ASPECTS OF SYMMETRY Contents 1. Introduction 1 2. Symmetry in ...

ASPECTS OF SYMMETRY
SAAD ALI, GARRISON BUSS, EMILY FIORILLI, AKSHI GANDHI, ZUBAIR ILYAS, CHO AH KIM, KAYLEIGH
KREMER, NATALIE KRZYZANOWSKI, TZIPPORA MENDELSBERG, ELLIOT PETROPULOS, JAMIE RAYAHIN,
JUSTIN SANDERS, AND VRUSHANK SHIRISHBHAI SHAH
Abstract. Applications of symmetry arguments in music, dance, Islamic art, molecular biology, chemistry,
and genetics, evolution and game theory, are presented and suggested for further studies.
Contents
1. Introduction 1
2. Symmetry in music - I 2
3. Symmetry in music- II 3
4. Symmetry in dance 5
5. Islamic patterns 6
6. Tilings of the plane 8
7. Symmetry, phylogeny, and evolution 9
8. Molecular symmetry 13
9. Symmetry and the genetic code 15
10. Structure of protein amino acid crystals 17
11. Symmetric games 20
References 22
1. Introduction
Symmetry is a language with which many aspects of science and the arts can be expressed, appreciated,
and eventually better understood and explained. The present paper is aimed at introducing the reader to a
number of those aspects found in music, dance, Islamic art, biology, chemistry, physics, and game theory. It
is an invitation to further applying symmetry as a language in those and other contexts.
Key words and phrases. Flow of time, rhythm, tonal symmetry, spatial symmetry, cyclicity, pitch-class symmetry, time
symmetry, bodily symmetry, fluctuating asymmetry, motion capturing, evolutionary studies, contemplation and philosophy,
Euclidian plane, directional asymmetry, group representation, characters, orbits, permutations, rotations, reversals, canonical
projections, structural symmetry, DNA, base pairs, circular permutations, crystals, payoff matrices, shuffling, game profiles.
This paper was jointly written by the students attending the HON 201 seminar Symmetry in Science and Applications at
the University of Illinois at Chicago Honors College. M. Viana, Instructor. Revised December 3, 2008.
1

2. Symmetry in music - I
Mankind has always had a strong connection with music. From the almost hypnotic tap-tap-tap of tribal
drums to intricate and harmonious arias and concertos, music touches people in a very unique way. What
is more interesting is the fact that all music seems to follow some universal rules of symmetry. These
symmetries, of course, go beyond the strict and static mathematical symmetries, however, because they
evolve both in musical time and space. This accounts for the numerous violations of symmetry that we see
in almost all music.
Music and the flow of time. While studying musical symmetry, one will find that music and time are inseparable,
and can almost be considered as one in the same thing. Music must adhere to the constant flow of
time. This, inevitably, forces any composition to contain certain structural elements, and these elements can
be arranged in a symmetrical manner. For example, a song is comprised of formal parts or sections; there is
an intro, chorus or some other type of variant, and an outro. If the intro and the outro are exactly the same,
we find that the song is perfectly symmetrical with respect to its structure; it has an A-B-A framework. This,
however, is usually not the case; the intro/outro are rarely identical. Musicians use variation to captivate
their audience, but variation is also important to further the development of any repetition. Repetition
requires contrast and -vice versa- contrast demands repetition [1].
Rhythm. Another association between music and time is rhythm; a song’s pace is influenced heavily by its
rhythm. A guitarist will use up and down strokes on the strings with a steady rhythm, using four strums
per second, eight strums per second, etc. Therefore, the symmetry of a song can be found in a change in this
rhythm. An example of this would be a change of 4st/s (four strums per second) to 8st/s, and then back to
4st/s. This is another form of the A-B-A style of symmetry.
This up and down strum will also create a symmetry within notes. The down strums produce a note
beginning from the E string and progressing in the following manner: E,A,D,G,B,E. An upward strum will
produce a mirror image (E,B,G,D,A,E) of this set. In this sense, any chords played with an alternating up
and down strum will have this mirror symmetry, along with the symmetry of the rhythm.
Tonal symmetry. Music also finds symmetry within its tonal structure. Musical tones vary from deep, muted
notes to high-pitch, resonating notes. Kempf [1] describes a song’s change in tone using a rainbow as an
example. The music within a song will vary from low tones to high ones, and then back down to low tones,
much like the bend in a rainbow. When listening to a composition, one can readily pick out the up and
down shifts in tone. Here we can discover another symmetry within the tones; a vertical symmetry. Again,
musical symmetry does not adhere to strict rules. A musician will attempt to vary the shifts in tone, based
on which sequence of notes is most pleasing to hear. For example, a shift from a G-chord to a D-chord may
sound more pleasing than a shift from a C-chord to a Bm-chord. These shifts in chords can be explained by
tonal distances.
Using the guitar as an example, one can see the physical distance between two chords. The F and G-
chords are placed two frets apart on the neck of the guitar, and can be played in an alternating fashion. In
doing so, a guitarist can create a symmetry of the notes(F-G-F-G), which have a tonal distance of two frets,
with respect to the guitar.

The reader will appreciate the opportunity of comparing the notions of horizontal (repetition) and vertical
(tonal) symmetries in music with the underlying mathematical structure (the Klein group of symmetries of
the planar rectangle) implicit in the narrative of [1].
3. Symmetry in music- II
It may seem counterintuitive to apply so potentially rigid a mathematical concept as symmetry to something
as flexible and progressive as form in tonal music. Yet, it proves instructive to measure degrees of
exact symmetrical correspondence. Symmetry, perhaps the most basic of what Hegel calls ”the relations
of the abstract understanding,” forms a virtually unavoidable constant against which we can evaluate the
boundaries of art and, indeed, life itself [2].
Spatial symmetry. In its modern definition, symmetry applies to the quality that allows certain objects,
patterns, or sets of relationships to remain invariant when subjected to a certain group of transformations.
In two-dimensional space this group comprises the geometric transformations of rotations, reflections, and
translations (displacement). For instance, a square has four axial reflections (four axes about which it can
be reflected) and fourfold rotational symmetry (four positions to which it can be rotated about a central
point).
Figure 3.1. Symmetry transformation of a square, from Symmetrical Form and Common-
Practice Tonality.
Pitch-class symmetry. It contains only twelve locations. Pitch-Class symmetry has significant symmetrical
properties of its own: twelvefold rotational and twelvefold transpositional symmetry. The rotational character
stems from cyclicity: each pitch class returns to its initial position after traversing the other eleven equidistant
positions by chromatic scale or circle of fifths. Cyclicity also makes translation equivalent to rotation, since
in this space the former also eventually leads back to its starting point. A pitch-class set with transpositional
symmetry, then, is one that divides pitch-class space into equidistant smaller segments, reducing its twelvefold
symmetry to a more limited order.
Whereas transpositional symmetry is derived from equal octave division, inversion symmetry is derived
from corresponding placement about a reflection line that divides the pitch-class space into two mirroring
halves. For example, (0,2,7) reflects about a line from 1 to 7, (1,2,4,7,9,10) about a line from 51/2 to 111/2
(recalling that reflections occur at half the rotational angle).

Figure 3.2. Symmetry transformation of a square, from Symmetrical Form and Common-
Practice Tonality.
Temporal symmetry. Temporal symmetry forms an essential component of all dynamic symmetrical systems,
such as planetary motions, or the rotations of hands on a clock face (as opposed to the static clock face
itself), or the passage of musical events (as opposed to atemporal pitch-sets).
A temporally symmetric system, then, is one in which an event occurs regularly, separated by a recurring
time-interval, and thus remains invariant under displacement by that interval. The basic idea is analogous
to that of two-dimensional translational symmetry: corresponding to the regular recurrence of patterns in
space, an event recurs at regular intervals in time. Modeling time as a horizontal line, temporal translation
can be represented as in Figure 3.3, corresponding to the spatial representation in Figure 3.1. From this it
follows that both spatial and temporal translation encompass two related components: a recurring unit of
space or time
Figure 3.3. Symmetry transformation of a square, from Symmetrical Form and Common-
Practice Tonality.
Conclusion and summary. The concept of symmetry can be applied to a wide range of phenomena, including
physical objects (elementary particles, geometric figures, planetary systems), abstract systems (laws
of physics, mathematical relations, pitch systems), and processes taking place in time (chemical reactions,
biological processes, a segment of music). Symmetry, even strictly defined, is essentially a matter of degree.
Absolute symmetry (infinite degrees of all possible symmetries) means absolute indistinguishability; and

while the idea provides an essential conceptual limit, it is of little practical value. Objects that are symmetrically
structured normally possess only partial symmetry which can be viewed as the result of a break in a
more encompassing symmetry.
4. Symmetry in dance
Introduction. The main hypothesis proposed in [3] is that quality of dance can reveal bodily symmetry, or,
using the terminology of evolutionary studies, fluctuating asymmetry (FA). This hypothesis was formulated
with basis on the theory that symmetry shown in dance movements can be considered as a measure of
developmental stability. The experiment described in [3] and conducted in Jamaica (a society in which
dancing is important in the lives of both sexes) found that there is a strong positive association between FA
and dancing ability. It was also found that these associations between FA and dancing ability were stronger
in men than in women.
Fluctuating asymmetry and genetic or phenotypic quality of the dancer. It was Darwin who first to suggested
that dance is a sexually selected courtship signal. The justification for studying the dancer’s FA
comes from the observations that (i) FA is inversely correlated with developmental stability, defined as “an
organisms ability to reach an adaptive endpoint despite ontogenetic perturbations” [3]; (ii) that increased
FA is associated with morbidity, mortality, poor fecundity, and other variables linked to natural and sexual
selection, and (iii) that FA is inversely associated with attractiveness based on a person’s odor, voice, and
facial appearance.
Motion capturing. The technique of motion-capturing is generally used in sports and medical studies to
assess potential confounds within dance evaluations such as clothing, culture, physical attractiveness and
fluctuating asymmetry. This technology is used to actively extract complex movement in the form of film
animation. From the animations, the investigators may attend to those traits intrinsic to dancing without
distractions from the dancer.
Experimental design. The investigators chose 40 dance animations from the motion-capturing of 183 human
dancers conducted in 2004 in Southfield, Jamaica, based on two measures of FA. Those in the top third of
both measures were classified as asymmetrical (n=20) and those in the bottom third (n=20) were categorized
as symmetrical.
The traits that were measured were the elbow, wrist, knee, ankle, foot, third digit, fourth digit, and ears.
These traits were chosen because they revealed true fluctuating asymmetry and have proven useful in past
studies [3]. When looking at the videos created, no dancers could be identified as a specific dancer in the
film, even sex was unable to be identified (although female evaluators were better at detecting the sex of the
dancer than male evaluators). These evaluators based their judgments on movements, rather than the sex
of the dancer.
Experimental results. The experiment showed that there is a significant effect of symmetry and sex on dance
ability. The experiment showed that symmetrical males were evaluated as significantly better dancers than
asymmetrical males (as were females). It was also found that symmetrical males were significantly better

dancers than symmetrical females. In contrast, there was no statistically significant difference in dance
ability between asymmetric males and females.
In this experiment, the relative preference for symmetrical dancers was also tested. More specifically,
to test for sex differences between evaluators and the strength of preference for symmetrical individual
dancers. Relative preference for symmetrical dancers was constructed by subtracting dance evaluations
given to asymmetrical dancers by dance evaluations given to symmetrical dancers. Therefore, higher scores
indicated stronger preferences for symmetrical dancers. From this it was determined that female evaluators
had stronger relative preferences for symmetrical male dancers but there was no sex difference in dance
ratings of symmetrical females. It was concluded that male evaluators gave higher ratings to the dances of
females.
Finally, it was found that FA in male evaluators was negatively associated with relative preference for
dances performed by symmetrical women. But, they came across no significant association between female
evaluator FA and preferences for symmetrical male dancers.
Conclusion. As motion-capture technology gets better, storing dance in a mathematical form, the investigators
hope to study in more detail which patterns of dance movement are associated with quality of dance and
fluctuating asymmetry of the dancer. A potential application of that methodological improvement would
be the testing the hypothesis of whether or not dance ability correlates with reproductive success in that
society.
5. Islamic patterns
Introduction. Islamic symmetrical patterns have a rich tradition that can be utilized in many ways, whether
it is simple enjoyment, an instrument of instruction for students ranging from toddlers to adults, and finally
a means of contemplation and philosophy [4].
Figure 5.1. The name Muhammad, showing symmetry in calligraphic writing. From [4].
Enjoyment. Symmetry plays an important role in Islamic culture, and one of the first places to look for it is
in specialized Arabic writing, called calligraphy. Arabs have raised their art form of writing in Arabic while

incorporating symmetry that a simple name can be made to stand out as an art piece by itself. For example,
Figure 5.1 shows the name Muhammad written in calligraphy.
Aside from calligraphy, many other examples relating to symmetry in Islam can be found. An important
one relates to home decoration in the form of carpet rugs. Arabs have a long history of carpet weaving, and
as such, the carpet is perhaps the most visible and most easily expressed form of decoration [4]. Oftentimes
the carpet will have symmetrical patterns on it, an example being tessellations where the entire carpet is
made up of patterns. This leaves the eye looking for visual start and finish points for the pattern that it
cannot find, and lends to the carpets appeal.
Instrument of instruction. Islamic patterns in architecture provide a great means of teaching for students of
all ages and areas of expertise. Starting from the younger age, toddlers can enjoy art by coloring in pictures
of Islamic patterns such as the one shown in Figure 5.2. Young students can therefore begin to appreciate
Figure 5.2. Stars, providing symmetrical images to coloring. [4].
at an early age the symmetry that is present in the world around them in its various forms. Furthermore,
teaching would not just be limited to the elementary grades. Students as advanced as undergraduates in
college can work with geometrical patterns found in Islamic architectural patterns to find a visual gateway
to the more conceptual notions of group theory.
Group theory involves mathematical processes present in the creation of symmetry transformations (rotations,
reflections, translations and glide reflections). It is also a tool of describing nature through mathematical
means, and while it is a deep and eventually abstract topic, Islamic patterns found in architecture
can provide that illustrative link that is often needed to understand those algebraic notions.
Contemplation and philosophy. Any discussions linking Islamic architectural patterns to contemplations of
philosophy should include the palace complex of Alhambra, in Granada, Spain [4]. Containing all seventeen
known types of symmetry, this building is a veritable source of Islamic patterns.
For anyone who has studied Islamic patterns, a common point to make may be that Muslims dedicate a
lot of their symmetrical art to depicting stars in various forms and sizes. A reasonable explanation to that
comes from a Quranic verse which states “Allah is the Light of the heavens and the earth.” From human
perception, the source of light is stars, and a logical step leads to the star having a significant place in Islamic

architectural patterns, especially in sacred buildings such as mosques. It would seem that Islamic patterns
borrowed heavily from a saying of Plato, “God ever geometrizes” [4]. Since Muslims believe in an abstract
God, philosophers agreed with Plato and believed that symmetrical geometry is a link between this tangible
world and the intangible world of the hereafter.
Summary. In this section we have briefly introduced some symmetry-related aspects of Islamic patterns
and how they can relate to a myriad of subjects. There is a rich tradition of symmetry that can be found
in Islamic art from the religions beginning. Islamic patterns can be used for many purposes, including
enjoyment, an instrument of instruction for students ranging from toddlers to adults, and finally a means of
contemplation and philosophy [4].
6. Tilings of the plane
Introduction. Tiling of geometric objects has been applied in art and aesthetics for thousands of years, and
only very recently have surprising findings of symmetric tiling groups has been verified. One of the most
interesting involved the combination of vectors connecting lattice points to form units which upon translation
cover the Euclidian plane.
Tiling the Euclidian plane. It was discovered that when attempting to construct repeating lattice units to
cover the Euclidian plane there is only a small set of angles between the basis vectors that allow for a complete
and gap-free covering of the plane. This theoretical result has since been named the Crystallographic
Restriction Theorem. It claims that the angles between the vectors generating the covering lattice can only
be of 60, 90, and 120 degrees.
Figure 6.1. Lattice unit with 60 deg, generating a tiling of the Euclidean plane by regular
triangles. From http://en.wikipedia.org Crystallographic restriction theorem.
Proof. we start with the matrices representing the planar rotations, specifically,
[ ]
cos θ sin θ
R(θ) =
, 0

Since we must cover the plane in all directions, after completing a full rotation around its center we must
neatly return to our starting point. However, we must to the turn in finitely many steps, so that we must
have
[ ]
cos(mθ) sin(mθ)
R(θ) m =
= I,
− sin(mθ) cos(mθ)
where I is the 2 × 2 identity matrix and m is a positive integer. Therefore we must have cos(mθ) =1(or
sin(mθ) =o), which implies θ =0, 60, 90, 120, 180 deg. Excluding the (null) arrays generated by 0, 180 we
obtain the proposed result.
□
The reader may also consult [5], for additional comments. The resulting tilings of regular triangles,
squares, and hexagons respectively as shown in 6.2, cover the entire Euclidian plane without gaps.
Figure 6.2. Tilings by regular triangles, squares, and hexagons. From
http://en.wikipedia.org Tiling by regular polygons.
7. Symmetry, phylogeny, and evolution
Introduction. Symmetry is something that can be seen throughout the world, whether it is music, art, or
in ordinary shapes. Furthermore, symmetry can be seen in nature. However, the types of symmetry seen
in everyday organisms are not the same as what they were originally. Over time, evolution has resulted in
changes in the structure of these organisms, with the consequence of different types of asymmetry. What has
concluded is symmetry, antisymmetry, and directional asymmetry. Through the course of this section, these
three categories of asymmetrical variation will be discussed, along with examples found in nature displaying
these specific variations.
Symmetry. Symmetry has a number of definitions, based on the context that it is being used in. In the
context of evolution, it can be described as when the exact form is present on the opposite side of some
boundary, whether it is a plane, line, point, or axis 1 . The organisms that exist today likely evolved from
symmetrical organisms, but over the years, many of them diverged into different forms. This evolution is
referred to breaking bilateral symmetry [6].
1 From The Free Dictionary, Farlex, Inc., Definition of Symmetry by the Free Online Dictionary, 2008

Forms. There are different types of symmetries, based on where the axis or center of symmetry exists.
Horizontal symmetry can be seen with respect to an horizontal (reflection) line, which results in the top and
bottom being equal. Vertical symmetry, similarly, is obtained with respect to a vertical axis of symmetry,
resulting in the left and right sides of the object being equal. Figures 7.1 and 7.2 show examples of horizontal
and vertical symmetries, respectively. Point symmetry exists when the object can be rotated around 180 deg,
and still appear to be the same. Equivalently, is has a center of reflection. Figure 7.3 shows an example of
this case. Identity symmetry, on the other hand, is as if nothing is done to the object. All objects have this
type of symmetry, since every object is equal to itself.
Examples. Symmetrical organisms are not found often in nature anymore because they have all evolved
over the generations. According to Figure 7.4, however, the Atherninomorph ancestor is one that was likely
symmetrical in its original form. According to this diagram, it had zero state changes, which signifies that
it remained symmetrical.
Figure 7.1. Horizontal symmetry, from www.arcytech.org.
Figure 7.2. Vertical symmetry, from http://en.wikipedia.org.
Antisymmetry. One of the types of asymmetry variations is called antisymmetry (AS). This occurs when
handedness within a single species is random, in the respect that either left or right handedness occurs at
approximately the same frequency [6]. It is important to note that “antisymmetric does not connote the
lack of symmetry” [7]. In fact, it means that both right-handed and left-handed symmetry exist, and that
the direction of handedness is different for each individual in that species [8].

Figure 7.3. Point symmetry, from http://www.regentsprep.org.
Figure 7.4. Phylogeny, from http://www.pnas.org/.
Development. It appears that antisymmetry patterns are caused by environmental factors. The difference in
environment results in natural selection favoring different patterns of development. This change in pattern
was likely caused during the post-larval stage, and therefore in a late part of development [6].
Examples. If the origin of asymmetry occurred in post larval times, then the type of asymmetry is likely
antisymmetry. An example of such a case is the male phallostethid fish, which had late development of
asymmetry. Further, this fish had antisymmetry persist through years of evolution events. In addition, the
flatfish Psettodes has antisymmetric patterns [9]. Other examples of anti symmetrical patterns in species
include lobster claws and male fiddler crabs. Antisymmetry also exists in the upper mandible of finches, as
well as in palm-tree trunks [8].

Directional asymmetry. It is said that there are two routes that lead to directional asymmetry. The first
route involves conventional evolution, by which a symmetrical ancestor evolves into a directional asymmetric
version. For example, this would be the case if a crab without any claws were to evolve into a form that had
only one claw. The second path involves two steps. The first step, known as random asymmetry, involves
the conversion of a symmetric ancestor into two directionally asymmetric forms. In the crab example below,
the crab ancestor without any claws evolves into two forms, one with the claw on its right side and the other
with the claw on its left side. These two forms collectively are called antisymmetry, since the two forms are
opposites of each other. In antisymmetric organisms, the two versions (right and left handed) are usually
found in roughly equal proportions of the population, as mentioned previously.
The second step of this second route is called genetic assimilation. During genetic assimilation, one of the
two antisymmetric forms that randomly developed disappears, and the other form predominates. The process
of genetic assimilation is aided by environmental factors and natural selection, which favor the propagation
of one of the forms. Figure 7.5 shows the evolutionary paths leading to the various symmetric forms 2 The
form that predominates takes on what is called a directional asymmetric form. The individuals of a species
that exhibits directional asymmetry will all experience that symmetry on the same side of their bodies[9].
Figure 7.5. Asymmetry paths. From http://scienceblogs.com.
There are two directionally asymmetric forms. The first one is called dextral asymmetry and the second is
sinistral asymmetry. Dextral and sinistral refers to the side where the asymmetry originates. Dextral means
“right-handed” and sinistral refers to “left-handed”.
Examples. Examples of a dextral asymmetric organism would be a crab that had a claw on the right side
of its body but not on the left. A sinistral crab would be the opposite. Humans experience directional
2 from http://scienceblogs.com/pharyngula.

asymmetry in many aspects of their bodies. For example, the heart is located on the left side, making this
specific organ sinistral asymmetric. On the phylogenetic tree (Figure 7.4), the bikolanus and posthon species
experience dextral asymmetry, while the trewavasae exhibits sinistral asymmetry.
Conclusion. In conclusion, there are three broad categories of symmetry present in organisms throughout
nature. The three states are symmetry, antisymmetry, and directional asymmetry, and all three states are
connected. Various processes such as random genetic mutation, natural selection, and genetic assimilation
all can have an effect in changing an organism from one type of asymmetry to the next. This constant
changing of symmetry can, eventually, lead to the creation of new species, as well as the development of new
types of symmetry.
8. Molecular symmetry
Introduction. Molecules and atoms both can have various sorts of symmetry. One way to explore this
symmetry is to use group theory, which is the branch of mathematics concerned with studying groups and
their properties. Group theory can lead to significant results in physics which are a consequence purely of
the symmetry in the system.
The notation of symmetry in Chemistry. The following is the basic notation for symmetry operations in
Chemistry:
• Identity operation, E - the identity operation appears in all symmetry groups, but has a ”do nothing”
effect.
• Rotation, C n - A rotation about an axis by a certain degree.
• Inversion, i - Every point (x, y, z) in the reference frame is transformed into (−x, −y, −z).
• Reflection, σ - A reflection with respect to a plane in the reference frame.
• Improper rotation, S n - A rotation C n that is followed by a reflection with respect to the plane
perpendicular to the rotation axis.
Group representation. The group C 3v is a group with six symmetry operations. It may also be called S 2 , the
group which contains the symmetries of a triangle, namely 3 rotations and 3 axial reflections. Its properties
can be inspected by manipulating a complete multiplication table of group operations. This shows the
general outcome of operations being multiplied, since data is arbitrary.
The operations which are multiplied (by each operation) are: E, C 3 , C3, 2 σ a , σ b , σ c . In other words, E
is multiplied by E, E is multiplied by C 3 , and so forth, until the next column (C 3 ) is multiplied by the
same components. To obtain a matrix representation for a row, the components of the row are multiplied
separately by the operation in each column. The position of the final outcome, which is determined from the
multiplication table itself, is then established by looking at where this final outcome appears in the column.
In the row of the matrix (the column of the multiplication table goes with the row of the matrix), a 1 is the
value placed in the position determined previously. The rest of the values are 0, since they do not match the
outcome.
By representing these components in a matrix form, further analysis can be easily made with the tools of
linear algebra. Each row has its own matrix. The first row, since it is operated on by E the identity yields

the 6 × 6 identity matrix. The resulting matrices, which correspond to the permutations in rows 1-6, are
shown below, respectively.
⎡
⎤ ⎡
⎤ ⎡
⎤
1 0 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
1 0 0 0 0 0
0 0 1 0 0 0
1 0 0 0 0 0
0 1 0 0 0 0
r 1 =
, r 2 =
, r 3 =
,
0 0 0 1 0 0
0 0 0 0 0 1
0 0 0 0 1 0
⎢
⎥ ⎢
⎥ ⎢
⎥
⎣ 0 0 0 0 1 0 ⎦ ⎣ 0 0 0 1 0 0 ⎦ ⎣ 0 0 0 0 0 1 ⎦
0 0 0 0 0 1
0 0 0 0 1 0
0 0 0 1 0 0
⎡
⎤ ⎡
⎤ ⎡
⎤
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
0 0 0 0 1 0
0 0 0 0 0 1
0 0 0 1 0 0
0 0 0 0 0 1
0 0 0 1 0 0
0 0 0 0 1 0
t 1 =
, t 2 =
, t 3 =
.
1 0 0 0 0 0
0 0 1 0 0 0
0 1 0 0 0 0
⎢
⎥ ⎢
⎥ ⎢
⎥
⎣ 0 1 0 0 0 0 ⎦ ⎣ 1 0 0 0 0 0 ⎦ ⎣ 0 0 1 0 0 0 ⎦
0 0 1 0 0 0
0 1 0 0 0 0
1 0 0 0 0 0
Canonical projections. The understanding of group theory is based on what is called the irreducible representations.
We use a character table corresponding to the specific group (defined as the sum of the diagonal
elements of the irreducible linear representations), which in our case is C 3v .
Much like we did with group orbits, by adding up matrices with different coefficients (which we obtain
from the character table), we can obtain results called canonical projections. The three arrangements
(irreducible characters) of coefficients for C 3v are A 1 = {1, 1, 1, 1, 1, 1}, A 2 = {1, 1, 1, −1, −1, −1}, and
E = {2, −1, −1, 0, 0, 0}. For each arrangement, the numbers correspond to the coefficient of each of the six
matrices shown above.
By doing such linear combination of those matrices with the characters as the coefficients, we get three
final matrices, namely
⎡
⎤ ⎡
1 1 1 1 1 1
1 1 1 1 1 1
P 1 = 1 1 1 1 1 1 1
, P 2 = 1 6
1 1 1 1 1 1
6
⎢
⎥ ⎢
⎣ 1 1 1 1 1 1 ⎦ ⎣
1 1 1 1 1 1
and
⎡
P 3 = 1 3
⎢
⎣
⎤
2 −1 −1 0 0 0
−1 2 −1 0 0 0
−1 −1 2 0 0 0
.
0 0 0 2 −1 −1
⎥
0 0 0 −1 2 −1 ⎦
0 0 0 −1 −1 2
⎤
1 1 1 −1 −1 −1
1 1 1 −1 −1 −1
1 1 1 −1 −1 −1
,
−1 −1 −1 1 1 1
⎥
−1 −1 −1 1 1 1 ⎦
−1 −1 −1 1 1 1
It is interesting to observe that these matrices have the following properties:

(1) P i P j = P j P i = 0 for i ≠ j (algebraically orthogonal);
(2) Pi 2 = P i (projections);
(3) P 1 + P 2 + P 3 = I (decomposing the identity matrix).
These canonical projections can be interpreted, in applications, as the averaging, the contrast between
rotations and reversals, and as the within-rotations and within-reversals variability, respectively [10].
Conclusion. Group theory can be applied to study most symmetry properties in chemistry, physics, and
their data-analytic interpretations as well. Like in the example shown with C 3v , this analysis is a general
rule for any group.
9. Symmetry and the genetic code
Introduction. DNA is an integral part of the human body and without this microscopic chain of base pairs,
the human species would be diminished to nothing. Propagation of humankind, in fact, the entire animate
world, depends on DNA’s maintenance, protection, and ultimate inheritance by the filial generations. Most
of the importance of DNA comes from the inherent symmetry in its structure. Therefore, the structure of
DNA stabilizes its functionality. Aside from the structural symmetry, it is interesting to focus attention on
the cyclical symmetry within DNA. Due to DNA’s inherent symmetry, base pair frequency within a DNA
molecule sheds light on the importance of a particular base pair strand in the function of DNA.
This is clearly noted in the Human immunodeficiency virus type 1 (HIV), were hyper-variable portion of
DNA are distinguished from conservative portions by the frequency of a specific 6-mer [11]. This distinction
plays out in functional strategic responses to positive and negative-selection pressures on the HIV strain[11].
Structural symmetry for DNA. The prime features of DNA include its two anti-parallel helices, its sugarphosphate
backbone, and the stacking and distances of bases (the bases lie perpendicular to the axis of
symmetry) from one another. The fact that DNA has two helices, which run anti-parallel to each other (a 5’
and 3’ strand), is an inherent symmetry of it and is important in the replication process of DNA. Without
this symmetry, mechanical problems would arise in the replication of DNA, and humans would not be able
to grow, or even survive for that matter. The two anti-parallel helices, now primarily known as the double
helix, contain pairs of complementary purines and pyrimidines, which are equally spaced from one another
by about 0.34nm. The bases are located innermost to the strand, and the sugars and phosphates are located
on the outside (backbone). This is true for each separate strand, illustrating their symmetry to one another.
Moreover, in both strands, these bases are stacked on top of one another. This stacking symmetry and
symmetry in the distances stabilizes the DNA molecule and allows it to proceed through replication without
destruction. Overall, with the characteristics described, DNA would be said to have dyadic symmetry about
the central axis. The importance of the physical structural symmetry of DNA is illustrated in Figure 9.1.
Base pairs. The bases that make up the inner portion of DNA are the fundamental aspect of DNA. These
base pairs are what code for the production of proteins, enzymes, and are integral for evolutionary purposes.
There are four nucleotides that are specific to DNA; adenine, guanine, cytosine, and thymine. These bases
pair in a specific order within the DNA molecule, where purines bind with pyrimidines. This pairing gives
a important principle called the parity rule. The parity rule, first introduced by Chargaff, states that the

Figure 9.1. Structure of DNA, from http://static.howstuffworks.com.
frequency of C will always be equal to that of G and A will always be as frequent as T. Interesting enough,
when the frequency of sets of nucleotide sequences is investigated, an important functional experience is
observed.
Circular permutation of base pairs. A useful tool in understanding the importance of particular base pair
sequence is the use of circular permutations and cyclic symmetry. When looking at a three nucleotide
sequence, such as ACT, the circular permutation is ACT, CTA, and TAC. This can be further permuted
to give a dihedral orbit [12] with the set {ACT, CT A, T AC, T CA, AT C, CAT }. The ACT sequence and
all subsequent permutations can be found within a genome in varying frequencies. By analyzing these
frequencies and their diversities a pattern of sequence importance can be determined.
The frequency diversity is the ratio between the highest and lowest frequency within an orbit. The closer
the ratio is to one the more likely the sequence and its permutations are integral to the genome [12]. Examples
of the relationship between frequency diversity and functionality is apparent in the HIV strain investigated
in [11]. The following table shows base pair sequence excesses in relating genes on a strand of DNA. Figure
9.2 shows an interesting relationship between hyper variable and conservative regions with respect to cyclic
sets of nucleotide sequences. It illustrates the importance of pyrimidine and purine frequency in coding for
specific genes, as in the pol, gag, etc. This further illustrates the apparent symmetry in the DNA molecule.
Conclusion. The importance of DNA is paramount to the survival of the human species. The inherent
symmetry in both its physical aspects and genetic code is important for the stability of this molecule and
hence, the stability of mankind. This aspect of DNA applies across the board. Certain nucleotide sequences
within a DNA determine the function of the relating gene. This allows DNA to protect a species, such as HIV,
whose DNA allows for continual variation of a particular gene coding for the protein coating. This tendency

Figure 9.2. Excess cyclic sets for five regions. From [11].
towards variation protects the HIV virus from positive-selection pressure, namely the immune system of the
host. While other regions of the gene coding for proteins required internally for the maintenance of the virus
have lower nucleotide frequencies the hyper-variable regions, because variations in these regions would alter
the internal functionality of the virus, limiting its ability to survive. This illustrates the importance of the
symmetry within the DNA molecule. The inherent structure of the physical model of DNA allows for smooth
replication and easy transcription in protein synthesis. The symmetry maintains the internal function of the
cell. It is within this area that the DNA molecule is conservative. When looking at the cyclic symmetry
of the genetic code, the apparent importance of the hyper-variability in protection of the cell is observed.
Taken as such, DNA’s importance to the breadth of the animate world is interconnected and tightly woven
with the highly symmetric order of its structure and the genetic code.
10. Structure of protein amino acid crystals
Introduction. All of the structures and functions of living organisms are the results of endless complex
chemical reactions of numerous molecules that play an imperative role in once survival. Proteins are one
of these molecules, which function along with other molecules in intensification and expansion of the living
organisms. Moreover, proteins have been selected from last several years to perform diverse tasks such as
collecting sun light, transporting materials, provide mechanical strength or even fighting viruses or bacteria
etc. Yet proteins are relatively simple molecules. Proteins are made up of number of amino acids (there are

20 types of proteins), which are connected into a specific sequences in the linear chains. The specifications
in amino acids sequences define and segregate one protein structure from another. With many protein
structures (now over 16,000 proteins) solved and a lot of experiments done over the past decade, it has
been found that most of proteins forms symmetrical complexes consisting of two or more identical subunits.
Hence, symmetry seems to be very important ingredient in understanding the structures of protein amino
acid crystals.
Protein amino acids structure. Crystals of amino acids are composed of Amino group (NH 2 ), Carboxylic
group (COOH), Hydrogen (H), and a distinct Radical (R) group, which are connected to central atom
Carbon (C) with four distinct and respective bonds. All the protein amino acids differ from each other
due this Radical ( R ), which varies from R=H in the simplest protein amino acid glycine to, for instance
phenylalanine CH 2 CH 6 H 5 . The basic molecular structure [13] of the protein amino acids is presented in
Figure 10.1.
Figure 10.1. Structure of protein amino acid, from http://web.ebscohost.com.
Classification of crystals by symmetry. All the crystals so far discovered can be described by crystal system,
which are groupings of crystal structures according to the axial system used to describe their lattices. Each
crystal system consists of a set of three axes in a particular geometrical arrangement. There are seven unique
crystal systems. They are as follows:
(1) Triclinic
(2) Monoclinic ( see Figure 10.2)
(3) Orthorhombic ( see Figure 10.3)
(4) Hexagonal
(5) Rhombohedral
(6) Tetragonal
(7) Cubic
The aggregate information from above types of crystal systems leads to the Bravais lattices, which are
the geometric arrangements of the lattice points, and thus the translational symmetry of the crystals. In

three dimensions, there are 14 distinct Bravais lattices that differ from one another in terms of translational
symmetry they contain. Moreover, often the crystal structure consists of identical group of atoms positioned
around each and every lattice points and thus repeats itself to a certain degree. These characteristic rotation
and mirror symmetries of the group of atoms are described by its crystallographic point group, which consists
of symmetry operations that leave at least one point unmoved and that leave the appearance of the crystal
structure unchanged. These symmetry operations are as follows:
(1) Reflection: it reflects the structure across a reflection plane.
(2) Rotation: it rotates the structure a specified portion of a circle about a rotation axis.
(3) Inversion: it changes the sign of the coordinate of each point with respect to a center of symmetry
or inversion point.
(4) Improper rotation: it consists of a rotation about an axis followed by an inversion.
Figure 10.2. Monoclinic crystals. From http://britannica.com.
Figure 10.3. Orthorhombic crystals. From http://www.silvershake.com.
Symmetry in protein amino acids. Crystals of all the protein amino acids, except glycine are optically
active and exist in the levorotated (L) and dextrorotated (D) molecular configurations. Proteins of the
living organisms are made up of only by L-amino acids, which are crystallize mainly in monoclinic C2 and
orthorhombic D2 point groups. As mention above, Monoclinic crystal system is one of the seven point
groups. In the monoclinic system, the crystal is described by three vectors of unequal length. They form
a rectangular prism with a parallelogram as its base. Hence two pairs of vectors are perpendicular, while

the third pair makes an angle other than 90 deg. There exist two such types of monoclinic Bravais lattices
and they are the simple monoclinic and the centered monoclinic lattices, with layers with a rectangular
and rhombic lattice, respectively. On the other hand, orthorhombic crystal system is also one of the seven
point groups, but differs from monoclinic crystal system. Orthorhombic lattices result from stretching a
cubic lattice along two of its orthogonal pairs by two different factors, resulting in a rectangular prism with
a rectangular base (a x b) and height (c ), so as a, b , c are different. Also, all the three bases intersect
at 90 degree angle and as a result, the three lattice vectors remain mutually orthogonal. There exist four
such types of orthorhombic Bravais lattices and they are simple orthorhombic, base-centered orthorhombic,
body-centered orthorhombic, and face-centered orthorhombic.
Conclusion. In this section we presented the fact that proteins in living organisms are made up only of
L-amino acids and that the structure of these amino acids mainly belongs to monoclinic and orthorhombic
crystal systems.
11. Symmetric games
Introduction. In game theory, special properties surround normal form symmetric games. In this section, we
will define what a symmetric game is, as well as how symmetric games have a reduced number of profiles.
The primary references are [14] and [15].
Basic definitions. Symmetric games occur when all players can choose from a single set of strategies, while
the outcome, or payoff, depends only on which strategies are played, not on who plays them. A profile is a
unique mapping of strategies for each player, and the number of profiles in a game is called the pay. In the
present section, we will use N to represent the number of players in a game and S to represent the number
of strategies.
Payoff matrices. The payoff matrix is a method of representing the payoffs for each player in a normal game.
Each cell within the payoff matrix represents the payoffs for each player using a single strategy profile,
with the coordinates of the cell corresponding to the strategy used by each player. During the round, the
players would choose their strategy without the knowledge of other players’ strategies. Without exploiting
symmetry, the payoff matrix requires S N cells.
To illustrate, consider a one-shot matching game, discussed in [15, p.11], where players independently
have to choose an object from the set
V = {◦,⊲,•},
consisting of a light disk, a light triangle, and a dark disk. Matrix (11.1) describes the payoff for player 1
(rows) depending on the actions of both players. Here, their actions depend on whether they attend to the
color or the shape of the objects.
(11.1)
◦ ⊲ •
◦ 1 − w c − w s 0 0
⊲ w s 1 − w c w s
• w c w c 1 − w s

In particular, the cells along the main diagonal of the matrix represent the outcomes (in probability terms)
when both players chose the same strategy.
Shuffling. Shuffling is a method of studying the effects of symmetry within a game. For instance, we may
allow the shuffling of players within a game and study its effect on the number of resulting profiles.
To illustrate, consider a game where two players chose to put either an apple or an orange in a bowl not
visible to a spectator. After the players each placed one of the fruits in the bowl, the spectator is allowed
to peek at the bowl. The spectator can tell whether there are two apples, two oranges, or an apple and
an orange, but in the final case, the spectator cannot tell which player placed which fruit, showing that
the outcome does not depend on shuffling their actions. The three relevant outcomes are the three unique
profiles present.
Reducing the number of profiles by symmetry. While payoff matrices require S N cells without exploiting
symmetry, a new equation dramatically decreases the number of unique profiles in symmetric games. The
equation
( )
N + S − 1 (N + S − 1)!
=
N N!(S − 1)!
produces the number of different profiles in a symmetric N-player game with S strategies.
Reducing the pay by symmetry. For this example, we will use a 2-player game with 3 strategies. The players
will be represented with {1, 2} and the strategies will be represented with {a, b, c}. The game profiles are
given by the set
V = {aa, bb, cc, ab, ba, bc, cb, ca, ac}
of all mappings from {1, 2} to {a, b, c}, with |V | =9=3 2 profiles, and hence possible payoff cells. Now we can
make the game symmetrical by letting the symmetries of S 2 = {1, (12)} shuffle the set of players appearing
in the definition of V . This symmetry means that the payoffs should be assigned up to interchanging of the
players, as they lose their unique identities. As such, {ab} should have assigned the same payoff as {ba},
{bc} the same as {cb}, and so on. Because of this, V only has
( )
N + S − 1
=6
N
distinct right-permutation orbits 3 or profiles, namely
{aa}, {bb}, {cc}, {ab, ba}, {ac, ca}, {bc, cb}.
The larger number of players and strategies, the greater the difference between the number of symmetric
and non-symmetric game profiles and payoff elements.
Conclusion. Within this section, we have defined symmetric games, payoff, pay, payoff matrices, and shuffling.
We have also provided examples to describe payoff matrices, and shuffling, to demonstrate how
symmetric games have a reduced pay compared to their non-symmetric equivalents.
3 Technically, the result of S2 acting on {1, 2}.

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