Quasi regular polygons and their duals with Coxeter symmetries D n. represented by complex numbers. Journal of Physics: Conference Series

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1 Joural of Physcs: Coferece Seres Quas regular polygos ad ther duals wth Coxeter symmetres D represeted by complex umbers To cte ths artcle: M Koca ad N O Koca J. Phys.: Cof. Ser Related cotet - 4d-polytopes descrbed by Coxeter dagrams ad quateros Mehmet Koca - h-fold Symmetrc quascrystallography from affe Coxeter groups N O Koca, M Koca ad S Al-Shdha - Quascrystallography from B lattces M Koca, N O Koca, A Al-Mukha et al. Vew the artcle ole for updates ad ehacemets. Ths cotet was dowloaded from IP address o 8/5/8 at 4:46

2 GROUP 8: Physcal ad Mathematcal Aspects of Symmetry Joural of Physcs: Coferece Seres 84 () 39 IOP Publshg do:.88/ /84//39 Quas regular polygos ad ther duals wth Coxeter symmetres D represeted by complex umbers M Koca ad N O Koca, Departmet of Physcs, College of Scece, Sulta Qaboos Uversty, P.O. Box 36, Al- Khoud, 3 Muscat, Sultaate of Oma E-mal: kocam@squ.edu.om ; azfe@squ.edu.om Abstract. Ths paper deals wth tlg of the plae by quas regular polygos ad ther duals. The problem s motvated from the fact that the graphee, fte umber of carbo molecules formg a hoeycomb lattce, may have states wth two bod legths ad equal bod agles or oe bod legth ad dfferet bod agles. We prove that the Eucldea plae ca be tled wth two tles cosstg of quas regular hexagos wth two dfferet legths (sogoal hexagos) ad regular hexagos. The dual lattce s costructed wth the sotoxal hexagos (equal edges but two dfferet teror agles) ad regular hexagos. We also gve smlar tlgs of the plae wth the quas regular polygos alog wth the regular polygos possessg the Coxeter symmetres D, =,3,4,5. The group elemets as well as the vertces of the polygos are represeted by the complex umbers.. Itroducto The graphee [], a fte sheet of carbo atoms [], tled wth regular hexagos has attracted much atteto. I ths paper we approach the problem from the mathematcal pot of vew ad deal wth tlg of the plae by quas regular polygos possessg dhedral symmetres. We use the rak- Coxeter dagrams to descrbe the symmetres of the polygos. The polygos are of two types: the sogoal polygos cosstg of two alteratg uequal edges wth equal teror agles; the sotoxal polygos cosstg of equal edges but wth alteratg uequal teror agles. The sogoal polygo wth sdes s vertex trastve uder the dhedral group D. Its dual polygo, the sotoxal polygo, s edge trastve uder the same symmetry. The tlg of the plae wth the sogoal ad the sotoxal hexagos s a terestg problem by tself. The problem ca be exteded to the tlg of the plae by the sogoal polygos wth the Coxeter symmetres D, =4 ad 5. We also gve the dual tlgs of the plae cosstg of sotoxal polygos as well as correspodg regular polygos. The tlg of the plae wth two sotoxal hexagos ad oe regular hexago at the same vertex suggests that the graphee may have a state wth equal bod legths but wth dfferet bod agles. Ths s a slghtly modfed verso of the hoeycomb model of graphee. Oe ca fd some quas regular D tles the referece [3]. I three dmesos a superspace-group approach has bee formulated for the descrpto of the composte crystals [4]. I secto we troduce rak- Coxeter dagrams [5] descrbg the root spaces as well as dual spaces by complex umbers. I secto 3 we costruct some eve-sded polygos wth alteratg two edge legths (sogoal polygos) havg equal teror agles. We also costruct ther dual polygos (sotoxal polygos) wth equal edge legths but wth alteratg two teror agles. Wth ths Publshed uder lcece by IOP Publshg Ltd

3 GROUP 8: Physcal ad Mathematcal Aspects of Symmetry Joural of Physcs: Coferece Seres 84 () 39 IOP Publshg do:.88/ /84//39 approach we observe that the case of W( A) D3symmetry the Eucldea plae ca be tled by two sogoal hexagos havg two dfferet edge legths ad a regular hexago all sharg the same vertex. The quas regular polygos wth dhedral symmetresw( B) D4 ad W( H) D5are also costructed ad the tlg of the plae wth the sogoal ad sotoxal octagos ad decagos are dscussed. Secto 4 s devoted to the cocludg remarks.. Coxeter dagrams wth complex umbers All Coxeter dagrams are represeted wth oe type of smple roots [6] cotrary to the Dyk dagrams represetg the Le algebra root systems havg log ad short roots. I D space we have a fte umber of Coxeter dagrams show fgure. It s customary to use the otatos I () for the rak- Coxeter dagrams but we wll cotue usg the otatos A A, A, B ad H for I (),whe =,3,4,5 respectvely. Fgure. Rak- Coxeter dagram I ( ). Fgure mples that the agle betwee the smple roots ad s ( ). We choose the orm of the roots to be cosstet wth the Dyk dagrams whe they cocde. The Carta matrx defed by the scalar productc (, ) of the rak- dagrams ad ts verse ( C ) (, ), (, j,) wll read j j C j j ( ) cos C, ( ) cos ( ) cos. () ( ) 4s ( ) cos The fudametal weghts are the bass vectors of the dual space defed by the relato (, j) j[6] where j s the Kroecker delta. The smple roots ad the fudametal weghts are related to each other by the relatos: ( C ) j j, j C. () j Summato over the repeated dex s mplct. Acto of the reflecto geerator o a arbtrary vector s defed by the relato r(, ) ( o summato over ). (3) They geerate the dhedral group D of order satsfyg the relatos r r ( rr ). Whe ad are represeted by complex umbers, equato (3) ca be wrtte as r

4 GROUP 8: Physcal ad Mathematcal Aspects of Symmetry Joural of Physcs: Coferece Seres 84 () 39 IOP Publshg do:.88/ /84//39 r. (4) Here the roots are complex umbers for rak- Coxeter dagrams, a subset of quateros, where the ut complex umbers / ad / geerate the cyclc group of order ; however, the geerators equato (4) geerate the dhedral group of order. Let exp( ) be k the ut complex umber. The all teger powers{, k,,..., } costtute a scaled copy of the root system of rak- Coxeter dagrams. If represets ay complex umber, the geerators act o as follows: ( ) r ; re. (5) Let us follow the Le algebrac techque to obta the orbts of the Coxeter groups. If g s the Le algebra of rak l the the hghest weght a a... a ( a, a,..., a ) (6) l l l s represeted by the l o-egatve tegers [7]. Applyg the Coxeter- Weyl group W( g) o the hghest weght oe ca geerate the orbt O( ) W ( g). I D space the orbts of the Coxeter group s ether regular polygos or eve-sded quas regular polygos. I what follows, we wll dscuss the orbts of the Coxeter groups wth,3, 4,5. The fudametal weghts terms of complex umbers ca be wrtte as ( ) cos ( ),. (7) ( ) ( ) s s A geeral vertex aas the a arbtrary complex umber whch ca be wrtte as ( ) [ a ( cos s ( ) a a )] (8a) ad the geerators act as follows,,,, r re rre rre. (8b) It s clear from equato (8b) that r r ( rr ) ( rr ). The vertces of the polygo ca be determed as k k k k ( rr ) e, ( rr ) ( r) e, k,,...,. (9) For a a a the vertces of a regular polygo of edge legth a are gve terms of complex umbers by a smple formula a ( 4 k) a (4 k) e, e, k,,...,. () ( ) ( ) cos( ) cos( ) 3

5 GROUP 8: Physcal ad Mathematcal Aspects of Symmetry Joural of Physcs: Coferece Seres 84 () 39 IOP Publshg do:.88/ /84//39 3. Costructo of the quas regular polygos ad the tlg of the plae 3.. wth D symmetry CC Two smple roots ad, represetg the Coxeter dagram A A, are orthogoal to each other where the geerators form the Kle s four-group D C C. A geeral orbt s obtaed by actg the group elemets o the vector ( a a ). For () ad (), the orbts are two s egmets of straght les perpedcular to each other. The orbt O( ) O() volves the vectors, whch form a square. For all rak- Coxeter groups whe the orbt s derved from the vector a( ) a(), where a s a arbtrary real umber, the the polygo has a addtoal symmetry. It s the symmetry of the Coxeter-Dyk dagram whch ca be defed by the geerator : leadg to a larger symmetry W( g): C where (: ) deotes the sem-drect product of two groups. I the above case the group s D4 ( CC): C C4 : C of order 8. Whe we cosder the most geeral case, amely, a a the orbt represets a rectagle of sdes a, a. The rectagle s a sogoal polygo wth the D symmetry. The dual of the rectagle s a rhombus (a sotoxal polygo) whose vertces ca be determed by takg the md pot of oe of the edge of the rectagle, say, the vector a. We take the other vector bsectg the edge of legth a of the rectagle. To determe the dual of the rectagle the le jog these vectors amust be orthogoal to the vector a awhch determes the scale a factor the vertces of whch cosst of two fudametal orbts Oa ( ) { a } ad a O () { } represetg a rhombus. The rectagle s vertex trastve uder the Kle s groupc C ad ts dual rhombus s edge trastve. For the values a ad a the rectagle ad ts dual rhombus are gve fgure. Fgure. The rectagle ad ts dual rhombus possessg the symmetry D C C wth W( A symmetry ) D3 S3 The Coxeter group W( A) D3 S3cossts of sx elemets. Three elemets r, r, rrr = rrr represet reflectos wth respect to the les orthogoal to the roots,, respectvely ad the rotatoal group elemets, rr ad ( rr ) represet the cyclc group C3 whch rotates the system by.we have two fudametal orbts O() {,, } ad () {,, }. orbt represets a equlatera O 4

6 GROUP 8: Physcal ad Mathematcal Aspects of Symmetry Joural of Physcs: Coferece Seres 84 () 39 IOP Publshg do:.88/ /84//39 tragle, dual to each other whch are trasformed to each other by the Coxeter-Dyk dagram symmetry. The orbt O() O( ) represets a regular hexago whch has a larger symmetry D3: C D 6 because of the dagram symmetry :. Now we dscuss the orbt obtaed from the vector a a where a a. The Coxeter group W( A ) geerates the sx elemets of the orbt O( ) O( a,a ) arraged the couter clockwse order as follows: where Oa (, a) {,,,,, } a a, a( aa), 3 ( aa) a a a, a ( a a ), ( a a ) a. () They ca also be obtaed as complex umbers from equato () by substtutg =3. These vectors represet the vertces of a sogoal hexago wth teror agles ad the alteratg edge legths a ad a. The dual of the sogoal hexago s a sotoxal hexago whch the edges are equal, however, t has two dfferet alteratg teror agles ad such that 4. The vertces of the sotoxal hexago le o two fudametal orbtsoa ( ) ad O().The scale factor s determed by the relato a ( a).( a a ) = ( a a). () a a The vertces of the sotoxal hexago ca the be wrtte the couter clockwse order as follows: { B a, B, B a ( ), B, B a, B ( )}. (3) Defg oe ca check that the edge legth of the sotoxal hexago s gve by a a [ ( )] (4) 3 wth two dfferet teror agles gve by cos, cos. (5) ( ) ( ) Oe ca prove that for ay value of, 4. The sogoal hexago ad ts dual sotoxal hexago are show fgure 3 for the values a ad a. Fgure 3. The sogoal hexago ad ts dual sotoxal hexago possessg the symmetry. D 3 5

7 GROUP 8: Physcal ad Mathematcal Aspects of Symmetry Joural of Physcs: Coferece Seres 84 () 39 IOP Publshg do:.88/ /84//39 Oe ca tle the plae wth regular ad sogoal hexagos provded two sogoal hexagos ad oe regular hexago share the same vertex regardless of the values of a ad a. Oe type of tlg of the plae s show fgure 4 wth the values a ad a. Aother tlg s show fgure 4 for the sogoal hexago for a ad a. I the lmt a ad a sogoal hexago turs out to be a equlateral tragle ad such tlg s depcted fgure 4(c). The other extreme lmt s show fgure 4(d) where a ad a. The hoeycomb lattce correspods to the case where a a whch s show the fgure 4(e). The hoeycomb lattce represetg the tlg of the plae wth regular hexagos possesses traslatoal varace, that s to say, t s varat uder the affe Coxeter group Wˆ ( A ). We atcpate that a eutral state of the graphee cosstg of fte umber of carbo atoms ca be represeted by a tlg smlar to the oe fgure 4 provded oe of the parameters represets the double bods (say ) ad the other s represetg the sgle bods ( ). Expermetally oe expects a a but evertheless they are early equal to each a other cotrary to the sogoal hexago fgure 4 whch s a exaggerated verso of ths quas crystal lattce. As we wll dscuss the paper [8] the molecule represets tlg of the sphere wth C 6 sogoal hexagos ad the petagos for t has two bod legths. a (c) (d) (e) Fgure 4. Tlg of the plae wth sogoal hexagos-regular hexagos (a-b), regular hexagos ad tragles (c), tlg wth tragles (d) ad the hoeycomb lattce (e). Tlg of the plae ca also be made wth the sotoxal hexagos by jog ts two vertces wth two alteratg agles ad so that oe obtas a exteror agle of at each vertex. Ths way we create a regular hexago surrouded by sx sotoxal hexagos as show fgure 5. Ths tlg s the dual of the tlg fgure 4. I ths tlg all hexagos have the same edge legths. However at each vertex the three agles satsfy, as expected, the relato 36. Here aga whe 6

8 GROUP 8: Physcal ad Mathematcal Aspects of Symmetry Joural of Physcs: Coferece Seres 84 () 39 IOP Publshg do:.88/ /84//39 we obta the hoeycomb lattce of fgure 4.Ths tlg ca be cosdered as a coductg graphee model allowg slght modfcatos of the bod agles. Fgure 5. Tlg of the plae wth sotoxal ad regular hexagos. 3.3 = 4 wth W( B) D 4 symmetry We ca repeat smlar argumets rased for the case =3. Here also we have two fudametal orbts O() {,,, } (6a) O() {,,, }. (6b) Each orbt represets a square. The orbt O( ) O() s a regular octago possessg the symmetry D4 : C D 8.Oe ca tle the plae wth regular octago ad the square as show fgure 6. Fgure 6. Tlg of the plae by regular octagos ad squares. The geeral orbt O( ) O( aa ) wth a a s a sogoal octago wth teror agles 35. The vertces of a sogoal octago ca be geerated from the complex umber a a [ ( ) ] a a a. The vertces of the sotoxal octagos are determed by a fdg the factor ( a a). The vertces of sotoxal octago s the uo of the orbts ( a a) Oa ( ) ad O(). The alteratg agles satsfy the relato 7. The sogoal octago wth values a ad a ad ts dual sotoxal octago are depcted fgure 7. 7

9 GROUP 8: Physcal ad Mathematcal Aspects of Symmetry Joural of Physcs: Coferece Seres 84 () 39 IOP Publshg do:.88/ /84//39 Fgure 7. The sogoal octago ad ts dual sotoxal octago. The tlg of the plae wth two sogoal octagos ad oe square at each vertex s show fgure 8. The tlg of the plae wth two sotoxal octagos ad oe square at oe vertex s show fgure 8. Fgure 8. Aperodc tlg of the plae by sogoal octagos ad sotoxal octagos wth squares. 3.4 =5 wth W( H ) D symmetry 5 Ths case correspods to the Coxeter groupw( H) D 5, the dhedral group of order. Here the Carta matrx ad ts verse correspodg to the Coxeter dagram ca be wrtte terms of the golde rato 5 5 ad as C ad C. (7) The orbt of a geeral vector aa ca be obtaed as the set of vectors Oa (, a) D( a, a). The fudametal orbts are the regular petagos whch are the duals of each 5 other. The regular decago s obtaed by lettg a ad a. The regular decago s both vertex ad edge trastve uder the dhedral groupw( H): C D because of the dagram symmetry. The tlg of the plae wth fve-fold symmetry s troduced by Perose [9] sce the plae caot be tled oly wth regular petagos. It has bee recetly show that the Islamc tlg of the plae [] wth fve-fold symmetry dates back to the medeval tme. The Islamc archtecture used fve dfferet tles, decago, petago, rhombus, oregular hexago ad bow te. Here we gve fgure 9 oe of those tlgs of the plae wth regular decagos ad bow tes. H 8

10 GROUP 8: Physcal ad Mathematcal Aspects of Symmetry Joural of Physcs: Coferece Seres 84 () 39 IOP Publshg do:.88/ /84//39 Fgure 9. The tlg of the plae by regular decagos ad bow tes. Oe ca also costruct a sogoal decago represeted by alteratg edge legths a ad a wth a a. Its teror agles are all equal to44. Oe such sogoal decago s show fgure correspodg to the alteratg edge legths for a ad a. The dual of the sogoal decago s the sotoxal decago wth equal edge legths but wth alteratg agles 88. The sotoxal decago s show fgure. Fgure. The sogoal decago ad ts dual the sotoxal decago. Oe type of aperodc tlg of the plae wth sogoal decagos s show fgure. Fgure. Tlg of the plae wth sogoal decagos. 4. Cocluso We have dsplayed a method to costruct the quas regular polygos ad ther duals usg the rak- Coxeter dagrams. The sogoal polygos wth sdes ad ther duals sotoxal polygos possess the dhedral symmetry D. It s temptg to suggest that the tlg of the plae by the sogoal hexago ad regular hexago may represet a state of graphee where double bod ad sgle bods ca be represeted by two edges of the sogoal hexagos. We have costructed a umber of sogoal 9

11 GROUP 8: Physcal ad Mathematcal Aspects of Symmetry Joural of Physcs: Coferece Seres 84 () 39 IOP Publshg do:.88/ /84//39 polygos wth ther duals possessg varous dhedral symmetres.the correspodg tlgs of the plae wth the tles chose as sogoal ad sotoxal polygos are studed. Refereces [] Novoselov K S, Gem A K, Morozov S V, Jag D, Zhag Y, Duboos SV, Gregoreva I V ad Forsov AA 4 Scece, ; Gem A K ad Novoselov K S 7 Nature Materals 6 83 [] Wallace P R 947 Phys. Rev. 7 6 [3] Seechal M 989 Itroducto to the Mathematcs of Quascrystals ed Marko V. Jarc (Academc Press) p [4] Jaer A ad Jasse T 98 Acta Cryst. A36 399; bd A36 48 [5] Coxeter H S M ad Moser W O J 965 Geerators ad relatos for dscrete groups (Sprger Verlag) ; Coxeter H S M 973 Regular complex polytopes (Cambrdge: Cambrdge Uversty Press) [6] Carter R W 97 Smple groups of le type (Joh Wley & Sos Ltd.); Humphreys J E 99 Reflecto groups ad coxeter groups (Cambrdge Uversty Press, Cambrdge) [7] Slasky R 98 Phys. Rep.79, [8] Koca M, Al-Ajm M ad Shda S Quas regular polyhedra ad ther duals wth Coxeter symmetres represeted by quateros II arxv:6.973 [9] Perose R 974 Bull.Ist. Math. Appl., 66; Perose R 989 Itroducto to the Mathematcs of Quascrystals vol ed Marko V. Jarc (Academc Press) p 53 [] Lu P J ad Stehardt PJ 7 Scece 35, 6