The Beauty of Everyday
Mathematics
by Norbert Herrmann

BERLIN, HEIDELBERG: SPRINGER-VERLAG, 2012, # XIII + 138 PP.,

US $19.95, ISBN 978-3-642-22103-3, E-ISBN 978-3-642-22104-0

REVIEWED BY PAMELA GORKIN

FF
irst-semester calculus used to be one of my favorite
courses to teach. After seeing the definition of
derivative and using it to calculate lots of derivatives,

my students were thrilled by the ease with which they
could take derivatives using the product rule. After strug-
gling to compute areas under a curve using Riemann sums,
my students were amazed by the simplicity the Fundamental
Theorem of Calculus provides. I could be misremembering
that, of course. But what is true now is that our students arrive
on campus knowing the product rule, but unable to recall the
definition of derivative. They know rules for integration by
heart, but don’t know what is ‘‘fundamental’’ about the
Fundamental Theorem of Calculus. It’s much more difficult
for these students to see the beauty of calculus, but my hope
is that if we find the right examples, we’ll still be able to grab
our students’ attention. The appearance of a new book aimed
at teachers and students of mathematics, The Beauty of
Everyday Mathematics by Norbert Herrmann, provides such
examples.

Translated from the German Mathematik ist überall,
Herrmann’s book contains unusual examples of mathe-
matics in everyday life. Here’s one that I can relate to: I
have two identical and perfectly round beer coasters that I
borrowed, permanently, from a bar I go to (occasionally, of
course). I put them next to each other and slide the leftmost
coaster over the top of the coaster on the right, keeping the
centers of the coasters on the line segment joining the old
centers. When the coasters are next to each other, there is
no overlapping area. When they are on top of each other,
the entire area of the coaster on the bottom is covered. So
here’s a natural question that two beer-drinking mathe-
matically inclined people might ask: When is half the area
of the bottom coaster covered? Now most mathematicians
will be able to find an equation that, when solved, will
provide the solution. If you spend a few minutes, use some
familiar formulas, and apply a double-angle formula, you’ll
probably come up with the following:

a � sinðaÞ¼ p=2;

where a is the angle between the two radii that are drawn
from the center to the two points of intersection of the
coasters. There are, however, a few things that you should
note here: First, this is pretty sophisticated stuff. If you are
thinking of purchasing this book for someone else, your
recipient needs to be pretty savvy, mathematically speaking.

The second thing you should notice is that it’s pretty unlikely
that a nonmathematician would look at a � sinðaÞ¼ p=2
and say, ‘‘Hey, I know how to show a is pretty darn close to
2.309881 radians.’’ But it’s a great way to introduce Newton’s
method, which is precisely what Herrmann does. Not only
that, he presents a second method to solve this equation,
which he calls ‘‘The Fixed Point Procedure.’’ It’s a clever
example that leads to some beautiful mathematics.

A second example, reproduced as it appears in the text,
sounds similar, but the solution is quite different.

‘‘A group of young people has decided to picnic in the
great outdoors and is now sitting on the grass some-
where, battling flies and, well, with this specific can of
soda. Because this stupid container doesn’t want to stay
upright on the grass; instead, it just wants to tip over and
spill its delicious contents so that the ants can enjoy it.
This is exactly the kind of task which makes physicists
and mathematicians roll up their sleeves together.’’
The English translation leaves much to be desired, but it’s

possible to overlook that and appreciate the author’s crea-
tivity and humor. Let’s focus, then, on the problem of interest.
What is it? Well, when the can is empty, the center of gravity is
obviously in the middle of the can. And when the can is full
the center of gravity is also obviously in the middle of the can.
Apparently, the center of gravity travelled down and then
back up again, but it pretty clearly never hit the bottom. So,
‘‘how much soda do you have to drink so that the center of
gravity reaches its lowest point?’’ You may have noticed a few
things here. The author has an entertaining style, and the
problem is again, I think, very interesting. The solution
begins by describing the important variables as well as
recalling how one computes the center of gravity. It’s non-
trivial, but presented at the right level for the intended
audience.

Some chapters focus on the challenges of parking your car:
How can you model, mathematically, the act of parallel
parking? What’s the right position to enter a parking space?
Other chapters present problems of a visual nature: How far
should one person walk behind another in order to have the
best possible visual angle of that person’s legs? There’s also a
chapter consisting of things we find amusing, such as Hilbert’s
Hotel Infinity or the fact that ‘‘from a mathematical point of
view, all numbers are interesting.’’ Hermann even manages to
introduce induction, using the toasting problem as his moti-
vation: if N people are in a room and everyone toasts everyone
else, how many times would the glasses clink?

Hermann must be a remarkable teacher. He chooses the
topics carefully, the problems are well motivated, and the
presentation is clever, entertaining, and clear. The English
translation, as well as some curious misprints, may interfere
with your enjoyment, but overall The Beauty of Mathe-
matics is a charming little book.

Department of Mathematics

Bucknell University

Lewisburg, PA 17837

USA

e-mail: pgorkin@bucknell.edu

80 THE MATHEMATICAL INTELLIGENCER � 2013 Springer Science+Business Media New York

DOI 10.1007/s00283-013-9372-x


	The Beauty of Everyday Mathematicsby Norbert Herrmann