Symmetry throughthe Eyes of a Chemist

Second Edition

István HargittalBudapest Technical University

and Hungarian Academy of SciencesBudapest, Hungary

Magdolna HargittaiHungarian Academy of Sciences

Budapest, Hungary

Plenum Press • New York and London

64 Chapter 2 Simple and Combined Symmetries65

Figure 2-44. (a) The centrosymmetric arrangement of acetanilide molecules in the crystal,resulting in a centrosymmetric crystal habit. Reprinted with permission from Ref. [2-321.Copyright (1981) Americal Chemical Society; (b) The head-to-tail alignment of p-acetanilidemolecules in the crystal, resulting in the occurrence of a polar axis in the crystal habit. Reprintedwith permission from Ref. [2-321. Copyright (1981) American Chemical Society.

behavior. In solid/gas reactions, for example, crystal polarity may be a sourceof considerable anisotropy.

There are also important physical properties characterizing poiar crystals,such as pyroelectricity and piezoelectricity and others [2-341. The primitivecell of a pyroelectric crystal possesses a dipole moment. The separation of thecenters of the positive and negative charges changes upon heating. In thisprocess the two charges migrate to the two ends of the polar axis. Piezoelec-tricity is the separation of the positive and negative charges upon expansion!compression of the crystal. Both pyroelectricity and piezoelectricity havepractical uses.

2.7 CHIRALITY

There are many objects, both animate and inanimate, which have nosymmetry planes but which occur in pairs related by a symmetry plane andwhose mirror images cannot be superposed. Figure 2-45 shows a buildingdecoration, a detail from Bach's The Art of the Fugue, a pair of molecules, and apair of crystals. The simplest chiral molecules are those in which a carbon atomis surrounded by four different ligands—atoms or groups of atoms—at thevertices of a tetrahedron. All the naturally occurring amino acids are chiral,except glycine.

b

rrrrTir r!rtCHO CHO

HZ\OH HOZ\.HCH2OH CH2OH

Figure 2-45. Illustrations of chiral pairs: (a) Decorations whose motifs (of fourfold rotationalsymmetry) are each other's mirror images; photographs by the authors; (b) J. S. Bach, Die Kunstder Fuge, Contrapunctus XVIII, detail; (c) glyceraldehyde molecules; (d) quartz crystals.

a

qoo

a

Contrapunetus XVIII

d

Chapter 2 Simple and Combined Symmetries 6766

C

/ - .—..

Figure 2-46. Heterochiral pairs of hands: (a) Tombstone in the Jewish cemetery, Prague;

photograph by the authors; (b) Albrecht Durer's Praying Hands on the cover of the German

magazine Der Spiegel, June 15, 1992; reproduced by permission; (c) Buddha in Tokyo;

photograph by the authors; (d) United Nations stamp. (Continued on next page)

Figure 2-46. (Continued) (e) heterochiral pair of hands and models of a heterochiral pair ofamino acid molecules [2-36]; reproduced by permission from R. N. Bracewell.

a

Figure 2-47. Homochiral pairs of hands: (a) Cover of the German magazine Der Spiegel, May18, 1992; reproduced by permission; (b) U.S. stamp; (c) logo with SOS distress sign at a Swissrailway station; photograph by the authors.

W H. Thomson, Lord Kelvin, wrote [2-35]: "I call any geometrical

figure or group of points 'chiral,' and say it has ëhirality, if its image in a plane

mirror, ideally realized, cannot be brought into coincidence with itself." He

called forms of the same sense homochiral and forms of the opposite sense

heterochiral. The most common example of a heterochiral form is hands.Indeed, the word chirality itself comes from the Greek word for hand. Figures

2-46 and 2-47 show some heterochiral and homochiral pairs of hands.A chiral object and its mirror image are enantiomorphous, and they are

a

iIiII IIIIItII 11lJ*j1. J. 1t(hIe

b

ii i. '1 '''II '.'P 'I.f . ..__.\'olunreerknd a hand

I 7 -

I. .5: . 4

I . •.. I-' 2US\2Oc b

C

68C

hapter 2Sim

ple and Com

bined Symm

etries69

Figure 2-48.authors.

Louis Pasteur's bust in front of the Pasteur Institute, Paris.

Photograph by the

each other's enantiomorphs. L

ouis Pasteur (Figure 2-48)first suggested that

molecules can be chiral. In his fam

ous experiment in 1848,

he recrystallized a

salt of tartaric acid and obtained two kinds of sm

all crystals which w

ere mirror

images of each other, as show

n by Pasteur's models in

Figure 2-49. The tw

o

kinds of crystals had the same chem

ical composition

but differed in theiroptical activity. O

ne was levo-active (L

), and the other was

dextro-actiVe (D

).

Since the true absolute configuration of molecules

could not be determined at

the time, an arbitrary convention w

as applied, which,

luckily, proved tocoincide w

ith reality. If a molecule or a crystal

is chiral, it is necessarilyoptically active. T

he converse is, however, not true.

There are, in fact

Figure 2-49.Pasteur's m

odels of enantiomeric crystals in the Pasteur Institute, Paris.

Photo-

graphs by the authors.

nonenantiomorphous sym

metry classes of crystals w

hich may exhibit optical

activity.W

hyte [2-37] extended the definition of chirality as follows: "T

hree-dim

ensional forms (point arrangem

ents, structures, displacements, and other

processes) which possess non-superposable m

irror images are called 'chiral'."

A chiral process consists of successive states, all of w

hich are chiral. The tw

om

ain classes of chiral forms are screw

s and skews. Screw

s may be conical or

cylindrical and are ordered with respect to a line. E

xamples of the latter are the

left-handed and right-handed helices in Figure 2-50. The skew

s, on the otherhand, are ordered around their center. E

xamples are chiral m

olecules havingpoint-group sym

metry.

From the point of view

of molecules, or crystals, left and right are

intrinsically equivalent. An interesting overview

of the left/right problem in

science has been given by Gardner [2-391. D

istinguishing between left and

right has also considerable social, political, and psychological connotations.For exam

ple, left-handedness in children is viewed w

ith varying degrees oftolerance in different parts of the w

orld. Figure 2-51a shows a classroom

at theU

niversity of Connecticut w

ith different (homochiral and heterochiral) chairs

Figure 2-50.L

eft-handed and right-handedhelix decorations from

Zagorsk, R

ussia [2-381.Photograph by the authors.

L—

70C

hapter 2Sim

ple and Com

bined Symm

etries71

A

Figure 2-51.C

lassrooms w

ith heterochiral and homochiral chairs: (a) C

hairs for both the right-handed and

left-handedstudents; (b) O

lder chairs, all for right-handed students only. Photo-graphs by the authors.

to accomm

odate both the right-handed and the left-handed students. Older

classrooms at the sam

e university have chairs for the right-handed only (Figure2-5

ib).

2.7.1A

symm

etry and Dissym

metry

Sym

metry

operations of the first kind and of the second kind aresom

etimes distinguished in the literature (cf. R

ef. [2-40}). Operations of the

first kind are sometim

es also called even-numbered operations. For exam

ple,the identity operation is equivalent to tw

o consecutive reflections from a

symm

etry plane. It is an even-numbered operation, an operation of the first

kind. Simple rotations are also operations of the first kind. M

irror rotationleads to figures consisting of right-handed and left-handed com

ponents andtherefore is an operation of the second kind. Sim

ple reflection is also anoperation of the second kind as it m

ay be considered as a mirror rotation about

a onefold axis. A sim

ple reflection is related to the existence of two enan-

tiomorphic com

ponents in a figure. Figure 2-52illustrates

these distinctions bya series of sim

ple sketches after Shubnikov [2-401. In accordance with the

above description, chirality is sometim

es defined as the absence of symm

etryelem

ents of the second kind.Som

etimes, the term

s asymm

etry, dissymm

etry, and antisymm

etry areconfused in the literature although the scientific m

eaning of these terms is in

complete conform

ity with the etym

ology of these words. A

symm

etry means

the complete absence of sym

metry, dissym

metry m

eans the derangement of sym

-m

etry, and antisymm

etry means the sym

metry of opposites (see Section 4.6).

Figure 2-52.E

xamples

of symm

etry operations of the first kind (a) and of the second kind(b),

after Shubnikov [2-40].

Pasteur used "dissymm

etry" for the first time

as he designated the absence ofelem

ents of symm

etry of the second kind ina figure. A

ccordingly, dissymm

e-try did not exclude elem

ents of symm

etry of the first kind. PierreC

uriesuggested an even broader application of this

term. H

e called a crystaldissym

metric in the case of the absence of those elem

ents ofsym

metry upon

which depends the existence of one

or another physical property in that crystal.In Pierre C

urie's original words [2-41], "D

issymm

etrycreates the phenom

e-non." N

amely a phenom

enon exists and is observable dueto dissym

metry,

i.e., due to the absence of some sym

metry elem

ents from the

system. Finally,

Shubnikov 12-40] called dissymm

etry the fallingout of one or another elem

entof sym

metry from

a givengroup. H

e argued that to speak of the absence ofelem

ents of symm

etry makes sense only w

hen thesesym

metry elem

ents arepresent in som

e other structures.T

hus, from the point of view

of chirality any asymm

etric figure ischiral,

but asymm

etry is not a necessary condition for chirality. All dissym

metric

fig-ures are also chiral if dissym

metry m

eans the absence of symm

etry elements of

the second kind. In thissense, dissym

metry is synonym

ous with chirality.

An assem

bly of molecules m

ay be achiral forone of tw

o reasons. Either

all the molecules present are achiral

or the two kinds of enantiom

orphs arepresent in equal am

ounts. Chem

ical reactions between achiral m

olecules leadto achiral products. E

ither all product molecules w

ill be achiralor the tw

o kindsof chiral m

olecules will be produced in equal

amounts. C

hiral crystals may

sometim

es be obtained from achiral solutions. W

hen this happens,the tw

oenantiom

orphs will be obtained in (roughly) equal num

bers,as w

as observedby Pasteur. Q

uartz crystals arean inorganic exam

ple of chirality (Figure2-45d). R

oughly equal numbers of left-handed and right-handed crystals

areobtained from

the achiral silica melt.

ab

a#b+*

£

72C

hapter2

Simple and C

ombined Sym

metries

73

Incidentally, Pierre Curie's teachings on sym

metry are probably not so

widely know

n as they should be, considering their fundamental and general

importance. T

he fact that his works on sym

metry w

ere characterized byextrem

e brevity may have contributed to this. M

arie Curie and A

leksei V.

Shubnikov [2-42, 2-431 have considerably facilitated the dissemination of

Curie's teachings. O

ur discussion also relies on their works. T

here is also acritical and fascinating discussion of Pierre C

urie's symm

etry teachings byStew

art and Golubitsky [2-44].

Pierre Curie's above-quoted statem

ent concerning the role of dissymm

e-try in "creating" a phenom

enon is part of a broader formulation. It states that

in every phenomenon there m

ay be elements of sym

metry com

patible with,

though not required by, its existence. What is necessary is that certain elem

entsof sym

metry shall not exist. In other w

ords, it is the absence of certainsym

metry elem

ents that is a necessary condition for the phenomenon to exist.

Another im

portant statement of Pierre C

urie's is that when several

phenomena are superposed in the sam

e system, the dissym

metries are added

together. As a result, only those sym

metry elem

ents which w

ere comm

on toeach phenom

enon will be characteristic of the system

.Finally, concerning the sym

metry relationships of causes and effects,

Marie C

urie [2-421 formulated the follow

ing principles from Pierre C

urie'steachings. (1) "W

hen certain causes produce certain effects, the elements of

symm

etry in the causes ought to reappear in the effects produced"; (2) "When

certain effects reveal a certain dissymm

etry, this dissymm

etry should beapparent in the causes w

hich have given them birth"; how

ever, (3) "The

converse of these two statem

ents does not hold..

. [and]the effects produced

can be more sym

metrical than their causes."

2.72R

elevance to Origin of Life

The

situation with respect to living organism

s is unique. Living organ-

isms contain a large num

ber of chiral constituents, but only L-am

ino acids arepresent in proteins and only D

-nucleotides are present in nucleic acids. This

happens in spite of the fact that the energy of both enantiomorphs is equal and

their formation has equal probability in an achiral environm

ent. How

ever, onlyone of the tw

o occurs in nature, and the particular enantiomorphs involved in

life processes are the same in hum

ans, animals, plants, and m

icroorganisms.

The origin of this phenom

enon is a great puzzle which, according to Prelog

[2-45], may be regarded as a problem

of molecular theology.

This problem

has long fascinated those interested in the molecular basis

of the origin of life (e.g., Refs. [2-46], [2-471). T

here are in fact two questions.

One is w

hy do all the amino acids in a protein have the sam

e L-cO

nfigUratiO

fl, orw

hy do all the components of a nucleic acid, that is, all its nucleotides, have the

same D

-configuration? The other question, the m

ore intriguingone, is w

hydoes that particular configuration happen to be L

for the amino acids and w

hydoes it happen to be D

fornucleotides in all living organism

s? This second

question seems to be im

possible to answer satisfactorily at the present tim

e.A

ccording to Prelog [2-451, a possible explanation is that the creation ofliving m

atter was an extrem

ely improbable event, w

hich occurred onlyonce.

We m

ay then suppose that if there are living forms sim

ilar toours on a distant

planet, their molecular structures m

ay be the mirror im

ages of the correspond-ing m

olecular structures on the earth. We know

ofno structural reason at them

olecular level for living organisms to prefer one type of chirality.(T

here may

be reasons at the atomic nuclear level. T

he violation of parity at the nuclearlevel has already been referred to in C

hapter 1.) Ofcourse, once the selection is

made, the consequences of this selection m

ust be examined in relation to the

first question. The fact rem

ains, however, that chirality is intim

ately associatedw

ith life. This m

eans that at least dissymm

etry and possiblyasym

metry are

basic characteristics of living matter.

Although Pasteur believed that there is a sharp gap betw

een vital andnonliving processes, he attributed the asym

metry of living m

atter to theasym

metry of the structure of the universe and not to a vital force. Pasteur

himself w

rote that he was inclined to think that life, as it

appears to us, must be

a product of the dissymm

etry of the universe (see Ref. [2-48]).

Concerning the first question, O

rgel [2-461 suggests that we

compare the

structure of DN

A to a spiral staircase. T

he regular DN

A right-handed double

helix is composed of D

-nucleotides. On the other hand, if a D

NA

double helixw

ere synthesized from L

-nucleotides, it would be left-handed. T

hese two

helices can be visualized as right-handed and left-handed spiral staircases,respectively. B

oth structures can perform useful functions. A

DN

A double

helix containing both D- and L

-nucleotides, however, could not form

a trulyhelical structure at all since its handedness w

ould be changing. Just considerthe analogous spiral staircase that O

rgel suggestedas show

n in Figure 2-53.If each com

ponent of a complex system

is replaced by its mirror im

age,the m

irror image of the original system

is obtained. How

ever, if onlysom

ecom

ponents of the complex system

are replaced by their mirror im

ages, achaotic system

emerges. C

hemical system

s that are perfect mirror im

ages ofeach other behave identically, w

hereas systems in w

hich onlysom

e, but notall, com

ponents have been replaced by their mirror im

ages have quite differentchem

ical properties. If, for example, a naturally occurring

enzyme m

ade up ofL

-amino acids synthesizes a D

-nucleotide, then the corresponding artificialenzym

e obtained from D

-amino acids w

ould synthesize the L-nucleotide. O

nthe other hand, a corresponding polypeptide containing both

D- and L

-amino

acids would probably lack the enzym

ic activity.R

ecently, the first enzymatically active D

-protein has been synthesized

Figure 2-53.A

spiral staircase which changes its chirality [2-461

Reproduced by perm

issiOn

from L

. E. O

rgel.

[2-49]. There are m

any potential applications, both therapeutic and nonthera-

peutic, that may open up w

ith such progress [2-50].It has been know

n for

some tim

e that the two enantiom

erSof drugs and pesticides m

ay havevastly

different responses in a living organism.

Natural products extracted from

plants and animals are enantiom

ericallY pure

while the synthesized ones are

obtained in a 1:1 ratio of the enantiomers. In som

e cases,the tw

in of the one

exerting the beneficial action is harmless. In

other cases, however, the drug

molecule has an "evil tw

in" [2-511. Atragic exam

ple was the thalidom

ide

case, in Europe, in

which the right-handed m

olecule was a

sedative and the

left-handed one caused birth defects. Other

examples include one enantiom

er

of ethambutol fighting tuberculosis w

ith itsevil tw

in causing blindness, and

one enantiomer of naproxen

reducing arthritic inflamm

ation with its evil tw

in

poisoning the liver. Bitter and sw

eetasparagine are represented by structural

formulas in Figure 2-54.Ibuprofen is a lucky case in w

hich the twin of

the enantiomer that provides

the curing is converted to the beneficialversion by the body.

Even w

hen the twin is

harmless, it represents w

aste anda potential

pollutant. Thus, a lot of efforts

are directed toward producing

enantiomerically

pure drugs and pesticides. The techniques

of asymm

etric synthesis(see, e.g.,

Refs. [2-53] and

[2-541), based on the strategy ofem

ploying chiral catalysts(see, e.g., R

ef. [2-55]),are used to this end. O

ne of the fascinatingpossibilities

is to produce sweets from

chiral sugars of the enantiomer

that would not be

capable of contributingto obesity yet w

ould retain thetaste of the other

enantiomer. C

hiral separationand purity is an increasingly

important question.

Worldw

ide sales of enantiopuredrugs topped $35 billion in

1993 and areexpected to reach about $40

billion in 1997 [2-561. There

is a rapidly growing

literature on the subject, with

even special journals exclusively dedicatedto

this topic. Production ofenantiom

erically pure substanceshas also becom

e atopic in investm

entreports and the daily press.

2.7.3La coupe du roi

Am

ongthe m

any chemical

processes in which chirality/achirality rela-

tionships may be im

portantare the fragm

entation of some m

oleculesand the

reverse process of the association ofm

olecular fragments. Such fragm

entationand association

can be considered generally andnot just for m

olecules. The

usual cases are those inw

hich an achiral object isbisected into achiral or

heterochiral halves. On the other

hand, if an achiral objectcan be bisected into

two hom

ochjral halves, itcannot be bisected into tw

o heterochiralones. A

relatively simple

case is the tessellation of planar achiralfigures into achiral,

heterochjral, and homochjral

segments. Som

e examples are show

n inFigure

2-55. For a detaileddiscussion, see R

ef [2-7].A

net et at. [2-5 71 have citeda French parlor trick called Ia

coupe du roi—or the royal section—

in which

an apple is bisected into two hom

ochiralhalves,as show

n in Figure 2-56. An apple

can be easily bisected into two achiral

halves. On the other

hand, itis im

possible to bisectan apple into tw

oheterochiral halves. T

wo heterochjral

halves, however, can be obtained

fromtw

o apples, both cut intotw

o homochiral halves in the

opposite sense (seeFigure 2-56).

74C

hapter 2Sim

ple and Com

bined Symm

etries75

HH

09

H,H

H2N

OC

CO

H/C

\\H

2NN

H2

Figure

2-54.B

itterand Sw

eet asparagine representedby structural form

ulas [2-521.

Chapter 2 Simple and Combined Symmetries 77

76

Figure 2-56. The French parlor trick ía coupe do roi, after Anet et al. [2-57]. An apple can becut into two homochiral halves in two ways which are enantiomorphous to each other. An applecannot be cut into two heterochiral halves. Two heterochiral halves originating from two differentapples cannot be combined into one apple.

In la coupe du roi, two vertical half cuts are made through the apple—onefrom the top to the equator, and another, perpendicularly, from the bottom to theequator. In addition, two nonadjacent quarter cuts are made along the equator.If all this is properly done, the apple should separate into two homochiralhalves as seen in Figure 2-56.

The first chemical analog of la coupe du roi was demonstrated by Cm-quini et al. [2-581 by bisecting the achiral molecule cis-3,7-dimethyl-1,5-cyclooctanedione into homochiral halves, viz., 2-methyl-l,4-butanediol. Thereaction sequence is depicted in Figure 2-57 after Cinquini et al. [2-58], whopainstakingly documented the analogy with the pomaceous model. Only exam-ples of the reverse coupe du roi had been known prior to the work of Cinquini eta!. Thus, Anet et cii. [2-571 had reported the synthesis of chiral 4-(bromo-methyl)-6-(mercaptomethyl)[2 . 2lmetacyclophane. They then showed that twohomochiral molecules can be combined to form an achiral dimer as shown inand illustrated by Figure 2-58.

aD -® C

a L7b

Figure 2-55. Dissection of planar achiral ügures into achiral (a), heterochiral (b), and homo-

chiral segments (c): some examples.

Figure 2-57.L

a coupe du roi and the reaction sequencetransform

ing cis-3 ,7-dimethyl-1 ,5-

cyclooctanedione into2.m

ethy11,4-bUtanedi0l A

fter Cinquini et al. [2-581.

Used by perm

is-

sion, Copyright (1988) A

merican

Chem

ical Society.

2.8P

OLY

HE

DR

A

+B

r

"A convex polyhedron is said to be regular if its faces are regular and

equal, while its vertices are all surrounded alike" [2-591. A

polyhedron isconvex if every dihedral angle is less than 1800. T

he dihedral angle is the angleform

ed by two polygons joined along a com

mon edge.

There are only five regular convex polyhedra, a very sm

all number

indeed. The regular convex polyhedra are called Platonic solids because they

constituted an important part of Plato's natural philosophy. T

hey are thetetrahedron, cube (hexahedron), octahedron, dodecahedron, and icosahedron.T

he faces are regular polygons, either regular triangles, regular pentagons, orsquares.

A regular polygon has equal interior angles and equal sides. Figure 2-59

presents a regular triangle, a regular quadrangle (i.e., a square), a regularpentagon, and so on. T

he circle is obtained in the limit as the num

ber of sidesapproaches infinity. T

he regular polygons have an n-fold rotational symm

etry

78C

hapter 2Sim

ple and Com

bined Symm

etries79

C0

CH

CH

3

C2

ii00

CH

3C

sC

H3

1/

0N

+

Figure 2-58.R

everse coupe du roi and the formation of a dim

er from tw

o homochiral 4-(brom

o-m

ethyl)-6-(mercaptom

ethyl)[2.2lmetacyclophane m

olecules. After A

net etal. [2-571. Used by

permission. C

opyright (1983) Am

erican Chem

ical Society.

C2

Cs

CH

3

OH

CH

3

OH

Cs

Cs

80C

hapter 2Sim

ple and Com

bined Symm

etries81

AH

O0000

Figure

2-59.R

egular polygons.

axis perpendicular to their plane and going through theirm

idpoint. Here n is

1, 2, 3,.

.up

to infinity for the circle.T

he five regular polyhedra are shown in Figure 2-60. T

heir characteristicparam

eters are given in Table 2-3. Figure 2-61 reproduces an

East G

erman

stamp w

ith Euler and his equation, V

—E

+ F =

2,w

here V, E

, and F are thenum

ber of vertices, edges, and faces. The equation is valid for

polyhedra

having arty kind of polygonal faces. According to W

eyl [2-9], the existenceof

Figure 2-60.T

he five Platonic solids.

Table 2-3.

Characteristics of the R

egular Polyhedra

Nam

ePolygon

Num

berof faces

Vertex

figureN

umber

of verticesN

umber

of edges

Tetrahedron

34

34

6

Cube

46

38

12

Octahedron

38

46

12

Dodecahedron

512

320

30

Icosahedron3

205

1230

the tetrahedron, cube, and octahedron is a fairly trivial geometric fact. O

n theother hand, he considered the discovery of the regular dodecahedron and theregular icosahedron "one of the m

ost beautiful and singular discoveries made

in the whole history of m

athematics." H

owever, to ask w

ho first constructedthe regular polyhedra is, according to C

oxeter [2-59], like asking who first used

fire.M

any primitive organism

s have the shape of the pentagonal dodeca-hedron. A

s will be seen later, it is not possible to have crystal structures having

this symm

etry. Belov [2-60] suggested that the pentagonal sym

metry of

primitive organism

s represents their defense against crystallization. Severalradiolarians of different shapes from

Häckel's book [2-13] are show

n in Figure2-62. A

rtistic representations of regular polyhedra are shown in Figure 2-63.

Figure 2-64 shows K

epler and his planetary model based on the regular

solids [2-611. According to this m

odel, the greatest distance of one planet fromthe sun stands in a fixed ratio to the least distance of the next outer planet fromthe sun. T

here are five ratios describing the distances of the six planets thatw

ere known to K

epler. A regular solid can be interposed betw

een two adjacent

planets so that the inner planet, when at its greatest distance from

the sun, lays

Figure 2-61.E

uler and his equation e —k

+f

2, corresponding to V —

E+

F =2,

where the

Germ

an e (Ecke), k (K

ante), and f (Fläche) correspond to vertex (V), edge (E

), and face (F),respectively.

82 Chapter 2

on the inscribed sphere of the solid, while the outer planet, when at its leastdistance, lays on the circumscribed sphere.

Arthur Koestler in The Sleepwalkers [2-62] called this planetary model "afalse inspiration, a supreme hoax of the Socratic daimon,...". However, theplanetary model, which is also a densest packing model, probably representsKepler's best attempt at attaining a unifed view of his work both in astronomyand in what we call today crystallography.

There are excellent monographs on regular figures, two of which areespecially noteworthy [2-59, 2-631. The Platonic solids have very highsymmetries and one especially important common characteristic. None of therotational symmetry axes of the regular polyhedra is unique, but each axis isassociated with several axes equivalent to itself. The five regular solids can beclassifed into three symmetry classes:

Figure 2-62. Radiolarians from Häckel's book [2-131.

b

c d

Figure 2-63. Artistic representations of regular polyhedra: (a) Sculpture in the garden of TelAviv University; photograph by the authors; (b) pentagonal dodecahedron by Horst Janssen,ChriStal/-Knechi (crystal slave); reproduced by permission; (c) Leonardo da Vinci's dodeca-hedron drawn for Luca Pacioli's De Divina Proporfione; (d) sculpture by Victor Vasarely in Pécs,Hungary; photograph by the authors.

84 Chapter 2 Simple and Combined Symmetries 85

lIt

II LIllt.1•

Tetrahedron

Cube and octahedron

3/2m = 3/4

3/4m = 6/4

Dodecahedron and icosahedron 3/5m = 3/10

It is equivalent to describe the symmetry class of the tetrahedron as 312mor3/4. The skew line between two axes means that they are not orthogonal. Thesymbol 3/2m denotes a threefold axis and a twofold axis which are notperpendicular and a symmetry plane which includes these axes. These threesymmetry elements are indicated in Figure 2-65. The symmetry class 3/2m isequivalent to a combination of a threefold axis and a fourfold mirror-rotationaxis. In both cases the threefold axes connect one of the vertices of thetetrahedron with the midpoint of the opposite face. The fourfold mirror-rotation axes coincide with the twofold axes. The presence of the fourfoldmirror-rotation axis is easily seen if the tetrahedron is rotated by a quarterrotation about a twofold axis and is then reflected by a symmetry planeperpendicular to this axis. The symmetry operations chosen as basic will thengenerate the remaining symmetry elements. Thus, the two descriptions areequivalent.

Characteristic symmetry elements of the cube are shown in Figure 2-65.Three different symmetry planes go through the center of the cube parallel toits faces. Furthermore, six symmetry planes connect the opposite edges andalso diagonally bisect the faces. The fourfold rotation axes connect themidpoints of opposite faces. The sixfold mirror-rotation axes coincide withthreefold rotation axes. They connect opposite vertices and are located alongthe body diagonals. The symbol 6/4 does not directly indicate the symmetryplanes connecting the midpoints of opposite edges, the twofold rotation axes,or the center of symmetry. These latter elements are generated by the others.The presence of a center of symmetry is well seen by the fact that each face andedge of the cube has its parallel counterpart. The tetrahedron, on the otherhand, has no center of symmetry.

The octahedron is in the same symmetry class as the cube. The antiparal-lel character of the octahedron faces is especially conspicuous. As seen inFigure 2-65, the fourfold symmetry axes go through the vertices, the threefoldaxes go through the face midpoints, and the twofold axes go through the edgemidpoints.

The pentagonal dodecahedron and the icosahedron are in the samesymmetry class. The fivefold, threefold, and twofold rotation axes intersect themidpoints of faces, the vertices, and the edges of the dodecahedron, respec-

Figure 2-64. Johannes Kepler on a Hungarian stamp and a detail of his planetary model basedon the regular solids [2-61].

Figure 2-65. Characteristic symmetry elements of the Platonic solids.

86 Chapter 2 Simple and Combined Symmetries 87

tively (Figure 2-65). On the other hand, the corresponding axes intersect thevertices and the midpoints of faces and edges of the icosahedron (Figure 2-65).

Consequently, the five regular polyhedra exhibit a dual relationship asregards their faces and vertex figures. The tetrahedron is self-dual (Table 2-3).

If the definition of regular polyhedra is not restricted to convex figures,their number rises from five to nine. The additional four are depicted in Figure2-66 (for more information, see, e.g., Refs. [2-59] and [2-63]—[2-66]). Theyare called by the common name of regular star polyhedra. One of them, viz.,the great stellated dodecahedron, is illustrated by the decoration at the top ofthe Sacristy of St. Peter's Basilica in Vatican City, and another, the smallstellated dodecahedron, by an ordinary lamp in Figure 2-67.

The sphere deserves special mention. It is one of the simplest possiblefigures and, accordingly, one with high and complicated symmetry. It has aninfinite number of rotation axes with infinite order. All of them coincide withbody diagonals going through the midpoint of the sphere. The midpoint, whichis also a singular point, is the center of symmetry of the sphere. The followingsymmetry elements may be chosen as basic ones: two infinite order rotationaxes which are not perpendicular plus one symmetry plane. Therefore, thesymmetry class of the sphere is co/m. Concerning the symmetry of thesphere, Kepes [2-671 quotes Copernicus:

The spherical is the form of all forms most perfect, having need ofno articulation; and the spherical is the form of greatest volumetriccapacity, best able to contain and circumscribe all else; and all theseparated parts of the world—I mean the sun, the moon, and thestars—are observed to have spherical form; and all things tend tolimit themselves under this form—as appears in drops of water andother liquids whenever of themselves they tend to limit themselves.

So no one may doubt that the spherical is the form of the world, thedivine body.

Artistic appearances of spheres are shown in Figure 2-68.In addition to the regular polyhedra, there are various families of poly-

hedra with decreased degrees of regularity [2-59, 2-63—2-661. The so-calledsemiregular or Archimedean polyhedra are similar to the Platonic polyhedra inthat all their faces are regular and all their vertices are congruent. However, thepolygons of their faces are not all of the same kind. The thirteen semiregularpolyhedra are listed in Table 2-4, and some of them are also shown in Figure2-69. Table 2-4 also enumerates their rotation axes.

The simplest semiregular polyhedra are obtained by symmetrically shav-

a

Figure 2-67. Examples of star polyhedra: (a) Great stellated dodecahedron as decoration at thetop of the Sacristy of St. Peter's Basilica, Vatican City; (b) small stellated dodecahedron as a lampin an Italian home, Bologna. Photographs by the authors.

b

a b c d

Figure 2-66. The four regular star polyhedra: (a) Small stellated dodecahedron; (b) the greatdodecahedron; (c) great stellated dodecahedron; (d) the great icosahedron. From H. M. Cundyand A. P Rollett [2-MI. Used by permission of Oxford University Press.

88C

hapter 2Sim

ple and Com

bined Symm

etries89

Figure 2-68.A

rtistic expressions of the sphere: (a) In front of the World T

rade Center, N

ewY

ork; (b) sculpture by J.-B. C

arpeaux in Paris. Photographs by the authors.

Table 2-4.

The T

hirteen Semiregula r Polyhedra

No.

Nam

e

Num

ber ofN

umber of

rotation axes

FacesV

erticesE

dges2-fold

3-fold4-fold

5-fold

I2345678910111213

Truncated tetrahedron"

Truncated cube"

Truncated octahedron"

Cuboctahedron"

Truncated cuboctahedron

Rhom

bicuboctahedronSnub cubeT

runcated dodecahedron"Icosidodecahedron"T

runcated icosahedron"T

runcated icosidodecahedronR

hombicosidodecahedron

Snub dodecahedron

8141414

262638323232626292

12

242412

482424603060

1206060

1836362472

4860906090

180

120

150

34

64

64

64

64

64

64

1510

1510

1510

1510

1510

1510

0333333000000

0000000666666

"Truncated regular polyhedron.

bQuasiregular polyhedron.

b .o,Figure 2-69.

Some of the sem

iregular polyhedra: the so-called truncated regular polyhedra andquasiregular polyhedra.

ing off the corners of the regular solids. They are the truncated regular

polyhedra and are marked w

ith a superscript a in Table 2-4.

One

of them is the

truncated icosahedron, the shape of the buckminsterfullerene m

olecule. Tw

osem

iregular polyhedra are classified as so-called quasiregular polyhedra. They

have two kinds of faces, and each face of one kind is entirely surrounded by

faces of the other kind. They are m

arked with a superscript b in T

able 2-4. All

these seven semiregular polyhedra are show

n in Figure 2-69. The rem

ainingsix sem

iregular polyhedra may be derived from

the other semiregular poly-

hedra. The structures of zeolites, alum

inosilicates, are rich in polyhedralshapes, including the channels and cavities they form

(see, e.g., Ref. [2-681).

One of the m

ost comm

on zeolites is sodalite, Na6[A

16Si6O24]2N

aCI, w

hosenam

e refers to its sodium content. T

he sodalite unit itself is represented by atruncated octahedron in Figure 2-70a, w

here the line drawing ignores the

oxygen atoms and each line represents T

..

.T

(T =

Al,

Si). The three

remaining nodels of Figure 2-70 (b, c, and d) represent different m

odes oflinkages betw

een the sodalite units.It is especially interesting to see the

different cavities formed by different m

odes of linkage [2-691. Some other

examples of sem

iregular polyhedra are shown in Figure 2-71.

The prism

s and antiprisms are also im

portant polyhedron families. A

prism has tw

o congruent and parallel faces, and they are joined by a set ofparallelogram

s. An antiprism

also has two congruent and parallel faces, but

they are joined by a set of triangles. There is an infinite num

ber of prisms and

antiprisms, and som

e of them are show

n in Figure 2-72. A prism

or an anti-prism

is semiregular if all its faces are regular polygons. A

cube can be consid-ered a square prism

, and an octahedron can be considered a triangular antiprism.

There are additional polyhedra w

hich are important in discussing m

olecu-lar geom

etries and crystal structures.

a

Figure 2-70.Z

eolite structures: the shape and various modes of linkage of the sodalite units,

after Beagley and T

itiloye [2-691. Reproduced by perm

ission. The line draw

ings ignore theoxygen atom

s and represent T .

. T(T

=A

l,Si); (a) Sodalite unit; (b) and (c) sodalite units

linked through double 4-rings; (d) Sodalite units linked through double 6-rings. In this model

double lines represent T .

.T

.

Figure 2-71.E

xamples of sem

iregular polyhedra: (a) Truncated octahedron in a T

el Aviv

playground; (b) truncated icosahedron as a lamp in an Italian hom

e, Bologna; (c) cuboctahedron

as a top decoration of a garden lantern in Kyoto, Japan. Photographs by the authors.

Simple and C

ombined Sym

metries

91

RE

FE

RE

NC

ES

Figure 2-72.Prism

s and antiprisms.

[2-1]T

. Mann, T

he Magic M

ountain. The cited passage on p. 21 is in French both in the

original Germ

an and its English translation. T

he English translation cited in our text w

askindly provided by D

r. Jack M. D

avis, Professor of English, T

he University of

Connecticut, 1984.

[2-21J. K

epler, Strena, seu De N

ive Sexangula (1611); English translation, T

he Six-Cornered

Snowflake, C

larendon Press, Oxford (1966).

[2-31I.

I. Shafranovskii, Kepler's crystallographic ideas and his tract "T

he Six-Cornered

Snowflake," in K

epler, Four Hundred Y

ears (A. B

eer and P Beer, eds.), Proceedings of

conferences held in honor of Johannes Kepler, V

istas Astron. 18 (1975).

[2-41H

. A. Sackeim

, R. C

. Gur, and M

. C. Saucy, Science 202, 434 (1978).

[2-5]S. P Springer and G

. Deutsch, L

eft Brain, R

ight Brain, W

. Freeman &

Co., San

Francisco (1981).[2-6]

J. B. H

ellige, Hem

ispheric Asym

metry. W

hat's Right and W

hat's Left, H

arvard University

Press, Cam

bridge, Massachusetts (1993).

[2-7]A

. V. Shubnikov and V

. A. K

optsik, Symm

etry in Science and Art, Plenum

Press, New

York (1974.); R

ussian original, Simm

etriya v nauke i isskusjve, Nauka, M

oscow (1972).

[2-81R

. Jastrow, R

ed Giants and W

hite Dw

arfs, W. W

. Norton, N

ew Y

ork (1979).[2-9]

H. W

eyl, Symm

etry, Princeton University Press, Princeton, N

ew Jersey (1952).

[2-101L

. R. H

. Appleton, A

merican Indian D

esign and Decoration, D

over Publications, New

York (1971).

[2-111W

. Herzfielde, John H

eartfield. Leben und W

erk dargestelit von seinem B

ruder, 3rdrevised and expanded ed., V

EB

Verlag der K

unst, Dresden (1976).

[2-121M

. Ickis, The Standard B

ook of Quilt M

aking and Collecting, D

over Publications, New

York (1959).

[2-131E

. Häckel, K

unsformen der N

atur, Vols. 1—

10, Verlag des B

ibliographischen Instituts,L

eipzig (1899—1904).

[2-141U

. Nakaya, Snow

, Iwanam

i-Shoten Publishing Co., T

okyo (1938) [in Japanese].[2-IS]

G. T

aubes, Discover 1984 (January), 75.

a

bc

d

ab

C

92C

hapter 2Sim

ple and Com

bined Symm

etries93

[2-16]D

. McL

achlan, Proc. Nati. A

cad. Sci. U.S.A

. 43, 143 (1957).[2-17]

W. A

. Bentley and W

. J. Hum

phreys, Snow C

rystals, McG

raw-H

ill, New

York (1931).

[2-181J. N

ittmann and H

. F. Stanley, Nature 321, 663 (1986).

[2-191S. K

ai (ed), Pattern Formation in C

omplex D

issipative Systems, W

orld Scientific,Singapore (1992).

[2-201S. O

hta and H. H

onjo, in Santa Fe Institutec Studies in the Sciences of Com

plexity,A

ddison-Wesley Publishing C

o. in press.[2-211

Y. Furukaw

a and W. Shim

ada, J. Cryst. G

rowth

128,234 (1993).

[2-221R

. Kobayashi, Physica D

63, 410 (1993).[2-23]

D. A

. Tom

alia and H. D

. Durst, T

op. Curr. C

hem. 165, 193 (1993).

[2-241A

ttributed toM. Polanyi. Private com

munication from

Professor W. Jim

Neidhardt, N

ewJersey Institute of T

echnology, 1984.[2-251

A. L

. Mackay, Jugosi. C

ent. Kristallogr. 10, 5 (1975); A

. L. M

ackay, personalcom

munication, 1982.

[2-261J. N

eedham and L

u Gw

ei-Djen, W

hether 16, 319 (1961).[2-271

G. H

ellmann, Schneekrystalle, B

erlin (1893).[2-281

T. Stam

p and C. Stam

p, William

Scoresby; Arctic Scientist, C

aedmon of W

hitby (1976).[2-291

J. Reston, International H

erald Tribune, M

ay 7, 1981, p. 4.[2-301

W. E

ngelhardt, Mathem

atischer Unterricht 9(2), 49 (1963).

[2-311E

. A. A

bbott, Flatland. A R

omance of M

any Dim

ensions, Barnes and N

oble Books, 5th

ed., New

York (1983).

[2-321D

. Y. C

urtin and I. C. Paul, C

hem. R

ev. 81, 525 (1981).[2-331

P. Groth, C

hemische

Kristallographie,

5 vols., Verlag ori W

ilhelm E

ngelmann, L

eipzig(1906—

1919).[2-341

G. N

. Desiraju, C

rystal Engineering: T

he Design of O

rganic Solids, Elsevier, A

mster-

dam (1989).

[2-351L

ord Kelvin, B

altimore L

ectures, C. J. C

lay and Sons, London (1904).

[2-36]R

. N. B

racewell, T

he Galactic C

lub. Intelligent Life

inthe O

uter Space, W. H

. Freeman

& C

o., San Francisco (1975).[2-371

L. L

. Whyte, L

eonardo 8, 245 (1975); Nature 182, 198 (1958).

[2-381I. H

argittai and M. H

argittai, Symm

etry: A U

nifying Concept, Shelter Publications,

Bolinas, C

alifornia (1994).[2-391

M. G

ardner, The N

ew A

mbidextrous U

niverse. Symm

etry and Asym

metry from

Mirror

Reflections to Superstrings, 3rd revised ed., W

. H. Freem

an and Co., N

ew Y

ork (1990).[2-40]

A. V

. Shubnikov, Simm

etriya I antisimm

etriya konechnykh figur, lzd. Akad. N

aukSSSR

, Moscow

(1951).[2-41]

"C'est Ia dissym

etrie qui crée le phénomène"; P C

urie, I Phys. Paris 3, 393 (1894).[2-42]

M. C

urie, Pierre Curie,

The

Macm

illan Com

pany, New

York (1929).

[2-43]A

. V. Shubnikov, in Shubnikov C

entennial Papers (I. Hargittai and B

. K. V

ainshtein,eds.), C

rystal Symm

etries p. 357, Pergamon Press, O

xford (1988). [Originally appeared

in Russian in U

sp. Fiz. Nauk 59, 591 (1956).]

[2-44]1. Stew

art and M. G

olubitsky, Fearful Symm

etry. Is God a G

eometer? B

lackwell, O

xford(1992).

[2-45]V

. Prelog, Science 193, 17 (1976).[2-46]

L. F. O

rgel, The O

rigins of Life.' M

olecules and Natural Selection, John W

iley & Sons,

New

York (1973).

[2-47]J. D

. Bernal, T

he Origin of L

ife, The W

orld Publishing Co., C

leveland (1967).[2-481

J. B. S. H

aldane, Nature 185, 87 (1960), citing L

. Pasteur, C. R

. Acad. Sci. Paris, June 1,

1874.

[2-49]R

. C. M

ilton, S. C. F M

ilton, and S. B. H

. Kent, Science 256, 1445 (1992).

[2-50]G

. Jung, Angew

. Chem

. mt. E

d.E

ngI. 31, 1457 (1992).[2-51]

K. F. Schm

idt, Sci. New

s143, 348 (1993).

[2-52]R

. Janoschek (ed), Chirality—

FromW

eak Bosons to the A

lpha-Helix, Springer-V

erlag,B

erlin (1991).[2-53]

M. N

ógriidj, Stereoselective Synthesis.A Practical A

pproach, VC

H, W

einheim(1995).

[2-54]R

. A. A

itken and S. N. K

ilényi(eds.), A

symm

etric Synthesis, Blackie A

cademic &

Professional, London (1992).

[2-55]H

. Brunner and W

. Zettlm

eierH

andbook of Enantioselective C

atalysis, Vols. I

and II,V

CH

, Weinheim

(1993).[2-56]

5. C. Stinson, C

hem. E

ng.N

ews, Septem

ber 19, 1994, p. 38.[2-57]

F A. L

. Anet, S. S. M

iura, J.Siegel, and K

. Mislow

, J. Am

. Chem

.Soc. 105, 1419

(1983).[2-58]

M. C

inquini, F Cozzi, F Sannicoló

and A. Sironi, J. A

m. C

hem. Soc.110, 4363 (1988).

[2-59]H

. S. M. C

oxeter, Regular Polytopes,

3rd ed., Dover Publications, N

ewY

ork (1973),[2-60]

N. V

Belov, O

cherkipo strukturnoi m

ineralogii, Nedra, M

oscow(1976).

[2-6 1]J. K

epler, Mysterium

cosmographicum

(1595).[2-621

A. K

oestler, The Sleepw

alkers, The

Universal L

ibrary, Grosset and D

unlap,N

ew Y

ork(1963), p. 252.

[2-63]L

. Fejes TO

th, Regular Figures,

Pergamon Press, N

ew Y

ork(1964).

[2-64]H

. M. C

undy and A. P. R

ollett,M

athematical M

odels, Clarendon Press,O

xford (1961).[2-65]

M. J. W

enninger, PolyhedronM

odels, Cam

bridge University Press, N

ew Y

ork(1971).

[2-66]P Pearce and S. Pearce, Polyhedra

Primer, V

an Nostrand R

einhold, New

York

(1978).[2-671

N. C

opernicus, De R

evolutionibusO

rbium C

aelestium (1543), as cited in G

. Kepes,T

heN

ew L

andscape in Art and

Science, Theobald &

Co., C

hicago (1956).[2-68]

W. M

. Meier and D

. H.

Olson, A

tlas of Zeolite Structure T

ypes, 3rdrevised ed.

Butterw

orthHeinem

ann, London (1992).

[2-69]B

. Beagley and J. 0. T

itiloye,Struct. C

hem. 3, 429 (1992).