : I : 5573. = 94 5' 0= 116 27' y= 88 7' INTRODUCTION. 9 If the angles between the crystallographic axes equal 90, they are not indicated. Therefore, in the tetragonal, hexagonal, and orthorhombic systems, the axial ratios alone constitute the elements of crystallization, while in the cubic system, there are no unknown elements. Fig. 23. Rationality of Coefficients. The parametral ratio of any face may be expressed in general by na : pb : me, where the coefficients n, p, m< are according to observation always rational. In figure 23, the inner pyramid is assumed to be the fundamental form, page 7, with the following value of the intercepts: oa=i.256, ob = i, oc .752. But the coefficients n, p, m, are obviously all equal to unity. The ratio is, Hence, a : b : c. The outer pyramid, however, possesses the intercepts, oa = 1.256, oJ3 = 2, oC=2.2$6. These lengths, divided by the unit 10 INTRODUCTION. lengths of each axis, as indicated above, determine the values of n, p, and m for the outer pyramid, namely: = 3. 1.256 i .752 These values of n, p, and ;w, are, therefore, rational. Such values as ,_J, , f, or f are also possible, but never 3.1416-!-, 2.6578-)-, 1/3, and so forth. Symbols. The parametral ratio of the plane ABM, figure 18, may be written as follows: na :pb : me. But since, in this case, n = i, p = i, m = 3, the ratio becomes: a : I : fa If, however, the coefficients had the values |, f , and J, respect- ively, the ratio would then read: \a : \b : fa This, when expressed in terms of b, becomes: \a : b : 2c. Hence, the ratio na : b : me expresses the most general ratio or symbol for forms belonging to the orthorhombic, monoclinic, and triclinic systems. In the hexagonal and tetragonal systems, since the a and b axes are equal, this general symbol becomes, a : na : me. Fig. 24. Figure 24 shows a form, the ditetra- g o n a 1 bipyramid, with the symbol a : 2a : \c. In the cubic system, all three axes are equal and the general symbol reads, a : na : ma. INTRODUCTION. 11 I ^^^ 1 1 ~! 1 1 1 1 1 1 =K 1 i i The ratio a : 00 a : a, for example, symbolizes a form in the cubic system consisting of six faces, which cut one axis and extend parallel to the other two. Such a form is the cube, figure 25. The ratio a:2a: CD a represents a form with twenty-four faces; each face cuts one axis at a unit 's length, the second at twice that length, but extends parallel to the third axis. Figure 26 shows such a form, the tetrahexahe- dron. This system of crystallographic notation is known as the Weiss system after the inventor Prof. C. S. Weiss. These symbols are most readily under- stood and well adapted for beginners. Naumann, Dana, and Miller have intro- duced modifications tending to shorten the symbols of Weiss. Since these shortened forms are employed quite extensively, it will be necessary to explain each briefly. Symbols of Naumann and Dana. In the notation of Naumann, O and P, the initial letters of the words, octahedron and pyramid, respectively, are used as a basis. O is used for forms of the cubic system, P for all other systems. The coefficient m, referring to the vertical axis, is placed before and the other coefficient n, after one of these letters. For example, na : TJ : m, becomes mPn, and a : ma : ooa, CQQm. Dana 's notation is similar to that of Naumanrr. Dana, however, substitutes a short dash for the letters O and P, also i or I for oo, the sign of infinity. Otherwise, the two systems are alike. The following table shows several Weiss symbols with the cor- responding Naumann and Dana modifications: Fig. 26. Weiss a b a I a 2a a oca a 2a a na ^ 2C \C oca oca ma Naumann 2Pf |Pa ocOoo OCO2 mOn Dana 3 - i-2 m n 12 INTRODUCTION. Miller's System. In this system, as is also the case with the notations of Naumann and Dana, th$ letters referring to the various crystallographic axes are not indicated. The values given being understood as referring to the a, b, and c axes respectively, page 5. The reciprocals of the Weiss parameters are reduced to the lowest common denominator, the numerators then constitute the Miller symbols, called indices^. For example, the reciprocals of the Weiss parameters 2a : b : ^c would be |, y, i- These, reduced to the low- est common denominator, are f , f, f . Hence, 362 constitute the corresponding Miller indices. These are read three, six, two. A number of examples will make this system of notation clear. Thus, a : dob : ccc, becomes 100; 2a : b : $c, 5.10.2; a : a : ?>c, 331 ; a : ooa : 2c, 201, and so forth. The Miller indices correspond- ing to the general ratios a : no, : ma and na : b : me are written hkl. The Miller indices are important because of their almost universal application in crystallographic investigations. The trans- formation of the Naumann or Dana symbols to those of Miller should, after the foregoing explanations, present no difficulty whatever. Elements of Symmetry. The laws of. symmetry find expres- sion upon a crystal in the distribution of similar angles and faces. The presence, therefore, of planes, axes, or a center of symmetry these are the elements of symmetry is of great importance for the correct classification of a crystal. Planes of Symmetry. Any plane, which passes through the center of a crystal and divides it into two symme- trical parts, the one-half being the mirror-image of the other, is a plane of symmetry. Figure 27 shows a crystal of gypsum (CaSO 4 .2H 2 O) with its one plane of symmetry. Every plane of sym- metry is parallel to some face, which is either present or possible upon the crystal. In figure 27 it is parallel to the face b. It is sometimes convenient to subdivide the planes of symmetry into principal and secondary or common planes, according to whether they possess two or more equivalent and interchangeable directions or not. Figure 28 illustrates a crystal of the tetragonal system with five planes of sym- Fig. 27. INTRODUCTION. 13 metry. The horizontal plane c is the principal, the vertical planes the second- ary planes of symmetry. Axes of Symmetry. The line, about which a crystal may be revolved as an axis so that after a definite angular revolution the crystal assumes exactly the same posi- tion in space which it originally had, is termed an axis of symmetry. Depending upon the rotation necessary, only four types of axes of symmetry are from the stand- point of crystallography possible. 1 ' a) Those axes, about which the original position is reassumed after a revolution of 60, are said to be axes of hexagonal, six-fold, or six-count 2 '* symmetry. Such axes may be indicated by the sym- bol . Figure 29 shows such an axis. b) If the original position is regained after the crystal is revolved through 90, the axis is termed a tetragonal, four-count, or four- fold axis of symmetry. These axes are represented by , as illus- trated in figure 30. Fig. 28. *s ! T* 1 1 i 1 I i 1 j i -k Fig. 29. Fig. 30. Fig. 31. c) Axes requiring an angular revolution of 120 are trigonal, three-fold, or three-count axes of symmetry and may be symbolized by A. Figure 31 illustrates this type of axis. 1) For proof, see Groth's Physikalische Krystallographie, 4te Auflage, 1905, 321; also Viola'* Grundziige der Kristallographie, 1904, 251. 2) Because in a complete revolution of 360 the position is reassumed six times. 14 INTRODUCTION. Fig 3*. d) A binary, two-fold, or two-count axis necessitates a revo- lution through 1 80. These are indicated by in figure 30. Center of Symmetry. That point within a crystal through which straight lines may be drawn, so that on either side of and at the same distance from it, similar portions of the -crystal (faces, edges, angles, and so forth) are encountered, is a center of symmetry. Figure 32 has a center of symmetry the other elements of symmetry are lacking. Angular Position of Faces. Since crystals are oftentimes mis- shapen or distorted, page 3, it follows that the elements of symmetry are not always readily recog- nized. The angular position of the faces in respect to these elements is the essential feature, and not their distance or relative size. Fig- ure 33 shows an ideal crystal of a u g i t e. Here, the presence Fig. 33. Fig. 34. of a plane of symme- try, an axis, and a center of symmetry is obvious. Figure 34 shows a distorted crystal of the same mineral, possessing however exactly the same elements of symmetry, because the angular position of the faces is the same as in figure 33. Classes of Symmetry. Depending upon the elements of sym- metry present, crystals may be divided into thirty-tivo distinct groups, called classes of symmetry. l) Only forms which belong to the same class can occur in combination with each other. A crystal system, however, includes all those classes of symmetry which can be referred to the same type of crystallographic axes, page 5. The l) Also termed classes of crystals. INTRODUCTION. 15 various elements of symmetry and, wherever possible, an important representative are given for each of the thirty-two classes in the tabular classification on page 150. Holohedrism. Figure 35 shows a form of the cubic system. The face a b c has the parametral ratio a : \a : 30. Forty-seven other faces, having this same ratio, are how- ever also possible and, when present, give rise to the hexoctahedron, as the complete form is called. These forty-eight faces are all equal, scalene triangles. Such forms, which possess all the faces possible having the same ratio, are called holohedral forms. Fig. 35. Fiar 36. Hemihedrism. If, however, one-half of the faces of the hexoctahedron be suppressed and the other half be allowed to expand, as shown in figure 36, the diploid with half of the faces possessed by the hexoctahedron, namely twenty-four, results. Forms of this character are said to be hemihedral. Tetartohedrism. Again, if only one quarter of the faces expand, all others being suppressed, the hexoctahedron then yields the tetrahedral pentagonal dodecahedron with but twelve faces, figure 37. This is a tetarto- hedral form. Ogdohedrism. There are still other forms, which possess but one-eighth of the number of faces of the original holohedral form, and these are termed og-dohedral forms. Figure 270, page 91, shows such ogdohedral forms in combination. It is evident that these complete and partial forms possess differ- ent elements of symmetry and, hence, only those forms which are of the same type, that is, possess the same elements of symmetry, can enter into combination with each other. Compare page 14. Correlated Forms. Obviously, by the application of hemihe- drism, tetartohedrism, and Ogdohedrism the original holohedral forms Fig. 37. 16 INTRODUCTION. yield two, four, or eight new correlated forms, respectively. Most of these correlated forms are congruent and differ from each other only in respect to their position in space. A rotation through some definite angle is all that is necessary to have such forms occupy the same position. Such forms are then designated as //^/sand minus, or positive and negative, forms, page 27. Others, however, are related to one another as is the right hand to the left, hence cannot be superimposed and these are said to be enantiomorphous. These are designated as right and left forms. Compare page 37. M Fig. 38. Fig. 39. Fig. 40. Hemimorphism. This is a peculiar type of hemihedrism. It is only possible upon those holohedral and hemihedral forms, or their combinations, which possess a so-called singular** axis of symmetry. By its application the forms occurring about one end of this axis differ from those about the other. Such forms are hemimorphic, and the singular axis of symmetry is said to be polar. Hence, the plane of symmetry perpendicular to the singular axis is lost. Compare figures 38, 39, and 40. !) An axis which occurs but once and differs from all others. For example, the axis of tetragonal symmetry in figure 30. OF THE 1 CUBIC SYSTEM" Crystallographic Axes. All forms which can be referred to three equal and perpendicular axes belong to this system. Figure 41 shows the axial cross. One axis is held A. A. vertically, a second extends from front to rear, and the third from right to left. These axes are all interchangeable, each . being designated by a. Since there are no unknown elements of crystallization in this system (page 9), all substances, regard- less of their chemical composition, crystal- p7* 41 lizing in this system with forms having the same parametral ratios must of necessity possess the same interfacial angles. Classes of Symmetry. The cubic system includes five groups or classes of symmetry. Beginning with the class of highest sym- metry, they are: 1) Hexoctahedral Class (Holohedrism) 2) Hextetrahedral Class \ 3) Dyakisdodecahedral Class > (Hemihedrisni) 4) Pentagonal icositetrahedral Class ) 5) Tetrahedral pentagonal dodecahedral Class ( Tetartohedrisrti) Of these classes, the first three are most important, since they possess many representatives among the minerals. /. HEXOCTAHEDRAL CLASS.*) }\ r tt\ o\\ v (Holohedrism.} Elements of Symmetry, a) Planes. Forms of this class are characterized by nine planes of symmetry. Three of these are par- allel to the planes of the crystallographic axes and, hence, perpendic- ular to each other. They are the principal planes of symmetry. They divide space into eight equal parts called octants. The six 1) Also termed the regular, isometric, tesseral, or tessular system. 2) Termed by Dana the normal group. 18 CUBIC SYSTEM. other planes are each parallel to one of the crystallographic axes and bisect the angles between the other two. These are termed the sec- ondary or common planes of symmetry. By them space is divided i i Fig. 42. Fig. 43. into twenty-four equal parts. The nine planes together divide space into forty-eight equal sections. These nine planes are often indicated thus: 3 -4- 6 = 9. Figures 42 and 43 illustrate the location of the principal and secondary planes, respectively. b) Axes. The intersection lines of the three principal planes of symmetry give rise to the three principal axes of symmetry. These are parallel to the crystallographic axes and possess tetragonal symmetry, as illustrated by figure 44. The four axes equally inclined i < oo. The symbols are, therefore, (a : ma : ma), mOm, \hll\, h > I. The ideal forms, figures 53, 2O2J2ii(, and 54, 4O4|4iiJ, bear some resemblance to the octahedron, each face of which has been replaced by three four-sided faces, trapeziums, of !) Also known as the trisoctahedron. 2 ) Also termed the trapezohedron, icositetrahedron, andleucitohedron. HEXOCTAHEDRAL CLASS. 21 equal size. The form is, therefore, termed the tetrag-onal trisoc- tahedron. The six tetrahedral angles 1 ) a indicate the position of the crystallographic axes. The trigonal axes of symmetry join opposite trihedral angles, while those of binary symmetry con- nect the tetrahedral angles 2 ) b. 6. Tetrahexahe- Fig. 53. Fig. 54. dron. In this form the faces cut one axis at a unit's distance, the second at the distance ma, where m>\< 00, and extend parallel to the third axis. The symbols are, therefore, (a : ma : do a), oo O m, \hko\. The twenty- four faces in the ideal forms, figures 55, ooO2[2ioS, and 56, OoO4| 41 o|, are equal isosceles triangles. Since this form may be Fig. 55. Fig. 56. considered as a cube, whose faces have been replaced by tetragonal pyramids, it is often called the pyramid cube or tetrahexahedron. The crystallographic axes are located by the six tetrahedral angles. The axes of trigonal symmetry pass through opposite hex- ahedral angles, while the binary axes bisect the long edges. 7. Hexoctahedron. As is indicated by -the name, this form is bound- ed by forty-eight faces. Each cuts one crystallo- graphic axis at a unit's distance, the other two at greater but unequal dis- Fig. 57. Fig. 58. 1 ) With four equal edges. 2) These have two pairs of equal edges. 22 CUBIC SYSTEM. tances na and ma, respectively; n is less than m, the value of m being, as heretofore, w>i+12*> - - Tetrahexahedron a : ma : oc a oo Om \hko\ 24 - 6 8 - Hexoctahedron a : na : ma mOn \hkl\ 48 - 12 8 6 From this tabulation we see that the ratios of the octahedron, dodecahedron, and hexahedron contain no variables and, hence, each is represented by but one form. These are often called singular or fixed forms. The other ratios, however, contain either one or two variables and, therefore, each represents a series of forms. Compare figures 51 to 58. !) These have four equal edges. 2 ) Two pairs of two equal edges each. HEXOCTAHEDRAL CLASS. 23 The relationship existing between the simple forms is well expressed by the following diagram: a : a : a a : a : oo a a : na : ma a : ma : oo a a : oca : ooa The three fixed forms are placed at the corners of the triangle and, as is obvious, must be considered as the limiting forms of the others. For example, the value of m in the trigonal trisoctahedron (a : a : md) varies between unity and infinity, page 20. Hence, it follows that the octahedron and dodecahedron are its limiting forms. The tetragonal trisoctahedron (a : via : md) similarly passes over into the octahedron or cube, depending upon the value of m. The limiting forms are, therefore, in every case readily recognized. Those forms, which are on the sides of the triangle 1 ^ lie in the same zone, that is, their intersection lines are parallel. Combinations. The following figures illustrate some of the combinations (page 7) of the simple forms, which are observed most frequently. Fig. 59. Fig. 60. Fig. 61. Figures 59 and 60. h = oo O oo,{ioo}; o=O, {in}. Com- 1) For example, the octahedron, trigonal- trisoctahedron, and dodecahedron. 24 CUBIC SYSTEM. monly observed on galena, PbS. In figure 59 the octahedron predominates, whereas in figure 60 both forms are equally developed. Figure 61. h = 00 O oc, {100} ; d = ooO, {no}. Copper, Cu, and fluorite, CaF 2 , show this combination. Fig. 62. Fig. 63. Fig. 64. Figures 62 and 63. h = ooO 00,{ 100}; o = O,{iu}; and d -= ocO, {no}. Also observed on galena, PbS. Figure 64. h = OoOoo,jioo|; e = OoO2, {210}. Observed on copper, Cu; fluorite, CaF 2 ; and halite, NaCl. Fig. 65. Fig. 66. Fig. 67. Figure 65. o = O, { 1 1 1 } ; e = 0062, J2io}. Fluorite, CaF 2 . Figure 66. d= ooO, {no}; e = 0062, J2io}. Copper, Cu. Figure 67. h = oo O oo,{iooj ; i = 2O2,\2ii\. Observed on analcite, NaAl(SiO 3 ) 2 .H 2 O; and argentite, Ag 2 S. Figure68. o = O, { 1 1 1 } ; *' = 303, 13"}- Spinel, MgAl 2 O 4 . HEXOCTAHEDRAL CLASS. Fig. 68. Fig. 69. Fig. 70. Figure6 9 . ,'= 2O2,{2ii}; O = O,{in}. Argentite, Ag 2 S. Figure 70. o = f {in}; ^=ooO, {nof; f=2O2,{2ii Observed on spinel, MgAl 2 O 4 , and magnetite, Fe 3 O 4 . Fig. 71. Fig. 72. Fig. 73. Figure 71. rf=ooO,{no}; f = 2O2,{2ii{. A common com- bination observed on the garnet, R'^'^SijO . Figure 72. ,'=202,{2ii{; / = |O,{332}- Garnet. Figure 73. d= ooO,{noJ; 5 = 3O|,J32i}. Garnet. Figure 74. rf= Garnet. Fig. 74. o}; / = 2O2,{2ii(; ^ = 26 CUBIC SYSTEM. 2. HEXTETRAHEDRAL CLASS. 1 ) ( Tetrahedral Hemihedrism. ) Elements of Symmetry, By decreasing the elements of sym- metry of the hexoctahedral class, page 17, the other classes of the cubic system may be deduced. In this class the principal planes of symmetry disappear, which necessitates not only the loss of the six axes of binary symmetry, but also the three of tetragonal symmetry parallel to the crystal- lographic axes, see page 18. The center of symmetry also disappears. Hence, the elements remaining are: six secondary planes of symmetry, and four trigonal axes, which are now polar. There are f mO -f- ; -f , K\hhl\ (figure 80; , *\hhl\ (figure 79). 1) The letfer K (KA.O/OS, inclined) is placed before the Miller indices of forms of this class because their faces are inclined. This type of hemihedrism is often known as the inclined-face hemihedrism. 28 CUBIC SYSTEM. The crystallographic axes pass through opposite tetrahedral angles, while the axes of trigonal symmetry join opposite trihedral angles, one of which is acute, the other obtuse. Trigonal tristetrahedron. The application of the tetrahedral hemihedrism to the tetragonal trisoctahedron produces two congruent half-forms, bounded by twelve similar isosceles triangles, figures* 82, 83, and 84. Since these new forms may be considered as tetrahedrons whose faces have been replaced by trigonal pyramids, they are termed trigonal tristetrahedrons^ or pyramid tetrahedrons. Sometimes, moreover, the term trigonal dodecahedron is also used. Fig. 82. Fig. 83. Fig. 84. The symbols are written: a \ma\ma , mQm mOm -. +/ -T-T-5 + -T~> {A//} (figure 84);- -j-, {*//} (figure 82). The crystallographic axes bisect the long edges. The trigonal axes pass from the trihedral angles to the opposite hexahedral. Hextetrahedron. In precisely the same manner the hexoctahe- dron, figure 86, yields two congruent forms of tetrahedral habit, bounded by twenty-four similar scalene triangles. These new forms are called hextetrahedrons. *) The following relationship between this and the preceding form and their respective holohedral forms should be carefully noted, viz: The trigonal trisoct?hedron (each face is triangular) yields the tetragonal tristetrahedron (faces of tetragonal outline), wh'le, vice versa, the tetragonal trisoctahedron furnishes the trigonal tristetrahedron. Compare figures 79 to 84. HEXTETRAHEDRAL CLASS. 29 Fig. 85. The symbols are: a \na\ma mQn \hkl\ (figure 87); - hkl\ (figure 85.) The crystallographic axes connect opposite tetrahedral angles. The trigonal axes of symmetry pass through opposite hexahedral angles, one of which is more obtuse than the other. Hexahedron, dodecahedron, and tetrahexahedron. As is evident from figure 7 5, no forms, geometrically new, can result from the application of this type of hemihedrism on the cube, dodecahedron, and tetrahexahedron. The faces of these forms belong simultane- ously to different octants and, hence, the suppression of a portion of a face in one octant is counterbalanced by a corresponding expansion in another. Therefore, no change in these forms can result. They are from a geometrical standpoint exactly similar to those of the hexoctahedral class. Their symmetry is, however, of a lower grade. This is not to be recognized on models. On crystals, however, the form and position of the so-called etch figures, as also the different physical characteristics of the faces, reveal the lower grade of symmetry 1 ). These forms are, therefore, only apparently holohedral. !) When crystals are subjected to the action of some solvent for a short time, small depressions or elevations, the so-called etch figures, appear. Being dependent upon the internal molecular structure, their form and position indicate the symmetry of the crystal. For instance, figure 88 shows the etch figures on a crystal (cube) of halite, NaCl. Here, it is evident, that the symmetry of the figures in respect to that of the cube is such as to place the crystal in the hexoctahedral class. Figure 81) represents a cube of sylvite, KC1, which geometri- cally does not differ from the crystal of halite. A lower grade of symmetry is, however, revealed by the position of the etch figures. This crystal belongs to the pentagonal icositetrahedral class, page 36, for no planes of symmetry can be passed through these Fig gg^ 30 CUBIC SYSTEM. The following table shows the important features of the forms of this class of symmetry. FORMS SYMBOLS Number of Equal Faces Number of Angles Trihedral Tetrahedral | | Hexahedral Weiss Naumann Miller Tetrahedron a: a: a \ + 2 0^ I 2 K\III\ K\HI\ K J 4 - 2 Tetragonal tristetrahedron a: a: ma mO *\hhl\ *\hhl\ 1 [l2 4+4 6 , 2 mO 2 I 2 Trigonal tristetrahedron a : ma : ma \ ( mO?n i K\kll\ K\hJl\ \12 4 4 H 2 mOm 2 ( 2 Hextetrahedron a:na: ma f , mOn K\hkl\ K\k~kl\ 1 \24 \ 6 4 + 4 1 2 mOn 2 2 Dodecahedron a: a: oca ocO K{l 10} Apparently holohedral J Tetrahexahe- dron a : ma ; oc a ocO;;/ *{hko} Hexahedron a: oca: cca ocO oc K{IOO} Fig. 90. Fig. 91. figures, which at the same time are planes of symmetry of the cube, as is the case with the crystal of halite, figure 88. Fora fuller discussion of these interesting figures consult Groth, Physi- kalische Krystallographie, 4te Auflage, 1905, 251; also Dana, Text-book of Mineralogy, 1898, 149. Figures 90 and 91 show two cubes repre- senting crystals of sphalerite, ZnS (figure 90), and pyrite, FeS2 (figure 91). Although these form? do not differ from the holohedral, it is at once seen HEXTETRAHEDRAL CLASS. 31 Combinations. Some of the more common combinations are illustrated by the following figures: Fig. 92. Fig. 93. Fig. 94. O Figure 92. o = -\ , KJIIIJ; h = OoO 00, K| iooj. Ob- served on sphalerite, ZnS. Figure 93. o = + , *{ 1 11 } ; o' = - , K{III(. 2 O Sphalerite. Figure 94. 0=+, "Jin!; h = ooO oo, *{ ioo[ ; d 00 O, K{IIO{. Tetrahedrite (Cu 2 , Fe, Zn) 4 (As, Sb) 2 S 7 . O Figure 95. o ~- -h , 2O2 = - - ie2ii-. Tetrahedrite. Fig. 95. 2O2 Figure 96. n = + , Fig. 96. Fig. 97. r = ooO, Tetrahedrite. that adjoining faces possess striations extending in different directions. Figure 90 possesses six secondary planes of symmetry, the principal are wanting, and, hence this cube is only apparently holohedral. It belongs to hextetrahedral class. Figure 91 possesses a still lower grade of sym- metry for here only three principal planes are present, and it must, therefore, be referred to the dyakisdodecahedral class, page 32, 32 CUBIC SYSTEM. Figure 97. d ooO, K{IIO|; m- Sphalerite. Figure 98. /i=ooOoc, K{IOO|; o O ,KJ3iiJ. + , K\\\\ o Fig. 98. ; d = z; = c O "^SSS 1 !* This combination occurs on boracite, Mg 7 Cl 2 B 16 O 30 . 3. DYAKISDODECAHEDRAL CLASSY (^Pyritohedral Hemihedrism.} Elements of Symmetry. Here the six secondary planes and the six axes of binary symmetry are lost. The elements of symmetry which remain are, there- fore, three principal planes parallel to the planes of the crystallographic axes, four trigonal axes, and also the center of symmetry. The three axes parallel to the crystallographic axes now possess binary symmetry. Figure 99 shows these elements as well as the application of the hemihedrism, which will be discussed in the next paragraph. Fig. 99. Pyritohedral hemihedrism 2 . In this class the forms may be assumed as derived from the holohedral types by the expansion of the faces lying wholly within alternate spaces formed by the intersection of the secondary planes of symmetry, page 18. Such 1) Called by Dana the pyritohedral group. 2) Also termed pentagonal or parallel-face hemihedrism, having reference to the pentagonal outline and parallel arrangement of the faces, respectively. DYAKISDODECAHEDRAL CLASS. 33 faces or pairs of faces are then symmetrical to the three principal planes of symmetry, figure 99. Pyritohedron. The application of this type of hemihedrism to the tetrahexahedron, figure 101, produces two correlated, congruent, hemihedral forms, figures 100 and 102, bounded by twelve similar faces. Each face is an unequilateral pentagon, four sides of which are equal. Since these forms are congruent, a rotation through 90 is all that is necessary to bring them into precisely the same position in space. They are, hence, designated as plus and minus forms. 1} Fig. 100. The symbols are: a: ma\ do a Fig. 101. Fig. 102. fa:ma:aca'} f ocOral | -J- J; +| -J- J, TJA0} (figure 102); (figure 100). Because this form occurs frequently on pyrite it is termed the pyritohedron, although pentagonal dodecahedron^ is quite often used. The axes of binary symmetry, hence, also the crystallographic axes bisect the six long edges. The trigonal axes pass through the trihedral angles, the edges of which are of equal lengths. 1) Also designated as right and left. It is preferable, however, to use those terms in reference to enantiomorphous forms only, page 16. 2) In this class, TT (TrapoAAryAos, parallel faced) is placed before the Miller indices, see page 27. To distinguish the other symbols from those of the preceding class, they are enclosed in brackets. 3 ) The regular pentagonal dodecahedron of geometry, bounded by equilateral pentagons intersecting in equal edges and angles is crystallographically an impossible form, the value of m being which is irrational, compare page 10. ^4 CUBIC SYSTEM. Dyakisdodecahedron. By the expansion and suppression of alter- nate pairs of faces crossed by the principal planes of symmetry, the hexoctahedron, figure 104, yields two correlated, congruent forms, each bounded by twenty-four similar trapeziums, figures 103 and 105. These forms are the dyakisdodecahedrons % also termed didodecahe- drons, or diploids. Fig. 103. Fig. 104. Fig. 105. The symbols are \a:na:ma |_ 2 -- ir\hlk\ (figure 103). ' -\ hkl \ (figure 105): mOn The crystallographic axes, also those of binary symmetry, pass through the six tetrahedral angles possessing two pairs of equal edges. The trigonal axes join opposite trihedral angles. Other Forms. The hexahedron, octahedron, dodecahedron, trigonal trisoctahedron, and tetragonal trisoctahedron, the other forms of the holohedral type, do not yield new forms by the applica- tion of this hemihedrism. This is clearly shown by figure 99, illus- trating the elements of symmetry of this class. Being apparently holohedral, these forms may occur independently or in combination, but always with the full number of faces. Their symmetry is, how- ever, less than that of the holohedrons. See page 29. In the following table -the important features of the forms of this class are indicated. DYAKISDODECAHEDRAL CLASS. 35 FORMS SYMBOLS Number of Faces Number of Angles Trihe- dral Tetra- hedral " en 3 V 0"&C '^ rnS + y{%75l- Sal Ammoniac, NH 4 C1. 5. TBTRAHEDRAL PENTAGONAL DODECAHE- DRAL CLASSY ( Tetartohedrism. ) Fig. 116. Elements of Symmetry. The nine planes, the three axes of tetragonal and the six of binary symmetry, as also the center, are lost. The elements of this class are, hence, four polar axes of trigonal and three of binary 1) The tetartohedral group of Dana. TETRAHEDRAL PENTAGONAL DODECAHEDRAL CLASS. 39 symmetry, the latter being parallel to the crystallographic axes. See figure 117. Tetartohedrism. The simple tetartohe- dral forms may be conceived as derived from the hemihedrons of any one of the three hemihedral classes tetrahedral, pyritohedral, or plagihedral by the subsequent applica- tion of one of the other two methods. Th^ result is in each case the same. Compare figures 118, 119, 120, 122, and 125. Fig. 117. Fig. 121. Fig. 122. Fig. 123. Fig. 124. Fig. 126. Fig. 125. Tetrahedral pentagonal dodecahedron. From the hexoctahe- dron by the successive application of two types of hemihedrism four 40 CUBIC SYSTEM. new forms of tetrahedral habit, bounded by twelve unsymmetrical pentagons, result. The derivation of these forms is easily to be seen from figures 121-126, also figure 37, page 15. In figure 122, the form derived by the expansion of the unshaded faces is known as the left positive tetrahedral pentagonal dodecahedron, whereas the one derived from the shaded faces alternating with these i. e., in the same octant is the rig-Jit positive form. From figure 125 the left negative and rig-Jit negative forms are likewise possible. 7 a\ na\ ma The symbols according to Weiss are -r r, - I . 4 According to Naumann and Miller they may be written: 1. -\- r - -, Kir\khl\*\ Positive right, figure 123. til O n 2. r - , KTr\hkl\, Negative rig Jit, figure 126. 4 3. -f- /- , KTr\hkl t \, Positive left, figure 121. 4 4. -/- , KTr\khl\, Negative left, figure 124. Forms i and 2, also 3 and 4, are among themselves congruent, whereas the pairs I and 3, and 2 and 4, are enantiomorphous. The crystallographic axes bisect the six equal edges, while the four trigonal axes join opposite trihedral angles of which one is more obtuse than the other. These trihedral angles possess equal edges. Other forms. If tetartohedrism be applied to the other holohe- drons, it follows that four of them will yield forms, as follows: Octahedron two tetrahedrons, figures 76 and 78, page 26. Trigonal trisoctahedron two tetragonal tristetrahedrons, figures 79 and 81, page 27. Tetragonal trisoctahedron two trigonal tristetrahedrons, figures 82 and 84, page 28. Tetrahexahedron two pyritohedroris, figures 100 and 102, page 33- The first three forms do not differ geometrically from those obtained by the tetrahedral hemihedrism, while the fourth is the same as that derived by the pyritohedral hemihedrism. They are, hence, i) Two letters, each representing one of the types of hemihedrism applied, are used with the Miller indices in this class. TETRAHEDRAL PENTAGONAL DODECAHEDRAL CLASS. 41 only apparently hemihedral forms. The cube and dodecahedron are unchanged geometrically and, therefore, apparently holohedral. The following table shows the principal features of the forms of this class: FORMS SYMBOLS Number of Faces Number of Angles \Veiss Naumann Miller Trihedral Equal Edges Unea 4 ual Edges Dodecahedron a : a : oo :t oc O KTT\IIO\ "j i Apparently holohedral Hexahedron a\ oca: GO .7 oo O oo KTT\IOO\ Tetrahedrons a: a: a I l ^ 2 KTT\III\ KTT\III\ 1 Ap t( h j parent ^trahec emihec iy Iral Irons i- 2 Tetragonal tristetrahe- drons a:a: ma i ^mO KK\hhl\ **\hhl\ 2 mO 2 2 Trigonal tristetrahe- drons a : ma : ma m Om \hll\ KTT\hn\ 1 2 m Om t 2 2 Pyritohedrons [a : ma : oo a ~| ' ,focOm^ KTT\/lkO\ KTT\khO\ 1 Apparently }- pyritohedral hemihedrons L - J roc<9w~i i L * J 2 J Tetrahedral pentagonal dodecahe- drons a:na: ma mOn 4 mOn r 4 . 7 mOn \khl\ KTT\hkl\ KTT\hkl\ M\khl\ 1 12 4 + 4 J) 12 4 r,J \ I 4 mOn L 4 1 ) One set is more obtuse than the other. 42 CUBIC SYSTEM. Combinations. In this class it is evident that the apparently holohedral and hemihedral forms of the various classes, as described on pages 40 and 41, may occur together on the same crystal. For example, figures 127 and 128 show crystals of sodium chlorate, NaClO 3 , where h = 00 O oo, 10 Fig. 127. Fig. 128. = OCO, KIT O K7T | I I I (figure 127) = - [ ^- 2 ], { I20} . p (figure I2g)= + F GC021 \ " ^* Fig. 129. Fig. 130. Fig. 131. R rtsrn N A I3 ' and ISI illustrate crystals of barium nitrate, Ba(N0 3 ) 2 , with the following forms: h = -- ooO oo, KTTJ ioo; ; o = --. w ir7Tl.^_ T^02 2 X= I {!{;#= -^-|, W jI20|; T-'l HEXAGONAL SYSTEM. -c Fig. 132. Crystallographic Axes. This system includes all forms which can be referred to four axes, three of which are equal and lie in a horizontal plane, and intersect each other at an angle of 60. These are termed the secondary or lateral axes, being designated by the letter a. These axes are inter- changeable. The fourth, a principal axis, is perpendicular to the plane of the second- ary axes and is termed the c axis. It may be longer or shorter than the secondary axes. The three equal axes, which bisect the angles between the secondary axes, are the intermediate axes. These may be designated by b. Figure 132 shows an axial cross of this system. In reading crystals of the hexagonal system, it is customary to hold the c axis vertical, letting one of the secondary or a axes extend from right to left. The extremities of the secondary axes are alternately characterized as plus and minus, see figure 132. In referring a .form to the crystallographic axes, it is common practice to consider them in the following order: # t first, then a 2 , thirdly 3 , and lastly the c axis. The symbols always refer to them in this order. It is also to be noted that in following this order, one of the lateral axis will always be preceded by a minus sign. * Since the lengths of the a and c axes differ, it is necessary to assume for each substance crystallizing in this system a fundamental form, whose intercepts are taken as representing the unit lengths of the secondary and principal axes, respectively. The ratio, which exists between the lengths of these axes is called the axial ratio and is always an irrational value, the a axis being assumed as unity, page ; ' [43] 44 HEXAGONAL SYSTEM. Classes of Symmetry. The hexagonal system includes a larger number of classes of symmetry than any other system, namely, twelve. The order in which they will be discussed is as follows: i. tt t * t* 2. 3- 4- 5- 6. 7- 8. 9- 10. ii. ( Holohedrism}. f Holohedrism and\ \hemimorphism. j ( Hemihedrism). (Hemihedrism and\ \hemimorphism. J ( Tetartohedrism.) 12. Trigonal pyramidal class Dihexagonal bipyramidal class Dihexagonal pyramidal class Ditrigonal bipyramidal class Ditrigonal scalenohedral class Hexagonal bipyramidal class Hexagonal trapezohedral class Ditrigonal pyramidal class Hexagonal pyramidal class Trigonal bipyramidal class Trigonal trapezohedral class Trigonal rhombohedral class Teta rtohedrism and hemimor- phism ( Ogdohedrism). Those classes marked with an * are the most important, for nearly all of the crystals of this system belong to some one of them. No representatives have as yet been observed for the classes marked by t. Those marked % are often grouped together and form the trigonal system. /. DIHEXAQONAL BIPYRAMIDAL CLASS.U (Holohedrism.} Symmetry, This class possesses the highest grade of symmetry of any in the hexagonal system. a) Planes. In all there are seven planes of symmetry. One of these, the -principal plane, is parallel to the plane of the secondary axes, hence, horizontal. The other planes are divided into two series of three each, which are termed the secondary and inter- mediate, respectively. Each of the secondary planes includes the c or principal axis and one of the secondary, a, axes. These planes are, therefore, vertical and perpendicular to the principal plane. The normal group of Dana. DIHEXAGONAL BIPYKAMIDAL CLASS. 45 Fig. 133. They intersect at angles of 60. The inter- mediate planes are also vertical and perpen- dicular to the principal plane^, for they bisect the angles between the secondary planes and, hence, each includes the.c and one of the intermediate axes. The secondary and principal planes divide space into twelve equal parts, called dodecants; the seven planes, however, into twenty-four parts, figure 133. These planes are often designated as follows: i Principal -f- 3 Secondary -f 3 Intermediate = 7 Plants. b) Axes. Parallel to the vertical or c axis is an axis of hexagonal symmetry, while the axes parallel to the secondary and intermediate axes possess binary symmetry. These axes are often indicated, thus, i*+3-h3 = 7 axes. c) Center. This element of symmetry is also present, requiring every face to have a parallel counter-face. Figure 134, the projection of the most complicated form upon a plane perpendicular to the vertical axis, shows the elements of symmetry of this class. i. Hexagonal bipyramid of the first order. From figure 132, it is obvious that any plane which cuts any two adjacent secondary axes at the unit distance from the center must extend parallel to the third. If such a plane be assumed to cut the c axis at its unit length from the center, the parametral ratio would then be Fig 134 According to the above elements of symmetry, twelve planes possessing this ratio are possible. They enclose space and give rise to the form termed the hexagonal bipyramid^ of the first order, 1) Since these are really double pyramids, the term bipyramid is employed. 46 HEXAGONAL SYSTEM. Fig. 135. figure 135. In the ideal form, the faces are all equal, isosceles triangles. The symbols are (aiOOa: a: c), P, jionf. 1 ) Because the intercepts along the c and two secondary axes are taken as units, such bipyramids are also known as fundamental or unit bipyramids, page 6. Planes are, however, possible which cut the two secondary axes at the unit distances, but intercept the c axis at the distance mc^ the coefficient m being some rational value smaller or greater than I, see page 6. Such bipyramids, accord- ing as m is greater or less than unity, are more acute or obtuse than the fundamental form. They are termed modified hexagonal bipyra- mids of the first order. Their symbols are (a: oca: a:mc), mP, \h o hl\, where m = y, also m > o < oc . The principal axis passes through the hexahedral angles, the secondary axes join tetrahedral angles, while the intermediate bisect the horizontal edges. Hence when such bipyramids are held correctly, a face is directed towards the observer. The various axes of symmetry are located by means of the above. 2. Hexagonal bipyramid of the second order. In form, this bipyramid does not differ from the preceding. It is, however, Fig. 136. Fig. 137. i) In this system it is advantageous to employ the indices as modified by Bravais (h ikl) rather than those of Miller, who uses but three. DIHEXAGONAL BIPYRAMIDAL CLASS. 47 to be distinguished by its position in respect to the secondary axes. The bipyramid of the second order is so held that an edge, and not a face, is directed towards the observer. This means that the secondary axes are perpendicular to and bisect the horizontal edges as shown in figure 136. Figure 137 shows the cross section including the second- ary axes. From these figures it is obvious that each face cuts one of the secondary axes at a unit distance, the other two at greater but equal distances. For example, AB cuts a z at the unit distance OS, and a l and a 2 at greater but equal distances OM and ON, respectively. The following considerations will determine the length of OM and ON, the intercepts on a l and 2 , in terms of OS = i. As already indicated, the secondary axes are perpendicular to the horizontal edges, hence OS and ON are perpendicular to AB and BC, respectively. Therefore, in the right triangleSpRB and NRB, the side RB is common and the angles OBR and NBR are equal. 1} Therefore, OR == RN. But OR = OS == i. Hence, ON = OR + RN == 2. In the same manner it can be shown that the intercept on a^ is equal to that along 2 , that is, twice the unit length. The parametral ratio of the hexagonal bipyramid of the second order, therefore, is (20: 2a: a: mc\ or expressed according to Naumann and 2/1 Bravais, mP2 and \hh2h l\, where -j- = m. -Figure 137 shows the positions of the bipyramids of both orders in respect to the secondary axes, the inner outline representing that of first, the outer the one of the second order. Fig. 138. i) For, angle ABC equals 120, angle NBR is then 60, being the supplement of ABC. But the intermediate axis OZ bisects the angle ABC, hence angle OBR is also 60. 48 HEXAGONAL SYSTEM. 3. Dihexagonal bipyramid. The faces of this form cut the three secondary axes at unequal distances. For example, in figure 138 the face represented by dB cuts the a^ axis at A, 2 at C, and a s at B. Assuming the shortest of these intercepts as unity, hence, OB = a i, we at once see that one of these axes is cut at a unit's distance from O, the other two, however, at greater distances. If we let the intercepts ,OA and OC be represented by n(OB) = na, and />(OB) = pa, respectively, the ratio will read na :pa : a : me, mPn, \hikl\. In this ratio p n Fig. 139. Twenty-four planes having this ratio are possible and give rise to the form called the dihexag'onal bipyramid, figure 139. In the ideal form the faces are equal, scalene triangles, cutting in twenty-four polar 2) , a and b, and twelve equal basa! 3) edges. The polar edges and angles are alternately dissimilar. This is shown by figure 140, where the heavy inner outline represents the form of the first order, the outer the one of the second, and the inter- mediate outlines the dihexagonal type in respect to the secondary axes. z4ro, i) From the above discussion, it follows that in figure 138 OA:OC: OB = :/:!. Draw XB parallel to a 2 . and then OA:XA = OC:XB. But the triangle OXB is equilateral, Hence, XB = XO = OB = 1. And XA = OA OX = OA 1. Therefore, OA : OA 1 = OC : 1. But OA = n, and OC =/, Hence, w:n !=/:!, r ^l = p - It can also be shown that the algebraic sum of the Miller- Bra vais indices -{ hik } is equal to h + i + ~k = o. For, OA = -4- OC = 4^ and OB n t 1 -i-> page 12. Therefore, -T-: =- r~ = 7- : r-, J_. J_ 1 . h ' h ~~ k Or, k = h + 1. Hence, h -\- i + k = o. Those joining the horizontal and principal axes. These lie in the principal plane of symmetry. DIHEXAGONAL BIPYRAMIDAL CLASS. 49 These three hexagonal bipyramids are closely related, for, if we suppose the plane represented by AB, figure 140, to be rotated about the point B so that the intercept along a 2 increases in length, the one Fig. 140. along j decreases until it equals oB' = oB = i. Then the plane is parallel to a 2 and the ratio for the bipyramid of the first order results. If, however, AB is rotated so that the intercept along a 2 is decreased in length, the one along a^ increases until it equals oC = 2oB' = 2a. When this is the case, the intercept on 2 is also equal to 2<7, for then the plane is perpendicular to # 3 . This gives rise to the ratio of the bipyramid of the second order. That the bipyramids of the first and second orders are the limit- ing forms of the dihexagonal bipyramid is also shown by the fact that p - . For, if n = i, it follows that/ = 00, hence, the ratio of the form of the first order. But, when n 2, *p = 2 also, there- fore, the ratio for the second order results. With dihexagonal bipyramids the following holds good : n > i < 2, and/ > 2 < 00. The dihexagonal bipyramid whose polar edges and angles are all equal is crystallographically not a possible form, because the value of n would then be ^(i + 1/3) = 1/2. sin 75 = 1.36603+, which of course is irrational. It also follows that in those dihexagonal bipyr- amids, where the value of n is less than 1.36603 + , for instance, t = i. 20, the more acute pole angles indicate the location of the secondary axes, the more obtuse that of the intermediate, and vice 50 HEXAGONAL SYSTEM. ^r 1 _J T\ i I. i 1 l l 1 1 1 1 1 1 v _ -J r- - k Fig, 141. versa, when n is greater than 1.36603 + , for example, |-=i.6o. This is clearly shown by figure 140. Hexagonal prism of the first order. This form is easily de- rived from the bipyramid of the same order by allowing the intercept along the c axis to assume its maximum value, infinity. Then the twelve planes of the bipyramid are reduced to six, each plane cutting two secondary axes at the unit distance and extending parallel to the c axis. The symbols are (a : oca : a: oor), OoP, |ioio|. This form cannot enclose space and, hence, may be termed an open form, page 6. It cannot occur indepen- dently and is always to be observed in combination, figure 141. The secondary axes join opposite edges, i. e., a face is directed towards the observer when properly held. Hexagonal prism of the second order. This prism bears the same relation to the preceding form that the bipyramid of the second order does to the one of the first, page 46. The symbols are (2a:2a :a : 00 ^), 00 P2, jii2o[. It is, hence, an open form consisting of six faces. The secondary axes join the centers of opposite faces, hence, an edge is directed towards the observer, figure 142. Dihexagonal prism. This form may be obtained from the corresponding bipyra- mid by increasing the value of m to infinity, which gives (na : pa : a : ocr), OoPw, \hiko\. This prism consist of twelve faces whose alternate intersection angles are dissimilar. This form, figure 143, is closely related to the corresponding bipyramid and, hence, all that has been said concerning Fig. 142. j j i i :C'.I!7 ' i x i i i 1 -i-- i -- i i _ Fig. 143. DIHEXAGONAL BIPYRAMIDAL CLASS. 51 the dihexagonal bipyramid, page 48, in respect to the location of the secondary axes and its limiting forms might be repeated here, substituting, of course, for the bipyramids of the first and second orders the corresponding prisms. 7. Hexagonal basal pinacoid. The faces of this form are parallel to the principal plane of symmetry and possess the following symbols (cca : oca : 0/oa: c), OP, joooij. It is evident from the presence of a center and principal plane of symmetry that two such planes are possible. This, like the prisms, is an open form and must always occur in combination. Figure 141 shows this form in com- bination with the prism of the first order. These are the seven simple forms possible in this system. Their principal features may be summarized as follows: FORMS SYMBOLS Number of Faces | Solid Angles T etrahedral Hexahedral Dodecahedral Weiss Naumann Miller- Biavais Unit Bipyramid First order a : 2 I 2 Prism First Order a : OCrt : a : O0 OOP OOP? j IOIO \ 6 1 Same as in holo- hedral class j Prism Second Order 20, : 20. : a : OOc {1120} 6- Dihexagonal Prism na : pa : a : OO c mPn \hi~k o\ 12 F Upper and Lower Basal Pinacoids O0 : OCa : OC : c ( P u I 1 0001 j- {0001 j 2 ",t !_,/ 1) The term pyramid in itself suggests hemimorphism, since it is not a doubly terminated form as is the bipyramid. HEXAGONAL HEM1HEDRISMS. 55 Combinations. OP Figure 153, <: = u, P I oooi j ; o = - u &/, jiouj & {lonf ; i = 2 p ip u & /, 52021! & 52021}; v = /, {2023 j, ip and /*=/, |ioi2c ; observed on lodyrite, Agl. Fig. 154. Figure 154, p - - w, {ioii[; w = ooP, {1010}, observed on Zincite, ZnO. HEXAGONAL HEMIHEDRISMS. In the hexagonal system four types of hemihedrism, as illustrated by figures 155 to 158, are possible. Fig. 155. Fig. 156. Fig. 157. Fig. 158. a) Trigonal hemihedrism. The three secondary planes of symmetry divide space into six equal sections. x) All faces in alternate sections are extended, the others suppressed, figure 155. b) Rhombohedral hemihedrism. The three secondary planes together with the principal plane divide space into twelve sections, called dodecants, page 45. Faces in alternate dodecants are subject to extension, figure 156. c) Pyramidal hemihedrism. By means of the three second- ary and three intermediate planes twelve sections, figure 157, result. Faces in alternate sections are suppressed. i) Liebisch, however, refers to the sections derived by the intersection of the intermediate instead of the secondary planes. The forms, which result, are in both cases identical. Their positions in respect to the secondary axes differ by an angle of 80. Compare Liebisch, Grundriss der Physikalischen Krystallographie, 1896, 112. 56 HEXAGONAL SYSTEM. d) Trapezohedral hemihedrism. All planes of symmetry possible in this system divide space into twenty-four sections. The extension of faces occurs in alternate sections of this character, figure 158. Of these hemihedrisms, the rhombohedral and pyramidal types are the most important. No representative of the trigonal hemihedrism has yet been observed. 3. DITRIOONAL BIPYRAMIDAL CLASS. ( Trigonal Hemihedrism.} Symmetry. From figure 155 showing this method of hemi- hedrism it is obvious that the secondary planes of symmetry are lost. With them the center and the axis of hexagonal symmetry also disappear. The elements of this class are, hence, one principal and three interme- diate planes, three binary axes parallel to the intermediate, and an axis of trigonal symmetry parallel to the c axis. The binary axes are polar. Figure 159 not only shows these Fig. 159. elements but the application of the hemihe- drism as well. Trigonal bipyramids. From the hexagonal bipyramid of the first order two correlated, congruent forms, each bounded by six equal isosceles triangles, result. These forms and their position in respect to the secondary axes are shown by figures 160-165. They , a: 00 a: a: me are the trigonal bipyramids. The symbols are j mP hohl\i figure 162; , \ohhl\, figure 160. - 2 2 The axis joining the trihedral angles is of one of trigonal symmetry, those passing through the centers of the horizontal edges to the opposite tetrahedral angles are of binary symmetry. These are parallel to the intermediate axes. Trigonal prisms. The corresponding hexagonal prism yields two trigonal prisms, 1} each bounded by three faces, as is shown by figures 163 to 168. Being open forms, they are shown in combination with the basal pinacoid. DITRIGONAL BIPYRAMIDAL CLASS. 57 Fig. 160. Fig. 163. Fig. 166. The symbols are : OOP Fig. 161. Fig. 162. Fig. 165. Fig. 167. Fig. 168. , a: oca: a: oo^l ooP - + I- I! + - , \hoho\, figure 1 68; ohho\, figure 166. The trigonal axis is parallel to the intersection lines of the prism faces, while those of binary symmetry pass from the centers of the faces to those of the opposite edges, figures 166 and 168. Ditrigonal bipyramids. Every dihexagonal bipyramid, when subjected to the trigonal hemihedrism, gives rise to two correlated and congruent forms, bounded by twelve scalene triangles, which are 58 HEXAGONAL SYSTEM. known as the ditrig'onal bipyramids. The derivation and position of these forms are illustrated by figures 169 to 174. _, \na : pa : a : me] mPn The symbols are: +_ -y- , -| , \hikl\, figure in P n . . 171; , \ihkl\, figure 169. The trigonal axis joins the hexahedral angles, those of binary symmetry extend from an obtuse to an opposite more acute tetrahe- dral angle. Fig. 172. I I j_- Fig. 175. Fig. 176. Fig. 177. DITRIGONAL BIPYRAMIDAL CLASS. 59 Ditrigonal prisms. The dihexagonal prism yields two ditri- gonal prisms, as shown by figures 172 to 177. The symbols are [na:pa\a\ oo^l ocP;; mPn 2 j , + -^ , \hiko\, figure 177; , \ihko\ figure 175. The trigonal axis is parallel to the intersection lines of the prism faces, those of binary symmetry are parallel to the intermediate axes. Other forms. The bipyramids and prism of the second order as also the basal pinacoid are not changed geometrically by the trigonal hemihedrism, for none of their faces lie wholly within the sections formed by the secondary axes. Compare figures 159 and 164. These forms are, therefore, apparently holohedral. No representative of this class has yet been discovered. The chief characteristic of the forms may be tabulated as follows: FORMS SYMBOLS 1 Number of Faces Solid Angles Trigonal Tetrahe- dral Hexahe- dral Weiss Naumann Miller- Bravais Trigonal Bipyramids First order _L_ ( a :OCa:a :mc~\ f+r ntP 2 {ho7ik} {ohlik} [ 6 j 2 3 L( 2 } Hexagonal Bipyramids Second order 2a\2a\a\ me mP2 {hh~2hl} Apparently holohedral Ditrigonal Bipyramids , [na \pa\a : mc~\ f + *5! 2 } mPn { hiKl] {i h ic/ } 1 [12 3+3 2 -LI 2 ] ( 2. Trigonal Prisms First order , fa:aca:a:OC^ r rjn p 1-4- (hoTio } {o hH. 0} 1. j 2 OOP -U 2 J ( - 2 Hexagonal Prism Second order 2 a : 20, : a : 00 c OCP2 \ I I 20\ Apparently holohedral Ditrigonal Prisms _L_ [na \pa: OCa:c^ r ,J~Pn \hfko \ \ih~ko \ j. \ ' ^ CfOPn 1 2 J { 2 Basal Pinacoid OCa : OOa :CCa \c OP {o o o i \ Apparently holohedral 60 HEXAGONAL SYSTEM. 4. D1TRIQONAL SCALENOHEDRAL CLASS.*) Fig. 178. (Rhombohedral ffemihedrism.} Symmetry. The principal and second- ary planes of symmetry together with the hexagonal axis are lost. The following elements are present: three intermediate planes, three axes of binary and one of trigonal symmetry, also the center. Figure 178 shows these elements and the applica- tion of the rhombohedral hemihedrism, page 55. Rhombohedrons. From the hexagonal bipyramid of the first order two new congruent forms are the result of this type of hemihe- drism, figures 181 to 183. In the ideal development each of these forms is bounded by six equal rhombs and are called positive and negative rhombohedrons. The lateral edges form a zigzag line about the torm. The six polar edges form two equal trihe- dral angles, bounded by equal edges. These may be larger or smaller than the other trihedral angles, according to the value of a:c. 2 ^ i) Dana terms this class the rhombohedral Fig. 180. Fig. 179. 2) The cube, when held so that one of its axes of trigonal symmetry, page IS, is vertical, may be con- sidered as a rhombohe- dron whose edges and angles are equal. "The ratio, a : c, in this case would be 1 : T/1.5 = 1 : 1.2247+. Those rhombohedrons, there- fore, whose c axes have a greater value than 1.2247+ have pole angles less than 90. When, however, the value is less than 1.2217+, the pole angles are then greater than 90 and, hence, such rhombohedrons may be spoken of as acute and obtuse, respectively, figures 179 and 180. DITRIGONAL SCALENOHEDRAL CLASS. 61 Fig. 181. Fig. 182. Fig. 183. ~ u (a:oca:a:mc] mP T The symbols are _ | ; + ,K\hohl\, figure 183; , K\ohhl\, figure 181. The principal crystallographic axis passes through the two equal trihedral angles, the secondary axes bisect opposite lateral edges. These axes indicate the directions of those of trigonal and binary symmetry, respectively. Scalenohedrons. Every dihexagonal bipyramid gives rise to two congruent forms, bounded by twelve similar scalene triangles, called scalenohedrons, of which one is positive and the other negative, figures 184 to 1 86. Each possesses six obtuse and six more acute polar edges, also six zigzag lateral edges. As is the case with the rhombohedrons, obtuse and acute scalenohedrons are possible, depending upon the value of a : c. Fig. 184. Fig. 185. Fig. 186, 62 HEXAGONAL SYSTEM. ._, . \na :pa : a\mc\ The symbols are : -f- m Pn , K\hikl\, figure 186; - , figure 184. Scalenohedrons with twelve equal polar edges are crystal- lographically impossible, see page 49. The axis of trigonal symmetry passes through the two hexahe- dral angles, while those of binary symmetry bisect the lateral edges. The other holohedral forms remain unchanged by the application of the rhombohedral hemihedrism, compare figures 178, 182, and 185. Abbreviated Symbols of Naumann. In order to more easily express some of the interesting relationships existing between forms of this class, which are quite common, Naumann substituted for P and-f the symbols of the rhombohedrons used above, +R and 4^ mR, respectively. For every rhombohedron there exists a series of scalenohedrons, whose lateral edges coin- cide with those of the rhombohedron, as shown in figure 187. The inscribed rhom- bohedron is known as "the rhombohe- dron of the middle edges. " The scalen- ohedrons may, therefore, be indicated in general by mRn, mR representing the "rhombohedron of the middle edges. " In mRn, n has reference to the value of the c axis 1} of the scalenohedron in respect to that of the rhombohedron mR. For example, in figure 187, the length of the c axis of the scalenohedron is three times that of the inscribed positive unit rhombohedron. The symbol for this scalenohedron is, hence, according to Naumann -{-R3. The prism of the first order OoP, also the basal pinacoid OP, are often written QoR and OR, respectively. The symbols for the other forms mP2, QoP2, ooPw, remain unchanged. Fig. 187. And not to the secondary axes, as is usually the case in the regular Naumann symbols. DITRIGONAL SCALENOHEDRAL CLASS. To transform the full into the abbreviated symbols and vice versa, the following formulae are useful: 2. n __^ 4PI = 2 n 2 mnP 2H . For example, 2R2 = -. 3. *\hikl\ = 4. mRn = K R Here, K = 2R2. I v 2k ti '. (n i). - (n -\- i). ~\- For instance, 2R2 = ^4131}. The principal features of the forms of this class are given in the following table: FORMS SYMBOLS * Number of Faces So'td Angles "3 T3 J^ H Tttrahedial 1 Hexahedral Weiss Naumann Miller- Bravais Rhombohe- drons f a : cr. a : a : me 1 j +mR I mR *\hohl\ \6 J -? ' (^ ' I -I '* J K\o/ihl\ 2 | O Hexagonal Bipyramid Second order 2a: 2a: a : me mP2 K\/l/l27ll\ Apparently holohedral Scalenohedrons , \nd\-pa\a\mc\ ' 1 \- + mRn | mRn *\hikl\ *\ihkl\ } 12 6 2 -I 2 j Hexagonal Prism First order a : oca : a : 00 c GOT? \-\hoho\ 1 Ap j parei holol Itl) tied T ral Hexagonal Prism Second order 20, : 2a : a : CCc CCP2 K \II20\ Dihexagonal Prism na : pa : a : GO c tePn K { hiko \ Basal Pinacoid ooa : cca : ooa : c OR K { OOOI \ 64 HEXAGONAL SYSTEM. Combinations. Many of the more common minerals crystallize in this class, for example, Calcite, CaCO 3 ; Hematite, Fe 2 O 3 ; Corun- dum, A1 2 O 3 ; and Siderite, FeCO 3 . Fig. 188. Fig. 189. Fi- 190. Fig. 191. Figure 188. r = R, K j 101 1 \ ; c = OR, K\OOOI}._ Calcite. Figure 189. e = ^R, K{oii.2} ; m = OoR, K \ ioTp}. Calcite. Figure 190. e = iR, K{oii2(; a = OoPa, K{ 1 120}. Calcite. Figure 191. r = R, K{IOII}; /= 2R, K {o22i\. Calcite. Fig. 192. Fig. 193. Fig. 194. Figure 192. v = RS, KJ2I3I}; r = R, KJIOII}. Calcite. Figure 193. r R, KJIOII}; v = R3, ^{21^1 \ ._ Calcite. Figure 194. c = OR, K{OOOI } ; m = OoR, K{ loiol ; s = R$ ; = QC-P2, K-Jii2o}. Calcite. Fig. 195. Fig. 196. Fig. 197. HEXAGONAL BIPYRAMIDAL CLASS. 65 Fig. 198. Figure 195. d = OR, KJOOOI|; r = R, K{IOII|. Corundum. Figure 196. d OR, K|OOOI|; r R, KJioiif ; n = |P2, KJ2243J; I = OoP2, K { 1 120}. Corundum. Figure 197. r = R, KJIOII}; r' = R, K\ioi4\', p = |P2, K 1 2243}. Hematite. Figure 198. r = R, KJIOII j; 4r = 4R, *J404il; w = R5, S325i}; * = R3 KJ2i3if; g- = ooR, K{ loioj. Calcite. 6. HEXAGONAL BIPYRAMIDAL CLASS.*) (Pyramidal Hemihedrism.) Symmetry. The secondary and intermediate planes and the six axes of binary symmetry are lost. The remaining elements are the principal plane, the hexagonal axis and the center of symmetry, figure 203, page 66. Hexagonal bipyramids of the third order. Figure 200 illus- trates the pyramidal method of extension and suppression of faces on the dihexagonal bipyramid. It is obvious that this bipyramid yields two new forms, each bounded by twelve equal isosceles tri- angles. They are termed positive (figure 201) and negative*^ (figure 199) hexagonal bipyramids of the third order. In form these bipyramids do not differ from those of the first and second orders. Figures 202 and 204 show the positions of these bipyramids in respect to the secondary axes. The inner and outer light outlines represent the horizontal cross-sections of the bipyramids of the first and second orders, respectively, while the heavy outlines show the cross-sections of those of the third order. These occupy an inter- mediate position. , {na\pa\a\ me] , mPn I ~ ' i~ The symbols are: + I 4 iare2Oi; , ir\ihkl\ figure 199. TT\/likl\ fig- I)" Called by Dana the pyramidal group, 2) The terms right and left are sometimes used. 66 HEXAGONAL SYSTEM. The axis of hexagonal symmetry passes through the hexahedral angles. The position of the secondary crystallographic axes is shown by figures 202 and 204. These do not join the tetrahedral angles or the centers of the basal edges, but some point between them, which is dependent upon the value of n. Compare figures 135 to 136. Fig. 199. Fig. 202. Fig. 200. Fig. 203. Fig. 201. Fig. 204. Fig. 205. Fig. 206. Fig. 207. HEXAGONAL BIPYRAMIDAL CLASS. 67 Hexagonal prisms of the third order. From the dihexagonal prism two congruent forms consisting of six planes, the positive (figure 207) and negative (figure 205) hexagonal prisms of the third order are derived. Figures 202 and 204 show these forms and their relation to the other hexagonal prisms. Their position in regard to the secondary axes is the same as for the bipyramids of this order. _, The symbols are: _ na :pa : a : ccc ;H ooPn \hiko figure 207; , Tt\ihko\ figure 205. The axis of hexagonal symmetry extends parallel to the edges. The other holohedral forms are unaltered by the application of the pyramidal hemihedrism, since the suppression effects only one half of each face. Compare figures 200, 203 and 206. They are, hence, apparently holohedral. The principal features of this class have been summarized in the following table: FORMS SYMBOLS "o |i- ^ Solid 'Angles u EC aJ-o H arenl lohe i la HTJ as iy dral Weiss Naumann Miller- Bravais Hexagonal Bipyramids First order a : O0a:a:mc mP 7f\hoHl\ i App ho J Hexagonal Bipyramids Second order 20, : 20, : a : me mP2 TT\hh2hl} Hexagonal Bipyramids Third order , f na : pa : a : mc\ f mPn TT\ hill \ ir\i h~k l\ u J 6 2 \ 2 mPn 1 l - J (' * Hexagonal Prism First order a : oca : a : OOr OOP TT | k O h 0| App 1 ho J arent lohe< iy iral Hexagonal Prism Second order 20. : 20, : a : OO c OOP2 7TJ7 / 20 \ Hexagonal Prisms Third order _j_ ( na : pa : a : OOc ] f CCPn ir\hiJo\ ir\ihk~o \ r + - OO Pn -{ ' J \ Basal Pinacoid CCa : OOa : OOa : c OP ir\o o 01 \ Apparently holohedral 68 HEXAGONAL SYSTEM. Fig. 208. Combinations. Figure 208. m = oop ? ; c = OP, TrjoooiS; x -- P, TrjiouJ; 5 = 2P2, irjll2l|; U = + , ir j I 231}. This combination has been observed on apatite, Ca 5 Cl(P0 4 ), HBXAQONAL TRAPBZOHEDRAL CLASSY) ( Trapezohedral Hemihedrism. ) Symmetry. The center and all planes of symmetry disappear. Hence, the ele- ments of this class are the hexagonal axis and six binary axes of symmetry, figure 209. Hexagonal Trapezohedrons. From figure 211 it is evident that by means of this hemihedrism, the dihexagonal bipyramid yields two correlated forms bounded by similar trapeziums. The forms are enan- tiomorphous and designated as the right and left hexagonal trapezohedrons. . 209. Fig. 210. Their symbols are: r, I _ in Pn figure 212; l,T\kihl\ figure 210. Fig. 211. f na : fia : a : me] * . Fig. 212. : , r\hikl\ 1) Trapezohedral group of Dana. HEXAGONAL TRAPEZOHEDRAL CLASS. 69 On the right trapezohedrons the six longer basal edges are inclined to the left of the observer and, vice versa, to the right in the case of the left form, compare figures 212 and 210. The axis of hexagonal symmetry joins the hexahedral angles. The secondary crystallographic axes connect the centers of the longer basal edges, the intermediate those of the shorter. These axes possess binary symmetry. Since this hemihedrism calls for the suppression of only portions of each of the faces of the other holohedrons, no new forms can be derived from them. They are apparently holohedral. The chief features of the forms of this class are given in the following table: FORMS SYMBOLS Number of Faces Sohd Angles Trihedral Hexahedral Weiss Naumann Miller-Bravais Hexagonal Bipyramids First order a : oca : a : me mP r\hohl\ Appar- ently holohe- dral Hexagonal Bipyramids Second order 2(i : 2a : a : me mP2 T\hh2~hl\ Hexagonal Trapezohedrons 1 ( na: pa: a: me] f mPn r\hikl\ T\kihl\ } 12 12 2 r ; mPn '"[ * J I/ 2 Hexagonal Prism First order a : oo a : a : oc c OOP T\hoho\ 1 AF ho j >pa ent loh di r- iy e- al Hexagonal Prism Second order 2a : 2a : a : occ 00 P2 T\II20\ Dihexagonal Prism na : pa : a : occ oo Pn r\hiko\ Basal Pinacoid oca : ooa : ooa : c OP r \OOOl} 70 HEXAGONAL SYSTEM. No crystals showing the occurrence of the hexagonal trapezo- hedrons have as yet been discovered. The double salt, barium stibio- tartrate and potassium nitrate, Ba(SbO) 2 (C 4 H 4 O 6 ) 2 + KN O 3 , and the corresponding lead salt are assigned to this class. The crystals are apparently holohedral, the etch figures, however, reveal the lower grade of symmetry. 7. DITRIQONAL PYRAMIDAL CLASS.V ( Trigonal Hemihedrism with Hemimorphism. ) Symmetry. If the forms of the ditri- gonal bipyramidal class, page 56, become hemimorphic, the principal plane of symmetry and, of course, the axes of binary symmetry also are lost. Three intermediate planes and a polar axis of trigonal symmetry parallel to the c axis are the only elements left. These symmetry relations are shown in figure 213. The forms of this class may, moreover, be considered also as derived from those of the ditrigonal scalenohedral class, that is, from those showing rhombohedral hemihedrism, page 60. A comparison of figures 159 and 178 reveals the fact that when hemimorphism becomes effective the resulting forms must in both cases be the same. Therefore, this class is often known as the rhombohedral hemi- morphic class. Trigonal pyramid of the first order. The trigonal bipyramid, page 56, the result of the application of the trigonal hemihedrism to the hexagonal bipyramid of the first order, yields upon the addition of hemimorphism upper and lower forms, consisting of but three faces each. Since there are two trigonal bipyramids, one positive and the other negative, it follows that each hexagonal bipyramid of the first order now yields four forms. 2) Being singly terminated, they are termed trigonal pyramids. 3) Fig. 213. !) Called by Dana the rhombohedral hemimorphic class. 2) The forms are, hence, sometimes said to be the result of tetartohedrism, namely the ditri- gonal Pyramidal tetartohedrism . 3) Compare footnote on page 51. The letters u and / designate the upper and lower forms, respectively. DITRIGONAL PYRAMIDAL CLASS. 71 Their symbols are: f a : oo a : a : mc\ m P Positive upper, + I u, ~u, \hohl\. f : oo a : a : me] . m P . 7 - Positive lower, -H /, + -/, {Ao A/}. 4 4 fa : oca : a : me] mP Negative upper, - - \u, - u, \o/i/il\. [a : oca : a : me] mP Negative lower, - /, - - /, \ohhl\. Hexagonal pyramids of the second order. The hexagonal bipyramid of this order remains unaffected by the trigonal hemihe- drism, page 59, but now yields an upper and lozuer form, each consisting of six faces, termed hexagonal pyramids of the second order. ( 2a : 2a : a : me^ mP2 The symbols are: \u, 1; ^-U 1 {hh2hl\, ?, \hh~2lil\. Ditrigonal pyramids. The dihexagonal bipyramid by means of the trigonal hemihedrism yields the positive and negative ditrigonal bipyramids, page 57. Each of these bipyramids now gives rise to an upper and lower form, termed ditrigonal pyramids. 1} These, like the preceding pyramids of this class, are open forms. They consist of six faces. The symbols are: [na : pa : a : me] mPn Positive upper, - \ u, -\ u, \hik l\. I 4 J 4 Positive lower, na: P a:a:me] ^ + mPn ^ -- !) Since, as said on page 70, these forms may be considered as derived from those showing rhombohedral hemihedrism, the terms rhombohedron- and scalenohedron-like faces are sometimes used for the trigonal and ditrigonal pyramids, respectively, the forms being assumed to be half of the rhombohedron and scalenohedron. 72 HEXAGONAL SYSTEM. \na : pa :a : me} mPn ,7-75 Negative upper, - I , -- u, \thkl\. I 4 J 4 ^ na : pa : a : mc\ mPn . ,.,--=, Negative lower, I - /, - - /, {thkl\. I 4 J 4 Trigonal prisms. Hemimorphism does not affect, morphologi- cally, the trigonal prisms, the hemihedrons of the hexagonal prism of the first order, page 56, for each face belongs alike to the upper and lower poles. a: aoa:a: ccc OoP The symbols are : + J - \nono\\ ~ Hexagonal prism of the second order. It will be recalled that the hexagonal prism of this order remains unchanged, geometrically, by the trigonal hemihedrism, page 59. Hence, as in the preceding case, each face belongs alike to the upper and lower poles, hemi- morphism also can not be effective in the production of new forms. This prism is, therefore, still apparently holohedral. Its symbols are: 2a : 2a : a : one, ooP2, \hh2ho\. Ditrigonal prisms. As in the case of the trigonal prisms, page 59, these forms remain unchanged. (na \ pa :a\ ace] ccPn The symbols are : -f^ I - - - I ; + , \htko\\ QQ Pn These, together with the trigonal prisms, are not to be dis- tinguished morphologically from those of the ditrigonal bipyramidal class. They are apparently hemihedral. Basal pinacoids. This form now evidently consists of two types, upper and lower, of one face each. f oca : oca : oca : c] OP The symbols are: - - (,/, : - u, 5oooi}; - /, 1 0001 1 . 2 In all the above forms the crystallographic axes are located as in the ditrigonal bipyramidal class. DITRIGONAL PYRAMIDAL CLASS. 73 The following table shows the principal features of the forms of this class: FORMS SYMBOLS Number of Faces Solid Angles 1 H Hexahedral Weiss Naumann Miller-Bravais Trigonal Pyramids First order _j_ 7y _,_y fa:OCa:a:mc] + mP u 4 mP 4 \ holil \ [hohl\ \ohhl\ "3 J 6 I - - ' ( 4 \ Hexagonal Pyramids Second order j\2a : 20, : a : mc\ mP 2 \hh2~hl\ \hh2h l\ - I I 2 m P2 It, [ 2 J 2 Ditrigonal Pyramids \ U \-l m Pn 4 mPn ~ u 4 mPn 4 \ihll\ \ihkl\ 6 1 '-( 4 \ Trigonal Prisms (a : CO a : a : OC<:1 r OOP | h o ~h o [ \ o h h o j Appar- ently of ditrigonal bipyra- midal class. 2 OCP -I 2 Ditrigonal Prisms f# : pa \ a : mc^ , &Pn (. j - ) t > 2 CCPn 1 2 J 2 Hexagonal Prism Second order 20. : 2a : a : OO c OO P2 \ h h 2 ti o \ Apparently holohedral Basal Pinacoids r OP OP { I } \OOOI \ I - - 1 2 J Combinations. The mineral tourmaline furnishes excellent combinations of the above forms. 74 HEXAGONAL SYSTEM. In the accompanying figures P - 214 and 215, P = 4- u, { ion } ; 4 P __ 2 P P = + - /, Join | ; o = -- w, 4 4 OP 50221 i; c - I, {oooij; n t = + Fig. 215. u, 2131; / = OOP f . . lOIIOj H20 4 2 As indicated on page 70, these forms may be assumed as derived from those of the ditrigonal scalenohedral class, i. e., showing rhombohedral hemihedrism, and, hence, are often designated as follows : = ; c= I ooR = ooP2; / -- ; o = 2Ru. 8. HEXAGONAL PYRAMIDAL CLASS. (Pyramidal Hemihedrism with Hemimorphism.) Symmetry. This class possesses but one element of symmetry, namely, a polar axis of hexagonal symmetry, as shown in figure 216. This is obvious, for when the forms of the hex- agonal bipyramidal class, page 65, become hemimorphic, the principal plane and center of symmetry are of necessity lost. Forms of exactly the same character are, moreover, to be derived by applying hemimorphism to those of the hexagonal trapezohedral class. Compare figures 203 and 209. Forms. The bipyramids of the first, second and third orders of the hexagonal bipyramidal class 1} now consist of upper and lower 1) If, however, hemimorphism be applied to the hexagonal trapezohedral forms, the hexagonal trapezohedrons and dihexagonal prism would yield forms corresponding precisely to the pyramids and prisms of the third order. The other forms being apparently holohedral in both classes would, of course, be affected alike by the introduction of hemimorphism. Fig. 216. HEXAGONAL PYRAMIDAL CLASS. 75 pyramids. The prisms, on the other hand, remain unaffected mor- phologically. The basal pinacoid, of course, is now composed of an upper and lower form. As in the preceding class, these are now open forms. Their position in respect to the crystallographic axes is the same as in the hexagonal bipyramidal class, page 65. The principal features are given in the following table: FORMS SYMBOLS FACES Weiss Naumann Miller- Bravais Hexagonal Pyramids First order .(a : OOa : a : mc\ u / f mP u mP { ~ l \ho~hl \ \ho~hl\ 1 ' I 2 J Hexagonal Pyramids Second order , ( 20, : 20, : a : me } ( mP 2 u { hh2hl \ \hh~ 2 Jil\ 6 J m 2 p 2 I 2 J ( 2 l Hexagonal Pyramids Third order + u + [{na:pa:a:mc} ( mPn \hi~kl] \hi~kl \ \k1Ji l\ \kihl \ } 6 + 4 U + mPn l mPn u 4 mPn 1 ( ~ 4 1 4 J Hexagonal Prism First order a : OC a : a : X c 00 P {ho~ho\ Appar- ently [ holohe- dral Hexagonal Prisms Second order 2(i : 20, : a : Qcr OOP 2 \ hh~ 2 Jil\ Hexagonal Prisms Third order fmz :pa:a:VOc'\ f OOPn 00 Pn I { hi~ko\ \kTko \ 1 6 1 2 J Basal Pinacoids , fOC0 : OCa : OC : c] ( OP u ( ?< { oooi \ \ ooo7\ i ' *' *( 2 J Combinations. By means of etch figures crystals of a compounds have been referred to this class. Among the number of minerals, 76 HEXAGONAL SYSTEM. nepheline, Na 8 Al 8 Si 9 O 34 , is to be mentioned. Figure 217 shows a crystal of strontium antimonyltartrate, Sr (SbO) 2 p = ~u, {ion -/, J202IL HEXAGONAL TETARTOHEDRISMS. The tetartohedral forms may be assumed as derived from the holohedral by applying simultaneously two types Fig. 217. of hernihedrism, page 55. Thus, by combining the tri- gonal and pyramidal hemihedrisms, the trig-onal tetarto- hedrism results, figure 218. A second type, known as the trapezo- hcdral tetartohedrism, is the result of the combination of the Fig. 218. Fig. 219. Fig. 220. rhombohedral and trapezohedral hemihedrism, figure 219. The simultaneous application of the rhombohedral and pyramidal hemihe- drisms gives rise to the rhombohedral tetartohedrism, figure 220. These three are the only types possible, for other combinations of the four methods of hemihedrisms do not satisfy the conditions of tetartohedrism. 9. TRIGONAL BIPYRAMIDAL CLASS. ( Trigonal Tetartohedrism. ) Symmetry, This class possesses two ele- ments of symmetry, namely, a principal plane Fig. 221 and an axis of trigonal symmetry, figure 221. TRIGONAL BIPYRAMIDAL CLASS. 77 Trigonal b i p y r a- mids and prisms of the first order. As can be readily seen from figures 222 and 223, the hexa- gonal bipyramids and prism of the first order now yield trigonal bipyramids and prisms, respectively. These forms do not differ morphologically from those of the ditrigonal bipyramidal class, compare figures 160 to 168. The symbols for the trigonal bipyramids of the first order are: a : oca : a : me] . mP ^~ -J, + , \hohl\\ and a : ooa : a : me] mP , J, - , {ohhl\. Those for the corresponding trigonal prisms being: a : oca : a : ooc ocP + \hoho\\ and Fig. 222. Fig. 223. a : cca : a : -, \ohho\. Trigonal bipyramids and prisms of the second order. These forms are obtained from the hexagonal bipyramid and prism of the Fig. 224. Fig. 225. Fig. 226. Fig. 227. Fig. 228. Fig. 229. 78 HEXAGONAL SYSTEM. second order as can be seen from figures 224 to 229. The position of these forms as well as those of the other two orders in respect to the crystallographic axes is shown in figure 237. The symbols for the trigonal bipyramids of the second order: ( 2a : 2a I me ?;z P 2 {h h 2 hl\, figure 226; and : 2a : a : me , - ^-, J 2 hhhl\, figure 224. Those of the corresponding trigonal prisms: 2a : 2a : a : ooc] ocP2 , \h1i2ho}, figure 229, and \2a : 2a : a : ooc] Gc?2 I , - , | 2 h h h o \ , figure 227. ( 2 J 2 Trigonal bipyramids and prisms of the third order. The dihexagonal bipyramid now furnishes four new forms, termed trig- onal bipyramids of the third order, while the dihexagonal prism yields the corresponding prisms. Figures 230 to 235 show the derivation of two forms of the bipyramid and prism. Figures 236 and 237, however, show the position of these forms in respect to the crystallographic axes as also to the forms of the other orders. Fig. 230. Fig. 231. Fig. 232. Fig. 233. Fig. 234. Fig. 235. TRIGONAL BIPYRAMIDAL CLASS. 79 The symbols of the bipyramids are: Positive right, { na : pa : a : me + r [~ ~2~ Positive left, ; ^na-.pa-.a-.mc^ . + . {hikl}, figure 232. . , \kihl\, figure 230. Negative right, I 2 Negative left, { na : pa : a : me Those of the prisms are: [ na : pa : a : Gc 2 _ J 4 -I- -; \khil\. ; \hiko\i figure 235. ; \kiho\, figure 233. na:pa:a:aoc] QcPw ( .,-r i r\- -j- I; r ~^ ; jiA^o}. Fig. 236. Fig. 237. The basal pinacoid, as is obvious from figures 223, ^28 and remains unaltered. 80 HEXAGONAL SYSTEM. The following table summarizes the principal features of each form. No representative of this class has yet been discovered. FORMS SYMBOLS Number of Faces Solid . Angles Trihedral H Weiss Naumann Miller-Bravais Trigonal Bipyramids First order . f# : QC # : a : mc^ f+ 1 _^/ ) L ^ { A^ J/} J0AA/ | 1- 2 3 ( * I Trigonal Bipyramids Second order _!_ f.?a : 2a \ a : ntc\ mP2 { AA J^/ \ \ 2hhhl \ Q 2 3 mPz I J~ I 2 J Trigonal Bipyramids Third order _j_^ + ,\na\pa\a\mc'\ ' + r=S 4 mPn \hiJl\ \kthl \ \ ih~kl\ \ kJTi \ ] Q J 2 3 \ + * - mPn -'- [ 4 J 4 t mPn ; ^ Trigonal Prisms First order ^{a: OOa :a:OOc] L^P 00 P l~"T^ \hoho\ \oh~ho] 3 -L 2 J Trigonal Prisms Second order _,_ \2a\2a : a: OOc] f oc/ 7 ^ \II~2 0\ { ^77o\ 3 . + ^ 00 P2 2 I 2 J Trigonal Prisms Third order , r | i\na\pa\a\&c\ f oc />,/ \hi~ko \ \ k~i~ho \ \ ihlio\ \ kh^o \ 3 J - - 4 ,r \0 I } 2 - - TRIGONAL TRAPEZOHEDRAL CLASS. 81 10. TRIGONAL TRAPEZOHEDRAL CLASS.*) ( Trapezohedral Tetartohedrism. ) Symmetry. A principal axis of trigonal symmetry and three polar axes of binary symmetry are the only elements in this class, figure 238. Rhombohedrons of the first order. From the hexagonal bipyramid of the first order, two rhombohedrons of the same order result as can be seen from figure 239. These forms are identical, morphologically, with those of the ditrigonal scalenohedral class, page 60. , {a:Cioa:a: me Their symbols are: Hr ' + mR, \ho7il\\ mR, \oh7il\. 2} Trigonal bipyramids of the second order. The hexagonal bipyramids of the second order, figure 240, yield two trigonal bipyramids of this order. They are identi- cal, geometrically, with those of the trigonal bipyramidal class, page 77. 3 I \ 2a\ ia \a\mc The symbols are: Fig. 238. Fig. 239. Trigonal Trapezohedrons. The dihex- agonal bipyramids yield four new forms, each bounded by six faces, which in the ideal development are equal trapeziums. Figure 242 shows the application of the Fig. 240. trapezohedral tetartohedrism so as to give rise to the positive right trigonal trapezohedron, figure 243. Fig- ure 241 shows the positive left form. These are enantiomorphous. 1) The trapezohedral group of Dana . 2 ) Sometimes the two Greek letters K T are placed before the Miller indices. See footnote on page 40. 3) Sometimes designated as right and left. 82 HEXAGONAL SYSTEM. The trigonal trapezohedrons may also be conceived as derived from the scalenohedron by the subsequent application of hemihe- drism. This is shown by figure 245, a negative scalenohedron, which now yields the negative right (figure 246) and negative left (figure 244) trigonal trapezohedrons, respectively. Fig. 244. Fig. 245. Fig. 246. The symbols may be written as follows: i) Positive right, na\pa\a\mc\ , ^mPn . r l -j- j , -h r - , \hikl\> figure 243- TRIGONAL TRAPEZOHEDRAL CLASS. 83 ) Positive left, f na : pa : a : me] . . mPn -h / , H- / - L 4 J 4 3) Negative right, na : pa : a : me] mPn * 4 4 4) Negative left, na : pa : a : me] _ mPn 4 J' 4 ' r \k i h l\ , figure 241. i h k /}, figure 246. k h i /}, figure 244. Forms i and 2, 3 and 4 are among themselves enantimorphous, while i and 3, 2 and 4 are congruent. The polar axes of binary symmetry bisect the zigzag edges. Trigonal prisms of the second order. Figure 247 shows that the hexagonal prism of this order yields two trigonal prisms. These are similar to those of the trigonal bipyramidal class, page 77. + The symbols are : -, \hh2ho\\ OCP2 2 h h h o Ditrigonal Prisms. The dihexagonal Fig 2 47. prism now gives rise to two ditrigonal prisms as shown in figures 248, 249 and 250. These are similar, morpho- logically, to those of the ditrigonal bipyramidal class, page 59, but differ from them in respect to the position of the crystallographic axes. A comparison of figures 251 with 172 and 174 shows the difference. Fig. 248. Fig. 249. Fig. 250. 84 HEXAGONAL SYSTEM. ,~, , \na : -ha : a :Ooc The symbols are : Hr - 1 ; -f- figure 250; - -, \ki7io\, figure , \hiko\t Fig. 251. Fig. 252. The other forms, the hexagonal prism of the first order and the basal pinacoid, remain unchanged, figure 252. The important features of this class are given in the fol- lowing table: FORMS SYMBOLS Number of Faces Solid Angles |1 si H^ Weiss Naumann Miller-Bravais Rhombohedrons First order fa:QOa:a: me f +mR \ mR {A o A /} {oh hi) \ App ty \ hed 2 irent- emi- 3 L 2 Trigonal Bipyramids Second order _L_ \2a : 20, : a : me] f mP2 { 2 h'hhl} \ ^ ^ I 2 } Trigonal Trapezo- hedrons fna:pa:a:mc] f mPn {hill} {kihl} [i hkl} 6 J 2 6 4 mPn '1 4 J 4 j mPn 4 Hexagonal Prism First order a : CQa : a : CCc OOP \hoHo\ 6 Appj he rent- holo- iral Trigonal Prisms Second order OO P2 {hh'zho} \2hh~ho\ F 2 2 (. 2 | Ditrigonal Prisms _l_ i L___! \ CCPn \hiko\ \kTJto] 6 2 j OOPn { 2 -( 1 J Basal PJnacoid CGa : CCa : OOa : c OP \O O O I\ 2 Apparent- ly holo- hedral *) Often designated as right and left forms. UNIVERSITY 1 C4L!? TRIGONAL RHOMBOHEDRAL CLASS. 85 Combinations. Quartz, SiO 2 , and Cinnabar, HgS, furnish excellent examples of minerals crystallizing in this class. Fig. 253. Fig. 254. Fig. 255. Figures 253 and 254. a oo R j ioio( ; r = + R { 101 1 j ; 2P2 = R{oiu}; $ (figure 254) = + -{ Il2 i}, (figure253) 21 fi}; * (figure 254) = + r Figure 255. r = - [ JS^il, (figure 253) + /- 4 4 ; w = OoR{ioio}; r = -f R{ ion | ; -IP4 g- = iR|oii2(; n' = 2R|o22ii; and x = + / ^-^ 18355. J Cin- nabar. //. TRIGONAL RHOMBOHEDRAL CLASS. 1 ) ( Rhombohedral Tetartohedrism. ) Symmetry. This class possesses a prin- cipal axis of trigonal symmetry. A center , of symmetry is also present, figure 256. Rhombohed r o n s of the first order. The Fig. 256. hexagonal bipyramid of the first order, figure 257, now yields two rhombohedrons which are identical, geometric- Fig. 257. The tri rhombohedral group oi Dana. HEXAGONAL SYSTEM. ally, with those of the ditrigonal scalenohedral class, page 60, figures 179 and 181. The symbols are: I f a : 00 a : a : mc\ \ mR{o/i/il\. Rhombohedrons of the second order. Figure 259 shows the application of rhombohedral tetartohedrism on the hexagonal bipyr- amid of the second order. This form now yields the rhombohedrons of the second order. Fig. 258. The symbols are: , \2a : 2a : a : me Fig. 259. Fig. 260. + - \hh2hl\ figure 260; 1 2 h h h l\ figure 258. Rhombohedrons of the third order. From the dihexagonal bipyramid four rhombohedrons of the third order result. Two of Fig. 261. Fig. 262. Fig. 263. TRIGONAL RHOMBOHEDRAL CLASS. 87 these are shown in figures 261, 262 and 263. The position of these rhombohedrons with respect to the crystallographic axes is shown by figures 236 and 237, which also indicate the relation existing between the rhombohedrons of the three orders. The symbols are: Positive right, na : pa : a : me} 4 Positive left, f na : pa : a : me ' 4 Negative right, f na : pa : a : me ~~4~~ Negative left, C w : ^> : : w<7 \k~hil\. mPn ( , . r ,) ^ r- , \hikl\, figure 263. 4 ,wPw r 4 mPn r \ki hl\, figure 261. \ihkl\. mPn 4 > Hexagonal prisms of the third order. Figure 264 shows that the dihexagonal prism now yields two hexagonal prisms of the third order. Compare figures 205 and 207. The symbols are: oo P^ \na \pa :a: ccPn r \hiko\-, Fig. 264. \kiho\. Other forms. The remaining holo- hedrons are unchanged by this tetartohedrism, see figures 265 and 266. Fig. 265. Fig. 266. 88 HEXAGONAL SYSTEM. The principal features may be tabulated as follows: FORMS SYMBOLS V) V u cS b Solid Angles Weiss Naumann Miller- Bravais Trihedral Rhombo- hedrons First order faiocaia: mc^\ [ +;/? -mR |A0/} 6 Apparently of ditrigonal scalenohe- dral class 1 * J |o^A/} Rhombo- hedrons Second order \2a : 2a : a : me] ' , wft {hh~2lil\ \2hhh l\ 1 -6 J__ 6 2 1 >-h 6 1 mP2 I 2 J 2 Rhombo- hedrons Third order , r , rfna-.pa-.a-.tnc} mPn \hikl\ \k~ih l\ \ihkl\ {khTl\ 2 l > + 6 1 / 4- mPn I / ^ mPn ~ L ( 4 J 7 4 ^/^^ / 4 Hexagonal Prism First order a : oca : a : aoc OOP \IOIO\ [6 J Apparently holohedral Hexagonal Prism Second order 2a : 2a : a : occ OOP2 \II20} Hexagonal Prisms Third order , \ na : pa : a : oo c "] . ocPn . + ~ ocPn \hiko\ \ki7io} I 6 JL 2 Apparently of hexagonal bipyramidal class 1 l * J I Basal Pinacoid ooa : oca : oca : c OP \000l] Apparently holohedral Three equal edges. TRIGONAL PYRAMIDAL CLASS. 89 Combinations. Dolomite, CaMg(CO 3 ) 2 , and dioptase, J^CuSiO^ crystallize in this class. Figure 267. m = OcP2, j 1 120} ; r = 2R, 5 022 1 j ; 5 = / * , {18. 17. 1.8}, dioptase. Figure 268. m = 4R, {4041 } ; VP 2 r = R, . jioiij; ?z = > 5i6.8-8.3j 1 >; *: = OR, {oooi}, dol- omite. Fig. 268. 12. TRIGONAL PYRAMIDAL CLASS. [Ogdohedrism. "] Trigonal Tetartohedrism with Hemimorphism.) Symmetry. The only element of symmetry is a polar axis of trigonal symmetry parallel to the c axis, figure 269. ^Mp Forms. When hemimorphism becomes effective on forms of the trigonal bipyramidal class, page 76, it is evident the various bipyramids and the basal pinacoid give rise to upper and lower forms. The prisms, however, remain unchanged since they belong alike to the upper and lower poles. The principal features of these forms are given in the following table: Rhombohedron of the second order. 90 HEXAGONAL SYSTEM. FORMS SYMBOLS FACES Weiss Naumann Miller-Bravais Trigonal Pyramids First order [- a : OCa : a : me ] J ^'u 4 -\ ~ l mP u [-?' \hoHl\ \ho~hl\ 3 .- 1 I 4 J "' \ok~hl] \oh~hl \ Trigonal Pyramids Second order _,_ f 20 : 20. : a : me ~] f+ S T" + f T' _^ a 4 mPa l~^~ ' \htT2~hl \ \hh~2~hl \ \2hhhl} \ 2 hhhl \ 3 J ~ I 4 J "' Trigonal Pyramids Third order +r _._/ r :>: a :^ , r w^w + r-y- w/^n + ' T - mPn - r -T , mPn -i _ +-T' + / W L^/ 6* - r wPn / "y~^ -/ "L^?/ P2 "I * J ( Trigonal Prisms Third order f:M:a: QCc 'j f+'^ QC/>n \hi~ko \ \kiho \ \hiko \ \kh~io \ ^ r ODP '^l ^ J 4 _/ OP** 1 * Basal Pinacoids f QCfl : QCa : QCa : c 1 f(?/> M j^ f \OOOI\ \ooo7} 1 1 * J"' TRIGONAL PYRAMIDAL CLASS. P Combination. Figure 270. r= + = - - w, J022I}; ^=/, _ 2134}. Sodium joooij; /= periodate, NaIO 4 + 3H 2 O. 91 Fig. 270. TETRAGONAL SYSTEMS Crystallographic Axes. The tetragonal system includes all forms which can be referred to three perpendicular axes, two of which are equal and lie in a horizontal plane. These are termed the secondary or lateral axes and are designated as the a axes. Per- pendicular to the plane of the lateral axes is the -principals? c axis, which may be longer or shorter than the a axes. The axes, which bisect the angles between the a axes, are the intermediate axes. They are designated as the b axes in figure 271. Crystals of this system are held so that the c axis is vertical and one of the a axes is directed towards the observer. Since the lengths of the a and c axes differ, it is necessary to know the ratio existing between these axes, that is, the axial ratio, as was the case in the hexagonal system. Compare pages 7 and 43. -c Fig. 271. Classes of Symmetry. symmetry, as follows: This system embraces seven classes of 1 . Ditetragonal bipyramidal class 2. Ditetragonal pyramidal class 3. Tetragonal scalenohedral class 1 4. Tetragonal bipyramidal class \ 5. Tetragonal trapezohedral class j 6. Tetragonal pyramidal class 7. Tetragonal bisphenoidal class (Holohedrism.) { Holohedrism and} \hemimorphism. J (Hemihedrism) . [Hemihedrism and\ \hemimorphism. J ( Tetartohedrism). Also termed quadratic or pyramidal systei [92] DITETRAGONAL BIPYRAMIDAL CLASS. 93 No representative Classes i , 3 and 4 are the most important, of class 7 has yet been observed. DITETRAQONAL BIPYRAMIDAL CLASS. 1 ) ( Holohedrism. ) Symmetry, a) Planes* In this class there are five planes of symmetry. The plane parallel to the plane of the secondary and intermediate axes is termed the principal plane. The vertical planes includ- ing the c and one of the a axes are called the secondary planes, while those which include one of the b axes are termed the intermediate planes, figure 272. The principal and secondary planes divide space into eight equal parts, termed octants. The five planes, figure 272, divide it into sixteen equal sections. The five planes may be designated as follows: i Principal -h 2 Secondary + 2 Inter- mediate = 5 Planes. b) Axes. Parallel to the c axis, there is an axis of tetrag-onal symmetry. The axes parallel to the secondary and inter- mediate axes possess binary symmetry, figure 273. These may be written: i| -}-2+20:=5 axes. c) Center. A center of symmetry is also present in this class. These elements of symmetry are shown in figure 274, which represents the projection of the most complex form upon the principal plane of symmetry. Flg * 2 ' 4 * Tetragonal bipyramid of the first order. This form is anal- ogous to the octahedron of the cubic system, page 19. But since Fig. 273. 1) The normal group of Dana. 94 TETRAGONAL SYSTEM. Fig 275. the c axis differs from the secondary axes, the ratio must be written (a : a : c\ which would indicate the cutting of all three axes at unit distances, 1 ) figure 275. But since the inter- cept along the c axis may be longer or shorter than the unit length, the general ratio would read (a : a : me), where m is some value between zero and infinity. Like the octahe- dron, this form, the tetragonal bipyramid,^ is bounded by eight faces which enclose space. The faces are equal isosceles triangles when the development is ideal. The Naumann and Miller symbols are: P, | in}; or raP, \hhl\. The principal crystallographic axis passes through the two tetra- hedral angles of the same size, the secondary axes through the other four equal tetrahedral angles, while the intermediate axes bisect the horizontal edges. Tetragonal Mpyramid of the second order. The faces of this form cut the c and one of the a axes, but extend parallel to the other. The parametral ra,tio is, therefore, a : O0 : me. This requires eight faces to enclose space and the form is the bipyramid of the second order, figure 276. The symbols according to Fig. 276. Naumann and Miller are: raPoc, \ohl\. In form this bipyramid does not differ from the preceding, but they can be readily distinguished on account of their position in respect to the secondary axes. In this form, the secondary axes bisect the horizontal edges and the intermediate axes pass through the four equal tetrahedral angles. This is the 1) Indicating a unit form, compare page 7. 2) The more the ratio a : c approaches 1 : 1, the more does this form simulate the octahedron. This tendency of forms to simulate those of a higher grade of symmetry is spoken of as pseudo symmetry. DITETRAGONAL BIPYRAMIDAL CLASS. 95 Fig. 277. opposite of what was noted with the bipyramid of the first order, compare figures 275 and 276. Hence, the bipyramid of the first order is always held so that an edge is directed toward the observer, whereas the bipyramid of the second order presents a face. In both bipyramids the principal axis passes through the two equal tetra- hedral angles. Di tetragonal bipyramid. The faces of this bipyramid cut the two secondary axes at different distances, whereas the intercept along the c axis may be unity or me. Sixteen such faces are possible and, hence, the term ditet- rag-onal bipyramid is used, figure 277. 1} The symbols are: (a\na:mc\ mPn, \hkl\. Since the polar edges 2) are alternately dissimilar it follows that the faces are equal, similar scalene triangles. The ditetragonal bipyramid possessing equal polar edges is crystallographically an impossible form, for then the ratio a : na : me would necessitate a value for n equal to the tangent of 67 30', namely, the irrational value 2.4i42 + . 3) From the above it follows that when n is less than 2.41424- the ditetragonal bipyramid simulates the tetragonal bipyramid of the first order, and finally when it equals I, it passes over into that form. On the other hand, if n is greater than 2.4142+ it approaches more the bipyramid of the second order, and when it is equal to infinity passes over into that form. Hence, n > I < oo. Figure 278 illus- trates this clearly. It is also to be noted, that when n is less than 2.4142-!- the second- ary axes pass through the more acute angles, whereas, when n is greater than 2.4142 + they join the more obtuse. In figure 278, outline I represents the cross-section of the tetragonal bipyramid of the first order, 2 that Fig. 278. *) Compare figure 24, page 10. 3 ) Compare footnote, page 48. 3) See also page 40. 96 TETRAGONAL SYSTEM. of the second order, and 3, 4 and 5 ditetragonal bipyramids where n equals f , 3, and 6, respectively. Tetragonal prism of the first order. If the value of the intercept along the c axis in the tetragonal bipyramid of this order becomes infinity, the number of the faces of the bipyramid is reduced to four giving rise to the tetragonal prism of the first order, figure 279. This is an open form and possesses the following symbols: (a : a : aoc), ooP, | noj. The secondary axes join opposite edges, hence, an edge is directed toward the observer. 1_ 1 JL-. Fig. 280. Fig. 281. Tetragonal prism of the second order. The same relationship exists between this form and its corresponding bipyramid as was noted on the preceding form. The symbols are: (a: oca: ooc), ooPoo, jioo}. This is also an open form consist- ing of four faces, figure 280. The sec- ondary axes join the centers of oppo- site faces. Hence, a face is directed toward the observer. Ditetragonal prism. As is obvious, this form consists of eight faces possess- ing the following symbols: a:na:Ooc, OoPw, \hko\. What was said on page 94 concern- ing the polar angles and the position of the secondary axes applies here also. Figure 281 represents a ditetragonal prism. DITETRAGONAL BIPYRAMIDAL CLASS. 97 Basal Pinacoid. This form is similar to that of the hexagonal system, page 51. It is parallel to the secondary axes but cuts the c axis. The symbols may be written : (ooa : oca : c], OP, jooif. This form consists of but two faces. They are shown in combi- nation with the three prisms in figures 279, 280, and 281. These are the seven simple forms possible in the tetragonal system. The following table shows the chief characteristics of each. FORMS SYMBOLS I I Solid Angles Tetrahedral Octahedral Weiss Nautnann Miller Unit Bipyramid First order a : a : c P {///} 8 2 + 4 Modified Bipyramids First order a : a : me mP \hhl\ 8 2 + 4 Bipyramids Second order a : oca : me mPoc \hol\ 8 2 + 4 Ditetragonal Bipyramids a : na : me mPn \hkl\ 16 4 + 4 2 Prism First order a : a : occ ocP \IIO\ 4 ' Prism Second order a : oca : occ ocPoc \IOO\ 4 . . Ditetragonal Prisms a : na : occ ocPn \hko\ 8 Basal Pinacoid oca : oca : c OP \OOI\ 2 TETRAGONAL SYSTEM. Relationship of forms. This is clearly expressed by the follow- ing diagram. Compare pages 23 and 52. oca : oca : c a : oca : me a : oca : occ a : 7ia : me a : na : Oc c a :a : occ Combinations. Some of the more common combinations are illustrated by the following figures: Fig. 282. Fig. 284. Fig. 285. Figures 282 to 286. m ^ooPjuoj, p Pjin}, a = OoPocJioo}, u = 3PS33M. v=~- 2P|22i|, and x - 3P3J3ii}. These combinations are to be observed on zircon, ZrSiO 4 . Fig. 286 Fig. 288. Fig. 289, DITETRAGONAL PYRAMIDAL CLASS. Figure 287. m< = OoP JioiJ, and 5 = Pjiiij. uoJ, a = OoPoo Cassiterite, Sn 2 O 4 . IOO 99 e = Figures 288 and 289. p = P\m 2 = |P|ii3}, i;=|PJii7}, w=OoP{uo}, a = ocPoojioo}, and e = P oo | 101 \. These combinations occur on Anatase, TiO 2 . , c = OPjooij, a = ooP oo j looi 2. DITBTRAQONAL PYRAMIDAL CLASS. 1 ) (Holohedristn with Hemimorphism.} Symmetry. Since hemimorphism is effective on forms of this class, the principal plane and the four axes of binary symmetry dis- appear. The remaining elements are, therefore, two secondary and two intermediate planes and a polar axis of tetragonal symmetry, figure 290. Fig. 290. Forms. The forms of this class are analogous to those of the dihexagbnal pyramidal class, page 53, and the ditetragonal bipyr- amids and tetragonal bipyramids of the first and second orders as well as the basal pinacoid are now to be considered as divided into upper and lower forms. For example, the ditetragonal bipyramid now yields an upper and a lower ditetragonal pyramid, each consisting of eight faces. The three prisms, however, remain unaltered; they are apparently holohedral. *) The hemimorphic group of Dana. 100 TETRAGONAL SYSTEM. The principal features are summarized in the following table: FORMS SYMBOLS in V % Solid Angles Tetrahedral Octahedral Weiss Naumann Miller Upper and Lower Pyramids First order \a\a\ me] u f mP u 2 mP 1 I I///J \i,7\ \hol\ \hol\ 1 4 I [ 2 \l 2 Upper and Lower Pyramids Second order \ a : ooa : mc\ u mPoo 4 I u 2 mPao l [ * \i I 2 Upper and Lower Ditetragonal Pyramids (a : na : me] u m Pn \ * mPn \iiki\ \hkl\ 8 I 1 * \i I * Tetragonal Prism First order a : a : cc'c OOP \uo\ 4 4 8 j Appar- ently holohedral Tetragonal Prism Second order a : oca,: 006 oo Poo \IOO\ Ditetragonal Prisms a : na : GCC ccPn \hko\ Upper and Lower Basal Pinacoids roo# : oca : c] u r OP \IOO\ \OOI\ i u 1 p l { * \l 2 l DITETRAGONAL PYRAMIDAL CLASS. 101 r ip Combinations. Figure 291. x = ^ u< |n 3 ; Silver fluoride, AgF.H 2 O. Figure 292. a = OoP 00 w = - /, 100 Fig. 291. Ill OP ^ r < ) X 7 c = u> }ooif ; w = - /, 1 1 1 Observed on penta-erythrite, C 5 H 12 O 4 . TETRAGONAL HBMIHEDRISMS. In this system three types of hemihedrisms are possible, figures 293, 294 and 295. a) Sphenoidal hemihedrism. The principal an^tw -second- ary planes divide space into octants. All faces in alternate octants are suppressed, the others expanded, figure 293.'-,' >' Fig. 292. Fig. 293. Fig. 294. Fig. 295. b) Pyramidal Hemihedrism. The two secondary and the two intermediate planes of symmetry divide space into eight equal sections, figure 272. All faces lying wholly within alternate sections are extended, the others suppressed, figure 294. c) Trapezohedral hemihedrism. By means of the five planes of symmetry sixteen equal sections result. In this type of hemihe- drism all faces lying wholly within such sections are alternately ex- tended and suppressed, figure 295. The pyramidal and sphenoidal types of hemihedrism are the most important. 102 TETRAGONAL SYSTEM. 3. TETRAGONAL SCALENOHEDRAL CLASS. 1 ) (Sphenoidal Hemihedrism.} Symmetry. In this class the elements of symmetry consist of two intermediate planes of symmetry, two axes of binary symmetry parallel to the secondary crystal- lographic axes, and one binary axis parallel to the principal or c axis, figure 296. The sphenoidal hemihedrism is analogous to the tetrahedral and rhombohedral hemihedrisms of the cubic and hexagonal systems, respect- ively; see pages 26 and 60. Tetragonal bisphenoids. Figure 298 shows the application of the sphenoidal hemihedrism to the bipyramid of the first order. It is obvious from this figure that the bipyramid of the first order now yields! two i>ew correlated forms, each bounded by four faces. When the development is ideal, these faces are equal isosceles triangles. On account of the resulting forms resembling a wedge, they are termed the -positive and negative tetragonal bisphenoids of the first order. Fig. 296. Fig. 297. The symbols are : f# : a : me ~*~ Fig. 298. Fig. 299. ; + m? , *\hhl\ figure 299; 8 8 2 i' - ] mPn [ 2 J ' I ' \ 2 Prism First order a : a : ccc ocP r\\\Q\ >, Dparei holol fitly ledral Prism Second order a : oca : occ oopoc T\ 100 \ Ditetragonal Prism a \ na : oo c oo Pn r\hko\ Basal Pinacoid Oc a : 00 a : c OP r\OOI\ Thus far, the tetragonal trapezohedron has not been observed on either minerals or artificial salts. There are, however, a number of compounds like sulphate of strychnine, (C 21 H 22 N 2 O 2 ) 2 H 2 SO 4 H-6H 2 O, and sulphate of nickel, NiSO 4 +6H 2 O, which have been referred to this class of symmetry by means of etch figures. 110 TETRAGONAL SYSTEM. 6. TETRAGONAL PYRAMIDAL CLASS. 1 ) ( Pyramidal Hemihedrism with Hemimorphism. ) Symmetry. If the forms of the tetra- gonal bipyramidal class, page 1 50, become hemimorphic, the principal plane and center of symmetry disappear. Hence, the only ele- ment of symmetry of this class is a polar axis of tetragonal symmetry which is parallel to the c axis, figure 322. Forms. The bipyramids of the tetragonal bipyramidal class now yield upper and /oz^r pyramids. The basal pinacoid is also divided into two forms. The prisms remain unaltered. Since the ditetragonal bipyramid is now divided into four pyr- amids, this class is sometimes considered a tetartohedral class. The chief features of the forms are given in the following table: Fig. 322. FORMS SYMBOLS FACES Weiss Naumann Miller Pyramids First order f a : a : me ] I T- J 7 f . t I \hhi\ j** 7 i }' Pyramids Second order f a '. QCi : me ] u 1 mPQC j*'j 5*<<( 4 - " />QC , [ 2 J / : ^ 7 Pyramids Third order _l_ f a : na : mc^\ u + f^] + ^J ; -f2!l. 1 4 J - [=^] ' l"'j {**/! l*i, j i **'~S * ~[ 4 J / Prism First order a : a : QCt ' - Apparently of tetragonal bipyramidal class Prism Second order a : QCa : Qc Prisms Third order i f a : na : QCc ] - I 2 J Basal Pinacoids f QCa : Ota : c ] u OP u k< \'\ ! 1 2 J / {"I The pyramidal-hemimorphic group of Dana. TETRAGONAL BISPHENOIDAL CLASS. Ill Combination. The mineral wulfenite, PbMoO 4 , crystallizes in this class, figure 323. ; p = P o = - in P o' ~- - /, 7. TETRAGONAL BISPHENOIDAL CLASSY ( Tetartohedrism. ) Symmetry. The only element of sym- metry remaining in this class is an axis of binary symmetry parallel to the c axis, figure 324- Tetartohedrism. The forms of this class may be conceived as derived from the holo- hedrons by the simultaneous occurrence of either the sphenoidal and trapezohedral, or the sphenoidal and pyramidal hemihedrisms; compare figures 293, 294, and 295. The simultaneous occurrence, however, of the pyramidal and trapezohedral hemihedrisms give rise to the forms as described in the pre- ceding class. Hi sphenoids of the first order. Figure 325 shows that the bipyramid of the first order now yields two bisphenoids of the same order. Compare figure 298. The symbols are: Fig. 325. Bisphenoids of the second order. The bipyramid of the second order, figure 327, gives rise to the positive and negative bisphenoids Dana terms this the tetartohedral group. 112 TETRAGONAL SYSTEM. of the second order. These bisphenoids bear the same relation to those of the first order as do the corresponding bipyramids. Fig. 326. The symbols are: , ( a : oca : mc\ ohl\ y figure 326. Fig 327. Fig. 328. mPoo . mPoo j\hol\, figure 328; Fig. 329. Fig. 330. Fig. 331. Fig. 332. Fig. 333. Fig. 334. TETRAGONAL BISPHENOIDAL CLASS. 113 .Bisphenoids of the third order. In a like manner, as is obvious, the ditetragonal bipyramid, figure 330, yields four bisphenoids of the third order. Figure 333 shows that these forms may also result when hemihedrism is applied to the tetragonal scalenohedron. The symbols are: [a : na : me] mPn . _ Positive right, + r\ "+*" \khl\^ figure 321. Positive left, -f / Negative right, r a : na : me , +/ a : na : me 4 mPn \hkl\, figure 329. , \khl\, figure 334. Negative left, / fa : na : me /- -,\hkl\, figure 332. The same relationship exists between the bisphenoids of the three orders which was noted with the corresponding prisms, page 105. In bisphenoids of the first order the secondary axes bisect the zigzag edges. They join the centers of opposite faces in forms of the second order, while in those of the third order they occupy an intermediate position. Fig. 335. Prisms of the third order. Figure 335 shows that the ditetra- gonal prism now yields two prisms of the third order, compare figures 314 and 316. OoPn . The symbols are: ( a: na : doe] -{ 2 j ' The other forms, prisms of the first and second orders and the basal pinacoid, remain unchanged. 114 TETRAGONAL SYSTEM. The principal features may be tabulated as follows: FORMS SYMBOLS 1 Trihedral Solid Angles Weiss Naumann Miller Bisphenoids First order + [a : a : mc~\ - mP \hhi\ } -4 \hhl\ 4 2 mP 2 \ * \ Bisphenoids Second order _ a: ooa: me 1 \ \ mPoc \hol\ \ohl\ 1 u J 4 2 mPoo 2 J 2 V Bisphenoids Third order r [ a : na : mc\ -< mPn \kht\ \hhl\ \hkl\ \h~kl\ 1 -4 > 4 fr 4 mPn T + l ^n l mPn -/i" 4 J l ^ 4 Prism First order a : a : ooc Appa rently h oloh edral Prism Second order a : ooa : ooc Prisms Third order 1 a : na : 00^:1 ] ' , mPoo \hko\ \h~ko\ ] i 4 2 mPtt 2 J 2 Pasal Pinacoid ooa : ooa : c Apparently holohedral representatives of this class of symmetry have yet been No observed, -C ORTHORHOMBIC SYSTEMS Crystallographic Axes. This system includes all forms which can be referred to three unequal and perpendicular axes, figure 336. One axis is held vertically, which is, as heretofore, the c axis. Another is directed ^ toward the observer and is the a axis, some- times also called the brachy diagonal. The third axis extends from right to left ~ and is the b axis or macrodiag'onal. There is no principal axis in this system, hence any axis may be chosen as the vertical Fig. 336. or c axis. On this account one and the same crystal may be held in different positions by various observers, which has in some instances led to considerable confusion, for, as is obvious, the nomenclature of the various forms cannot then remain constant. In this system the axial ratio consists of two unknown values, viz: a\b : c = .8130: 1 : I -937 compare page 7. Classes of Symmetry. The orthorhombic system comprises three classes of symmetry, as follows: i. Orthorhombic bipyramidal class (Holohedrism.) Holohedrism and 1 2. Orthorhombic pyramidal class , . ... [ hemimorphism. 3. Orthorhombic bisphenoidal class (Hemihedrism.) Numerous representatives of all these classes have been observed among minerals and artificial salts. The first class is, however, the most important. /. ORTHORHOMBIC BIPYRAMIDAL CLASS.*) (Holohedrism.} Symmetry, a) Planes. There are three planes of symmetry. These are perpendicular to each other and their intersection gives rise !) Sometimes termed the trimetric, rhombic, or prismatic system. 2) The normal group ol Dana. 116 ORTHORHOMBIC SYSTEM. Fig. 337. to the crystallographic axes, figure 337. Inasmuch as these planes are all dissimilar, they may be written: 1 + 1 + 1=3 planes. b) Axes. Three axes of binary sym- metry are to be observed, figure 337. They are parallel to the crystallographic axes and indicated thus: !+!+!= 3 axes. c) Center. This element of sym- metry is also present and demands parallel- ism of faces. Figure 338 shows the above elements of symmetry. Orthorhombic bi pyramids. The form whose faces possess the parametral ratio, a : b : c, is known as the unit or fundamental orthorhombic bipyramid. It consists of eight similar scalene triangles, figure 339. The symbols according to Naumann and Miller are as follows: P, {nij. Fig. 338. Fig. 339. Fig. 340. Fig. 341. The outer form, in figure 340, possesses the ratio a : b : me (in > o < oc). In this case m = 2. This is a modified orthorhom- bic bipyramid. Its symbols are : mP, \hhl\. In figure 341, the heavy, inner form is the unit bipyramid. The lighter bipyramids intercept the b and c axes at unit distances but the ORTHORHOMBIC BIPYRAMIDAL CLASS. 117 a axis at distances greater than unity. Their ratios may, however, be indicated in general as, na : b : me, (n > I ; m > o < 00 ). These are the brachybipyramids, because the intercepts along the br achy diagonal are greater than unity.' The Naumann and Miller symbols are: mPn, \hkl\. Fig. 342. Fig. 343. Figure 342 shows two bipyramids (outer) which cut the a axis at unity but intercept the b axis at the general distance nb, (n > i). The ratios would, therefore, be expressed by a : nb : me. Since the intercepts along the macrodiagonal are greater than unity, these are called macrobipyramids* The symbols according to Naumann and Miller are mPn, \khl\. Figure 343 shows the relationship existing be- tween the unit, macro-, and brachy bipyramids, while figure 344 shows it for the unit, modified, and macro- bipyramids. Prisms. Similarly there are three types of prisms, namely, the unit, macro-, and brachyprisms . Fig. 344. Each consists of four faces, cutting the a and b axes, but extending parallel to the c axis 118 ORTHORHOMBIC SYSTEM. Figure 345 repre- sents a tmit-prism with the following symbols: a \b : act, ooP, {no}. The brachyprism is shown in figure 346. Its symbols are: na\b\ oo^, \hko\ In figure 347, there is a unit prism surrounded by a macroprism, whose symbols may be written: a : nb : ccc, ocPn, \kho\. For the relationship existing between these three prisms compare figure 343. Domes. These are horizontal Fig. 347. prisms and, hence, cut the and one of the horizontal axes. Domes, which are parallel to the a axis or brachydiagonal, are called brachy- domes. Their general symbols are: oca : b : me, mP$o, \ohl\, figure 348. Those, which extend parallel to the macrodiag- onal, are termed macro domes, figure 349. Their symbols are: Fig. 348. Fig. 349. \hol\. . As is obvious, prisms and domes ORTHORHOMBIC BIPYRAMIDAL CLASS. 119 are open forms and, hence, can only occur in combination with other forms. Pinacoids. These cut one axis and extend parallel to the other two. There are three types, as follows: Basal pinacoids, ooa: Oo5 : c, OP, {001 j. Brachypinacoids, 11 \ohl\ \ohJ\ 2 \ * \ mPac 7 [ 2 \l 1 2 l Macro \a : oo5 : me] u (mPoo Ju \hol\ \hol\ *r j mPoo r [ 2 \l ( 2 l Pinacoids Basal f ooa : QOb :c] u r OP u \ p l \OOI\ \OOI\ 1 I J [ 2 -J? 1 2 l Brachy ooa : b : ooc ooP66 \OIO\ \ Apparently \ holohedral Macro a : oob : ooc oo Poo \IOO\ ORTHORHOMBIC BISPHENOIDAL CLASS. 123 Combinations. Figure 361. a = ooPoo, |ioo}; g- = ooP, }noj; b =ooPo6, joio}; o = 3 Poo 101 u t J3Oi}; ; r = -u, ou ; o Clino hemi-pyramids, (na' : b : me), n>\; Ortho hemi-pyramids, :nb \mc\ n>i; + nt?7i [hk7\\ - -mV T i \hkl\\ h>k. m?n' \hkl\\ hkl\, hi, h i , h>k. Domes. In this system two types of domes are also possible, namely, those which extend parallel to the a' and b axes, respectively. Those, which are parallel to a\ are termed clinodomes and consist of four faces, figure 375. The general symbols are: oca' : b : me, mPcc', \ohl\. Since the a' axis is inclined to the c, it follows that the domes which are MONOCLINIC PRISMATIC CLASS. 129 parallel to the b axis consist of but two faces. Figure 376 shows such faces enclosing the positive angle and are termed the positive hemi-orthodomc, whereas in 377 the negative hemi-orthodome is Fig. 376. Fig. 31 Fig. 378. represented. It is evident that the faces of the positive form are always the smaller. Figure 378 shows these hemidomes in com- bination. Their general symbols are: Positive hemi-orthodome, a' : ccZ> : me, + ;wPoc, \hol\. Negative hem i-orth odome, a' : GC& : m, mPtt, \hol\. Pinacoids, There are three types of pinacoids possible in the monoclinic system, namely, Basal pinacoids, oca' : ccb : c, OP, jooi \. Clinopinacoids, oca' : b : 00 r, ooPoo', joio}. Orthopinacoids, a' : ccb : occ, OOPOC, \100\. These are forms consisting of but two faces. Figure 379 shows a combi- nation of these pinacoids. All forms of the monoclinic system are open forms and, hence, every crystal of this system is a combination. Fig. 379. A summary of the forms of this class is given as follows: 130 MONOCLINIC SYSTEM. FORMS SYMBOLS I s. Weiss Naumann Mi ler Hemi- pyramids Unit (a' : b : c) ( +J> I ~ P [//}} \III\ ' . ^4 ^4 Modified (a f :b :mc) j -h mP \ mP \hhl\ \hhl\ Clino- (na' : b : me) j +mPn' ( mPri \hkl\ \hhl \ Ortho- _(a' :nb\mc) 5 +mPn \ - mPn \khl\ \khl\ Prisms Unit a' :5:ooc OOP \I,0\ Clino- no! : b : ooc aoPn' \hko\ Ortho- a' : nb : ooc ooPn \kho\ Clinodome ooa' :b : me mPoo' \ohl\ 4 Hemi- orthodomes Positive a' : oob : me + mPo5 \hol\ }> Negative a' : oob : me -mPoo \hol\ Pinacoids Basal ooa' : oob \ OP \OOI\ 2 Clino- aoa f : b : voc 00 POO' \OIO\ Ortho- a' : 00^ : ooc oo Poo \IOO\ Relationship of forms. This is clearly shown by the following diagram : OP OP OP OP OP 1 i 1 1 1 -4- l P + ' PTi 1 m m \ \ Pm" -1- P*' ~r r ~r f)l m m \ \ A- P -1- P7} ' 1 + Poo " i 1 "1 1 A 1 i tl "1 1 r^PryV -4- i r^ P*? "1 _ rmPr/S MONOCLINIC PRISMATIC CLASS. 131 a TTl Fig. 381. Fig. 382. Fig. 383. Combinations. Figure 380. m =ooP, jnof; b= ooPoo', Joio}; p=P, jmf. Gypsum, CaSO 4 .2H 2 O. Figure 381. c = OP, |ooj|; m ooP, jno|; =ooPoo', {010} ; j = 2Pob, J20i{. Orthoclase, KAlSi 3 O 8 . Figure 382. m=aoP, juof; = ooPoo', {010} ; u = P, {in}; c=OP, \ooi\- m = ooP, {no}; / =oo?3, {310}. Diopside. Figure 386. c OP, jooi}; a = ocPoo, {iooj; r = Poo, jioi}; o = P, {111}; = +P, {in}; g = -hPoo, { ioT( ; 132 MONOCLINIC SYSTEM. {201 J ; q ---- Poc' f jouS ; TT == 2 P2, J2if}; I = S 3 1 1 } I = P3, I 3 i 3 j ; * = 3P3, i 3 1 1 ! Praseodymium sulphate, Pr 2 (S0 4 ) 8 .8H 8 0. 2. MONOCLINIC DOMATIC CLASS. 1 ) ( Hemihedrism. ) ^ ^ -*-_ Symmetry. This class possesses but one element of symmetry, namelv, a plane of symmetry as shown in figure 387. Tetra-pyramids. Since the axis of Fig" 387. binary symmetry is lost, it follows that the hemi-pyramids now yield two tetra-pyramids. These forms consist of but two faces which are situated symmetrically to the plane of symmetry. For example, in figure 371 the positive hemi-pyramid yields the lower positive tetra-pyramid , which con- sists of the two faces in front, and the upper positive tetra-pyramid made up of the other two faces. In like manner the negative hemi-pyramid yields the upper and lower negative tetra-pyramids. Their symbols are given in the tabulation on page 133. Hemi-clinodomes. Obviously, the clinodome is now divided into an upper and a lower hemi- clinodome. Tetra-domes. The hemi-orthodomes yield upper and lower tetra-domes. Hemi-prisms. Each prism now yields two hemi-prisms, desig- nated as front and rear forms. Pinacoids. The clinopinacoids must on account of the sym- metry occur with the full number of faces, namely, two. The basal and orthopinacoids each yield two forms. In the case of the basal pinacoid, they are designated as upper and lower forms, while the terms front and rear are employed for the orthopinacoids. The forms and their general symbols may be tabulated as follows : *) Clinohedral group of Dana. MONOCLINIC DOMATIC CLASS. 133 FORMS SYMBOLS Weiss Naumann Miller Tetra- pyramids Unit and Modified | K : * : *} u 7 ,mP \hh~t \ \hhi\ \hhl\ \ithi\ I 2 mP 2 U mP u 2 mP 2 -( 2 J Clino , (na' : b : me: mPn' {hkl} \liki\ \hki\ \hkl\ i 2 . mPn' \ 2 u mPn' 1 2 "> l 2 U mPri I 2 Ortho . (a' \ nb \ m] r mPn . ?nPn + u mPn \khl\ \khi\ \khi\ \khl\ { 2 \ 11 ' 1 u 2 mP~ii ~~T~ l Hemi- prisms Unit r':Z:*l OOP \IIO\ \1 IO \ 2 f OOP 1 * V' 1 T 2 Clino {na' : 5: 00^1 , f oo /V \hko\ \liko\ ; ' f CcPn' 1 , K'' \ 2 ' Ortho f a' : nb : oo^l f ( ccPn \kho\ \kho\ \ 2 f oo Pn I * P I * ' 134 MONOCLINIC SYSTEM. FORMS SYMBOLS Faces Weiss Naumann Miller Hemi-clinodomes foo' : b : mc\ r i | 1 mPte' \ohl\ \ohl ! 2 j 7( 2 mPac' [ 2 J "' l L 2 Tetra-orthodomes (a':&l:mt\ , mP^z i ~, \hol\ \hol\ \hol\ \liol\ 1 . I 2 U mPoo 1 2 l mPtt -( * r 1 2 ^ mP& 2 Clinopinacoid O0' : b : aoc Apparently holohedral Pinacoids - Basal f aoa' : oob : c] 7 r OP 1 ^, \OOI\ \ ooi \ I U y / L 2 ) Ortho f': 005:00*] ccPac \IOO\ \IOO\ 2 S oo Poo ( 2 \S'> T 2 Fig. 3.8. OP Combination. Fig. 388. c 21, {ooi}; ooPoo a = ocP /, jioo|, and OOPX) r, j 100 J ; ;;/ = OOP - . Pet' /, {-no}, and r, JUO{; q = u, {on}; v = + -y- /, Tetrathionate of potassium, K a S 4 O e . MONOCLINIC SPHENOIDAL CLASS. 135 3. MONOCLINIC SPHENOIDAL CLASS. 1 ) ( Hemimorphism. ) Symmetry. A polar axis of binary symmetry is the only element of symmetry / present, figure 389. The forms of this class are hemimorphic along the b axis, which is now a singular axis. XN X / Those forms which extend parallel to Fig. 389. b axis, the orthodomes, basal and orthc- pinacoids, are unaltered. The others yield right and left forms, for example, the unit prism yields the right and left hemi-prisms. The forms and their symbols are tabulated as follows: SYMBOLS \V\ks Naumann Miller j mces mP \ i \ h h 7( - mP \ /i 11 i j Unit and , . f a' : fi : me 1 ) \ I 2 \in ., Modified 2 \ mP \ h h l\ 2 mP 2 ) itf Itf lr\ \hhl\ Tetra- Clino 4- : m\ -h T / 1 ' * j>^/?/ | I / , 1 \ ( 2 } 2 vi F n 2 i^y^/S \klil\ Dan.i terms this class the hemimorphic group. 136 MONOCLINIC SYSTEM. FORMS SYMBOLS ] Faces 2 Weiss Naumann Miller Hemi- prisms Unit f a'\b\ Cfoc } 1* J r ' oi"7 TABULAR CLASSIFICATION SHOWING THB ELEMENTS OF SYMMETRY AND THE SIMPLE FORMS OF THB THtRTY-TWO CLASSES OF CRYSTALS. CUBIC SYSTEM. Classes of Crystals, I to 5. (Page 150.) TABULAR CLASSIFICATION SHOWING THE ELEMENTS OF SYMMETRY A* /. CUBIC SYMMETRY Planes Axes ci \ ci '. a ft * n ' QC n CLASS _* >, IM _a rt T3 a a> ocO ~ ^H w^ u a, 1 i/7/i j//o|- \W- " I Vw ^ t r > t- i. Jttexoetahedral ( Holohedrisni] 3 6 3 4 6 I Octahedron Ht Dodecahedron Tetrahe- 2. Hextetrahedral __ 6 - 4 3 _ drons ( Tetrahedral Hemihedrisni} (Polar, 3. Dyakisdodecahedral 4 3 I ( Pyritohedral^ Hemihedrism ) 4. Pentagonal Icositetrahedral 3 4 6 ( Plagihedral Hemihedrism ) 5. Tetrahedral Tetrahe- Pentagonal Dodecahedral 4 3 . drons (Polar) ( T~ ) ( Tetartohedrisni} 1) The blank spaces indicate that the forms are apparently holohedral. [1501 9 THE SIMPLE FORMS OF THE THIRTY-TWO CLASSES OF CRYSTALS. SYSTEMS j FORMS REPRESENT- ATIVE DOrt : OOrt TOOOC 100 \ a ' : a : ma mO \hhl\ a : ma : ma mOm \hu\ a : ma : ooa acOm \hko\ a : na : ma mOn \hki\ Cu** xahedron I / ^Tetragonal Trisoctahe- dron \\ Tetrahexa- hedron ! Hexoctahe- dron Galena (PbS) Trisoctahe- dron i I Tetragonal Tristetrahe- drons () Trigonal Tristetrahe- drons () Hextetrahe- drons () Tetrahedrite (Cu 2 ,Fe,Zn) 4 (AS,Sb) 2 S 7 i Pyritohe- drons () Dyakis- dodecahedrons () Pyrite (FeS 2 ) Pentagonal Icositetrahe- drons (r,D Sal Ammoniac (NH 4 C1) Tetragonal Tristetrahe- drons () Trigonal Tristetrahe- drons () Pyritohe- drons () Tetrahedral Pentagonal Dodecahedrons r, 1 Sodium Chlorate (NaClO 3 ) [150] HEXAGONAL SYSTEM. Classes of Crystals, 6 to II. (Page 151.) 151 2. HEXAQO CLASS SYMMETRY - c Planes Axes A a : oca: a:mc mP yr j/O/7 { 2 : 2r/ : (t : me mP2 \hh2hl\ \na Principal Secondary Intermediatt * 6. Dihexagonal Hexagonal Hexagonal ! Bipyramidal I 3 3 I - 3 + 3 i Bipyramids Bipyramids R | ( Holohedrism ) First order Second order 7. Dihexagonal Pyramidal 3 ^? Hexagonal Pyramids Hexagonal Pyramids Di i: f Holohedrism with 1 (Polar) First order Second order ( Hemimorphism j (, fl (*, /) 8. Ditrigonal Bipyramidal I i Trigonal Bipyramids E Bi f Trigonal 1 ' (Pc/ar) First order / Tj) Q ' ( ) 9. Ditrigonal Rhombohe- Sc Scalenohedral ? 1 3 J drons f Rhombohedral 1 First order rt L HemjJtedr^sm } () ) 10. Hexagonal Bipyramidal I I _ i Bi f Pyramidal 1 ri / ^ Hemihedrism j n. Hexagonal H Trapezohedral _ _ I 3 + 3 ' Ti (Trapezohedral 1 Hemihedrism j 5" - - II. SYSTEM. i FORMS REPRESENTATIVl a, \a\nu ocP ocP? na : pa :a : cce oca :jCa:Oca :c ooP?^ 9 OP kl\ \IOIO\ '~ \II20\ \hiko\ \OOOI \ agonal ramids Hexagonal Prism First order Hexagonal Prism Second order Dihexagonal Prisms i Basal Pinacoid Beryl (Be 3 Al 2 (Si0 8 ) fl :agonal trnids Basal Pinacoids O, /) Zincite (ZnO) gonal amids h) Trigonal Prisms First order Ditrigonal Prisms nohe- ons t) Calcite (CaCO 3 ) gonal amids order 1) Hexagonal Prisms Third order Apatite (Ca 5 Cl(P0 4 ) 3 ) Barium Stibio igonal ;zohe- - tartrate a n < Potassium Ni trate /) f Ba (SbO) 2 ' KN0 3 J HEXAGONAL SYSTEM. (Continued) Classes of Crystals, 12 to 17. (Page 152.) 152 2. HEXAGC (Co, CLASS 12. Ditrigonal Pyramidal Trigonal Hemi- hedrisnt wifli Hemimorphism 1V"T 13. Hexagonal Pyramidal f Pyramidal Heini- 1 // c drism with i A- [^ Hemimorph 14. Trigonal Bi pyramidal r Trigonal "I L Tetartohedrism } 1 5. Trigonal Trapezohedral r Trabezohedral 1 r 7 \Jfcta 1 6. Trigonal Khombohedral f Rhombohedral \ L, jfTetartohedrism j 17. Trigonal Pyramidal ( Ogdohedrism ) SYMMETRY Planes Axes Center a: do a: a: me mP \IOII\ 2a : 2a : a : Die / mP2 \ h h~2~hl\ Piincipal X rt o c o Intermediate A ^P - i (Polar) - Trigonal Pyramids First order Hexagonal Pyramids Second order I i (Polar.) - - Hexagonal Pyramids First order Hexagonal Pyramids Second order (". I - Trigonal Bipyramids First order Trigonal Bipyramids Second order - i 3 ( Polar) - Rhombohe- drons First order Trigonal Bipyramids Second order - - I. j , > 1 I Rhombohe- drons First order Rhombohe- drons Second order - - - I (Polar) - Trigonal Pyramids First order (+, 1) Trigonal Pyramids Second order i : \L SYSTEM. 152 'ted} FORMS a : a : me a : oca : a : oc r 2a : 2 a :a:Occ na \pa : a : ccc OC:(X:00:6- REPRESENTATIVE iPn OOP OOP2 acPn OP ikl\ \IOIO\ \II20\ \hiko\ 0001 \ T o ur mal in e -igonal amids Trigonal Prisms Ditrigonal Prisms Basal Pinacoids A1 3 B 2 5 H 2 fAl Mg Fe ] v, +/) First order (, /) 3 ' 2 ' 2 > Li, Na, HJ 9 (SiOj 4 agonal amids d order - Hexagonal Prisms Third order Basal Pinacoid //, / Strontium Anti- monyl - tartrate Sr(SbO) 2 (C 4 H 4 gonal Trigonal Trigonal Trigonal ramids Prisms Prisms Prisms d order First order Second order Third order igonal >ezohe- rons Trigonal Prisms Second order Ditrigonal Prisms Quartz (Si0 2 ) nbohe- Hexagonal rons Prisms Dioptase d order Third order (H 2 CuSiO 4 ) ', 1 () gonal amids d order r, u; r, I; Trigonal Prisms First order Trigonal Prisms Second order Trigonal Prisms Third order /I 1 7\ Basal Pinacoids Sodium Periodate (NaI0 4 -f3H 2 0) /, u; ( ~r r, ~ri) '' TETRAGONAL SYSTEM. Classes of Crystals, 18 to 24. (Page 153.) 153 3. TETRA CLASS SYMMETRY Planes 'Axes u 1 I a : a : me ?;/P \hhl\ a : oca : > ;;/Poo \hol\ 1 o e I N) Secondary Intermediate 18. Ditetragonal Bi pyramidal ( Holohedrism ) . 2 I 2 + 2 Tetragonal Bipyramids First order Tetragoi Bipyram Second 01 19. Ditetragonal Pyramidal { Holohedrism with"\ [ Hemimorphism j _5* 2 2 I (Polar) Tetragonal Pyramids First order (", i) Tetragoi Pyrami( Second or (", i) 20. Tetragonal Soalenohedrai f Sphenoidal 1 i Hemihedrism i _*. 5 2 I +2 Tetragonal Bisphenoids First order (+) 21. Tetragonal 15 i pyramidal 7i |' Pyramidal 1 i Hemihedrism \ ^ I I 22. Tetragonal Trapezohedral f Trapezohedral} L Hemihedrism j / ; I 2 + 2 23. Tetragonal Pyramidal r Pyra midal Hem ih edrism 1 (^ zt'zV// Hemitnorphism u I ( Polar) Tetragonal Pyramids First order (, i) Tetragoi Pyrainic Second or (". /) 24. Tetragonal Bisphenoidal ( Tetartohedrism) 7 I Tetragonal Bisphenoids First order (+) Tetragoi Bispheno Second or () MAL SYSTEM. 153 FORMS 7 : na : me niPit \hki\ a \a : ooc ocP iiio| a : oc a : 00 c \ ocPoo |ioo| a : na : GO c 00 Pfl \hko\ oo a : 00 a : c OP \00l\ REPRESENTATIVE )itetragonal Bipyramids Tetragonal Prism First order Tetragonal i Prism Second order Ditetra- gonal Prisms Basal Pinacoid Cassiterite (Sn0 2 ) )itetragonal Pyramids (, Basal Pinacoids (, Silver Fluoride (AgF + H 2 0) Tetragonal Scalenohe- drons () Chalcopyrite (CuFeS 2 ) Tetragonal Bipyramids Third order (+) Tetragonal Prisms Third order () Scheelite (CaWOJ Tetragonal Trapezohe- drons (r, /) Nickel Sulphate (NiSO, + 6H 2 O) Tetragonal Pyramids Third order w > l Tetragonal Prisms Third order () Basal Pinacoids (w, /) Wulfenite (PbMoOj Tetragonal Bisphenoids Third order r, 1 Tetragonal Prisms Third order () ORTHORHOMBIC SYSTEM. Classes of Crystals, 25 to 27. (Page 154.) 154 4. ORTHOR CLASS SYMMETRY i S3 E A L 1 ^ Center na : b : me mPn \hkl\ nd : b : occ 00 Pn \hko\ o 25. Orthorhombic Bipyramidal (Holohedrism} I + I + I l + I +1 I Orthorhombic Bipyramids Orthorhombic Prisms B 26. Orthorhombic Pyramidal r , t f Holohedrism 1 [ with Hemimorphism j I (P.tlar) Orthorhombic Pyramids (, i) B 27. Orthorhombic Bisphenoidal (Hemihedrism] I + I+I Orthorhombic Bisphenoids (r, I) WIC SYSTEM. 154 FORMS _ : me X l\ a : oc5 ': me mPoo \hol\ OO<2 : & : ODC OcPob JOIOJ : oc^ : ace OoPoo i iooj oca : cob : c OP JooiJ REPRESENTATIVE domes Macrodomes Brachy- pinacoid Macro- pinacoid Basal Pinacoid Barite (BaSOJ lomes /) Macrodomes (*,/) Basal Pinacoids (. /) Calamine (Zn 2 (OH) 2 SiO,) Epsomite (MgSO t + 7 H a O) MONOCLINIC SYSTEM. Classes of Crystals, 28 to 30. (Page 155.) 155 5. MONOC CLASS SYMMETRY 0) c ri Ou M V c5 i na' : b : me mPn' \hkl\ uu' : b : ocr OcP;/' \hko\ Oca' : b : me mPoo' \ohl\ 28. Monoclinic Prismatic (Ho/ohedrism] I I I Hemi-pyramids () Prisms Clinodomes 29. Monoclinic Domatic (Hetnihedrism] Tetra-pyramids (. Hemi-prisms (J\r) Hemi- clinodomes (, i) 30. Monoclinic Sphenoidal ( Hemimorphism ) I (/War) Tetra-pyramids ( Sphenoids) (n Hemi-prisms (r, I) Hemi- clinodomes (r, /) '1C SYSTEM. 155 FORMS REPRESENTATIVE : ocb : me wPoo SAO/! oc' : ^ : ocr 00 POO' jo/oj a': oob : ccc ooPoo \IOO\ oca' : oo^ : OP {oo/j Hemi- rthodomes () Clinopinacoid Orthopinacoid Basal Pinacoid Gypsum (CaS0 4 + 2H 2 0) Tetra- rthodomes + w, + /) Orthopinacoids (f,r) Basal Pinacoids (, /) Tetrathionate of Potassium (KAO.) Clinopinacoids (r,/) Tartaric Acid (C t H 6 O s ) TRICUNIC SYSTEM. Classes of Crystals, 31 and 32. (Page 1 56.) 156 6. TRIG* CLASS SYMMETRY 1 ^e (A < 3 I na \ b \ me mPn \hkl\ wa \ b :C6c mPao' \ohl\ U 31. Triclinic Pinacoidal ( Holohedrisni) Tetra-pyramids Hemi-prisms (r,/) Hemi- brachydomes C, Ogc io-pyram ?^ ids 32. Triclinic Asymmetric ( Hemihedrism ) ?^ n Tetra-prisms f r , r 1 [7 / '7 r ] Tetra- brachydomes r ?/ u I u w -< 7 SYSTEM. 156 ^ORMS : mt OOtf : & : oc^ a : 006 : ooc a: 005:00^ REPRESENTATIVE zPoc OoPdc oo Poo OP H6l\ So/oj j/oo( \OOI\ lemi- rodomes Brachy- pinacoid Macro- pinacoid Basal Pinacoid Albite (NaAlSi 3 O 8 ) retra- rodomes Brachy- Macro- Basal Strontium bitartrate pinacoids pinacoids Pinacoids fSr(C 4 H t 6 H) 2 + ] /M d,r) I 4H S INDEX. Acid, tartaric, 137, 155 Albite, 8, 142, 148, 156 Albite Law, 148 Alum, 2 Amorphous structure, 1 substances, 2 Analcite, 24 Anatase, 99 Anglesite, 8 Angular position of faces, 14 Anorthic system, 138 Apatite, 68, 151 Apparently holohedral forms, 29 Aragonite, 8, 120, 147, 148, 149 Argentite, 24, 25 Asymmetric class, 142, 156 group, 142 system, 138 Axes, crystallographic, 4, 17, 43, 92, 115, 126, 138 of symmetry, 13 Axial cross, 4 Axial ratio, 7 Axinite, 142 Axis, polar, 16 singular, 16 Barite, 154 Barium nitrate, 42 Barium stibiotartrate and potassium nitrate, 70, 151 Beryl, 53, 151 Bisphenoids, orthorhombic, 124 tetragonal, first order, 102, 111 second order, 111 third order, 113 Boracite, 32 Brachy bipyramids, 117 diagonal, 115, 138 domes, 118 prisms, -117 Brazilian law, 147 Brookite, 120 Calamine, 123, 154 Calcite, 64, 65, 144, 146, 151 Cassiterite, 99, 147, 153 Celestite, 121 Center of symmetry, 14 Chalcocite, 120 Chalcopyrite, 105, 153 Chemical crystallography, 2 Chrysoberyl, 120 Cinnabar, 85 Classes of crystals, 14, 150 Classification of the thirty-two classes of crystals, 150 Clino-axis, 126 domes, 128 hemi-pyramid, 128 prism, 128 rhombic system, 126 rhomboidal system, 138 Clinohedral group, 132 Closed forms, 6 Cobaltite, 36 Coefficients, rationality of, 9 Combinations, 7 cubic, 23, 31, 35, 38, 42 hexagonal, 52, 55, 64, 68, 74, 76, 85, 89, 91 tetragonal, 98, 101, 104, 108, 111 orthorhombic, 120, 123, 125 monoclinic, 131, 134, 136 triclinic, 142, 143 Common twinning laws, 145 Composition plane, 144 Compound crystals, 144 Congruent forms, 16 Constancy of interfa^ial angles, 2 Contact twins, 145 Copper, 24 Correlated forms, 15 Corundum, 64, 65 Crystal faces, 2 forms, 6 habit, 4 [157] 158 INDEX. Crystal system, 14 systems, 5 Crystalline structure, 1 Crystallization, elements of, 8 Crystallized substances, 2 Crystallographic axes, 4 cubic, 17 hexagonal, 43 tetragonal, 92 orthorhombic, 115 monoclinic, 126 triclinic, 138 Crystallography, 2 Crystals, 1, 2, distorted, 3 formation of, 1, 2 Cube, 19 Cubic system, 5, 17, 150 Cyclic twins, 148 Dana's symbols, 11 Deltoid, 27 Diamond, 145 Didodecahedron, 34 Dihexagonal bipyramid, 48 bipyramidal class, 44, 151 prism, 50 pyramidal class, 53, 151 pyramids, 54 Diopside, 131 Dioptase, 89, 152 Diploid, 34 Distortion, 3 Ditetragonal bipyramid, 95 bipyramidal class, 93, 153 prism, 96 pyramidal class, 99, 153 Ditrigonal bipyramidal class, 56, 151 bipyramids, 57 prisms, 59, 72 pyramids, 71 pyramidal class, 70, 152 pyramidal tetartohedrism, 70 scalenohedral class, 60, 151 Dodecahedron, 19 deltoid, 27 pentagonal, 33 rhombic, 19 tetrahedral pentagonal, 39 Dolomite, 89 Domatic class, 132, 155 Dyakisdodecahedral class, 32, 150 Dyakisdodecahedron, 34 Elements of crystallization, 8 symmetry, 12 Epsomite, 125, 154 Etch-figures, 29 Faces, angular position of, 14 Fivelings, 148 Fluorite, 24, 145 Form, crystal, 6 Formation of crystals, 1, 2 Forms, congruent, 16 closed, 6 correlated, 15 Enantiomorphous, 16 fundamental, 6 hemi-morphic, 16 open, 6 unit, 7 Fourlings, 148 Fundamental forms, 6 Galena, 24, 150 Garnet, 25 Geometrical crystallography, 2 Gypsum, 8, 131, 147, 155 Gyroidal hemihedrism, 36 Gyroids, 37 Habit, crystal, 4 prismatic, 4 tabular, 4 Halite, 24, 29 Hematite, 64, 65 Hemi-bipyramid, 127 clinodomes, 132 domes, 140 Hemihedrism, 15 cubic, 26, 32, 36 hexagonal, 55 tetragonal, 101 orthorhombic, 123 monoclinic, 132 triclinic, 142 INDEX. 159 Hemimorphic forms, 16 group, hexagonal, 53 monoclinic, 135 orthorhombic, 121 tetragonal, 99 Hemimorphism, 16 hexagonal, 53, 70, 74, 89 monoclinic, 132 orthorhombic, 121 tetragonal, 99, 110 Hemimorphite > 103 Hemi-orthodomes, 129 Hemiprisms, monoclinic, 132 triclinic, 139 Hemiprismatic system, 126 Hemipyramids, 127 Hexagonal basal pinacoid, 51 bipyramidal class, 65, 151 bipyramid, first order, 45 second order, 46 third order, 65 hemihedrisms, 55 prism, first order, 50 second order, 50, 72 third order, 67, 87 pyramid, first order, 54 second order, 54, 71 third order, 75 pyramidal class, 74, 152 system, 5, 43, 151, 152 tetartohedrisms, 76 trapezohedral class, 68, 151 trapezohedrons, 68 Hexahedron, 19 Hexoctahedral class, 17, 150 Hexoctahedron, 21 Hextetrahedral class, 26, 150 Hextetrahedron, 28 Holohedral, apparently, 29 Holohedrism, 15 Icositetrahedron, 20 Incline-face hemihedrism, 27 Indices, Miller's, 12 lodyrite, 55 Iron cross, 146 Isometric system, 17 Juxtaposition twins, 145 Karlsbad law, 148 Leucitohedron, 20 Limiting forms, 23 Macrobipyramids, 117 diagonal, 115, 138 domes, 118 prisms, 117 Magnetite, 25, 145 Miller's indices, 12 system, 12 Mimicry, 149 Modified hemi-pyramid, 128 pyramid, 7 Monoclinic domatic class, 132, 155 domes, 128 prismatic class, 126, 155 prisms, 128 sphenoidal class, 135, 155 system, 5, 126, 155 Monoclinohedral system, 126 Monosymmetric system, 126 Naumann symbols, 11, 62 Nepheline, 76 Nickel sulphate, 109, 153 Normal group, cubic, 17 hexagonal, 44 monoclinic, 126 orthorhombic, 115 tetragonal, 93 triclinic, 138 Oblique system, 126 Octahedron, 19 Octants, 17 Ogdohedrism, 15, 89 Ogdo-pyramids, 142 Open forms, 6 Ortho-axis, 126 Orthoclase, 131, 147, 148 Orthohemi-pyramid, 128 Ortho-prism, 128 Orthorhombic bipyramidal class, 115, 154 bipyramids, 116 bisphenoidal class, 123, 154 bisphenoids, 124 domes, 118 160 INDEX. Orthorhombic pinacoids, 119 prisms, 117 pyramidal class, 121, 154 system, 5, 115, 154 Parallel-face hemihedrism, 32 grouping, 144 Parameters, 5 Parametral ratio, 5 Penetration twins, 145 Penta-erythrite, 101 Pentagonal dodecahedron, 33 hemihedrism, 32 icositetrahedron, 37 icositetahedral class, 36, 150 Pericline law, 148 Physical crystallography, 2 Pinacoidal class, 138, 156 Plagiohedral group, 36 hemihedrism, 36 Planes of symmetry, 12 Polar axes, 16 Pole, 53 Polysynthetic twins, 148 Potassium tetrathionate, 134, 155 Praseodymium sulphate, 132 Prismatic habit, 4 system, 115 Pseudosymmetry, 94, 149 Pyramid cube, 21 Pyramidal group, hexagonal, 65 tetragonal, 105 hemihedrism, hexagonal, 55, 65 tetragonal, 101, 105 system, 92 Pyrite, 30, 33, 35, 36, 146, 150 Pyritohedral group, 32 hemihedrism, 32 Pyritohedron, 33 Quadratic system, 91 Quartz, 8, 85, 144, 146, 147, 152 Ratio, axial, 7 parametral, 5 Rationality of coefficients, 9 Re-entrant angles, 148 Reflection twins, 145 Regular system, 17 Repeated twinning, 148 Rhombic dodecahedron, 19 Rhombic system, 115 Rhombohedral group, 60 hemimorphic, 70 hemihedrism, 55, 60 hemimorphic class, 70 tetartohedrism, 76, 85 Rhombohedron of the middle edges, 62 like forms, 71 Rhombohedrons, first order, 60, 81, 85 second order, 86 third order, 86 Rotation twins, 145 Rutile, 149 Salammoniac, 38, 150 Scalenohedron like forms, 71 Scalenohedrons, hexagonal, 61 tetragonal, 103 Scheelite, 108, 153 Siderite, 64 Silver fluoride, 101, 153 Singular axis, 16 Sodium chlorate, 42, 150 periodate, 91, 152 Sphalerite, 30, 31, 32 Sphenoidal group, orthorhombic, 123 tetragonal, 102 hemihedrism, 101, 102 Spinel, 24, 25, 145 law, 145 Staurolite, 144, 147 Steno, Nicolas, 4 Striations, 31, 149 Strontium antimonyl-tartarte, 76, 152 bitartrate, 143, 156 Structure, amorphous, 1 crystalline, 1 Struvite, 123 Strychnine sulphate, 109 Sulphur, 7, 120 Supplementary twins, 146 Sylvite, 29 Symbols, 10 Weiss, 11 Symmetry, axes of, 13, 14 INDEX. 161 Symmetry, center, 14 classes of, 14 elements of, 12 planes of, 12 Tabular habit, 4 Tartaric acid, 137, 155 Tesseral system, 17 Tessular system, 17 Tetartohedral group, cubic, 38 tetragonal, 111 Tetartohedrism, 15 cubic, 38 hexagonal, 76 tetragonal, 111 Tetra-bipyramids, 139 domes, monoclinic, 132 triclinic, 142 pyramids, monoclinic, 132 triclinic, 139 Tetrathionate of potassium, 134, 155 Tetragonal basal pinacoid, 97 bipyramidal class, 105, 153 bipyramids, first order, 93 second order, 94 third order, 105 bisphenoidal class, 111, 153 bisphenoids, first order, 102, 111 second order, 111 third order, 113 hemihedrisms, 101 prisms, first order, 96 second order, 96 third order, 105 pyramidal class, 110, 153 pyramids, first order, 110 second order, 110 third order, 110 scalenohedral class, 102, 153 scalenohedrons, 103 system, 5, 92, 153 trapezohedral class, 108, 153 trapezohedrons, 108 trisoctahedron, 20 tristetrahedron, 28 Tetrahedral group, 26 hemihedrism, 26 pentagonal dodecahedral class, 38, 150 dodecahedron, 39 Tetrahedrite, 31, 150 Tetrahedron, 26 Tetrahexahedron, 21 Thirty-two classes of crystals, 14, 150 symmetry, 14, 150 Thorium sulphate, 131 Threelings, 148 Topaz, 8, 121 Tourmaline, 74, 152 Trapezohedral group, hexagonal, 68 tetragonal, 108 trigonal, 81 hemihedrism, hexagonal, 56, 68 tetragonal, 101, 108 tetartohedrism, 76 Trapezohedron, 20 Trapezohedrons, hexagonal, 68 tetragonal, 101, 108 Triclinic system, 5, 138, 156 Trigonal bipyramidal class, 76, 152 bipyramids, first order, 56, 77 second order, 77, 81 third order, 78 hemihedrism, 55, 56 prisms, first order, 56, 72, 77 second order, 77, 83 third order, 78 pyramids, first order, 70, 90 second order, 90 third order, 90 pyramidal class, 89, 152 rhombohedral class, 85 tetartohedrism, 76 trapezohedral class, 81, 152 trapezohedrons, 81 trisoctahedron, 20 tristetrahedron, 28 Trillings, 148, 149 Trimetric system, 115 Trisoctahedron, 20 tetragonal, 20 trigonal, 20 Tri-rhombohedral group, 85 Tristetrahedron, tetragonal, 27 trigonal, 28 Twin crystals, 144 162 INDEX. Twinning axis, 145 law, 145 plane, 144 Twins, 144 contact, 145 juxtaposition, 145 penetration, 145 reflection. 145 rotation, 145 Unit form, 7 Urea, 104 Weiss, Prof. C. S., 11 symbols, 11 Wulfenite, 111, 153 Zincite, 55, 151 Zircon, 8, 98, 147 UNIVERSITY OF CALIFORNIA LIBRARY This book is DUE on the last date stamped below. Fij NOV 10 1947 LIBRARY USE JUL 2 1959 REG'D LD MAR161959B JUL JUN 81953LU JAN 171954 LU LD 21-100m-12,'46(A2012si6)4120 16Ju!'5SHj JUL 16 LIEHASY 'J AUG LD AUG REC'D AUG LD V. .. UNIVERSITY OF CALIFORNIA LIBRARY