Symmetry by Weyl

8/13/2019 Symmetry by Weyl

http://slidepdf.com/reader/full/symmetry-by-weyl 1/94











C~pyri t Igp?, by PIbICx?lOU U n i d l y

second Printing 198

Third Prinfiog I

Founh hindcg I

Fifth PIbIlilIg 1966

Rinted in the United S t a t a hmk

PREFACE

AND BIlBLIOGMPHICAL REMARKS

S T R ~ Grom the me w ha t vague notion

of symmetry harmony of pmportkns,

these four lecture8 gradually dGvdop irst the

goomebic concept of symmetry in its sever l

forms, as bilateral h'analatory, rotati

ornamental and erysrallographic symmetry

etc, andy

ise to the general deaunder-

lying all these special Forms namely that of

invariance of a umfqumtion elements

under 4 group o automorphic ta*-tions. 1aimattwotbinga:ontheonthand

to diil ay the g ~ tariety of applications

of the priaciple of symm try in the arts in

inoipnic and org nic nature, on the o t k

hand to cl rify step by step the philmophieo-

mathematical dgnhhnce of the idca of pym-

metry. The IsWw purpose m kes it neots-nary to wnfmnt the notions and theorLen of

symmetry and relativity, while numerow

illustrations supgmrting the text hdp to

accomplish the former.

s readersof this bouJt I had a wider circle

in mind than that of leacud speeialista It

does not shun nathematiw that would defeat

its purpose), but daailed treatment of mwt

o the problems it deals with in p rticul rbomplete rnatbnatical treatment, is beyond

its scope. To the lectures, which reproduce

in slightly d i e d ersion the Louis lark

Vanuxan Lectuns given by the author at

rinceton University in February 1951, tw

appendices eontaiaing mathematical proofs

have been added.

Other books in the field, as for instanceF. aeger's daasical ecttacs on th p r i ~ I e



o ym y and its a icutims in natural m i m e

(Amsterdam and Lwdon 1917), or the

much smaller and r ore recent booklet by

Jacque N La p E at su applicatim(Paris, Albin Michtl, 1950) cover only part

of the mate though in a mom detailed

fathion, Symmetry is but a W i e n

D A q Thompson s magni6-t work 8

Orour off m New edition, Cambridge

En@. and New York, 1948). Andma

Speiads Tkmis ar ru~panuan sndliukrOrvlnuffg (3. A d . Berlin 1937) and other

publieatiom by the same author an mpor-tant for the synopb of the aesthetic and

mathematical sw of the subject JayHambidge s D ic . pm d t ~ Yale Uni-

versity Prrsq 1920) has little more th n he

name n commonwith th present book. Itsclosest dative is perhaps theJuly 1949 num-

b on symmetry of the Ge-rman periodicalSrudim awde( I., pp. 203-278: q u d

as .stuU?rn G%iMrnl ).

A comple-te Iist of m- for the ilhrstra

tions is to be found at th end of the book

To the Frh-ton University reaa nd ts

editorsIwiahtoexpreaswarm~foathe

inward and outward carethey have lavished

on this little volume; to the authoridea of

Rineeton Univasity no less Bincen thadksfor the oppochmity they gave me to deliver

thkswansongonthee-veofmyrerinment

from the Institute for Advanced Study.

5zemMmw~nzDs nrb 7W

CONTENTS

Bilateral sym~~etry 3

Translatory, rotational, and related

symmetries 4

Ornamental symmetry 83

Crystals. The general mathematicalidea of symmetry I 9

PPENDICES

A. Determination of a l l finite groups

of proper rotations in g-space 49

B Inclusion of improper rotations IS5

Acknowledgments in

Index 6



IUTER L SYMMETRY



BILATERAL SYMMETRY

IQ I A M N Q P M I S T A K E N ~ ~ C ~ ~ ~ ~

used inoureverydaylanguageintwomean-

ings. In the one smse syqmetrie meanssomething like well-propwtioned, well-bal-anced, and symmetry denotes thatWI? ofam-

d a n c e of several partr bywhich theyhte

grate into a whole. Bmty is bound upwith

symmetry. Thus Polyklcitos, who wrote a

book on proportton and whomthe

ancientspraised for the harmoniow perfeetion of his

wdptum, uses the word, nd UIWfollow

him in setting down a canon of proportions

for thehuman figrm1 In this the idea

is by no means restricted to apatial objcns;

thesynonym harmony pointsmoretoward

its coustic l andmusical th n ts geom

I pplications. h m s a good German

q u i d e a t for the Greek symmetry; for like

this t carries also the emmotationof middle

'D u, V i iBi hn mn r n e n d k k Anpwrias 1528.Tobcaraa,D%uhimrltdoanotiratlxword

Igmmehy, but the a&u&d'' Latin tran*lthnby

Ida M a dJoachb Camemdu~1592) beam thc t tk

t k ~ p l r r i r r r r T o ~ t c 8 t h e s t s t c m e n t l a.aaibcd 4Mc.ml&u, wp3) tbat rJm empbymmt

a p t many ntnnba would almost esgm er

c o r r s m a i n l e u t p a n r S e e * ~ b a f ~ A usuja dc 1 mppwph dmu DiOdOIT 4 98

5-9 fn Cbuiqm 26 1951), pp. 63 66.

Vibuviua d c h 'sSpmcuy NRUp q m

tion RDportionirthcmmmcnamtitiondthc

variow d m e n t pam with the whdr For a

marclaboratemDdanattcmptintheaame~

k CCOt 8G David Bkk l d Aeuwlie ln8asUI4, am-

bridgqU,HmvdUnivaaityReas 1933,andthelcchmrbgtheaamelllthorm Amamrmatieal

theap of aatktb llld it a*tioIla to pomgand mu&, ics u t i w Pm &t, Vol. 19 (July,

1932). pp 189 942



measure))) th mean toward which the

virtuous should strive in their actions accord-

ing to Aristotle's Nicomachean Ethics, and

which Galen in De &mpatmmtix d d b e s

as hat stateofmind which isequally removed

from both exaemes: w W p o v rrp ~ ~ 6 7 b

TGV U ~ VW x e ~ .The image of th balance provides a

naturallinktothesecondsenseinwhichthe

word symrn hy is used in modern time8:

bilata.l spnwly th thesymmetry of left and

right, which isa ow~picuousn th structure

of the higher anLnaLs especially th human

body. Now this bilateral symmetry is a

strictly geometric and, in contrast to th

vague notion of symmetry d k c d Maran absolutely pr concept. body, a

spatial codgumtion is eymmetric with re-

spect to a given pknc E if it is d e d nto

itself by reflection in E. Take any line 1p

pmdieular to E and any point p on t there

Qdas one and only one point 8 on I which

has th same dhtance from E but lies on the

other aide. The pointp' coincideswithp ody

ifp is on E. Reflectionin E is that mapping

FIG. 1

R in E.

of space upon it , S: p +p , that carries

thearbimrypointpinmthisitsmirrOrimage

p' with rrspect to E. mapping is defined

w h~ e ve r hlle isestablishedby which every

point p is associated with an image '. An-

other example: a rutation a d papen-dicularaxk, say by30°,camiesBachpodntp

of sp ce into a point p and thus deb- a

mapping. figurehas rota- symme*

m u d anaxislifitiscarriedintoitrdfby

ll rotations m d . B i l a d symmetry

a p p m thus as the int of a geometricwncq?t of symmetry that m to aueh

operations as relkdons or rotations. Be

came of their omnplete rota- wthednleintheplane,thesphdeinspace

were amsidered by the the

most perfect geometric f i p e s , and Asktotle

asaibedsphaicalshapetotheeeleatialhodies

becauseanyotherwoulddctcaetfromthcir

heavenly perfection. It L n this tradition

that a.modem pxt edQesses th Divine

B d n g a s ' T h o u g r e a t ~ :

a hgr.6otmeW f m ~ o i hglustinns

Frmn wbum my som*os ,Fa dl 8It8fn' tamdrEcgsTI b pmt in sha drFs q s

G ~ J ~ ~ E B M ~ Q E ~ ~ .

S ymm e a y , a s w i d e o r a s ~ w a a y o um a ydeflneitsmewing s one dea by which man

throughtheagabastriedto compreheddand

createorder, beauty, and perfection.

T h e c o u r a t h ~ ~ e l e c t u n s w i l l ~ i s a s

follows. P i w ll d i i ilateral sym-

metry in some detail and its mle in art as

* A ~ u a W i o L h a m , ~ ~ f i m mk-

pony, Hamurr, B ~ ~ G undCb. 1 ~ 1 .



well asorganic and inorganic nature. Then

we shall generalize this concept gtadually,

in the direction indited by our example of

rotational symmetry, irst staying within the

d e s geometry, but then going bqrond

th s limits through the pmxs o mathe-

matical abstraction along a road that will

y ead us to a mathematical idea of

peat generality, the Platonic idea as it were

behind l l the speci l appearances and ap-

plications of symmetry. To a certain degree

this scheme ia typical for ll theoretic Lnowl-edge: We begin with some general but vague

principle symmetry in the first sense , then

find an important case where we c n give

that notion a concrete precise meaning bi-

lateral symmetry), and £ru that c se wegradually riae again to generality, guided

more by mathematical mmu tion and

abstractionthan by themiragenof philasophy;

and ifwe re lucky we end up with an idea

no less universal than the ane from which we

started. Gone may emuch of its emotional

appeal, but it has the same or even greater

unifying power in the realm of thought and

h exact instead of vague.

open the discussion on bilateral sym-

metry by using this noble Greek sculpturefmm the fourth century B.c. the statue of a

praying boy Fig. 2 , to let you feel as in a

symbol th great significance of this type of

symmetry bothfor life and art. Onemay sk

whether the acsthetickalue of symmetry de-

pends on its vital value: Did the arrist din

cover the symm try with which nature ac-

cording to some inherent law haa endowed

its creatures, and then copied and p e t f d

what nature presented but in imperfect

r d k a t i o ~ ~ r haa the aesthetic value of

symmetry an independent source I am in-

FIG 2



dined to think with Plato that the mathe-

matical id- is tho o G m rigin both:the mathematical laws govaning nature re

the ori in of apm y in nature the in

t u i t i v c ~ t i o n o f ~ i d e a i n t h e ~ d v es m i n d i t a E l r i g i n i n a r t , ~ I a m

ready to +t that lte arts the fact he

bilateral symmetry of rhe human body in ita

outward appsaranee bas cted as an addi-

t i o n a l ~ ~ ~

ofallandtnt~theSutnerians rm

to have been pgrdhll lrty fond to d strict bilateralmheFatdiesgmmerry. A t v p i d d b

~onthefam0rmEilvervaaofKin~En-

temena,*rukdinthedtgdr;lgash

FIG

around 2700 B.c. shows a lion-headed eagle

with spreadwings ace, each whose clawsgrip8 a stag in side view, which in its turn is

frontally attacked by a lion (the stags in the

upper design are replaced by goats in the

Iowa) (Fig. 3). Extension of he exactsym

metry of the eagle to the other beasts ob-

viously enforces their duplication. Not much

later the eagle is given tw heads facing in

either direction, the formal principle ofsym

metry thua completely overwhelming th

imitative principle of aud m nature. This

heraldic design can then be followed toPersia, Syria, later to Byzantium, and anyone

who lived before the First World War will

remember the double-headed eagle in the

coats-of-arms of ~ z a k s tRussia and the

Austm-Hungarian monarchy.

Look now at this Sumerian picture Fig.4 .

The two eagle-headed men are nearly but

not quite symmetric; why not? In plane

geometry reflection in a vertical line can

lso be bmught about by rotating the plane

in space around the xis by 180. If you

look at their rms you would say these two

FIG



monsters rise from each other by such rota-

tion; th overlappings depicting their@

tion n spaceprevent the plane picture fromhaving bilateral symmetry. Yet the artist

aimed at t symmetry by giving both

figures a half turn toward the o h e z and

also by the arrangement of feet and wings:

the droopii wing is the right one in the

left fiseun. the left one in the right figme.

im designs on the cyhdriwl Ba b y l w h

se l stmles are frequently ruled by -die

symmetry. I member seeing in thedh

tion of my form r coihgue the l te Emst

Henfad, samples where for m w ' s lre

not the head, but the lower bull-shaped part

of a god's body, rendered in p f i l e , w s

doubled and given four instead of two hind

legs I n c h r i a t i a n t im e s o n em yw e a nanaIogy in c rt in qnwmtations of tf

Eu a s on Byaantiae patsn Fig.5 ) , w h e m t ~ m ~ p ~ m C h r i s F s @ f a d D gthe disciples. But here symm y id not

~eteandbaadeady-thanfarmal

f o r C t a s i s t o n o n e s i d e ~t brard,ontheorherpomsthewlne.

Betwen Snmeria and Bymmiwn let meins t P A hese n mled s (Fig.

6)arefiomDmius'palaeeihguSab~ilti~



thedays o Marathon. h d n g he

we find them hr -H Fig,7) at theMegaron in Tiryns, late h dic about 1200

3 0 Who elieves strongly in tristoric e m

tindty and d q ~ d c n e t in trace the gr80b

tl desigss o e life, dolphin and

o c t o p u s , b i c k t o ~ M i n ~ t n c u l t u r e o f ~

the hg ldio symmeay to oriental, in the

last .hstancc Summisn, htlumm Skip

t h ~ o f f l c a t s w e s t i l l s t e t h e s am c i n f l u -enceaatworkinthkplaque~i. )fromthe

altar en hute in the dom o T d 4Idy, leventh century A.D. The peawdw

drin ingfrom a pine well amongvlne leave

a r e a n a n d i e n t C l d t b s y m b d d b

ralitp the t h e a l dk symmetry is

Oric~tat

For in contrast to the orient, occidental

art, like life itself, is inclined to mitigate, to

I w m , to modify, even to break strict s v -

metry. But seldom s asymmetry merely the

absence of symmetry. Even in asymmetric

designs one feels symmetry as the norm from

which one deviates under the influence of

forees o uon-formal character. I think the

riders from the famous Etruscan Tomb of the

Triclinium at Corneto (Fig. 9 pmvide a good

example. I have already mentioned repre-

sentationsof the Eucharist with Christ dupli-

cated handing out bread and wine. The

central gmup, Mary M e d by two angels,

in t is mosaic of the Lord s Ascension (Fig.

10 in the cathedral at Monreale, Sicily

(twelfth century), has almost perfeet sym-

metry. [The band omammt's above and

below the mosaic will demand our attention

in the second lecture.] The principle of

symmetry is somewhat less strictly observed

in an earlier mosaic from San ApoUinare in



Ra- Fig. I , siwwin i wmundcdbyanan~guardoflnmor. ForlnnanceMaryintheMmvealemossicr?liees

b a t h h e n d s ~ y h t h e o r o r m r s g e s -tme;hrrconlytherigbhandsanraiad-tt~ h s made f rther into ds in thenext picture Fig. 12 , a yzanfhe relief

ikon fromSanMar- Venice. It is ahi? ?,and o cmm the tw figwen praying foxmercy s the Lord s about to pronounce the

last judgment cannot be mirror im ge o

e cb other;for to the right st nds hisV i



uem1y we touch &round hem w h a s the

p.eci.c gBOrn @ of Gitatgal sym-

m e t r y b G g i a s l t o ~ ~ ~ ~ € h e v a g u e w t i o n

o A ~ d G s i p p w i f h *

w started. SymmGPrp.3 w Dagt)$atF r e y i E , a n ~ O n ~ . d ~ i nI b t , ~ ' ~ ~ n s t a n d ~ ~motionand awmhg, the me order aad law,

the other arbitdmssand aeddtitt, the~ ~ ~ i y a a d ~ t h e ~ ~ l i f c ip I a y a s d U . ' wkamecGalor-

are r e p m a d as symbols for wm s h o r ~ d r a y s w g i v c ~ l i n h e y r a -l e viaw not inp . ohablyl o a ~ E e a a o n s p * ~ d~ d ~ , w ~ l t h c y ~ ~ ~ -

~ p l e s o r ~ ~ a n d ~ a t e ,rn bila* nymm@ic t is, hw-r>p u e t k t n o t ~ ~ t l y h ~omas o

God&catbdra ared&rent, asforinst nce

inc2ePms. ButinpFad*every-~ ~ ~ e e m s t o d ~ b o t 1 3 C ~ C C t h ecathefltai,dyto*factthat*-

were built io dWerent pesiadk It is unda

standable&at a rttrr imewasno1ong.bsatis

& d w i t h * ~ o f a n w t i b ~ ~ ~ ~o n e m a y ~ b n e o f ~ a r y r r z m c t r -

M i f i m h a g e a o c c n r w e r c : r h e r c i s a ~ ,b e i r a f a r k a d e & g r r ~ o t a ~~ i n t o w h k h a ~ ~ .aturea s d a a p a i n t e n l a a L c u s t e B ~ m o t i f . 1~ ~ w i l l ~ y h ~ n r e t o y o o ~ ~

T e ~ ~ ~ ~ f a m i I E B T t o n w j ~ * I k e La t i t i a m y e . t u d - e % r y d a y , i a W ~ -

ofsilaapkuur.~ w e a r e a h D u t t o ~ o a a M T O a

natart, 1st m

eanside whatphilo~& I iu right. To the adareas fihunk p. 276.

mind them is no innerdiikmx no polarity

betweenleftand right, a s t h e m i s i s f o r ~

inthecontrast~matcand mak,orofthe

ant& aod p d o r nds aaanimal. Itreq an arbitrary act of choice to detar-

mine what is left a d what is right Butafter it ia made for one body it is

for every body. I must ixy to make this alittle clearer. In space the distinction o left

;andright eon- the orientation ofa screw.

IfyouspeakofturningIeftyoumeantbatthe

s n in which you turn eombined with the

upward dire tion from foot to had o you~

body forms a left screw. The daily rotationofthe earthtqetherwiththeMondits

axis&om South to North Pole isa left scrav,

it i a right saew if you give the ash €he

opposite W t ion . Therearecertaincrystab

line substanca called optically active which

betray theinnerasymmea~f theird t u -

tion by turning €he pohmatm.

plane ofp o k h d light sent through them to

thel a~totiaGright;bythia,of-,we

meank t hesut c nwhichtheplane-tea

while the l travels in a definite diredon

combined with that forms a left

a e r e w ( o r a r i g h t e n e , a a t h e e a s e m a y b e ) .H e n c e w h m w e a a i d a b a r e ~ n o w r e p c a t

in a t e m h h g y due to Lei-, that left

a n d r i g h t a n i m b m i * w e m t t o ~

thattheinnershuchlreofspacedoennot

p s m i t w , e r w p t b y a r ~ C h o i e q m d i a -

tingukhaleftfmmarightm

I wish to make tbis und-tal mrtienstill

more preoise for on it depends the entire

theory of d v i t y which is but

v t f symmary ceding to Eudidone can describe the s t . f apace. by anumber of basic relatiom between points

such aa ABC ie on a straight line, ABCD lie



in a plane AB iscongruentCD Perhapsthe

aat way of dacribing the spucture of spa

is the one Hehnhdtz adopted: by the singln&Oll of COBpWU ofm A Dapp@ S

of space d t e s ith every point a point

pl:p +pf. A pair of mappings S Sf:@-+d,

~'+p,ofdchtheoneistheinvera~ofthe

other, so that if S c rries p into p then Sf

FIG 13

~ p f b a i n t o p a a d p i o G v ~sof as a pair ofo~lb$o-memappings or imcrp

fanrarions A ~ ~ t i o o . w h i o h ~ ~ ~

thestrue~eof-ifwedefinethisstructuruintheH- way, that

mean that it carriesany tw m w t igurinto two w on& called an d g

m p h by thc ntathema-. dbnie

that thia is the idsa undalyhgthe gemmetric conoqt Mwa u ~ & v j a f i g u r e i a t o t m e f h a t

in mwbnizs .-itife e c h o f t h e n v o ~ i s w d d a a l b y i ~ ~ 'What we mean then by stating thatl t a d

rightareofthesameeostnceistheh~th~r ediar n splaar i s an a&mw hidnn.

Spaceasmichisaudiedbygeom~. But

spaceisalaothemadiumofa~physteaI00

nnrrn The strnenne of the physical

world is rweated by the general laws of na-

ture. They are formulated in temw of cer-

tain bslsic quantities which are functions in

spaceand time. We would d u d e hat the

~hys ical structure of sp ce "contains a

screw, to use a suggestive ftgure of speech, if

these lawsw e ot invariant throughout with

nspect to reflection. Emst Mach ells of heintellectualabockhemeivedwhen hele rnsd

asa boy that a magnetic needle isdeflected in

a cer ainsense to the left o the right, if

suspended pardel to a wire through which

an electric - is sent in a de6nite direc-

tion (Fig. 14 . Since the whde geometricand physical wntiguration, including the

electric current and thcm t h and north poles

of the magnetic needle, to all appearances,

are symmetric with respect to the plane Ehid through the wire and the needle the

needle should react likeBuridan's a s etween



equal bundles of hay and reluse to decide

betwealeftandright, just as sa ho fe qu al

armswith equal weights neither go down on

their left nor on their right side but stgy

horizontal. But ppe r nces are sometimes

deceptive. Young Mach's dilemm was the

result of a too hasty amamption con-g

the effect of reflection in E o n the dectric

current and the positive and negative mag-

netic poles of the needle: while we knew a

how goometric entities fare under

refloetion, we have to learn from nature howthe physical quantities behave. And this in

what we find: under rdieetion in the plane

theelectriccurrentpreewea its direction,butthe magnetic muth a d orth pdes are nter

changed. Of course thisway out, which re-

estaMishen the equivalenceof eft and rig& in

pmible only becauseof the-tid equality

of positive and negative majl;netism. AU

doubts were d i e d when one found that

themagnetimn oftheneedlehasitsariginin

rmleculardectricc-tseireulating~~)und

the needle's direction; it is dear that

reflection in the plane E such m t s

hange the m e n which they flow.

Thenet t is that in aY physieanothing

has shown up indicating an intrinak diffs-

enceofleftandright. Justasallpointsand

alldire nsinsgaceareeqaivalent, mare

left and right. Position, direetioh lcft and

rightarenlutk-concepts. Inlanguagetirrg dwith theology this ssue of relativieywaa din-

cused at great length in a famousoontmvauy

berwaen bnia and Clarke, the t o

clergyman acting as the spokamau for

NRaton.' ewtonwith his belief n abaolute

stc G w. Lcilrmk, Pidhphi SI* cd.

Oshardt 1875 scq.), w pp. 352-440, in

Lribnil hirdkit8r. 5.

space and time d d w motion a p d

of the creation of the world out of God'sarbitrary will fm o t h d t would be in-

explicable *hy matter moves in t is rather

than in any other d rectton. Leibniz in loath

to burden God with such d & a rm lacksufficientreason. Says he, U n b he as-

swnption that space be somefbing in itself it

in impwible togivea reasonwhy Godshould

have put the bodies (withoutampaingwith

their mutual distances and dative positions)

just at this particular place m d not nomewhem else 6r nstance, why He should not

have arranged everythkg in the opposite

order by turning East and Went about. If,onthGotherhand,spaceisnothingagthan

the spatial ords and relation of things then

the two tea supposed above, the ctu l

one and its transposition,are in no way dif-

ferent from each other and th m lo n it

is a quite inad-ble question to ask why

one srate was pmtared to the other. By

pondering the problem of I& and right b t

was first.led to hia conception of space and

t h e as f s f intuition.s k t ' s opinion

s ms to have en this: If the first creative

act of God had been theformingofaleft

hand then this hand,even

at the time whenit could be wmpated to nothing dsa, d

the d i v e haracter of ft, which can

only intuitively but never conecptually be a p

prehended. Lei- contradicts: Accordingto him it would have made no difference if

Gad had created a right hand first rather

left one. m e must foIlow the

world's creation a step iiarther before a differ-

ence can appear. ad God, rarha than

~ M s ' * K l i t i l r d a r r i n m v ~ ~ & ~

pxiaUy 1 3 o l t h c ~ ~ + e r j ~ i . m ~ mMu @ . .



making irst a lefr and then a right hand,startad with a right hand and th o f d

another rigat hand,He wm l havech ng d

theplanoftheuniveawaotinthefirstbutinthereoondact,bybFiaBlngforthahandwhich wss equally rather than oppositely

ral to the ht-mated s p e

scientific thinking lea with IAbnk.Myrbiaal W&bg bss always t lrrn the wn-

-view asiserincod by itswageofrightnd ldt ar symbols for nuch pola~pposites

an goodand evil. Younecd d y hin d thedouble m e q d theword ri ht itaelf. In

t h i s ~ f r o r n h d i c h e l a n a a l a ' ~ f ~ C ~ ~ n -tim of dem fFom the S i e dling (Fig.

15) God2@ight h a d , on the right touehea

life into Adam's I .

Pwpk shakt right hands. Sinida is the

L a t i n w d f o r M t , a n d ~ d S p e a l r sof thekftsideoftheshiotdasitasinirapside.

But s n srnrm is at the same time that d o h

isevil,andin~~nmonEngJinhodytbisfigwative - of the Lath ur-

vives.@ Of the r ~ ealefactm who wcre

crudficdwithChrist,theonewb oks*Himtopar*isonHisright.StMatrhm,~hepta5, a d b e s the last judgment as

f 0 l l ~ ~ ~ : A n d t l e s h a l l s e t t h e s h a e p o n B i sright hand but the gQaa on the eft. Then

shall the King ray lmto h m n his right

hand, ome ye, bkmed d my Father, in-h e r i t t h G ~ ~ m ~ h p -thefou~datio~ofdieworkl.. . Theahh ay also unto them on the left hand,

D e p a r t f r o m m e y e e u r s a d i ~ - ~~pnpemtfarthedevilandhisang&.

FIG IS

G ~mmemh~ra~actcne~dnrich-

o n c e d d i ~ i n z u r i e h 0 n ' ~ a n d h e nin painting$'; togemcr with an artkle on

The p b lem d Pvasion b

Rap$ae17stap is try^ mnv find it

print&in&breviatedf&minhis&bbn~ u K i e s h i c h t c , l 9 4 1 . B y a n e d w

amples, ar Raphael's isliw Mabum and

Rembmdt's etching rmaPsrpps Bgifk fh 66

maW6lBin alea to shmv that right in

p a i n t i q h a s a n o t h e r ~ r t t h a n M .Ractieallyallmethods ofrepduction inter-

changelek andright, anditseemthat rormSrtimcsweremuchlemsmsitivethanweare

i n v c d m . E v a Rm~brarrdtdid not hesitate to b&g r Descent f%wmthe me as a converse etcbiug upon the

d e t . Ccmaklechg thatwedo a lotmore

reading than thepeople say, of the

mtury t is suggests the hypoahesis that the

sa



d i i m c e pointed out byW8lfflinis CO~Ccted

with our habit of reading from left to right.

Asfar as I remember e himselfrejcctedthis

s well as a number of other psychological

explanations put forward in the disc

after his lecture. The printed text concludes

with the remark hat the problem obviously

has deep mots m e hicb reach down to

thevayfoundatiomofour~~~~nature?

I for my par4 m dkidhed to take the

maiter that Saioltsly.'

In &ce the belid in the equiv lence ofleft and right has bem upheld even in the

face of e t in biological facts p-tly to

be mentioned which seem to augge8t theirinequivalence even more strongly than doos

the deviation of the magnetic needle whichshwkd young Mach. The same

of equivalence arises withrcapecftopnstd

fume, which are intedmqed by i n d g

the d ii t i o n of time, and with respect topoJihhw and ncgnlr'w id .In then CaSeq

especially in the second, it is p h a p d-th n for the pair kft-right that a evi-

dence is not m&Xent to settle the qu d o a;

the empirical facts have to be consulted

To be sure the role which past and future

play in our wnsdowwa would inc i tetheir intrinsic difference-the paat knowable

and mcbqeable, the future unknown and

still alterable by decisiom ken n o w 4

one would expect that this difference bas itsbasis in the physical laws of nature. But

t h e awsofwhieh we canboast a reasonably

certainknowledge are invariant with respecta l s ~A ~ a t t a ~ , und knBiide, Amicia, J h h dm 8slnrdkhm GaMa

1926 p.n uliua . schl-, ' ' III~IW alk h a

dd quadri, Ctiticn 28 1930 p. 7 Paul Opp€,

Right and kft in Raphad s cammu, j% bnaI

th W 0 k g nd Courtm II d ,194, p. 82.

to thehvmion of h a they arc withre-

spect to the intaehange of left and rlght

Leibnizmade it clear that tbt emporalmodi

paatandfutuxe&tothecdsftuskmcf

thewwld. ' E v e n i f i t ~ ~ e h t t h e g a o t

-wave laws formulatedby qu emphya r e n o t a l t a e d b y l e t t i n g ~ f i o w b ~

them e t a p h w idea ausation, aod wi th

itthcenewaycharacteroftime,mslyeater

phy through the rtatisieal in

oftlaose laws in rmll ofprobabilitp and

particlea our present physical k n o w

leaves us even lnorr nncatain about theequidcaec or nOn valence of positmand m v e eopidy I tseem di&dt to

devisephysicallawainwhichthaparcnot

iny w but the ne@%tive

part of the positively chargfd proton till re-

mainstobedisawsnd.

This M-phiiomphical excursion was

netdedasabaekgroundhrthetheof

the I&--right symmetry in nature; we had

t o u n d d t h a t t h o m # t i o n

ofnannc posscsai that s w j . &t one

W i U n o t ~ t h a t a n y l l p c d a l ~ n a -tureshowsit to p e r b t h . Evenw itissur-

lnisiag to what eat- it p a i k l?memustbeareas~lfwthiqanditisnotfarto

stek:aatateofequilibriumis~ytobeaytn-metric. More preciacy under conditbna

which dstermint a unique a t e of cquilib-rium the symmetry of th benditions m ft

carryover totheatateof equilibrium. T h e

fore tennis balk and starr arc sphww the

ear thw~bGasphsTe twi f i td idno tro ta te

amuadaaaxis. T h c F a t a ~ ~ i t a t

tbepoles but the o ta r i dw cyhbi cal sym

metry around i&ads s prcscnnd. The fea-

nnc that xplanation is, thacforenot the mtational symmetry ofits shape but





















































































































































Documents

Symmetry by Weyl