Symmetry Discovered: Concepts and Applications in Nature and Science

by Joe Rosen

Overview

Symmetry provides an insight into the way nature works and is often used by scientists and technologists to help solve problems. Symmetry has numerous other applications as well — with more being discovered all the time in science, the arts and other fields of human endeavor.
This classic work provides an excellent introduction to the basic concepts and terminology (including, optionally, group theory), as well as lucid discussions of geometric symmetry, other symmetries and appropriate symmetry, symmetry in nature, uses of symmetry in science and much more.
Readers wishing to pursue specific topics will find many references that reflect the author's wide reading in the subject and his own obvious enthusiasm. For this edition, Dr. Rosen has provided a new preface, solutions to the problems, and an addendum to the bibliography.

Product Details

ISBN-13:	9780486145006
Publisher:	Dover Publications
Publication date:	09/18/2012
Series:	Dover Books on Mathematics
Sold by:	Barnes & Noble
Format:	eBook
Pages:	176
File size:	6 MB

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Symmetry Discovered

Concepts and Applications in Nature and Science

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By Joe Rosen

Dover Publications, Inc.

Copyright © 1998 Joe Rosen
All rights reserved.
ISBN: 978-0-486-14500-6

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CHAPTER 1

Symmetry: what? where? how?

Sometimes he thought sadly to himself, 'Why?' and sometimes he thought, 'Wherefore?' and sometimes he thought, 'Inasmuch as which?'-and sometimes he didn't quite know what he was thinking about. (A.A. Milne: Winnie-the-Pooh)

What is symmetry?

Consider a square of definite size, located at a definite position in space, and having a definite orientation (fig. 1.1). Among all possible actions that can be performed on this square, there are some that will leave it in a condition indistinguishable from its original condition (fig. 1.2). Which are these?

Since the square must remain a square, all changes of shape are eliminated (fig. 1.3). The square must retain its size, so size changing is forbidden (fig. 1.4).

The position of the square in space cannot be altered. We must therefore reject any movement which displaces its center (fig. 1.5).

Let us rotate the square. To preserve position the axis must pass through the center. If the axis is perpendicular to the plane of the square, any rotation will leave the square in its original plane, and, of these, three are objects of our search: rotation by 90º, 180º or 270º leaves the square in a condition indistinguishable from its original one (fig. 1.6). (Rotation by 360º also fulfills this requirement, but is equivalent to no rotation at all and can be ignored. Any rotation of more than 360º is equivalent to one of less than 360º, which can be found by subtracting off 360º a sufficient number of times.) No other rotation does this.

Are other rotation axes passing through the center allowable? Most are not. Those that are lie in the plane of the square. There are four of them: the two diagonals and the two lines parallel to one or the other pair of opposite edges. Only a rotation by 180º is acceptable here, otherwise the square will not be brought back into its original plane (fig. 1.7).

Rotate in plane, about center? Yes, only by 90º, 180º, 270º Finally, we consider mirror reflections. To keep the center fixed, the plane of the mirror must pass through the center. (The mirror is considered to be two-sided.) It is easily seen that the only mirror orientations that will produce reflections indistinguishable from the original are: the plane of the mirror is perpendicular to the plane of the square and passes through any one of the four rotation axes of the preceding paragraph (fig. 1.8).

This square and all these actions that change it but leave it looking unchanged are an example of symmetry. The square is symmetric with respect to the actions, which are symmetry transformations of the square. The square itself is an example of a system. A system is whatever it is that we wish to examine with regard to its symmetry properties. The more different symmetry transformations a system has, the higher its degree of symmetry.

Lack of symmetry is asymmetry. Our square is asymmetric under change of size, for instance. An example of general asymmetry is the system consisting of our square with the digit 5 drawn on it (fig. 1.9). The addition of 5 certainly does not add new symmetries to the square, and it is easy to see that none of the symmetries that we found for the square remain applicable to the square plus 5. This system is asymmetric

We emphasize that the system called the 'square' that we investigated so thoroughly is characterized by more than just squareness. Its definition also includes having definite size, position in space, and orientation. If we reduced the strictness of our definition by giving up one or more of these conditions, the degree of symmetry of the system would obviously increase. In the extreme, a system characterized solely by its squareness, that is, a square with no additional specifications, is symmetric under everything but change of shape, the only action capable of modifying its single property. In contrast to our square, this one is symmetric under change of size, displacement of its center, any rotation about any axis (whether passing through its center or not, or not even passing through the square), and reflection by a mirror having any position and orientation. Various intermediate cases can also be considered, and they will have correspondingly intermediate degrees of symmetry. These examples show how important it is to be fully aware of what is and what is not essential to the system whose symmetry is being considered.

These points will be considered in detail in the following chapters.

Where is symmetry?

What can be a system and have symmetry? Anything can. And not only things. To emphasize the generality of the concept we present a few examples: a system could be a geometric figure, like our square, or a physical entity such as an elementary particle, an atom, a molecule, a crystal, a plant, an animal, the earth, the solar system, our galaxy, or the whole universe. It could be a process taking place in time: the scattering of elementary particles by each other, a chemical reaction, the fall of a stone, a beam of light, biological growth, a piece of music, the flight of men to the moon, the evolution of the solar system, or the development of the universe. The system might even be abstract: the laws of physics, an idea or concept, a mathematical relation, a feeling. In fact, I cannot find a system to which the concept of symmetry is inapplicable.

So where, then, is symmetry? Symmetry can be anywhere!

How is symmetry?

What are the possible symmetry transformations? These can be as diverse and imaginative as the possible systems upon which they might act. The first to come to mind are the geometric transformations considered above: change of shape, change of size, displacement, rotation, reflection. A symmetry transformation might concern time: displacement in time, reversal of chronological order of events, change of size of time intervals. A system might be symmetric with respect to interchange of its parts. Physicists work with symmetry under transformations such as interchange of positive and negative electric charge, interchange of particles and antiparticles, change of velocity (this has to do with special relativity), and various abstract symmetry transformations best defined in the context of the physical system to which they are relevant.

These and other examples of symmetry transformations and systems on which they act are presented and discussed in later chapters.

CHAPTER 2

The language of symmetry

Concepts and terminology

'Well,' said Owl, 'the customary procedure in such cases is as follows.'

'What does Crustimoney Proseedcake mean?' said Pooh. 'For I am a Bear of Very Little Brain, and long words Bother me.' (A.A. Milne: Winnie-the-Pooh)

After making a brief initial acquaintance with the concept of symmetry and its general applicability, we now examine the concepts more closely and familiarize ourselves with some important terms, which we need for our discussions.

A system, as mentioned in the previous chapter, is any object of interest with regard to its symmetry properties. The examples of the previous chapter illustrate the generality of the concept of a system; it can be abstract or concrete, microscopic or macroscopic, static or dynamic, finite or infinite.

A possible condition of a system is referred to as a state. For example, the system of the square that we investigated in some detail in the previous chapter has many (actually an infinite number of) states. Each state is determined by specifying the size of the square, its location in space, and its orientation. In contrast, the system whose sole characteristic is squareness, so that properties such as size, location and orientation are not relevant to it, has only one state, since there are not various degrees of squareness.

(Concerning this latter system, the following discussion might be of interest. It is valid to claim that the system has two states, squareness and nonsquareness. This is because we consider the possibility of changing its shape (but see that this is obviously not a symmetry transformation), and how can such a possibility be considered, unless it has a state of nonsquareness? We could even generalize and allow the system to have an infinite number of states-all possible shapes. We could then define the system as 'a geometric figure having a definite shape'; it would not have to be specifically a square. The symmetry of this system is the same as that of the system characterized solely by squareness: any transformation that does not change shape is a symmetry transformation. Therefore, the symmetry of a system characterized solely by shape is symmetric under change of shape! If that was too twisted a piece of reasoning, just ignore it and continue.)

We have used the term transformation without really defining it. It is common to think of a transformation as an action that changes a system from some initial state to some other final state. This is not wrong, but it is too limited a definition for our purpose.

First, we would prefer reducing any emphasis on the action that causes the change of the system while increasing the importance of the relationship 'initial state -> final state'. The final state, called the image, should be considered as derived from the initial state and related to it. We should think less of performing an actual action, which changes the system from the initial state to the image state, and more of setting up a correspondence- to the initial state of the system we make correspond another state as its image.

Second, such a correspondence should be set up, not for just a single state of the system, but for all states. A transformation acts on a system no matter what state it is in. So a transformation is a rule of some kind, whereby the appropriate image can be derived for every state of the system. Any 'action' involved might easily differ according to which state the system happens to be in, and indeed it is hardly reasonable to think in terms of actually performing 'actions' for all states simultaneously. The correspondence picture is by far the most suitable, and this is the meaning we shall reserve for transformation.

The rule of correspondence, by which an image state is associated with every state of a system, is almost completely arbitrary, limited almost solely by the imagination of the inventor of the transformation. (Of course, some transformations are more useful than others.) This correspondence might be expressed either as a general prescription, in which the relation between every state and its image is described in general terms (fig. 2.1), or it might be exhibited as a double list, like a translating dictionary, with all states of the system listed in the first column and their corresponding images listed respectively across from them in the second column (fig. 2.2). For a system with an infinite number of states, the general prescription is the only possible one. The only limitation on this correspondence is that every state of the system must appear in the set of image states and must appear only once, so that the images of different states are always different. This is called a one-to-one correspondence and completes the definition of a transformation.

Returning to our square, for example, we have a system with an infinite number of states, and any transformation must be described in general terms. The transformation 'rotation by 90º about the axis through its center and perpendicular to its plane' is just such a prescription, even though it is formulated in terms of an 'action'. As a 'state -> image' correspondence this transformation means that, whatever the size of the square, wherever it is located in space, whatever the orientation of the plane in which it lies, and whatever its own orientation in that plane, the appropriate image has the same size and location in space and lies in the same plane, but its orientation in this plane differs from that of the original state by an angle of 90º.

To show an example of the double list kind of transformation, we need a system with a finite number of states. Let this consist of three depressions in the sand and a ball lying in any one of them. The system has three states, which we label A, B, C according to which depression contains the ball. A transformation for this system is a specification of where the ball is to be finally placed if it is initially found in each possible depression. The following is a possible transformation:

Initial state (depression) Image state (depression)

A
B
B
C
C
A

Another possible transformation is:

Initial state (depression) Image state (depression)

A
B
B
A
C
C

Another possibility is:

Initial state (depression) Image state (depression)

A
C
B
B
C
A

Other transformations are easily set up.

In brief then, a transformation is a one-to-one assignment of an image state to each state of a system.

A transformation that completely cancels the effect of another transformation is called the inverse of the latter. The inverse transformation acts as follows. A transformation assigns an image state to every state of the system. Its inverse transformation also does this, but in general makes a different assignment, such that the image which it assigns to each image of the original transformation is just the corresponding initial state (fig. 2.3).

The inverse of the transformation of rotation by 90º about a given axis, for example, is rotation by an additional 270º about the same axis, producing a total rotation of 360º which is no rotation at all (fig. 2.4). (The transformation of rotation by 90º the other way is also a valid inverse, but it is equivalent to rotation by 270º and so can be ignored.)

The inverse of the transformation for the system of three depressions and a ball is the transformation

If the system is initially in state A, the original transformation puts it in state B. The inverse then returns it to state A. If it is initially in state B, it is transformed to C, then back to B. And similarly, from C to A, then back to C.

The inverse of the transformation happens to be just this same transformation itself. If the system is in state A, the transformation puts it in state B. Another application of the transformation returns it to state A. State B becomes A, then turns back into state B. State C remains state C in both steps.

So the inverse of any transformation is found simply by reversing the arrow of the correspondence.

When it happens that a transformation affects a system in such a way that all images are indistinguishable from their respective initial states, the system is said to be invariant or symmetric under the transformation. This transformation is then called an invariance transformation or a symmetry transformation of the system (fig. 2.5). A square, for example, is invariant under rotation by 90º about an axis through its center and perpendicular to its plane, since its shape is such that the result of this rotation is indistinguishable from the initial state for every possible initial state.

The general term symmetry means invariance under one or more transformations. The more different transformations a system is invariant under, the higher its degree of symmetry.

A system is asymmetric under a transformation if it is not invariant under the transformation, and is completely asymmetric if it has no symmetry transformations at all. For example, a square with the digit 5 added (chapter 1) is asymmetric under rotations and reflections.

The set of all symmetry transformations of a system comprises the symmetry group of the system. The term group is used here in its precise mathematical sense and implies that this set has certain very definite properties. For the interested reader a brief introduction to group theory is presented in the next section of this chapter. At this point we shall only look at some of the more important (for the purpose of our discussion) features of the symmetry group. Some references are Weyl (SYM, 1), Bell and Fletcher (GRP, 4), Bell (GRP, 3), Linn (GEO, 2), and Shubnikov (COL, 1).

Most worthy of note is the property that the transformation consisting of the consecutive application of two symmetry transformations, the second acting on the image of the first, is also a symmetry transformation. This is easily seen as follows. Denote the initial state of the system by A. The first symmetry transformation transforms it to some state B, which is indistinguishable from A. The second symmetry transformation transforms state B into some state C, which is indistinguishable from B and therefore also from A. The combined transformation brings the system from state A to state C, and, since these states are indistinguishable, it is a symmetry transformation (fig. 2.6).

(Continues...)

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Excerpted from Symmetry Discovered by Joe Rosen. Copyright © 1998 Joe Rosen. Excerpted by permission of Dover Publications, Inc..
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.

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Table of Contents

Foreword
Preface
Preface to the Dover edition
1 Symmetry: what? where? how?
What is symmetry?
Where is symmetry?
How is symmetry?
2 The language of symmetry
Concepts and terminology
Group theory (optional)
3 Geometric symmetry
Linear symmetry
Planar symmetry
Spatial symmetry
4 Other symmetries and approximate symmetry
Temporal symmetry
Permutation symmetry
Color symmetry
Analogy
Approximate symmetry
5 Symmetry in nature
Symmetries with which we are acquainted
Symmetries we have no yet studied
6 Uses of symmetry in science
The symmetry principle
The symmetry approach
Symmetry in research
Conclusion
Bibliography
Addendum to Bibliography
Solutions to problems
Index