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Vol. 3, No. 1, March 2014 | At Right Angles 73

S

ymmetry is a topic that

resonates with audiences

of varied backgrounds and

levels of mathematical knowledge.

One can ask a layperson to explain

what symmetry means to them

and there would invariably be a

fairly accurate response from an

informal or non-mathematical

point of view. However, symmetry

has deep roots in mathematics

and in some sense pervades most

areas of study in mathematics. The

mathematical study of symmetry

though has its primary residence

in an area of abstract algebra

called ‘group theory’. What is remarkable about Marcus Du Sautoy’s

book on symmetry is that the mathematics underlying the study of

symmetry is explained without recourse to technical mathematical

language. While the word ‘group’ makes its debut on page 9 of the

book, it is only much later that its mathematical context is explained.

By then the reader has had sufficient foundation laid to absorb the

mathematical context.

Of Monsters and

Moonshine

A review of ‘Symmetry’

by Marcus Du Sautoy

Geetha Venkataraman

Keywords: symmetry, patterns, Marcus du Sautoy, reflection, rotation, tiling, group, permutation

review

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74 At Right Angles | Vol. 3, No. 1, March 2014

As a mathematician there was a special joy in

realising that it was possible to talk about areas of

research in a language that would find consonance

with the interested reader. The book begins with

notions of symmetry that are commonplace or

intuitive notions. Through the course of the book

the reader is taken on a journey that explores the

connections of symmetry with nature, evolution,

psychology, music and even mathematics at the

research level. There is also a conscious attempt

to illustrate mathematical ideas with concrete

examples from everyday life.

The book starts with the author on the shore

of the Red Sea contemplating the fact that he

has turned 40. The number 40 is important in a

mathematician’s life. The Nobel Prize equivalent

in Mathematics is the Field’s medal. In a way it

is tougher to get a Field’s medal than a Nobel

Prize as at most four are awarded every four

years and only to mathematicians who have done

outstanding work and have not yet attained the

age of 40.

The chapters in the book traverse the months of

a calendar year beginning with the first chapter

titled August: Endings and Beginnings and

finishing with July: Reflections. Thus the book

intertwines a year of Marcus Du Sautoy’s life, his

forays into searching for symmetrical objects

that are part of his research, his encounters as a

mathematician with ‘symmetry seekers’; a term

used for the mathematicians trying to classify

and quantify ‘indivisible collections of symmetry’,

with the story being told of the main protagonist,

namely, symmetry.

Since a year of Marcus’s life is interlinked with

the story of symmetry we learn about how

the author got interested in mathematics at

the age of 12 because of a schoolteacher who

encouraged him. This might indeed be the case

for many a mathematician. A book the author

was encouraged to read, as a schoolboy was The

Language of Mathematics by Frank Land. If one

thinks about it with some care, mathematics as

a language is particularly efficient in expressing

precisely and concisely the statements that one

wishes to make. The problem though is that it is

not always an easy language to master.

There are several intersecting strands that are

covered in the book. One strand represents

the usual story that one expects while learning

about symmetry: reflections and rotations

of regular geometric figures like the square,

equilateral triangle to those of the five platonic

solids, symmetries of infinite figures like wall- paper patterns and tilings. The chapter October:

The Palace of Symmetry, discusses the search

by the author and his son Tomer for the 17

wallpaper patterns or tessellations that exist

in the Alhambra Palace in Granada, Spain, built

around 1300 by Spain’s Muslim rulers. Another

strand brings to fore the life histories and works

of the mathematicians of the Renaissance period

leading to those from the early 19th century who

were responsible for creating the mathematical

language of group theory to analyse symmetries.

These stories are the fodder for Chapters 5-8 from

December: connections to March: indivisible shapes.

April: Sounding Symmetry, as the title hints at,

discusses the links between western classical

music and symmetry. It also points out the

opposing philosophy, between when musicians

use symmetrical object as a basis for creating their

music but keep them secret from the audience,

and the task undertaken by a mathematician of

laying bare all the facts logically about the objects

of study.

The strand where the book goes beyond the

expected is when it moves into the difficult

territory of describing one of the mammoth

tasks that occupied the symmetry seekers for a

large part of the previous century, namely, ‘the

classification of simple finite groups’. This history

is explained entirely in terms of ‘indivisible

symmetry groups’ with many anecdotes and

mathematical experiences thrown in. While this

thread is woven into several chapters, the last

three are primarily devoted to this twentieth

century tale.

There are strands that are entirely missing,

though. The book is deficient in telling the story

of symmetry of non-western cultures. The Asian

experience with symmetry finds no place. Indeed

there is hardly any mention of the role played by

symmetry in ancient or even medieval India. It is

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Vol. 3, No. 1, March 2014 | At Right Angles 75

a lacuna that is not even acknowledged in passing

by the author. Euro-centrism is the lens used.

Let us think of symmetry as ‘a magic trick move’

that keeps an object looking exactly like it did

and in the exact same position as it occupied to

start with. For example, as explained in the book,

if we take a 50 pence coin (which is shaped like a

regular heptagon or seven-sided figure) and draw

its outline on a piece of paper, then a symmetry

of the coin is any move or action that can be

performed on the coin which brings the coin back

into the outline drawn. In other words if someone

had closed their eyes while the symmetry was

being performed on the coin then they would

assume that nothing had taken place. If we forget

the markings on the coin and use this definition of

symmetry then it is not too difficult to see that a

regular heptagon has 14 symmetries in all.

An easier example to work with is the equilateral

triangle. It has six symmetries. Three are mirror

symmetries or reflections, each about a line

joining a vertex to the mid-point of the opposite

side. These three lines of reflection meet at a

point that is like the ‘centre’ of the equilateral

triangle. The other three symmetries of the

equilateral triangle are rotations through 0°

,

120°

and 240°

respectively, say in the counter- clockwise direction, about an axis passing through

the centre and perpendicular to the plane of

the triangle. For example, if the three vertices

of an equilateral triangle are marked A, B, C in

the counter-clockwise direction then the 120°

counter-clockwise rotation will take A to B, B to C

and C back to A. The figures show the effect on the

vertices after a reflection or rotation symmetry

has been performed.

A six-pointed star with no reflection symmetries

(see figure) also has exactly six symmetries;

these are rotations through 0°

, 60°

, 120°

, 180°

,

240°

and 300°

. Here too, we can keep track of the

symmetries by assuming a marking of the vertices

and noting the effect of the respective symmetries

on the vertices.

But is the collection of six symmetries of an

equilateral triangle the same as the collection

of six symmetries of a six-pointed star with no

mirror symmetries? The answer is No: the two

collections of symmetries are not the same when

we consider them as ‘groups’. Here is one way to

see this: for the six-pointed star, if we take any

two symmetries, it does not matter in which order

we apply them; the final result is the same. But

in the case of the equilateral triangle, if we apply

a reflection followed by (say) a 120°

rotation, we

get a different symmetry than when we first apply

a 120°

rotation followed by the reflection. If we

let the symbol M denote the reflection and R the

rotation through 120°

, then in the language of

mathematics we write: M * R ≠ R * M. (The symbol

M * R denotes that R is applied first and then M.)