Point Groups and Space Groups in Geometric Algebra


 Emery Richard
 2 years ago
 Views:
Transcription
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
EQUATIONS OF LINES AND PLANES
EQUATIONS OF LINES AND PLANES MATH 195, SECTION 59 (VIPUL NAIK) Corresponding mteril in the ook: Section 12.5. Wht students should definitely get: Prmetric eqution of line given in pointdirection nd twopoint
More informationReasoning to Solve Equations and Inequalities
Lesson4 Resoning to Solve Equtions nd Inequlities In erlier work in this unit, you modeled situtions with severl vriles nd equtions. For exmple, suppose you were given usiness plns for concert showing
More informationA.7.1 Trigonometric interpretation of dot product... 324. A.7.2 Geometric interpretation of dot product... 324
A P P E N D I X A Vectors CONTENTS A.1 Scling vector................................................ 321 A.2 Unit or Direction vectors...................................... 321 A.3 Vector ddition.................................................
More informationLINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES
LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES DAVID WEBB CONTENTS Liner trnsformtions 2 The representing mtrix of liner trnsformtion 3 3 An ppliction: reflections in the plne 6 4 The lgebr of
More informationPHY 140A: Solid State Physics. Solution to Homework #2
PHY 140A: Solid Stte Physics Solution to Homework # TA: Xun Ji 1 October 14, 006 1 Emil: jixun@physics.ucl.edu Problem #1 Prove tht the reciprocl lttice for the reciprocl lttice is the originl lttice.
More informationAppendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered:
Appendi D: Completing the Squre nd the Qudrtic Formul Fctoring qudrtic epressions such s: + 6 + 8 ws one of the topics introduced in Appendi C. Fctoring qudrtic epressions is useful skill tht cn help you
More informationVectors 2. 1. Recap of vectors
Vectors 2. Recp of vectors Vectors re directed line segments  they cn be represented in component form or by direction nd mgnitude. We cn use trigonometry nd Pythgors theorem to switch between the forms
More informationSymmetry in crystals National Workshop on Crystal Structure Determination Using Powder XRD
Symmetry in crystls Ntionl Workshop on Crystl Structure Determintion Using Powder XRD Muhmmd Sbieh Anwr School of Science nd Engineering Lhore University of Mngement & Sciences (LUMS) Pkistn. (Dted: August
More informationMath 314, Homework Assignment 1. 1. Prove that two nonvertical lines are perpendicular if and only if the product of their slopes is 1.
Mth 4, Homework Assignment. Prove tht two nonverticl lines re perpendiculr if nd only if the product of their slopes is. Proof. Let l nd l e nonverticl lines in R of slopes m nd m, respectively. Suppose
More information. At first sight a! b seems an unwieldy formula but use of the following mnemonic will possibly help. a 1 a 2 a 3 a 1 a 2
7 CHAPTER THREE. Cross Product Given two vectors = (,, nd = (,, in R, the cross product of nd written! is defined to e: " = (!,!,! Note! clled cross is VECTOR (unlike which is sclr. Exmple (,, " (4,5,6
More informationSection 54 Trigonometric Functions
5 Trigonometric Functions Section 5 Trigonometric Functions Definition of the Trigonometric Functions Clcultor Evlution of Trigonometric Functions Definition of the Trigonometric Functions Alternte Form
More informationRegular Sets and Expressions
Regulr Sets nd Expressions Finite utomt re importnt in science, mthemtics, nd engineering. Engineers like them ecuse they re super models for circuits (And, since the dvent of VLSI systems sometimes finite
More information9.3. The Scalar Product. Introduction. Prerequisites. Learning Outcomes
The Sclr Product 9.3 Introduction There re two kinds of multipliction involving vectors. The first is known s the sclr product or dot product. This is soclled becuse when the sclr product of two vectors
More informationSection 1: Crystal Structure
Phsics 927 Section 1: Crstl Structure A solid is sid to be crstl if toms re rrnged in such w tht their positions re ectl periodic. This concept is illustrted in Fig.1 using twodimensionl (2D) structure.
More informationFactoring Trinomials of the Form. x 2 b x c. Example 1 Factoring Trinomials. The product of 4 and 2 is 8. The sum of 3 and 2 is 5.
Section P.6 Fctoring Trinomils 6 P.6 Fctoring Trinomils Wht you should lern: Fctor trinomils of the form 2 c Fctor trinomils of the form 2 c Fctor trinomils y grouping Fctor perfect squre trinomils Select
More informationMatrix Algebra CHAPTER 1 PREAMBLE 1.1 MATRIX ALGEBRA
CHAPTER 1 Mtrix Algebr PREAMBLE Tody, the importnce of mtrix lgebr is of utmost importnce in the field of physics nd engineering in more thn one wy, wheres before 1925, the mtrices were rrely used by the
More informationQuadratic Equations  1
Alger Module A60 Qudrtic Equtions  1 Copyright This puliction The Northern Alert Institute of Technology 00. All Rights Reserved. LAST REVISED Novemer, 008 Qudrtic Equtions  1 Sttement of Prerequisite
More informationVector differentiation. Chapters 6, 7
Chpter 2 Vectors Courtesy NASA/JPLCltech Summry (see exmples in Hw 1, 2, 3) Circ 1900 A.D., J. Willird Gis invented useful comintion of mgnitude nd direction clled vectors nd their higherdimensionl counterprts
More information4.11 Inner Product Spaces
314 CHAPTER 4 Vector Spces 9. A mtrix of the form 0 0 b c 0 d 0 0 e 0 f g 0 h 0 cnnot be invertible. 10. A mtrix of the form bc d e f ghi such tht e bd = 0 cnnot be invertible. 4.11 Inner Product Spces
More informationContent Objectives: After completing the activity, students will gain experience of informally proving Pythagoras Theorem
Pythgors Theorem S Topic 1 Level: Key Stge 3 Dimension: Mesures, Shpe nd Spce Module: Lerning Geometry through Deductive Approch Unit: Pythgors Theorem Student ility: Averge Content Ojectives: After completing
More informationA Note on Complement of Trapezoidal Fuzzy Numbers Using the αcut Method
Interntionl Journl of Applictions of Fuzzy Sets nd Artificil Intelligence ISSN  Vol.  A Note on Complement of Trpezoidl Fuzzy Numers Using the αcut Method D. Stephen Dingr K. Jivgn PG nd Reserch Deprtment
More informationSPECIAL PRODUCTS AND FACTORIZATION
MODULE  Specil Products nd Fctoriztion 4 SPECIAL PRODUCTS AND FACTORIZATION In n erlier lesson you hve lernt multipliction of lgebric epressions, prticulrly polynomils. In the study of lgebr, we come
More informationNull Similar Curves with Variable Transformations in Minkowski 3space
Null Similr Curves with Vrile Trnsformtions in Minkowski spce Mehmet Önder Cell Byr University, Fculty of Science nd Arts, Deprtment of Mthemtics, Murdiye Cmpus, 45047 Murdiye, Mnis, Turkey. mil: mehmet.onder@yr.edu.tr
More informationHomework 3 Solutions
CS 341: Foundtions of Computer Science II Prof. Mrvin Nkym Homework 3 Solutions 1. Give NFAs with the specified numer of sttes recognizing ech of the following lnguges. In ll cses, the lphet is Σ = {,1}.
More informationAssuming all values are initially zero, what are the values of A and B after executing this Verilog code inside an always block? C=1; A <= C; B = C;
B26 Appendix B The Bsics of Logic Design Check Yourself ALU n [Arthritic Logic Unit or (rre) Arithmetic Logic Unit] A rndomnumer genertor supplied s stndrd with ll computer systems Stn KellyBootle,
More informationPentominoes. Pentominoes. Bruce Baguley Cascade Math Systems, LLC. The pentominoes are a simplelooking set of objects through which some powerful
Pentominoes Bruce Bguley Cscde Mth Systems, LLC Astrct. Pentominoes nd their reltives the polyominoes, polycues, nd polyhypercues will e used to explore nd pply vrious importnt mthemticl concepts. In this
More informationGeneralized Inverses: How to Invert a NonInvertible Matrix
Generlized Inverses: How to Invert NonInvertible Mtrix S. Swyer September 7, 2006 rev August 6, 2008. Introduction nd Definition. Let A be generl m n mtrix. Then nturl question is when we cn solve Ax
More informationClassification of Solid Structures
Clssifiction of Solid Structures Represents n tom or molecule Amorphous: Atoms (molecules) bond to form very shortrnge (few toms) periodic structure. Crystls: Atoms (molecules) bond to form longrnge
More informationVectors and dyadics. Chapter 2. Summary. 2.1 Examples of scalars, vectors, and dyadics
Chpter 2 Vectors nd dydics Summry Circ 1900 A.D., J. Willird Gis proposed the ide of vectors nd their higherdimensionl counterprts dydics, tridics, ndpolydics. Vectors descrie threedimensionl spce nd
More informationCONIC SECTIONS. Chapter 11
CONIC SECTIONS Chpter 11 11.1 Overview 11.1.1 Sections of cone Let l e fied verticl line nd m e nother line intersecting it t fied point V nd inclined to it t n ngle α (Fig. 11.1). Fig. 11.1 Suppose we
More informationFactoring Polynomials
Fctoring Polynomils Some definitions (not necessrily ll for secondry school mthemtics): A polynomil is the sum of one or more terms, in which ech term consists of product of constnt nd one or more vribles
More informationPure C4. Revision Notes
Pure C4 Revision Notes Mrch 0 Contents Core 4 Alger Prtil frctions Coordinte Geometry 5 Prmetric equtions 5 Conversion from prmetric to Crtesin form 6 Are under curve given prmetriclly 7 Sequences nd
More informationMathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100
hsn.uk.net Higher Mthemtics UNIT 3 OUTCOME 1 Vectors Contents Vectors 18 1 Vectors nd Sclrs 18 Components 18 3 Mgnitude 130 4 Equl Vectors 131 5 Addition nd Subtrction of Vectors 13 6 Multipliction by
More informationVectors and dyadics. Chapter 2. Summary. 2.1 Examples of scalars, vectors, and dyadics
Chpter 2 Vectors nd dydics Summry Circ 1900 A.D., J. Willird Gis proposed the ide of vectors nd their higherdimensionl counterprts dydics, tridics, ndpolydics. Vectors descrie threedimensionl spce nd
More informationExample 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers.
2 Rtionl Numbers Integers such s 5 were importnt when solving the eqution x+5 = 0. In similr wy, frctions re importnt for solving equtions like 2x = 1. Wht bout equtions like 2x + 1 = 0? Equtions of this
More informationSquare Roots Teacher Notes
Henri Picciotto Squre Roots Techer Notes This unit is intended to help students develop n understnding of squre roots from visul / geometric point of view, nd lso to develop their numer sense round this
More informationA new algorithm for generating Pythagorean triples
A new lgorithm for generting Pythgoren triples RH Dye 1 nd RWD Nicklls 2 The Mthemticl Gzette (1998); 82 (Mrch, No. 493), p. 86 91 (JSTOR rchive) http://www.nicklls.org/dick/ppers/mths/pythgtriples1998.pdf
More informationVectors. The magnitude of a vector is its length, which can be determined by Pythagoras Theorem. The magnitude of a is written as a.
Vectors mesurement which onl descries the mgnitude (i.e. size) of the oject is clled sclr quntit, e.g. Glsgow is 11 miles from irdrie. vector is quntit with mgnitude nd direction, e.g. Glsgow is 11 miles
More informationSolving Linear Equations  Formulas
1. Solving Liner Equtions  Formuls Ojective: Solve liner formuls for given vrile. Solving formuls is much like solving generl liner equtions. The only difference is we will hve severl vriles in the prolem
More informationChapter 9: Quadratic Equations
Chpter 9: Qudrtic Equtions QUADRATIC EQUATIONS DEFINITION + + c = 0,, c re constnts (generlly integers) ROOTS Synonyms: Solutions or Zeros Cn hve 0, 1, or rel roots Consider the grph of qudrtic equtions.
More informationSolutions to Section 1
Solutions to Section Exercise. Show tht nd. This follows from the fct tht mx{, } nd mx{, } Exercise. Show tht = { if 0 if < 0 Tht is, the bsolute vlue function is piecewise defined function. Grph this
More informationMATH 150 HOMEWORK 4 SOLUTIONS
MATH 150 HOMEWORK 4 SOLUTIONS Section 1.8 Show tht the product of two of the numbers 65 1000 8 2001 + 3 177, 79 1212 9 2399 + 2 2001, nd 24 4493 5 8192 + 7 1777 is nonnegtive. Is your proof constructive
More information10.5 Graphing Quadratic Functions
0.5 Grphing Qudrtic Functions Now tht we cn solve qudrtic equtions, we wnt to lern how to grph the function ssocited with the qudrtic eqution. We cll this the qudrtic function. Grphs of Qudrtic Functions
More informationSirindhorn International Institute of Technology Thammasat University at Rangsit
Sirindhorn Interntionl Institute of Technology Thmmst University t Rngsit School of Informtion, Computer nd Communiction Technology COURSE : ECS 204 Bsic Electricl Engineering L INSTRUCTOR : Asst. Prof.
More informationHomework Assignment 1 Solutions
Dept. of Mth. Sci., WPI MA 1034 Anlysis 4 Bogdn Doytchinov, Term D01 Homework Assignment 1 Solutions 1. Find n eqution of sphere tht hs center t the point (5, 3, 6) nd touches the yzplne. Solution. The
More informationSection 74 Translation of Axes
62 7 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY Section 74 Trnsltion of Aes Trnsltion of Aes Stndrd Equtions of Trnslted Conics Grphing Equtions of the Form A 2 C 2 D E F 0 Finding Equtions of Conics In the
More informationBasic Math Review. Numbers. Important Properties. Absolute Value PROPERTIES OF ADDITION NATURAL NUMBERS {1, 2, 3, 4, 5, }
ƒ Bsic Mth Review Numers NATURAL NUMBERS {1,, 3, 4, 5, } WHOLE NUMBERS {0, 1,, 3, 4, } INTEGERS {, 3,, 1, 0, 1,, } The Numer Line 5 4 3 1 0 1 3 4 5 Negtive integers Positive integers RATIONAL NUMBERS All
More informationEquations between labeled directed graphs
Equtions etween leled directed grphs Existence of solutions GrretFontelles A., Misnikov A., Ventur E. My 2013 Motivtionl prolem H 1 nd H 2 two sugroups of the free group generted y X A, F (X, A). H 1
More informationCS99S Laboratory 2 Preparation Copyright W. J. Dally 2001 October 1, 2001
CS99S Lortory 2 Preprtion Copyright W. J. Dlly 2 Octoer, 2 Ojectives:. Understnd the principle of sttic CMOS gte circuits 2. Build simple logic gtes from MOS trnsistors 3. Evlute these gtes to oserve logic
More informationSCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics. Basic Algebra
SCHOOL OF ENGINEERING & BUILT ENVIRONMENT Mthemtics Bsic Alger. Opertions nd Epressions. Common Mistkes. Division of Algeric Epressions. Eponentil Functions nd Logrithms. Opertions nd their Inverses. Mnipulting
More informationIn the following there are presented four different kinds of simulation games for a given Büchi automaton A = :
Simultion Gmes Motivtion There re t lest two distinct purposes for which it is useful to compute simultion reltionships etween the sttes of utomt. Firstly, with the use of simultion reltions it is possile
More information4.0 5Minute Review: Rational Functions
mth 130 dy 4: working with limits 1 40 5Minute Review: Rtionl Functions DEFINITION A rtionl function 1 is function of the form y = r(x) = p(x) q(x), 1 Here the term rtionl mens rtio s in the rtio of two
More information5.2. LINE INTEGRALS 265. Let us quickly review the kind of integrals we have studied so far before we introduce a new one.
5.2. LINE INTEGRALS 265 5.2 Line Integrls 5.2.1 Introduction Let us quickly review the kind of integrls we hve studied so fr before we introduce new one. 1. Definite integrl. Given continuous relvlued
More informationPhysics 6010, Fall 2010 Symmetries and Conservation Laws: Energy, Momentum and Angular Momentum Relevant Sections in Text: 2.6, 2.
Physics 6010, Fll 2010 Symmetries nd Conservtion Lws: Energy, Momentum nd Angulr Momentum Relevnt Sections in Text: 2.6, 2.7 Symmetries nd Conservtion Lws By conservtion lw we men quntity constructed from
More informationP.3 Polynomials and Factoring. P.3 an 1. Polynomial STUDY TIP. Example 1 Writing Polynomials in Standard Form. What you should learn
33337_0P03.qp 2/27/06 24 9:3 AM Chpter P Pge 24 Prerequisites P.3 Polynomils nd Fctoring Wht you should lern Polynomils An lgeric epression is collection of vriles nd rel numers. The most common type of
More informationMODULE 3. 0, y = 0 for all y
Topics: Inner products MOULE 3 The inner product of two vectors: The inner product of two vectors x, y V, denoted by x, y is (in generl) complex vlued function which hs the following four properties: i)
More informationConic Sections. The coefficient of x 2 is positive, so the parabola opens. y up. The vertex is the lowest point on the parabola, or
SECTION. Conic Sections SECTION. Conic Sections OBJECTIVE A To grph prol The conic sections re curves tht cn e constructed from the intersection of plne nd right circulr cone. The prol, which ws introduced
More informationGeometry and Measure. 12am 1am 2am 3am 4am 5am 6am 7am 8am 9am 10am 11am 12pm
Reding Scles There re two things to do when reding scle. 1. Mke sure you know wht ech division on the scle represents. 2. Mke sure you red in the right direction. Mesure Length metres (m), kilometres (km),
More informationSection 2.3. Motion Along a Curve. The Calculus of Functions of Several Variables
The Clculus of Functions of Severl Vribles Section 2.3 Motion Along Curve Velocity ccelertion Consider prticle moving in spce so tht its position t time t is given by x(t. We think of x(t s moving long
More informationUse Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions.
Lerning Objectives Loci nd Conics Lesson 3: The Ellipse Level: Preclculus Time required: 120 minutes In this lesson, students will generlize their knowledge of the circle to the ellipse. The prmetric nd
More informationPolynomial Functions. Polynomial functions in one variable can be written in expanded form as ( )
Polynomil Functions Polynomil functions in one vrible cn be written in expnded form s n n 1 n 2 2 f x = x + x + x + + x + x+ n n 1 n 2 2 1 0 Exmples of polynomils in expnded form re nd 3 8 7 4 = 5 4 +
More informationLecture 2: Matrix Algebra. General
Lecture 2: Mtrix Algebr Generl Definitions Algebric Opertions Vector Spces, Liner Independence nd Rnk of Mtrix Inverse Mtrix Liner Eqution Systems, the Inverse Mtrix nd Crmer s Rule Chrcteristic Roots
More informationaddition, there are double entries for the symbols used to signify different parameters. These parameters are explained in this appendix.
APPENDIX A: The ellipse August 15, 1997 Becuse of its importnce in both pproximting the erth s shpe nd describing stellite orbits, n informl discussion of the ellipse is presented in this ppendix. The
More informationRadius of the Earth  Radii Used in Geodesy James R. Clynch February 2006
dius of the Erth  dii Used in Geodesy Jmes. Clynch Februry 006 I. Erth dii Uses There is only one rdius of sphere. The erth is pproximtely sphere nd therefore, for some cses, this pproximtion is dequte.
More informationOperations with Polynomials
38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: Write polynomils in stndrd form nd identify the leding coefficients nd degrees of polynomils Add nd subtrct polynomils Multiply
More information1.2 The Integers and Rational Numbers
.2. THE INTEGERS AND RATIONAL NUMBERS.2 The Integers n Rtionl Numers The elements of the set of integers: consist of three types of numers: Z {..., 5, 4, 3, 2,, 0,, 2, 3, 4, 5,...} I. The (positive) nturl
More informationBayesian Updating with Continuous Priors Class 13, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom
Byesin Updting with Continuous Priors Clss 3, 8.05, Spring 04 Jeremy Orloff nd Jonthn Bloom Lerning Gols. Understnd prmeterized fmily of distriutions s representing continuous rnge of hypotheses for the
More informationAnswer, Key Homework 8 David McIntyre 1
Answer, Key Homework 8 Dvid McIntyre 1 This printout should hve 17 questions, check tht it is complete. Multiplechoice questions my continue on the net column or pge: find ll choices before mking your
More informationand thus, they are similar. If k = 3 then the Jordan form of both matrices is
Homework ssignment 11 Section 7. pp. 24925 Exercise 1. Let N 1 nd N 2 be nilpotent mtrices over the field F. Prove tht N 1 nd N 2 re similr if nd only if they hve the sme miniml polynomil. Solution: If
More informationAll pay auctions with certain and uncertain prizes a comment
CENTER FOR RESEARC IN ECONOMICS AND MANAGEMENT CREAM Publiction No. 12015 All py uctions with certin nd uncertin prizes comment Christin Riis All py uctions with certin nd uncertin prizes comment Christin
More informationScalar and Vector Quantities. A scalar is a quantity having only magnitude (and possibly phase). LECTURE 2a: VECTOR ANALYSIS Vector Algebra
Sclr nd Vector Quntities : VECTO NLYSIS Vector lgebr sclr is quntit hving onl mgnitude (nd possibl phse). Emples: voltge, current, chrge, energ, temperture vector is quntit hving direction in ddition to
More informationQuadrilaterals Here are some examples using quadrilaterals
Qudrilterls Here re some exmples using qudrilterls Exmple 30: igonls of rhomus rhomus hs sides length nd one digonl length, wht is the length of the other digonl? 4  Exmple 31: igonls of prllelogrm Given
More informationIntegration by Substitution
Integrtion by Substitution Dr. Philippe B. Lvl Kennesw Stte University August, 8 Abstrct This hndout contins mteril on very importnt integrtion method clled integrtion by substitution. Substitution is
More informationGENERALIZED QUATERNIONS SERRETFRENET AND BISHOP FRAMES SERRETFRENET VE BISHOP ÇATILARI
Sy 9, Arlk 0 GENERALIZED QUATERNIONS SERRETFRENET AND BISHOP FRAMES Erhn ATA*, Ysemin KEMER, Ali ATASOY Dumlupnr Uniersity, Fculty of Science nd Arts, Deprtment of Mthemtics, KÜTAHYA, et@dpu.edu.tr ABSTRACT
More informationMath 135 Circles and Completing the Square Examples
Mth 135 Circles nd Completing the Squre Exmples A perfect squre is number such tht = b 2 for some rel number b. Some exmples of perfect squres re 4 = 2 2, 16 = 4 2, 169 = 13 2. We wish to hve method for
More informationThe Math Learning Center PO Box 12929, Salem, Oregon 97309 0929 Math Learning Center
Resource Overview Quntile Mesure: Skill or Concept: 1010Q Determine perimeter using concrete models, nonstndrd units, nd stndrd units. (QT M 146) Use models to develop formuls for finding res of tringles,
More information6.2 Volumes of Revolution: The Disk Method
mth ppliction: volumes of revolution, prt ii Volumes of Revolution: The Disk Method One of the simplest pplictions of integrtion (Theorem ) nd the ccumultion process is to determine soclled volumes of
More informationPROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY
MAT 0630 INTERNET RESOURCES, REVIEW OF CONCEPTS AND COMMON MISTAKES PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY Contents 1. ACT Compss Prctice Tests 1 2. Common Mistkes 2 3. Distributive
More informationLECTURE #05. Learning Objective. To describe the geometry in and around a unit cell in terms of directions and planes.
LECTURE #05 Chpter 3: Lttice Positions, Directions nd Plnes Lerning Objective To describe the geometr in nd round unit cell in terms of directions nd plnes. 1 Relevnt Reding for this Lecture... Pges 6483.
More informationBrillouin Zones. Physics 3P41 Chris Wiebe
Brillouin Zones Physics 3P41 Chris Wiebe Direct spce to reciprocl spce * = 2 i j πδ ij Rel (direct) spce Reciprocl spce Note: The rel spce nd reciprocl spce vectors re not necessrily in the sme direction
More informationChapter 2 Symmetry. 2.1 Point Group Symmetry Symmetry Elements and Operations
Chpter Symmetry The concept of symmetry is eqully importnt for understnding properties of individul molecules, crystls nd liquid crystls [1]. The symmetry is of specil importnce in physics of liquid crystl
More informationCalculus of variations with fractional derivatives and fractional integrals
Anis do CNMAC v.2 ISSN 1984820X Clculus of vritions with frctionl derivtives nd frctionl integrls Ricrdo Almeid, Delfim F. M. Torres Deprtment of Mthemtics, University of Aveiro 3810193 Aveiro, Portugl
More informationLines and angles. Name. Use a ruler and pencil to draw: a 2 parallel lines. c 2 perpendicular lines. b 2 intersecting lines. Complete the following:
Lines nd s 1 Use ruler nd pencil to drw: 2 prllel lines 2 intersecting lines c 2 perpendiculr lines 2 Complete the following: drw in the digonls on this shpe mrk the interior s on this shpe c mrk equl
More informationIntegration. 148 Chapter 7 Integration
48 Chpter 7 Integrtion 7 Integrtion t ech, by supposing tht during ech tenth of second the object is going t constnt speed Since the object initilly hs speed, we gin suppose it mintins this speed, but
More information1 Numerical Solution to Quadratic Equations
cs42: introduction to numericl nlysis 09/4/0 Lecture 2: Introduction Prt II nd Solving Equtions Instructor: Professor Amos Ron Scribes: Yunpeng Li, Mrk Cowlishw Numericl Solution to Qudrtic Equtions Recll
More informationr 2 F ds W = r 1 qe ds = q
Chpter 4 The Electric Potentil 4.1 The Importnt Stuff 4.1.1 Electricl Potentil Energy A chrge q moving in constnt electric field E experiences force F = qe from tht field. Also, s we know from our study
More informationMultiplication and Division  Left to Right. Addition and Subtraction  Left to Right.
Order of Opertions r of Opertions Alger P lese Prenthesis  Do ll grouped opertions first. E cuse Eponents  Second M D er Multipliction nd Division  Left to Right. A unt S hniqu Addition nd Sutrction
More informationSect 8.3 Triangles and Hexagons
13 Objective 1: Sect 8.3 Tringles nd Hexgons Understnding nd Clssifying Different Types of Polygons. A Polygon is closed twodimensionl geometric figure consisting of t lest three line segments for its
More information4: RIEMANN SUMS, RIEMANN INTEGRALS, FUNDAMENTAL THEOREM OF CALCULUS
4: RIEMA SUMS, RIEMA ITEGRALS, FUDAMETAL THEOREM OF CALCULUS STEVE HEILMA Contents 1. Review 1 2. Riemnn Sums 2 3. Riemnn Integrl 3 4. Fundmentl Theorem of Clculus 7 5. Appendix: ottion 10 1. Review Theorem
More informationAnnouncements. Image Formation: Outline. Earliest Surviving Photograph. Compare to Paintings. How Cameras Produce Images. Image Formation and Cameras
Announcements Imge Formtion nd Cmers CSE 252A Lecture 3 http://cseweb.ucsd.edu/clsses/f4/cse252ab/ Pizz Course reserves vilble Instructor office hours TBD Homework 0 is due tody by :59 PM Wit list Red:
More informationAREA OF A SURFACE OF REVOLUTION
AREA OF A SURFACE OF REVOLUTION h cut r πr h A surfce of revolution is formed when curve is rotted bout line. Such surfce is the lterl boundr of solid of revolution of the tpe discussed in Sections 7.
More informationIn this section make precise the idea of a matrix inverse and develop a method to find the inverse of a given square matrix when it exists.
Mth 52 Sec S060/S0602 Notes Mtrices IV 5 Inverse Mtrices 5 Introduction In our erlier work on mtrix multipliction, we sw the ide of the inverse of mtrix Tht is, for squre mtrix A, there my exist mtrix
More informationIntroduction to Mathematical Reasoning, Saylor 111
Frction versus rtionl number. Wht s the difference? It s not n esy question. In fct, the difference is somewht like the difference between set of words on one hnd nd sentence on the other. A symbol is
More informationWhat is crystallography? Crystallography. Reflection. Symmetry
Wht is crystllogrphy? Crystllogrphy Chpter 2 Dels with the symmetry of crystls nd crystl structures Provides descriptive method of descriing the symmetry of crystls Wrning: Perkins hs condensed this mteril
More informationto the area of the region bounded by the graph of the function y = f(x), the xaxis y = 0 and two vertical lines x = a and x = b.
5.9 Are in rectngulr coordintes If f() on the intervl [; ], then the definite integrl f()d equls to the re of the region ounded the grph of the function = f(), the is = nd two verticl lines = nd =. =
More information11. Fourier series. sin mx cos nx dx = 0 for any m, n, sin 2 mx dx = π.
. Fourier series Summry of the bsic ides The following is quick summry of the introductory tretment of Fourier series in MATH. We consider function f with period π, tht is, stisfying f(x + π) = f(x) for
More informationUnion, Intersection and Complement. Formal Foundations Computer Theory
Union, Intersection nd Complement FAs Union, Intersection nd Complement FAs Forml Foundtions Computer Theory Ferury 21, 2013 This hndout shows (y exmples) how to construct FAs for the union, intersection
More informationReview guide for the final exam in Math 233
Review guide for the finl exm in Mth 33 1 Bsic mteril. This review includes the reminder of the mteril for mth 33. The finl exm will be cumultive exm with mny of the problems coming from the mteril covered
More informationOn the Meaning of Regression Coefficients for Categorical and Continuous Variables: Model I and Model II; Effect Coding and Dummy Coding
Dt_nlysisclm On the Mening of Regression for tegoricl nd ontinuous Vribles: I nd II; Effect oding nd Dummy oding R Grdner Deprtment of Psychology This describes the simple cse where there is one ctegoricl
More informationBabylonian Method of Computing the Square Root: Justifications Based on Fuzzy Techniques and on Computational Complexity
Bbylonin Method of Computing the Squre Root: Justifictions Bsed on Fuzzy Techniques nd on Computtionl Complexity Olg Koshelev Deprtment of Mthemtics Eduction University of Texs t El Pso 500 W. University
More information