An Approach n Colorng Sem-Regular Tlngs on the Hyperbolc Plane Ma Louse Antonette N De Las Peñas, mlp@mathscmathadmueduph Glenn R Lago, glago@yahoocom Math Department, Ateneo de Manla Unversty, Loyola Heghts, Quezon Cty, Phlppnes René P Felx, rene@math0updeduph Math Department, Unversty of the Phlppnes, Dlman, Quezon Cty, Phlppnes Abstract A colorng of a sem-regular tlng s perfect f every symmetry of the tlng permutes the colors of the tlng In ths paper, an approach to the constructon of perfect colorngs of sem-regular tlngs on the hyperbolc plane s presented Introducton In [], a method for colorng symmetrcal patterns was presented where a fundamental doman of the pattern s assgned exactly one color In ths paper, we present a general framework for colorng planar patterns where a fundamental doman of the pattern may be assgned more than one color We apply the framework to construct perfect colorngs of sem-regular tlngs on the hyperbolc plane We wll use the subgroup structure of the symmetry group of the tlng to systematcally construct the colorngs An edge-to-edge tlng s a plane tlng where the corners and sdes of the polygonal tles form all the vertces and edges of the tlng and vce versa A vertex of an edge-to-edge tlng s sad to be of type p p p q f the polygons about ths vertex n cyclc order are a p -gon, a p -gon,, and a pq - gon An edge-to-edge tlng havng regular polygons as ts tles wth vertces all of the same type, and where the symmetres of the tlng act transtvely on the vertces s called sem-regular We denote the sem-regular tlng as p p pq dependng on ts vertex type p p pq If the polygons n the q tlng are of the same type, partcularly a p-gon meetng q at a vertex, we denote the tlng as p In ths paper, we wll present an approach to color perfectly the 8 8 and 0 8 hyperbolc sem-regular tlngs General Framework for Colorng Planar Patterns The followng general framework for colorng planar patterns shall be used to obtan colorngs of semregular tlngs Let X be the set of tles n the tlng to be assgned colors; G be the symmetry group of the uncolored tlng; H be the subgroup of elements of G permutng the colors; C be the set of colors
Let O ( I ) correspondng to ths set s the set { hj X h H} be the H-orbts of colors and c a color n O = hc : h H and :, where J s the stablzer n H of the color c and X conssts of representatves of each H-orbt of elements of X wth the representatves colored c The followng are true: O Then { } X The acton of H on hj : I by left multplcaton In O, the number of colors s [ H : J ] If x X then StabH ( x) J, where Stab H ( x) = { h H : hx = x} If x X then Hx = [ H : J ] [ J : StabH ( x) ] The number of H-orbts of colors s less than or equal to the number of H-orbts of elements of O s equvalent on ts acton on { } The followng steps, based on the general framework, shall be used to obtan the requred colorng of the tlng orbt Determne the fnte group S of sometres n G that stablzes a representatve tle t from an Determne all subgroups J of G such that S < J If tle t has color c, apply c to all tles n the set Jt Ths makes J the stablzer of the color c nsde G If [ J ] k G : =, then Jt s of the tles n the class where t belongs k : One element of ths set has color c, whch s Jt There should be k other elements or colors To complete the colorng, assgn a color to every element of the set { gjt g G} Hence, the ndex k of the subgroup J n G s the number of colors that can be used to perfectly color the orbt of tles contanng t Colorng the Hyperbolc Plane Tessellatng the Hyperbolc Plane In [], Azz created a computer program Colorng the Hyperbolc Plane (CHP) that tessellates the hyperbolc plane wth congruent trangles of nteror angles π π π π π π,, and, where + + < π Denote by e one of the trangles of the tessellaton and call t the p q r p q r fundamental trangle Let P be the reflecton on the sde of the trangle opposte the angle p π, Q as the reflecton on the sde of the trangle opposte the angle q π, and R the reflecton on the sde of the trangle opposte r π The symmetry group G of the tessellaton s generated by and R, denoted by pqr Gven the fundamental trangle e, the tessellaton may be recovered by gettng the mages of e under R, and ther products There s a one-to-one correspondence between the elements of G and the trangles n the tessellaton Each trangle n the tessellaton can then be labeled by the correspondng
element of G The acton of G on the trangles of the tessellaton, where g G acts on a trangle by sendng t to ts mage, s equvalent to the acton of G on tself by left multplcaton R Q P Fgure: () Labelng the trangles n the tessellaton Colorng Usng Rght Cosets If S s a subgroup of G of ndex n, a colorng usng rght (or left) cosets of S refers to a bjectve map from the set of rght (or left) cosets of S to a set of n colors Trangles labeled by elements of a rght (or left) coset are colored usng the color assgned to the coset In Fgure, we gve a colorng of the hyperbolc plane usng rght cosets of the subgroup S and n Fgure, we gve another colorng usng left cosets of the subgroup S, where S represents a subgroup of ndex of the hyperbolc trangle group 6 Fgures: () Rght coset colorng usng S; () Left coset colorng usng S The rght coset colorngs of a gven subgroup S of the symmetry group G of the tessellaton plays an mportant role n studyng the subgroup structure of G S turns out to be the symmetry group of the colored tessellaton and S fxes the colors of the tessellaton In ths paper, we wll use the rght coset colorngs generated by CHP to determne the subgroups of G that contan the stablzer of the tles n the gven sem-regular tlngs Perfect Colorngs of Sem-Regular 8 8 and 0 8 Tlngs In ths part of the paper, we llustrate how to obtan perfect colorngs of sem-regular 8 8 and 0 8 tlngs usng the gven framework Both tlngs have symmetry group G = ; G contans rotatons of order,, wth centers of the correspondng rotatons lyng on mrror lnes In colorng the sem-regular tlngs, we wll make use of the subgroups of G GAP [8] s used to generate a lstng of the subgroups of G shown n Table For the purposes of ths paper, and due to colorng constrants, we wll only consder subgroups up to ndex
Lst of Subgroups of * of Index <= : Number of Subgroups = 7 Group( [ R, P ] ) Group( [ R, PRP ] ) Group( [ R P ] ) Group( [ R PQ ] ) Group( [ R PRPQ ] ) Group( [ RQRPRQR ] ) Group( [ RPQR ] ) Table: () Subgroups of of ndex less than or equal to The generators R, P appearng n Table are mrror reflectons wth axes shown n Fgures and for the respectve tlngs 8 8 and 0 8 Q R P R P Q Fgures: (-) Generators R, and P Sem-Regular 8 8 Tlng The sem-regular 8 8 tlng has two orbts of tles: the orbt of 8- gons and the orbt of -gons To construct the perfect colorngs, we color each orbt of tles separately We frst color the orbt of 8-gons Frst, note that the fnte group that stablzes an 8-gon s of type D, the dhedral group of order 8 We need to select the subgroups J that contans D The condton that J contans the stablzer s always satsfed by G To fnd other subgroups contanng D, we wll use the rght coset colorngs of the subgroups of G To obtan the rght coset colorngs, we use the CHP program From the program, Fgure 6 shows the rght coset colorng usng the subgroup J = RQRPRQR Note that the subgroup D generated by the o 90 rotaton about the ndcated pont x and mrror about the horzontal lne through x fxes the color of the gven rght coset colorng Thus, the subgroup J = RQRPRQR contans the group of type D and can now be used to color the orbt of 8-gons for the 8 8 tlng Fgure shows the rght coset colorng usng the subgroup J RPQR Smlarly, the o subgroup D generated by the 90 rotaton about x and mrror about the horzontal lne through x fxes the colors of the colorng Thus, the subgroup J also contans D We are now ready to color the orbt of 8-gons We wll use the subgroups J, namely J, J, and J = G Usng J = G, we color all 8-gons usng one color to obtan the colorng n Fgure 8
Next, we color the orbt of 8-gons usng J RQRPRQR To obtan a perfect colorng usng J, we frst choose a representatve tle t from the 8-gons We then color J t wth black, as seen n Fgure 9 To color the rest of the orbt, we apply the -fold rotaton wth center A lyng on mrrors R and Q on J t to obtan a colorng of fve colors gven n Fgure 0 Lastly, we color the orbt of 8-gons usng J RPQR Colorng all tles n J t black, we obtan Fgure Then we assgn dfferent shades and textures of gray to the tles n the other orbts by applyng the -fold rotaton about A to obtan Fgure Next, we color the orbt of -gons The fnte group that stablzes a -gon s of type D, the dhedral group of order 0 We now select the subgroup J ' that contans the stablzer Asde from G, our choce for J ' s the subgroup Q, R, PRP Fgure shows the rght coset colorng usng Q, R, PRP The subgroup generated by through x fxes the colors of the colorng and s of type D o 7 about x and mrror reflecton about the horzontal lne To color the orbt of -gons, we let obtan Fgure J ' = G to obtan Fgure and let J = ' R, PRP to To color the entre sem-regular tlng, we combne all the colorngs of each orbt of tles above Thus, the resultng perfect colorngs of the 8 8 tlng are shown n Fgure 6 x x t A 6 7 8 9 0 t A x Fgures: (6-7) Rght coset colorngs of J and J respectvely; (8) Perfect colorng of the orbt of 8-gons usng J = G ; (9) J t ; (0) Perfect colorng of the orbt of 8-gons usng J ; () J t ; () Perfect colorng of the orbt of 8-gons usng J ; () Rght coset colorngs of J ' ; (-) Perfect colorng of the orbt of -gons
Fgure: (6) Perfect colorngs of the 8 8 tlng Sem-Regular 0 8 Tlng The sem-regular 0 8 tlng has three orbts of tles: the orbt of -gons, 0-gons, and 8-gons We follow the steps gven n and color each orbt of tles separately where To color the -gons, we use J = G, J RQRPRQR, and J RPQR, D < J We have the three colorngs n Fgures 7, 8, and 9 Next, we use Fgures 0 and J ' = G and J = ' R, PRP, where D < J ', to color the 0-gons shown n Lastly, we use J " = G, J " RQRPRQR, and " RPQR, where D < J ", to color the 8-gons n Fgures,, and Next, we combne all these colorngs to obtan the perfect colorngs of the 0 8 sem-regular tlng as seen n Fgure 7 8 9 0 Fgures: (7-9) Perfect colorng of the orbt of -gons; (0-) Perfect colorng of the orbt of 0-gons; (-) Perfect colorng of the orbt of 8-gons
Fgure: () Perfect colorngs of the 0 8 tlng where the orbts do not share colors Observe that f J s used to color one orbt of tles, t can also be used to color a second orbt of tles as long as J contans the stablzer of a tle n the second orbt of tles Moreover, f a color used to color tle t n the frst orbt of tles s to be used to color tles n the second orbt, then the tle t that wll be assgned the same color as tle t should have a stablzer contaned n J In colorng the 0 8 tlng, the orbt of -gons and the orbt of 8-gons can share the same color These colorngs appear n Fgure 6 The colorngs A and B are obtaned usng J = G to color both orbts of -gons and 8-gons The colorngs n C and D are obtaned usng J RQRPRQR whle the colorngs n E and F are obtaned usng J RPQR A B C D E Fgure: (6) Perfect colorngs of the 0 8 tlng where the orbts share colors F Concluson In ths note, we gve an approach to color sem-regular tlngs on the hyperbolc plane We use the general framework for colorng planar patterns where an orbt of tles n the gven tlng s colored usng a subgroup of the symmetry group G of the tlng contanng the stablzer of the tle We use the GAP program to generate the subgroups of G whle a helpful tool n studyng more closely the subgroup structure of G s the CHP program
We ntend that the approach provded here n obtanng perfect colorngs of sem-regular tlngs wll provde a sprngboard n the constructon of colorngs (both perfect and non-perfect) of tlngs n general on the hyperbolc plane References [] Azz, Shahd A Computer Algorthm for Colorng A Hyperbolc Tessellaton A Masteral Thess, The Unversty of the Phlppnes Dlman, 996 [] Coxeter, HSM, and WO Moser Generators and Relatons for Dscrete Groups nd ed New York: Sprnger-Verlag, 96 [] De Las Peñas, Ma Louse Antonette N, René P Felx, and M V P Qulngun A Framework for Colorng Symmetrcal Patterns n Algebras and Combnatorcs: An Internatonal Congress, ICAC 97 Hong Kong Sngapore: Sprnger-Verlag 999 [] Felx, René P A General Framework for Colorng Planar Patterns, a paper presented n Natonal Research Councl of the Phlppnes (NRCP) Conference Phlppnes February, 00 [] Grünbaum, B, and GC Shephard Tlngs and Patterns New York: WH Freeman and Company, 987 [6] Hernandez, Nestne Hope Sevlla On Colorngs Induced by Low Index Subgroups of Some Hyperbolc Trangle Groups A Masteral Thess, The Unversty of the Phlppnes Dlman 00 [7] Mtchell, Kevn J Constructng Sem-Regular Tlngs, a paper presented n The Sprng 99 Meetng of the Seaway Secton of the Mathematcal Assocaton of Amerca 99 http://peoplehwsedu/mtchell/tlngs/parthtml [8] The GAP Group GAP Groups, Algorthms, and Programmng, Verson, 00, http://wwwgap-systemorg