

Research Article
Spatial (Artistic) Networks: From Deconstructing
Integer-Functions to Visual Arts

Ernesto Estrada 1 and Puri Pereira-Ramos2

1Department of Mathematics & Statistics, University of Strathclyde, 26 Richmond Street, Glasgow G1 1HQ, UK
2PeRArt Studio, Murgas 4, 15822 A Coruna, Spain

Correspondence should be addressed to Ernesto Estrada; ernesto.estrada@strath.ac.uk

Received 19 September 2017; Accepted 9 December 2017; Published 17 January 2018

Academic Editor: Gerard Olivar-Tost

Copyright © 2018 Ernesto Estrada and Puri Pereira-Ramos. This is an open access article distributed under the Creative Commons
Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is
properly cited.

Deconstructivism is an aesthetically appealing architectonic style. Here, we identify some general characteristics of this style, such
as decomposition of the whole into parts, superposition of layers, and conservation of the memory of the whole. Using these
attributes, we propose a method to deconstruct functions based on integers. Using this integer-function deconstruction we generate
spatial networks which display a few artistic attributes such as (i) biomorphic shapes, (ii) symmetry, and (iii) beauty. In building
these networks, the deconstructed integer-functions are used as the coordinates of the nodes in a unit square, which are then
joined according to a given connection radius like in random geometric graphs (RGGs). Some graph-theoretic invariants of these
networks are calculated and compared with the classical RGGs. We then show how these networks inspire an artist to create artistic
compositions using mixed techniques on canvas and on paper. Finally, we call for avoiding that the applicability of (network)
sciences should not go in detriment of curiosity-driven, and aesthetic-driven, researches. We claim that the aesthetic of network
research, and not only its applicability, would be an attractor for new minds to this field.

1. Introduction

There are multiple connections between networks and the
visual arts. The study of graph drawing is an old topic in
computer sciences and one of its main goals is the repre-
sentation of networks in aesthetically appealing ways [1, 2].
In modern network theory, there have been extraordinary
advances in the visualization of giant complex networks,
which can be considered as pieces of art by themselves [3]. A
different direction is the use of networks as an artistic mean of
expression. The artistic work of Tomás Saraceno is an example
of this kind of symbiosis where the author has used spider
webs to create a universe of expressions [4]. Other artists melt
networks into evocative images of the real-world to produce
artistic designs. This is the case of the artist J. K. Rofling
who has produced many of these symbiotic images [5]. Some
examples of the work of J. K. Rofling are illustrated in Figure 1.

Here, we explore a different approach to connect net-
works and the visual arts. Essentially, we start from the con-
struction of spatial networks based on simple rules, namely,

the location of points in a unit square. However, the coor-
dinates of these points are generated by a mathematical
transformation of integer numbers that generates artistic
patterns on the plane. The inspiration for such transformation
of integers and functions based on them comes from the
“poststructuralist” school of philosophy and literary criticism
known as deconstruction. This school started in the late 1960
after the influential book De La Grammatologie (1967) by
the French philosopher Derrida [6]. This school of philo-
sophical thinking influenced any areas of intellectual and
creative activity including novels, poetry, architecture, the
fine arts, and music. In architecture in particular, the term
“deconstructivism” was adopted since the end of the 1980s
[7]. According to Derrida this architectural style “is not simply
the technique of an architect who knows how to deconstruct
what has been constructed but a probing which touches
upon the technique itself, upon the authority of the architec-
tural metaphor and thereby constitutes its own architectural
rhetoric” (cited by Hoteit in [7]).

Hindawi
Complexity
Volume 2018, Article ID 9893867, 8 pages
https://doi.org/10.1155/2018/9893867

http://orcid.org/0000-0002-3066-7418
https://doi.org/10.1155/2018/9893867


2 Complexity

(a) (b)

Figure 1: Two of the works produced by J. K. Rofling and taken from [5] with permission of the artist. (a) The Trand. (b) The Guys.

(a) (b)

Figure 2: (a) City of Capitals in Moscow IBC, Russia. (b) Diagram of City of Culture of Galicia, Santiago de Compostela, Spain, by Peter
Eisenman.

We do not pretend here to make a complete analysis of the
deconstructivism in architecture but mainly of having a basic
idea of its principles to be applied beyond its original fron-
tiers. We then notice that the architectural deconstructivism
looks initially as a fragmentation of the buildings which
lack any visual logic. However, the deconstructing work
accounts not only for this fragmentation but also for keeping
a “memory” of the original composition in such a way that it
“remembers” what it was in the beginning, that is, a building.
In the City of Capitals in Moscow IBC, Russia, which is
illustrated in Figure 2(a), the building is deconstructed into
its unit block, that is, a cube, which is them “multiplied” to
create again a tower with a different shape as the traditional
ones. Another characteristic of deconstructivism is that the
whole work must superimpose elements in such a way that
“the design is produced, and the idea follows as its result.” As
described by Hoteit [7] one of these examples is the City of
Culture of Galicia, Santiago de Compostela, Spain, designed

by Eisenman. According to Hoteit [7] “Eisenman was mostly
known for using the superimposition of layers.” In his creation
of the City of Culture of Galicia “Eisenman determined the
following four local traces: The downtown’s historical street
grid; the typography of a hill; the abstract Cartesian grid; and
the symbol of the city of Santiago, which is the scallop shell.
Then, he superimposed these four abstracted traces to create
an imaginary site condition, which became a real site for his
project” (see also [8]). This idea is illustrated in Figure 2(b).

The connection with mathematical ideas here is evident.
The superposition of layers can be imitated by the sum
of parts and the compositional part can be obtained by
multiplying the deconstructed parts such that we can recover
certain “memory” from the original object. There are of
course several ways of imitating these two characteristics of
deconstructivism, but we have selected these two for the sake
of mathematical convenience. This idea attempts to follow the
existing line of connection between mathematical objects and


Complexity 3

visual arts. This includes among others knots [9, 10], mosaics
and tiles [11, 12], Fourier series [13], topological tori [14], and
fractal curves [15], all of which produce artistic patterns of
undoubtful beauty by themselves.

It can also be argued that some works in the cubism move-
ment show elements of deconstruction. Indeed, analytic cub-
ism is seen as an influential stream for deconstructivism via
the work of Frank Gehry. Analytical cubism includes impor-
tant paints by Picasso, Braque, Metzinger, and others [16, 17].
Here, again, the principles of fragmenting, integrating, and
superimposing are relevant in the analysis of these works [17].
Focusing only on these three principles to understand decon-
struction is a clear oversimplification. However, we consider
them here as the angular stone for what we will consider in
the current work. Here, we are concerned with a formula-
tion of deconstruction principles in mathematics.

2. Deconstructing Integer-Based Functions

Formulating deconstructivist principles for the whole of
mathematics is a too ambitious project for a single paper.
Instead, we focus here on integers and functions of integers.
Then, the question is how to deconstruct an integer? The first
idea should be to consider the individual digits of an integer
as its building blocks. That is, for an integer 𝑥 written in a
given base 𝑏, it is represented by

𝑥 = 𝑎1𝑏𝑛+𝑎2𝑏𝑛−1+𝑎3𝑏𝑛−2+ ⋅ ⋅ ⋅ +𝑎𝑛−1𝑏+𝑎𝑛, (1)
where 𝑎𝑖 ∈ Z are nonnegative integers, which can be con-
sidered as the building blocks of 𝑥. For instance, the building
blocks of𝑥 = 2018are 2, 0, 1, and 8. Here the “whole” is repre-
sented by the integer, which in architecture should be the
tower. The blocks are the digits forming that whole, like the
cubes in the tower.

Now, we should proceed to the “superposition of layers”
part. Here, we simply consider the function that sums the
digits of the integer 𝑥 in the base 𝑏 [18]:

𝑆𝑏 (𝑥) =
𝑛

∑
𝑖=1

𝑎𝑖 =
⌊log
𝑏
𝑥⌋

∑
𝑘=0

1
𝑏𝑘
(𝑥 mod 𝑏𝑘+1−𝑥 mod 𝑏𝑘). (2)

For instance, for 𝑥 = 2018, the integration will produce
𝑆10(𝑥) = 11. These sequences for different bases 𝑏 are stored
in the On-Line Encyclopaedia of Integer Sequences [19, 20]; for
instance, A007953 is the sequence for 𝑏 = 10.

In order to complete the deconstruction of the integer we
need the “recovery of the memory” of the original object. That
is, we consider the product of the integer 𝑥 by 𝑆10(𝑥) as the
final deconstruction of the integer 𝑥 [21]:

𝑥𝑏 = 𝑥
𝑛

∑
𝑖=1

𝑎𝑖. (3)

In this way, we have that a given integer is first dismem-
bered into its digits; then the digits are superimposed to each
other as the different layers of the integer using the digit-sum
function. Finally, we “recover” the memory of the original
number by multiplying the integer by its digit-sum. Hereafter,

we consider only the base 𝑏 = 10; thus 𝑥 = 𝑥𝑏. Using this
approach, the deconstructed integers “remember” something
about their original numbers. For instance, 1̂9 = 190, 2̂8 =
280, 3̂7 = 370, 4̂6 = 460, 5̂5 = 550, 9̂1 = 910, and 8̂2 =
820(see sequence A117570 in [19]). However, it does not mean
that 𝑥𝑏 is different for each integer. For instance, 7̂5 = 1̂50 =
900.

Let us now extend this approach to any function based on
integers. Let𝑓(𝑥)be a function of the number𝑥, for example,
sin(𝑥). Then, the sum of digit-functions 𝑓(𝑥) : Z → R as the
function defined on the integers, such that

𝑓(𝑥) = (𝑓(𝑎1)+𝑓(𝑎2)+ ⋅ ⋅ ⋅ +𝑓(𝑎𝑛))𝑓(𝑥) . (4)
For negative integers −𝑥, if the function 𝑓(−𝑥) exists, we
define

𝑓(−𝑥) = (𝑓(𝑎1)+𝑓(𝑎2)+ ⋅ ⋅ ⋅ +𝑓(𝑎𝑛))𝑓(−𝑥). (5)

We then consider the plot of pairs of functions 𝑓(𝑡) and
𝑔(𝑡) for the integers 𝑡 ≤ 𝑛/2 such that

𝑥 = 𝑓(𝑡),

𝑦 = 𝑔(𝑡).
(6)

If the functions 𝑓(𝑡) and 𝑔(𝑡) are also defined for
negative arguments we obtain the corresponding transforms
for −𝑛/2 ≤ 𝑡. We are going to use these functions to build
spatial networks as described in the next section.

3. Building Spatial Networks

In this section, we define our strategy for building spatial
graphs based on the deconstruction of integer-functions. This
strategy is based on the random geometric graphs (RGGs).
Thus, we first explain the way in which RGGs are built. The
RGG is defined by distributing uniformly and independently
𝑛points in the unit𝑑-dimensional cube[0,1]𝑑 [22]. Hereafter
we consider only the 2-dimensional case. Then, two points are
connected by an edge if their Euclidean distance is at most 𝑅,
which is a given fixed number known as the connection radius.
That is, we create a disk of radius𝑅centered at each node, and
every node inside that disk is connected to the central node as
illustrated in Figure 3. A few important structural parameters
of RGGs have been determined analytically in the literature
(see, e.g., [22]).

Now, let us consider the process 𝑇 that generates 𝑛 points
in the unit square according to the transforms of integer-
functions defined in the previous section. For instance, let us
consider −1000 ≤ 𝑡 ≤ 1000 and make the following trans-
formation.

Transform 1 (𝑇1).
𝑥 = �̂�,

𝑦 =
{
{
{

ŝin𝑡, 𝑡 ≤ 0
−ŝin𝑡, 𝑡 > 0.

(7)


4 Complexity

0.2 0.4 0.6 0.8 10
0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 3: Illustration of a RGG created with 250 nodes embedded
into a unit square where the nodes are connected if they are at a Eu-
clidean distance smaller than or equal to 𝑅 = 0.15.

Notice that we consider the trigonometric functions of
the numbers in degrees not in radians. For instance, sin𝑡
means “sine of 𝑡 degrees.” Then, we plot every point on the
unit square according to its coordinates (𝑥,𝑦) defined before
as illustrated in Figure 4(a). Using the approach to construct
RGGs described before we construct the network for a given
value of 𝑅. That is, after placing the points in the unit square
we center a disk of radius 𝑅 on each point and connect to
it every other point which is inside the corresponding disk.
Here we will use radii which guarantee the connectivity of
the graph—the study of the connectivity of these graphs
is beyond the scope of the current work. For instance, in
Figure 4(b) we illustrate the network created by using 𝑅 =
0.075.

4. Spatial (Artistic) Networks

It is straightforward to realize that the previously obtained
spatial graph (Figure 4(b)) displays a few artistic attributes:
(i) biomorphic shape, that is, suggestive in shape of a living
organism (a butterfly in this case); (ii) symmetry; and (iii)
beauty, just to mention three. The appearance of a biomorphic
shape here is just by chance and we have selected in this work
only those transforms of integer-functions which produce
artistically appealing shapes. However, it must be emphasized
that both—beauty and interpretation of shapes—are on the
eyes of the beholder, and different observers can see different
things in these and other spatial networks created from
integer-functions. Here we coin the name spatial “artistic”
networks (SANs) for the spatial networks created using the
previously described method.

Let us now consider other alternatives to the integer-
function deconstruction to see which artistic objects we
can obtain. Artistic composition is the result of artist cre-
ativity and it includes a series of general rules that can be

implemented computationally. Here,wemainly follow a hand-
made compositional creation in order to glue series of inte-
ger-function transforms into single art works. For instance,
let us consider the following parametric equations.

Transform 2 (𝑇2).
𝑥 = �̂�,

𝑦 = ĉos𝑡+ �̂� ⋅ ŝin𝑡.
(8)

The resulting network for 𝑅 = 0.085 with 𝑛 = 1000 points
is illustrated in Figure 5 where we have used −500 ≤ 𝑡 ≤
500 and the nodes are colored according to their closeness
centrality.

Transform 3 (𝑇3). Another example is obtained by transform-
ing the Astroid curve using the integer-digit transform. First,
let us remind the reader that the Astroid is the curve: 𝑥 =
cos3(𝑡) and 𝑦 = sin3(𝑡). Then, we make the transformation of
the coordinates as explained before, such that we have

𝑥 = (ĉos𝑡)3 ,

𝑦 = (ŝin𝑡)3 .
(9)

The corresponding SAN is illustrated in Figure 6, where
we have used again −1000 ≤ 𝑡 ≤ 1000 and the nodes are
colored according to their closeness centrality.

Transform 4 (𝑇4). The involute of the circle—𝑥 = cos𝑡 +
𝑡sin𝑡;𝑦 = sin𝑡−𝑡cos𝑡—can also be transformed accordingly
for −1000 ≤ 𝑡 ≤ 1000 such that we obtain the following
parametric equations:

𝑥 = ŝin𝑡− 𝑡 ĉos𝑡,

𝑦 = −ĉos𝑡− �̂� ŝin𝑡,
(10)

which produce the network illustrated in Figure 7.

Transform 5 (𝑇5). Finally, we obtain the integer-function
transformation of the cardioid curve, such that

𝑥 = 2ĉos𝑡+ ĉos(2𝑡),

𝑦 = 12 ĉos𝑡−5 ŝin(2𝑡),
(11)

where

ĉos(2𝑡)

= (cos(2𝑎1)+ cos(2𝑎2)+ ⋅ ⋅ ⋅ + cos(2𝑎𝑛))cos(2𝑡)

ŝin(2𝑡)

= (sin(2𝑎1)+ sin(2𝑎2)+ ⋅ ⋅ ⋅ + sin(2𝑎𝑛))sin(2𝑡) ,

(12)

with 𝑡 = 𝑎1𝑏𝑛+𝑎2𝑏𝑛−1+𝑎3𝑏𝑛−2+ ⋅ ⋅ ⋅+𝑎𝑛−1𝑏+𝑎𝑛 represented
in the decimal basis 𝑏 = 10. The spatial graph based on this
transformation is illustrated in Figure 8 where we have used
−1000 ≤ 𝑡 ≤ 1000.


Complexity 5

−0.5

0

0.5

0 2 4−2−4
×10

4

(a) (b)

Figure 4: Illustration of the process to build a spatial network based on integer-function deconstruction. (a) Distribution of the points
obtained from the transform 𝑇1 on a square. (b) Construction of the spatial graph using a connection radius 𝑅 = 0.075 with 𝑛 = 2,000 points
and coloring the nodes according to their closeness centrality.

Figure 5: Spatial network constructed from the distribution of
points in a unit square according to the transform𝑇2using a connec-
tion radius 𝑅 = 0.085 with 𝑛 = 1000 points and coloring the nodes
according to their closeness centrality.

5. Network Invariants of SANs

Here we consider a few invariants of the networks con-
structed by using the five transformations previously studied
and compare them with the same invariants for the analogous
RGG. That is, we construct RGGs with the same number
of nodes and connection radius than the SANs created by
the previously defined transforms. These invariants are as

Figure 6: Spatial network constructed from the distribution of
points in a unit square according to the transform𝑇3using a connec-
tion radius 𝑅 = 0.1 with 𝑛 = 2000 points and coloring the nodes
according to their closeness centrality.

follows: the number of nodes 𝑛, the number of edges 𝑚, the
edge density𝛿, the maximum degree𝑘max, the average Watts-
Strogatz clustering coefficient 𝐶, the global transitivity index
𝐶, average shortest path distance 𝑑, network diameter 𝑑max,
and the degree assortativity𝑟(for definitions and meaning see
[23]). In Table 1, we give the values of these graph-theoretic
invariants for the SANs and RGGs studied here.


6 Complexity

Table 1: Graph-theoretic invariants of the spatial “artistic” networks described in Section 4.

𝑇1 𝑇2 𝑇3 𝑇4 𝑇5
SAN RGG SAN RGG SAN RGG SAN RGG SAN RGG

𝑅 0.075 0.085 0.1 0.085 0.1
𝑛 2,000 1,000 2,000 2,000 2,000
𝑚 30,158 33,128 32,286 10,440 129,656 57,287 198,951 42,113 61,217 57,287
𝛿 0.015 0.017 0.077 0.021 0.065 0.029 0.099 0.021 0.031 0.029
𝑘max 117 55 205 39 299 87 581 64 150 87
𝐶 0.618 0.613 0.665 0.608 0.727 0.622 0.642 0.614 0.695 0.622
𝐶 0.217 0.200 0.249 0.198 0.247 0.202 0.264 0.202 0.225 0.202
𝑑 11.16 8.22 7.38 7.59 7.45 6.14 6.08 7.266 11.90 6.14
𝑑max 46 21 39 19 23 15 34 18 33 15
𝑟 0.65 0.60 0.75 0.59 0.74 0.61 0.79 0.60 0.68 0.61

Figure 7: Spatial network constructed from the distribution of
points in a unit square according to the transform𝑇4using a connec-
tion radius 𝑅 = 0.085 with 𝑛 = 2000 points and coloring the nodes
according to their closeness centrality.

In general, the graph-theoretic properties of SANs are
relatively similar to those of the RGGs. However, there
are some differences, particularly for the maximum degree
and maximum distance. That is, the SANs always have
significantly larger 𝑘max and 𝑑max than the corresponding
RGGs. These two parameters are larger in the SANs as a con-
sequence of the higher concentration of points in the center
of the figure in relation to their peripheries. This situation is
avoided in the RGG due to the random and homogeneous
distributions of the points in the unit square. The similarities
in terms of clustering coefficients and assortativity—notice
that all networks are degree assortative—between SANs and
RGGs are remarkable. We, however, are not claiming any
application of these graphs for solving problems in the real-
world, apart from being a source of artistic inspiration. Then,
the analysis of these properties is mostly a curiosity-driven
one and not the search for useful properties of these graphs.
In the next section, we explore how these networks inspire
some art.

Figure 8: Spatial network constructed from the distribution of
points in a unit square according to the transform𝑇5using a connec-
tion radius 𝑅 = 0.1 with 𝑛 = 2,000 points and coloring the nodes
according to their closeness centrality.

6. Artistic Inspiration

Science is sometimes seen as a dry and cold activity, such
that it is not able to inspire those which are not involved in
it. In earlier definitions of the humanities as “the branches
of polite learning, especially the ancient classics and literature
of aesthetics, as distinguishes from informational or utilitarian
values” the sciences are marginalized as “informational but
unaesthetic, that is, as useful but grubby” [24]. Many efforts
are currently done for attracting the attention of the general
public to the beauty of scientific discoveries. In mathematical
sciences, for instance, there are initiatives, such as Bridges
[25, 26], which bring together mathematicians and artists
to produce artistic works from, or inspired by, mathematics.
The Journal of Humanistic Mathematics [27] has also been
launched to fill the gap between the humanities and math-
ematics.


Complexity 7

(a) (b)

(c) (d)

Figure 9: Photograph of four artistic works of artist Puri Pereira. (a) “Butterfly # 1” painted in acrylic, ink, and watercolor on paper of
dimensions 30 × 40.5 cm. (b) “Fire on Water” painted in acrylic on canvas of dimensions 45.7 × 61 cm. (c) “Butterfly # 2” painted in acrylic
on canvas of dimensions 45.7 × 61 cm. (d) “Birds” painted in ink, watercolor, and acrylic on paper of dimensions 30 × 40.5 cm.

In this part of our work, we present a few snapshots of
what an artist can bring from the visual images produced by
the SANs obtained from the integer-functions deconstruc-
tion presented here. That is, the SANs previously described
have been the source of artistic inspiration for the production
of purely aesthetic works outside the constraints of (network)
sciences. The results are illustrated in Figure 9.

7. On the Artistic Value of (Network) Sciences

Network sciences have an important impact on our under-
standing of nature and modern society. Its practical impor-
tance has been documented in many papers in the last few
years. But network science is also driven by aesthetic criteria.

Sometimes it is the mathematical beauty of the equations
describing the structure of, or the dynamics on, the networks.
Sometimes it is the beauty of the embedding of the network
into certain space that produces outstanding visualizations.
Other times it is the result of the application of network
theory to a particular problem that produces an aesthetic
feeling due to the beauty of the findings or what is unexpected
of the connections found. Then, the importance of the
applications of networks to solve practical problems should
not hide its inherent beauty. The applicability of (network)
sciences should not go in detriment of curiosity-driven, and
aesthetic-driven, researches. We should find a compromise
between application-driven and curiosity-driven researches.
Abraham Flexner [28]—who was a founder of the Institute of


8 Complexity

Advanced Studies in Princeton and its Director from 1930 to
1939—stressed that “institutions of learning should be devoted
to the cultivation of curiosity and the less they are deflected by
considerations of immediacy of application, the more likely they
are to contribute not only to human welfare but to the equally
important satisfaction of intellectual interest which may indeed
be said to have become the ruling passion of intellectual life in
modern times.” Obviously, there are many pressing problems
in modern society that we are aimed to solve using network
methods and approaches, and we should never forget our
social responsibility. But our institutions should not forget
either that as Flexner remarked “a poem, a symphony, a
painting, a mathematical truth, a new scientific fact, all bear
in themselves all the justification that universities, colleges, and
institutes of research need or require” [28]. Thus, we should
be reminded that (network) science has a humanistic side,
which is as important as the many applications that it has
found. Forgetting this side of it—its beauty and capacity of
surprising—is similar to tear a wing to a bird. We all know
that birds with only one wing cannot fly.

8. Conclusions

The spatial artistic networks (SANs) created here are the
product of a curiosity-driven process more than of any prac-
tical necessity or real-world application. Thus, the value of
these networks does not reside in their usefulness as a math-
ematical tool for modeling reality but as a source of inspira-
tion of artistic work as well as attractive objects per se. We do
not discard, however, that such networks can find some appli-
cations for modeling spatial processes in the real-world, due
to their similarities with RGGs as well as by the fact that
the points here are not randomly distributed in space but by
using well-defined mathematical rules. The type of high-
density core and very sparse periphery reminds one with the
situation frequently found in many spatial networks like
cities.

Many chemistry students have been motivated to their
subject by the beauty of the representations of the molecular
structure. In physics, a similar situation exists when we
consider the aesthetic of cosmic landscapes and the structure
of the universe. Mathematicians always claim to be seduced
by the beauty of mathematical equations. Can we attract
students into network sciences by appealing to the aesthetic
beauty of networks? The only way to know it is by trying. We
hope that the current work contributes to this goal, either by
attracting curious minds to the field or by inspiring other re-
searchers in the field to explore the beauty of networks per se.

Conflicts of Interest

The authors declare that they have no conflicts of interest
regarding the publication of this paper.

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