Point Groups and Space Groups in Geometric Algebra


 Emery Richard
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1 Point Groups nd Spce Groups in Geometric Alger Dvid Hestenes Deprtment of Physics nd Astronomy Arizon Stte University, Tempe, Arizon, USA Astrct. Geometric lger provides the essentil foundtion for new pproch to symmetry groups. Ech of the lttice point groups nd 0 spce groups in three dimensions is generted from set of three symmetry vectors. This gretly fcilittes representtion, nlysis nd ppliction of the groups to moleculr modeling nd crystllogrphy.. Introduction Symmetry is fundmentl orgniztionl concept in rt s well s science. To develop nd exploit this concept to its fullest, it must e given precise mthemticl formultion. This hs een primry motivtion for developing the rnch of mthemtics known s group theory. There re mny kinds of symmetry, ut the symmetries of rigid odies re the most importnt nd useful, ecuse they re the most uiquitous s well s the most ovious. A geometric figure or rigid ody is sid to e symmetricl if there exist isometries which permute its prts while leving the oject s whole unchnged. An isometry of this kind is clled symmetry. The symmetries of given oject form group clled the symmetry group of the oject. Oviously, every symmetry group is sugroup of the group of ll such isometries, known s the Eucliden group E(). As is well known, every symmetry S cn e given the mthemticl form S : x x = R x +, () where x designtes point in the oject, R is n orthogonl trnsformtion with the origin s fixed point, nd the vector designtes trnsltion. In most pplictions the opertor R is represented y mtrix [R], so composition of trnsformtions is chieved y mtrix multipliction. This prctice hs two drwcks, however. First, use of mtrices requires introducing coordinte system, nd tht rings in ritrry fetures tht complicte prolems. Second, mtrix elements re usully difficult to interpret geometriclly. Geometric lger voids these drwcks with the coordintefree cnonicl form R x = ±R xr, () where R is n invertile multivector, clled versor, with even (odd) prity corresponding to the plus (minus) sign. The versor in () hs een normlized to unity, so its reverse R is equl to its inverse R. When R is even, eqution () descries rottion, nd R is often clled rotor or spinor. The reder is presumed to e fmilir with eqution () nd the versor representtion of orthogonl trnsformtions; the suject hs een thoroughly treted in (Hestenes 98) with mny pplictions to mechnics. Surprisingly, this pproch hs not heretofore een een pplied to systemtic tretment of discrete symmetry groups in the pulished literture. To rectify tht deficiency is the first of two mjor ojectives for this pper. The first hlf of the pper provides complete tretment of the point groups in two nd three dimensions. As this mteril is not to e found elsewhere, the min ides re illustrted with exmples, nd sutle points tht re esily overlooked re thoroughly discussed. Aside from the mthemticl definition of group, no prior knowledge out group theory is presumed. The min result is tht ech of the point groups in three dimensions cn e generted from set of t most three symmetry vectors tht re tied directly to fetures of the oject. This leds to new systemtic nottion nd clssifiction scheme for symmetry groups from which one cn directly write down the genertors for ny point group. The point groups determine the clsses of mthemticlly possile lttices known s crystl systems, s explined in Section. This shows how geometric lger cn simplify theoreticlly crystllogrphy
2 considerly for the symmetry vectors generting the point group cn e identified with lttice vectors tht generte the lttice. In other words, the point group cn e generted multiplictively from the ojects on which it opertes. Contrst this with the usul pproch which develops the group elements nd the lttice s seprte entities relted only indirectly. The point group of lttice leves lttice point fixed. To get the complete symmetry group of lttice, one needs to comine the point group with trnsltions. This rises nother prolem with the stndrd representtion for symmetry y (), nmely: The orthogonl group is multiplictive while the trnsltion group is dditive, so comining the two destroys the simplicity of oth. The source of this prolem cn e trced to the fct tht eqution () singles out one point, the origin, for specil tretment. The second mjor ojective of this pper is to show how geometric lger provides n elegnt solution of this prolem with simple new multiplictive representtion for the spce groups generted directly from lttice vectors. Sections nd 5 introduce the essentil mthemticl pprtus to chieve this ojective. Section 5 introduces new homogeneous formultion of Eucliden geometry tht trets ll points eqully nd genertes n lger of points, lines nd plnes. The formlism is pplied in Section to crete the desired multiplictive model of the Eucliden group. This model provides precise lgeric formultion of the geometric notion tht ll symmetries cn e generted from reflections in plnes. Therey, it provides new lgeric foundtion for geometric intuition nd powerful tool for computtionl geometry. The rest of the pper is devoted to systemtic presenttion of genertors for the 0 spce groups. Although spce limittions preclude tretment of ll the groups, we do show how to construct ny of the genertors from symmetry vectors, nd we introduce new scheme of spce group symols tht fcilittes construction of group genertors. All this is illustrted in complete tretment of the 7 plnr spce groups. The techniques nd results in this pper hold gret promise for simplifying nd enriching the use of symmetry groups in crystllogrphy nd moleculr modeling. There re mny fine ooks on crystllogrphy (O Keefe nd Hyde 99) tht cn serve s guide to prcticl pplictions of the method. For n exhustive description of the 0 spce groups, the stndrd reference is the Interntionl Tles for XRy Crystllogrphy (99). It is widely used in mteril science to chrcterize complex crystl structures, for which the identifiction of the symmetry, clss nd spce group continues to e nontrivil tsk. The Interntionl Tles re huge nd cumersome, so the simplifictions offered here would e of gret vlue in mteril science reserch nd engineering. Moreover, the method hs potentil for much wider ppliction.. Point Groups in Two Dimensions As usul in mthemticl nd physicl prolems, the est strtegy is to study the simplest cses first, nd therefrom discover results which re needed to hndle the most complex cses. So let us egin y exmining the dimensionl symmetry groups with fixed point. The fixed point condition elimintes trnsltions, so ll the symmetries re orthogonl trnsformtions. Consider, for exmple, the enzene molecule shown in Fig.. This molecule hs the structure of regulr hexgon with cron tom t ech vertex. Evidently, the simplest symmetry of this molecule is the rottion R tking ech vertex x k into its neighor x k+ s descried y x k+ = R x k = R x k R = x k R. () A sixfold repetition of this rottion rings ech vertex ck to its originl position so R stisfies the opertor eqution R =. () This reltion implies tht the powers of R compose group with six distinct elements R, R, R, R, R 5, R =. This group, the rottionl symmetry group of hexgon, or ny group isomorphic to it, is clled (or the) cyclic group of order nd commonly denoted y C. The group C is finite group, soclled ecuse it hs finite numer of elements. The order of finite group is the numer of elements it contins. The element R issidtoegenertor
3 c x x x x 5 x / π π / x = Fig.. Plnr enzene (C H ), showing genertors of the symmetry group. (Hydrogen toms not shown.) of C, ecuse the entire group cn e generted from R y the group opertion. The group C is completely determined y the condition R = on its genertor, with the tcit understnding tht lower powers of R re not equl to the identity element. Any such condition on the genertors of group is clled reltion of the group. A set of reltions which completely determine group is clled presenttion of the group. For C the presenttion consists of the single reltion R =. It is computtionlly dvntgeous to represent rottions y versors rther thn liner opertors, so we look for representtion of C y versors. According to (), the opertor R corresponds to unique versor S = R, so the opertor reltion R = corresponds to the versor reltion S =. (5) This presenttion of C hs the dvntge of dmitting the explicit solution S = e πi/ = e iπ/, () where i is the unit ivector for the plne of rottion. The representtion () shows explicitly tht the genertor of C is rottion through ngle π/. Now, we know from eqution () tht to every rottion there corresponds two rotors differing only y sign. Consequently, to every finite rottion group there corresponds rotor group with twice s mny elements. In the present cse the genertor R of the rotor group is relted to the genertor S of the cyclic group y S = R. Tking the negtive squre root of the reltion S =(R ) =(R ) =, we get the new reltion R =. (7) This is the presenttion for the dicyclic group of order generted y R. Strictly speking, we should include the reltion ( ) = in the presenttion of the group since it is not one of the group properties. However, this is tken cre of y the understnding tht the group elements re versors. Since the dicyclic group presented y (7) is the versor group of C, let us denote it y C. The dicyclic group ctully provides more complete description of rottionl symmetries thn the cyclic group, ecuse, s first explined in (Hestenes, 98), the pir of rotors ±R distinguish equivlent rottions of opposite senses. The cyclic group does not ssign sense to rottions. This importnt fct is illustrted in Fig. nd explined more fully elow. We hve seen how the rottionl symmetries of hexgon cn e chrcterized y the single eqution S = or etter y R =. However, hexgon hs reflectionl s well s rottionl symmetries. From Fig. it is evident tht the hexgon is invrint under reflection long ny digonl through vertex or the midpoint of side. For exmple, with = x, the reflection is symmetry of Fig., s is the reflection Ax = x, (8) Bx = x, (9)
4 R = = e i π ( /) { > < < > R = = ( ) = e = e i( π/ π) i( 5 π/ ) } Fig.. Illustrting the interprettion of the spinors ±R = ± = (±) s equivlent rottions with opposite sense generted y reflections with different senses. where is directed towrds the midpoint of side djcent to the vertex, s shown in Fig.. These reflections generte symmetry group of the hexgon which, for the time eing, we denote y H. This group is sometimes clled the dihedrl group of order, ut tht nme will e reserved for geometriclly different group isomorphic to it. To void introducing new nme, let us e content with the symol H. Now, to get on with the study of H, note tht the product BAx =() x() (0) is rottion; in fct, it is the rottion R which genertes C. Therefore, C is sugroup of H.From this we cn conclude tht the opertor equtions A = B =(BA) = () provide n strct presenttion of H. The rotor group H corresponding to H is generted y the vectors nd normlized to unity. Since R = must stisfy (7), the presenttion of H is the set of reltions = =, () () =. () According to (9), the two vectors ± in H correspond to the single reflection B. Physiclly, however, one cn distinguish two distinct mirror reflections in given plne y imgining the plne surfce silvered on one side or the other. Thus, we hve two distinct reflecting plnes (or mirrors) with opposite orienttions distinguished y the signs on their norml vectors ±. An oriented reflection in one of these oriented (silvered) plnes mintins the physicl distinction etween n oject nd its reflected imge. So the two oriented reflections specified y ±, descrie the two possile plcements of n oject on opposite sides of the reflecting plne. The (unoriented) reflection B in (9) mkes no distinction etween ojects nd reflected imges. The notion of oriented reflection is consistent with the notion of oriented rottion. For the products of oriented reflections designted y ± with n oriented reflection designted y the vector will produce the spinors representing equivlent rottions with opposite senses, s illustrted in Fig.. Thus, ech element of H chrcterizes some oriented symmetry of hexgon. The group H is the multiplictive group generted y two vectors, with the properties () nd (). The distinct elements in the group re exhiited in Tle. Note tht the geometricl interprettion given to in Fig. permits the ssignment of definite sense to the unit versor, s indicted in Tle. So the versor = e i0 represents rottion of zero ngle in the positive sense, while the versor =e iπ =e i( π) represents rottion of π with the opposite sense.
5 Six distinct rottions with "positive sense" represented y = = () () () () 5 Six distinct rottions with "negtive sense" represented y  = () = ( )  = ( ) = ( )  () = ( )  () = ( )  () = ( ) 5  () = 5 Twelve distinct reflections represented y +_ +_ +_ = +_ () +_ +_ +_ = +_ () Tle. The distinct elements of the group H. Ordinrily, the group H is regrded s the symmetry group of regulr hexgon. But we hve seen tht the corresponding versor group H provides more sutle nd complete chrcteriztion of the symmetries. Since the two groups re so closely relted, it mtters little which one is regrded s the true symmetry group of the hexgon. The versor group, however, is esier to descrie nd work with mthemticlly. Consequently, s we shll see, it will e esier to generlize nd relte to other symmetry groups. Our results for the hexgon generlize immeditely to ny regulr polygon nd enle us to find nd descrie ll the fixed point symmetry groups of ll twodimensionl figures. We merely consider the multiplictive group H p generted y two unit vectors nd relted y the dicyclic condition () p =, () where p is positive integer. The vectors nd determine reflections (8, 9) which generte the reflection group H p. The dicyclic group C p is sugroup of H p generted y = e iπ/p = e i(π/p) (5) the rotor for rottion through n ngle of mgnitude π/p. The corresponding rottion genertes the cyclic group C p. The versor group H p or, if you will, the reflection group H p is the symmetry group of regulr polygon with p sides. The group is well defined even for p =, though two sided polygon is hrd to imgine. When p =, (5) implies tht =, soh is the group consisting of the four elements ± nd ±. Thus, the group H is the group generted y single reflection. The group H consists of the two elements ± while the corresponding rottion group C contins only the identity element. Either of these lst two groups cn e regrded s the symmetry group of figure with no symmetry t ll. A symmetry group with fixed point is clled point group. The groups H p nd C p, for ny positive integer p, re point groups in two dimensions. The groups H p nd C p re oriented point groups. Besides H p nd C p, there re no other point groups in two dimensions. This cn e proved y considering the possiility of group generted y three distinct vectors,, c in the sme plne. If they re to e genertors of symmetry group, then ech pir of them must e relted y dicyclic condition like (). It cn e proved, then tht one of the vectors cn e generted from the other two, so two vectors suffice to generte ny symmetry group in two dimensions. Although it tkes us outside the domin of finite groups, it is worthwhile to consider the limiting cse p =. With incresing vlues of p, regulr psided polygon is n incresingly good pproximtion to circle, which cn e regrded s the limit t p =. Therefore, the complete orthogonl group O() in two dimensions cn e identified s the symmetry group of circle, the rottion sugroup of O + (). It cn e regrded s the symmetry group of n oriented circle. Note tht reflection will reverse the orienttion, so O() is the group of n unoriented circle. Note further, tht even for finite p, C p is the group of n oriented polygon while H p is the group of n unoriented polygon. 5
6 . Point Groups in Three Dimensions We hve seen how every finite sugroup of the orthogonl group O() cn e generted y one or two reflections. One might guess, then, tht no more thn three reflections re required to generte ny finite sugroup of the orthogonl group O(). So we shll see! If three unit vectors,, c re to e genertors of finite multiplictive group, then ech pir of vectors must generte finite sugroup, so we know from our preceding nlysis tht they must stisfy the dicycle conditions () p =(c) q =(c) r =, () where p, q, ndrre positive integers. If r =, then () implies c =, ndp=q, so () reduces to reltion etween two vectors, the cse we hve lredy considered. Therefore, if the vectors,, nd c re to e distinct, then ech of the integers p, q, ndrmust e greter thn. The three genertors of rottions in () re not independent, for they re relted y the eqution ()(c)=c. (7) This eqution reltes the sides of sphericl tringle with vertices,, ndc. This reltion restricts the simultneous vlues llowed for p, q, ndrin (). The precise nture of the restriction cn e scertined y writing () in the equivlent form = e ic π/p, c = e i π/q, (8) c = e i π/r. The unit vectors,, c re poles (or xes) of the rottions generted y, c, c, so the sphericl tringle they determine is ptly clled the polr tringle of the generting tringle {,, c}. From (8) it follows tht the interior ngles of the polr tringle re equl in mgnitude to corresponding sides of the generting tringle nd they hve the vlues π/p, π/q nd π/r. Therefore, ccording to the sphericl excess formul (Hestenes 98), the re of the polr tringle is given y = π ( p + q + r ). (9) This is the desired reltion mong p, q, ndrin its most convenient form. From (9) we cn determine the permissile vlues of p, q, ndr. Since the re must e positive, eqution (9) gives us the inequlity p + q + r >. (0) The integer solutions of this inequlity re esily found y tril nd error. Trying p = q = r =,we see tht there re no solutions with p>q>r>. So, without loss of generlity, we cn tke r = so (0) reduces to p + q >. () Requiring p q, we see tht ny vlue of p is llowed if q =, nd if q =, we find tht p =, or 5. This exhusts the possiilities. It is not difficult to prove tht no new point groups with four or more generting vectors re possile. For every suset of three vectors must generte one of the groups we hve lredy found, nd it follows from this tht if we hve four genertors, then one of them cn e generted from the other three. All we need now is suitle nomenclture to express our results in compct form. Since ech of the multiplictive groups generted y three unit vectors is distinguished y the vlues of p, q nd r = in the presenttion (), ech of these finite diorthogonl groups cn e identified y the symol [pq ]. Let us use the simpler symol pq for the corresponding orthogonl groups, ecuse they re more
7 Oriented Point Group Symol [ pq ] [ pq ] [ pq ] [ pq ] [ pq ] [ p ] [ p ] or or Hp D p Genertors,, c, c, c, c c, Point Group Symol pq pq pq pq pq p or H p p or D p Tle. Symols for the doule point (diorthogonl) groups in three dimensions nd their corresponding point (orthogonl) groups. The groups generted y three unit vectors hve the presenttion () p =(c) q =(c) =, with 5 p q. The groups generted y two unit vectors hve the presenttion () p =. prominent in the literture of mthemtics nd physics. The groups pq re usully clled point groups y physicists, who usully refer to the groups [pq ]sdoule point groups, though considering the geometricl reson for the douling, it might e etter to cll them oriented point groups. The usul derivtion of the doule groups is fr more complicted thn the one presented here. Consequently, the doule groups re seldom mentioned except in the most esoteric pplictions of group theory to physics. Of course, we hve seen tht there is mple reson to regrd the diorthogonl groups s more fundmentl thn the orthogonl groups. Even so, we hve lerned tht the diorthogonl nd orthogonl groups re so simply nd intimtely relted tht we hrdly need specil nottion to distinguish them. Without ltering the group presenttion (), we get sugroups of [pq] y tking the vrious poducts of the vectors,, c s genertors. To denote these groups, let us introduce the nottion p to indicte genertor stisfying the reltion () p =. Accordingly, [p q ] denotes the dirottion group generted y nd c,ndpqdenotes the corresponding rottion group. The nottion is explined further nd the vrious groups it denotes re listed in Tle. Now tht we hve compct nottion, we cn list in Tle ll the point groups in three dimension, tht is, ll the finite sugroups of O(). We egin y listing the groups pq for the llowed vlues of p nd q determined ove. Then we pply the overr nottion to generte list of cndidte sugroups p q, p q, pq, pq. Finlly, we check the cndidtes to see if they re new symmetry groups. The groups pq re sid to e finite reflection groups, ecuse they re generted y reflections. All the finite groups re reflection groups or sugroups thereof. The groups pq generted y two pirs of reflections re finite rottion groups. Tle shows tht the only finite rottion groups re the cyclic groups p = C p, the dihedrl groups p =D p, the tetrhedrl group =T, the octhedrl group =Ond the icoshedrl group 5 =I. These re the only finite groups with widely ccepted nmes. The lst three of them re symmetry groups of the fmous Pltonic solids, the five regulr solids discovered y the ncient Greeks. The tetrhedrl group is the rottionl symmetry group of tetrhedron. The octhedrl group is the rottionl symmetry group of oth the (8sided) octgon nd the (sided) cue. The icoshedrl group 5 is the symmetry group of oth the (0sided) icoshedron nd the (sided) dodechedron. The nottion 5 indictes the fivefold symmetry t ech vertex (fce) nd the threefold symmetry t ech fce (vertex) of the icoshedron (dodechedron). The 7
8 Symol Geometric Schoenflies Nme Order p C p (di)cyclic ()p p C pv = H p ()p p = (n) S p ()p = ()n p D p (di)dihedrl ()p p = (n) D nd ()p = ()n p C pv ()p p D ph ()p T (di)tetrhedrl () = T d () T h () O (di)octhedrl () = O h ()8 5 I (di)icososhedrl ()0 5 I h ()0 Tle. The (doule) point groups in E. As indicted y prentheses in the tle, for oriented point groups the order is doule nd the prefix di is dded to the nme for the corresponding orthogonl groups. The groups p nd p exist only for vlues of p, s indicted in the tle y writing p =n, where n is positive integer. The symols,, 5 do not pper, ecuse they do not descrie relizle symmetry groups. nottion nd hve similr interprettions for the other regulr solids. From the fct tht there re no other rottionl symmetry groups esides those we hve mentioned, it is not difficulty to prove tht there re no regulr convex polyhedr esides the Pltonic solids. There exist, however, some regulr solids which re strshped nd so not convex. The lrgest symmetry groups of the Pltonic solids re ctully the reflection groups, nd 5 rther thn their rottionl sugroups, ut this ws not pprecited when nmes were hnded out, so they re without specil nmes. The cyclic nd dihedrl groups re symmetry groups for vrious prisms or prismtic crystls rther thn polyhedr. However, in physics they pper most frequently s symmetry groups for molecules. We re now in position to see tht the dihedrl group D =, rther thn the cyclic group C =,is the rottionl symmetry group for the Benzene molecule (Fig. ) in spce of three dimensions rther thn two. Furthermore, it is redily verified tht the rottion group D = is isomorphic to the reflection group H =, nd they hve identicl effects on the plnr Benzene molecule; nevertheless, they hve different geometricl effects on three dimensionl ojects. In three dimensions the complete symmetry group of the Benzene molecule is the reflection group D h =, which is formed y using the generting vector c long with the reflection genertors nd of H =, s illustrted in Fig.. 8
9 ' c' c ' π / π / π/ Fig.. Genertors,, c for the doule point group [] of cue or n octgon. Vertices,, c of the polr tringle (or fundmentl region) specify xes of threefold, twofold, nd fourfold symmetry, s indicted y the tringle, lense, nd squre symols. Besides the groups pq generted y reflections nd the groups p q generted y rottions, Tle lists mixed groups pq, pq nd pq generted y comintions of rottions nd reflections. Some of the mixed groups re identicl to reflection groups. For exmple, the equivlence = mens tht,, c generte the sme group s, c; in other words, the group generted y three reflections cn lso e generted y one rottion nd one reflection. Some of the cndidtes for mixed groups must e rejected ecuse they do not stisfy the condition for symmetry group. To see why, consider the rotryreflection group pq. The corresponding diorthogonl group [pq] hs the sme genertor c. Since represents rottion nd c represents reflection, the product c represents comined rottion nd reflection, tht is, rotryreflection. The quntity R =(c) is n even versor generting dirottionl sugroup of [pq], so it must stisfy the dicyclic condition R n =(c) n (for some integer n) if[pq] is to e symmetry group. This condition must e evluted seprtely for ech group. For exmple, for the group [p], the vector c is orthogonl to oth vectors nd, hence c = c nd But () p =, so R =(c) =(). () R p =(c) p =() p. () Therefore, the dicyclic condition R n = cn e met only if p =n, tht is, only if p is n even integer. Thus, we hve proved tht the group p is symmetry group only if p is even, s stted in Tle. The sme rgument proves tht p is symmetry group only for even p. In similr wy, it cn e proved tht, nd 5 re not symmetry groups, ut the lger required is little trickier. Our geometric nottion for the finite groups is unconventionl, so Tle reltes it to the widely used Shoenflies nottion to fcilitte comprison with the literture on crystllogrphy nd group theory. The rtionle for the Schoenflies nottion need not e explined here. However, it should e noted tht our geometric nottion hs the gret dvntge of enling us to write down immeditely the genertors nd reltions for ny finite group y employing the simple code in Tle. Thus, for the group [], the ngle etween genertors nd is π/, the ngle etween nd c is π/, nd the ngle etween nd c is π/. Figure shows three such vectors in reltion to cue whose reflection group they generte. According to (8), the lgeric reltions mong the genertors re fully expressed y the equtions = e ic π/, () 9
10 Fig.. Fundmentl regions for the reflection group = O on the surfce of cue, n octgon, or sphere. c = e i π/, (5) c = e i π/ = i. () The poles,, c re lso shown in Fig., It should e evident from Fig. tht every reflection symmetry of the cue is generted y vector directed t the center of fce (like ) or t the midpoint of n edge (like or c). Furthermore, every one of these vectors is lso the pole of fourfold rottion symmetry (like c or ) or of twofold rottion symmetry (like, or c) ut not of threefold symmetry (like ). Indeed, we see from Fig. tht cn e otined from c y rottion generted y () = e ic π out the c xis, so we cn directly write down the reltion Similrly, y rottion out the xis, =() c(). (7) c =(c)(c) =cc. (8) This illustrtes how lgeric reltions in the group [] cn e written down directly nd interpreted y referring to some model of cue like Fig.. A threedimensionl physicl model of cue is even more helpful thn figure. The polr tringle with vertices,, c determines tringle on the surfce of cue, s seen in Fig.. This tringle is clled fundmentl region of the group for the following reson. Notice tht ech of the three genertors,, c is perpendiculr to one of the three sides of the tringle, so reflection y ny one of the genertors will trnsform the tringle into n djcent tringle of the sme size nd shpe. By series of such reflections the originl tringle cn e rought to position covering ny point on the cue. In other words, the entire surfce of the cue cn e prtitioned into tringulr fundmentl regions, s shown in Fig., so tht ny opertion of the group simply permutes the tringles. Fig. shows n lterntive prtition of the octhedron nd the sphere into fundmentl regions of the group. In completely nlogous wy, the tetrhedron nd the icoshedron (or dodechedron) cn e prtitioned into fundmentl regions of the groups nd 5 respectively. Given one fundmentl region of group, there is one nd only one group opertion which trnsforms it to ny one of the other fundmentl regions. Consequently, the order of group is equl to the numer of distinct fundmentl regions. Thus, from Fig. we see tht there re eight fundmentl regions on the fce of cue, so there re 8 = 8 elements in the group. To get generl formul for the order of finite groups, it is etter to consider fundmentl regions on unit sphere. Then the re of ech fundmentl region is equl to the re of the polr tringle given y (9), so the order of the group is otined y dividing this into the re π of the sphere. For exmple, tking r = nd q = in (9), we find tht the orders of the reflection groups p re given y π δ = p p. (9) This is twice the order of the rottion groups p, ecuse ll rottions re generted y pirs of reflections. The orders of the other finite groups nd their sugroups cn e found in similr wy. The results re listed in Tle. 0
11 . The Crystl Clsses nd 7 Crystl Systems A crystl is system of identicl toms or molecules locted ner the points of lttice. A  dimensionl lttice is discrete set of points generted y three linerly independent vectors,,. These vectors (nd their negtives,, ) generte discrete group under ddition known s the trnsltion group of the lttice. Ech element cn e ssocited with lttice point designted y n nd cn e expressed s liner comintion of the genertors with integer coefficients, tht is, n = n + n + n, (0) where n, n, n re integers. Given the generting vectors, ny set of integers n = {n,n,n } determines lttice point, so the lttice is n infinite set of points. Of course, ny crystl consists of only finite numer of toms, ut the numer is so lrge tht for the nlysis of mny crystl properties it cn e regrded s infinite without significnt error. Our im here is to clssify crystls ccording to the symmetries they possess. The symmetries of crystl depend only on the loctions of its toms nd not on the physicl nture of the toms. Therefore, the nlysis of crystl symmetries reduces to the nlysis of lttice symmetries, welldefined geometricl prolem. Like ny finite oject, the symmetry of lttice is descried y its symmetry group, the complete group of isometries tht leve it invrint. However, unlike the group of finite oject, the symmetry group of lttice includes trnsltions s well s orthogonl trnsformtions. Before considering trnsltions, we determine the conditions for lttice to e invrint under one of the point groups. Lttice clcultions re gretly fcilitted y introducing the reciprocl frme { k }. Although reciprocl frmes re fmilir tools in crystllogrphy, it is worth mentioning tht geometric lger fcilittes their definition nd use (Hestenes 98, Hestenes nd Soczyk 98). Presently, ll we need re the reltions j k = δ jk, () for j, k =,,, which determine the reciprocl frme uniquely. Now, ny fixedpoint symmetry R of lttice trnsforms lttice points k (k =,,) into new lttice points s k = R k = j j s jk, () where the mtrix elements s jk = j s k = j (R k ) () re ll integers. Consequently, the trce of this mtrix s kk = k (R k ) () k k is lso n integer. This puts significnt restriction on the possile symmetries of lttice. In prticulr, if R is rottion symmetry generting rottion sugroup, then it stisfies cyclic condition R p =, nd it rottes the lttice through n ngle θ =π/p. It cn e shown tht Tr R = k k (R k )=+cosθ. (5) This hs integer vlues only if which hs the solutions θ =0, cos θ =0, ±, ±, () π, π, π, π, π, π, 5π, π. (7)
12 Tle. The crystl clsses (point groups). System Triclinic Monoclinic Geometric = Clss C S = C i C C = h C s C h Schoenflies Interntionl m /m Order Numer of Spce Groups Orthorhomic D = V C v D = h V h mm mmm Tetrgonl C S C h D C v D = d V d D h /m mm m /mmm Trigonl (Rhomohedrl) C S = Ci D C v D d m m 7 Hexgonl Cuic = = C C h C h D C v D h D T T h h O T d O h /m mm m /mmm m m mm
13 order 8 8 _ Fig. 5. Sugroup reltions mong the crystllogrphic point groups. Drk lines connect groups in the sme crystl system. Consequently, the order p of ny cyclic sugroup of lttice point group is restricted to the vlues p =,,,,. (8) This is known s the crystllogrphic restriction. The point groups stisfying crystllogrphic restriction re clled crystllogrphic point groups. There re exctly of them. They re listed in Tle. Crystls re ccordingly clssified into crystl clsses, ech one corresponding to one of the point groups. Besides our geometric symols for the crystl clsses (point groups) nd the symols of Schoenflies, Tle lists symols dopted in the Interntionl Tles of XRy Crystllogrphy (99), n extensive stndrd reference on the crystllogrphic groups. It is conventionl to sudivide the crystl clsses into seven crystl systems with the nmes given in Tle. This sudivision corresponds to n rrngement of the point groups into fmilies of sugroups, s indicted in Fig. 5. The lrgest group in ech system is clled the holohedry of the system. Reltions of one system to nother re descried y the sugroup reltions mong their holohedry, s shown in Fig.. From the symols, it is esy to produce set of genertors for ech of the seven diholohedry (the versor groups of the holohedry). Figure 7 hs sets of such genertors rrnged to show the simple reltions mong them. Note tht the orthogonl vectors, c cn e chosen to e the sme for ech system, nd there re three distinct choices for the remining vector. Actully, from the genertors for [ nd [] the genertors of ll other crystllogrphic point groups cn e generted, ecuse ll the groups re sugroups of [] or [], s shown in Fig. 5. We hve determined ll possile point symmetry groups for dimensionl ojects. There re, however, n infinite numer of different ojects with the sme symmetry group, for symmetry group
14 order 8 tetrgonl cuic hexgonl _ trigonl 8 orthorhomic _ monoclinic triclinic Fig.. Sugroup reltions for the seven holohedry. descries reltion mong identicl prts of n oject without sying nything out the nture of those prts. 5. Homogeneous Eucliden Geometry As n ren for Eucliden geometry we employ the metric vector spce R, with Minkowski signture (,) nd its geometric lger R, = G(R, ). The Minkowski signture implies the existence of cone of null vectors similr to the light cone in spcetime. A vector x issidtoenull vector if x = x x =0.The set of ll null vectors in R, is clled null cone. Remrkly, the d Eucliden spce E cn e identified with the set of ll null vectors in R, stisfying the constrint x e =, (9) where e is distinguished null vector clled the point t infinity. This constrint is the eqution for hyperplne with norml e. Thus, we identify E with the intersection of hyperplne nd the null cone in R,, s expressed y E = {x x =0, x e =}, (0) where ech x designtes point in E. This is clled the homogeneous model of E, ecuse ll points re treted eqully. In contrst, the usul representtion of Eucliden points in R = R,0 is n inhomogeneous model of E, ecuse it singles out one point, the origin, s specil. The gret dvntge of the homogeneous model is the simplicity nd fluidity tht geometric lger gives to the reltions, constructions nd inferences of Eucliden geometry. For use in crystl geometry we record some of the sic definitions nd results without elortion. More detils re given in (Hestenes 00, 00 nd 99), including proofs of some results tht re just stted here. The primry fct is tht the squred Eucliden distnce etween ny two points x nd y is given y (x y) = x y () Thus, Eucliden distnces cn e computed directly from inner products etween points. The oriented line (or line segment) determined y two distinct points p nd q is represented y the trivector P = p q e, () known s line vector or sliding vector in clssicl prlnce. All geometric properties of the line (segment), including its reltion to other lines, points nd plnes, cn e computed from trivector P y lgeric mens. The tngent vector n for the line is n (p q) e = p q. ()
15 System Cuic Hexgonl Diholohedry [] [] Genertors c π / π/ c π / Tetrgonl [] c π/ Trigonl [] c c π / Orthorhomic [] c Monoclinic [] c c Triclinic [] c c = i Fig. 7. Genertors for the seven diholohedry. One of the genertors of [] nd [] is ivector c, nd the genertor of [] is the unit trivector c = i. All other genertors re vectors. nd the length of the line segment is given y P = n =(p q) = p q. () Apointxlies on the line P if nd only if x P = x p q e =0. (5) This is nonprmetric eqution for the line. To relte our homogenous method to the vst literture on geometry nd mechnics, we need to relte our homogeneous model for E to the stndrd vector spce model. Hppily, this cn e done in strightforwrd wy with n elegnt device clled the conforml split. The essentil ide is to prmetrize ll the points in Eucliden spce y the fmily (or pencil) of lines through single point. The pencil of lines through fixed point e 0 cn e chrcterized y the vrile line vector x = x e 0 e = x E. () This cn e inverted to express x s function of x: x = xe x e + e 0, (7) 5
16 where E e 0 e = E =. (8) Thus, we hve onetoone correspondence etween Eucliden points nd line segments ttched to given point. The line vectors specified y () form dimensionl vector spce R = {x}, (9) which cn e identified with the stndrd vector spce model of E, wherein the distinguished point e 0 is represented y the zero vector. The mpping of Eucliden points onto vectors in R defined y () nd (7) is clled conforml split. The conforml split of Eucliden points genertes split of the entire geometric lger into commuttive product of sulgers: R, = R R,, (50) where R = G(R ) s efore, nd R, is the Minkowski geometric lger generted y the vectors e 0 nd e. The chief use of the conforml split is to relte homogeneous geometry to stndrd vector spce geometry. In prticulr, it enles smooth connection etween the inhomogeneous tretment of point groups in the first prt of this pper nd the homogeneous tretment of the crystllogrphic groups in the second. Two points determine plne s well s line. For distinct points p nd q with n p q, the eqution for the oriented plne isecting the line etween them is n x =0. (5) The plne is the set of ll points x tht re equidistnt from the two points, s expressed y p x = q x The direction (sign) of n ssigns n orienttion to the plne. From (9) it follows tht every norml hs the property n e =0. (5) We dopt this s the defining property of norml (vector), ecuse every vector tht hs it determines unique plne defined y eqution (5). Every norml determines the loction s well s the orienttion of the plne. It is not essentil to specify the norml s difference etween two points, though it is often useful. The reltion of one plne to nother is completely determined y the lgeric properties of their normls without reference to ny points. To formlize tht fct, it is convenient to define the meet n m for plnes with normls m nd n y n m n (mi) =(n m)i, (5) where I is the unit pseudosclr for R,. The meet determines line vector representing the intersection of the two plnes. Indeed, the right side of (5) expresses the meet s the dul of ivector, so it is trivector, s required for line. The condition for point x to lie on this line is x (n m) =[x (n m)]i =[(x n)m (x m)n]i=0. (5) This condition is met if nd only if x n = x m = 0. In other words, x must lie in oth plnes. There re three distinct wys tht the plnes might intersect, depending on the vlue of n m. If n m= 0 the plnes coincides. Otherwise, (n m) = (n m) = n m (n m) 0. (55) If this quntity is positive, the plnes intersect in finite rel line. If it vnishes, the plnes re prllel, nd we my sy tht the lines intersect in line t infinity. The concept of line t infinity is introduce so we cn stte without exception tht every pir of plnes intersect in unique line. The null cse in (55) tells us tht the line vector for line t infinity must e the dul of null ivector.
17 . Symmetries from Reflections It hs een known for more thn century tht every symmetry in E cn e generted from reflections in plnes (Coxeter 97). In prticulr, ny rottion out given line cn e reduced to product of reflections in two plnes tht intersect in tht line, nd ny trnsltion cn e reduced to product of reflections in two prllel plnes. At long lst, geometric lger mkes it possile to cst this powerful geometric insight into simple lgeric form the fcilittes the composition of symmetries. By definition, symmetry S in E is trnsformtion tht leves invrint the Eucliden distnce etween points, expressed y (x y) = x y in our homogeneous model. Invrince of the inner product x y is the defining property of orthogonl trnsformtions on the vector spce R,. It is generl theorem of geometric lger (Hestenes 99, Hestenes nd Soczyk 98) tht every such trnsformtion S tking generic point x 0 to the point x cn e expressed in the cnonicl form x = S x = S xs (5) where S is n invertile multivector in R, nd, s efore, S = ±S ccording to the prity of S. To preserve our definition of homogeneous Eucliden spce, the point t infinity must e n invrint of the symmetry, s expressed y S es = e or Se = es = ±es. (57) Every such versor cn e expressed s product of vectors: S = n n...n k, (58) where n k e =0 or n k e= en k. (59) Moreover, for given S, the n k cn e chosen so tht k 5. The gret power of this theorem is tht it reduces the composition of symmetries, s expressed y the opertor eqution S S = S, (0) to geometric product of their corresponding symmetry versors: S S = S. () Thus, the Eucliden group E() is reduced to multiplictive group of versors. Compring (59) with (5), we see tht every symmetry vector n is norml for some plne in E. It follows tht the symmetry nx = n xn () is reflection in the nplne. Indeed, if x is ny invrint point of the symmetry, then x = n xn, so nx + xn =x n=0, () which is the norml eqution for the nplne. The composite symmetry S = mn of reflections in two distinct plnes is completely chrcterized y the geometric product mn of their normls. The symmetry S cn e generted from mny different reflections, so it is desirle to express its versor in cnonicl form independent of the choice of m nd n. For simplicity we impose the normliztion n = m =, though we will hve good reson to drop tht condition lter on. In this cse, we hve the identity: m n =(m n) (m n). () Also, the versor constrint (57) tkes the form (m n) e =0. (5) There re two different cses to consider, s specified y the conditions (55). 7
18 When the plnes re prllel, we hve (m n) = 0, so the constrint (5) llows us to define vector y writing e = e =m n. () Therefore, the versor mn =+m nis equivlent to versor T defined y A little lger shows tht this versor genertes trnsltion T + e. (7) T x = T xt =x+. (8) Squring this eqution, we see tht = x (9) hs the sme vlue for every point x. It cn e shown tht the right side of (8) hs conforml split of the form x + =(x+)e (x+) e+e 0, (70) in greement with the usul representtion for trnsltion in eqution (). The conforml split of the trnsltion vector in terms of the plne normls is otined directly from (): = E = e 0 e =m n e 0 =[(m n)e 0 ]E, (7) where (5) ws used to get the form on the right. The mgnitude of the trnsltion is therefore which holds for ny point e 0 chosen s origin. Also, using to show tht = = (m n) e 0, (7) ee = e = Ee (7) e = e = e = e, (7) we cn put (7) in the form T =+ e = e T, (75) where, despite the conforml split of vector, the right side is independent of the choice of origin. When the two plnes intersect, we hve (m n) < 0, nd the line vector for the intersection is L =(m n)i. If the origin e 0 is chosen to lie on the line, we hve e 0 L = e 0 (m n)i =0, (7) so the normls hve the conforml splits m = me, n = ne, nd the symmetry versor mn = mn (77) hs exctly the rotor form tht we studied in Sections nd. Since the choice of e 0 is ritrry, we cn conclude tht the symmetry m n is rottion out the line L =(m n)ithrough hlf the ngle etween m nd n. One other symmetry of specil interest is spce inversion t point p, definedy I p x=i p xi p, where I p =(p e)i=i p. (78) This is equivlent to reflection in three mutully orthogonl plnes t p. Representing inversion y the trivector versor I p voids choosing reflecting plnes (s done for the triclinic cse in Fig. 7). 8
19 7. The Spce Groups We hve seen tht there re point groups tht leve some lttice invrint. The complete symmetry group of crystl is clled its spce group. Ech element of spce group cn e written s n orthogonl trnsformtion comined with trnsltion, s represented y (). Consequently, every spce group cn e descried s point group comined with trnsltion group, nd we cn determine ll possile spce groups y finding ll possile comintions. An enumertion of the spce groups is of gret interest ecuse it chrcterizes the structure of ny regulr crystl tht might e found in nture. Our purpose now is to see how tht cn e done. The trnsltion group of crystl is n dditive group generted y three vectors,,, while the doule point group is multiplictive group generted y t most three vectors,, c. Consequently, the spce group cn e chrcterized y set of reltions mong these two sets of genertors. Indeed, we cn choose three linerly independent vectors from the two sets nd write the others in terms of them. Thus, every element of spce group cn e expressed in terms of three vectors which generte trnsltions y ddition nd orthogonl trnsformtions y multipliction. For the three symmetry vectors generting spce group, we choose the set,, c used to generte point groups in Section, ut we djust their lengths nd directions to generte the shortest trnsltions in the lttice comptile with their function s spce group genertors. Next we use the results of Sections 5 nd to express the versor genertors of spce group in terms of its symmetry vectors. Since ll lttice points re equivlent, it is convenient to select one of them, sy e 0, s the origin for conforml split nd relte the irreducile genertors to tht point. The symmetry vectors,, c of Section re trnslted into norml vectors,, c for plnes through e 0 y the conforml split = E, = E, c = ce. (79) The condition tht e 0 is t the intersection of the three plnes is e 0 = e 0 = c e 0 =0. (80) In the homogeneous model, reflections re generted y the normls,, c rther thn,, c, nd trnsltions re generted y versors T = T, T = T nd T c = T c, s defined in (75). This mkes it possile to compose reflections nd trnsltions y versor multipliction. Note tht =, (8) so nd generte the sme rottions s nd. The mgnitude of is djusted so tht T moves ech lttice point to the next one in the direction of. Likewise for nd c. The inverse trnsltion T = T moves the points ck. For integer n, n nfold ppliction of T is equivlent to single trnsltion y n lttice points, s expressed y the eqution T n = T n. (8) Actully, this formul holds for ny sclr vlue of n, lthough it connects lttice points only when n is n integer. We need frctionl vlues for some spce groups. In consonnce with eqution (0), ny trnsltion in the spce group cn e derived from the irreducile trnsltions y T (n+n +n c) = T n T n Tc n, (8) where the n k re ny integers. Now we re prepred for detiled nlysis of the spce groups. We cn determine ll the spce groups y tking ech of the point groups in turn nd considering the vrious wys it cn e comined with trnsltions to produce spce group. Thus, the spce groups fll into clsses determined y the point groups. The numer of spce groups in ech clss is given in Tle. There re 0 in ll. This is too mny to consider here, so let us turn to the simpler prolem of determining the spce groups in two dimensions. 9
20 7. Plnr Spce Groups In two dimensions there re 7 spce groups. Genertors for ech group re given in Tle 5 long with Geometric Symol designed to descrie the set of genertors in wy to e explined. For reference purposes, the tle gives the short symols for spce groups dopted in the Interntionl Tles for Xry Crystllogrphy. Finlly, the tle shows tht the spce groups fll into crystl systems distinguished y their symmetry vectors, in reltion to lttice. To see how every dimensionl spce group cn e descried in terms of two symmetry vectors, let us exmine representtive smple of the groups nd genertors in Tle 5. The reder is dvised to refer continully to the tle while the groups re discussed. In the geometric symol for ech group, the clss is indicted y the clss (point group) symol devised erlier. The spce group symol includes the slsh symol / to distinguish it from the point group symol nd to indicte tht trnsltions must e included mong the genertors. The numer of trnsltions tht must e included is not specified, s tht is esily inferred from the point group. The symol fter the slsh indictes some fusion mong reflections nd primry trnsltions, s explined elow. In the group /, the vectors nd generte trnsltions only. Since the point group contins only the identity element, it does not imply ny reltion etween the directions of the trnsltion vectors, so the lttice they generte is sid to e Olique. As shown in Tle 5, the group / hs four genertors: two irreducile trnsltions nd their inverses. In d the point group is generted y the ivector, which produces rottion y π in the plne. Note tht the group / hs only three genertors insted of the four in /, ecuse the inverse of ny trnsltion is generted ccording to T = T =( ) T ( ). (8) The symol indictes tht the groups / nd / contin single reflection versor, sy. Since reflection y is required to leve the lttice invrint, it must trnsform trnsltion genertors into trnsltion genertors. By considering the lterntives, one cn see tht this cn e done in the following wys. In the group /, the reflection is long the direction of one of the trnsltions, so the trnsltion cn e reversed y T =. T = T, (85). where the symol = indictes equlity up to n irrelevnt scle fctor ( in this cse). If the other trnsltion vector is orthogonl to, then T. = T. (8) Since nd determine rectngle, the lttice they generte is sid to e Rectngulr. Another reltion of reflection to trnsltions rises in the Rhomic cse. The / in the groups / nd / nd the / in / indictes frctionl comintion of primry trnsltions T,T. The primry trnsltions cn e derived therefrom: for exmple, (T / + ) T / + =. T. (87) Note in Tle 5, tht the choice of symmetry vectors, is different for the group / thn for the groups / nd /, though they pertin to the sme lttice. The in the group symol / specifies reflectiontrnsltion comintion, such s T / = T / T /, (88) which represents reflection in line (or plne in d) displced from the origin y /. The numer specifies the reltion (T / ). =, (89) The g in the group symol /g designtes glidereflection with versor genertor of the form where must e orthogonl to. Note tht G = T / = T /, (90) G = T /. = T, (9) 0
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