International Journal of Mathematical Archive-5(6), 2014, Available online through ISSN

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1 Iteratioal Joural of Mathematical Archive-5(6), 4, 64-7 Available olie through wwwmaifo ISSN GRAM SPECTRUMS OF TRIANGLES WITH TRIANGLE GROUPS Praab Kalita* Departmet of Mathematics, Gauhati Uiversity, Guwahati-784, Assam, Idia (Received o: -6-4; Revised & Accepted o: -6-4) ABSTRACT I this article, the coxeter diagrams, ad their associated groups of the triagle groups-euclidea, Spherical ad Hyperbolic have bee studied The tessellatios of triagles with the triagle groups have also bee discussed, ad fially foud the gram spectrums of triagles with triagle groups usig Maple Software Keywords: Triagle Groups, Coxeter diagram, Gram Matrix, Spectrum MSC Codes: 5A45, 5M5, 5A4, 5C5 INTRODUCTION AND PRELIMINARIES Simplex (i plural, simplexes or simplices) is the buildig block of space i geometry It is basically a polytope which is a geeralizatio [] of the otio of a triagle or tetrahedro to arbitrary dimesios A -dimesioal polytope P i X = E / S / H, with >, is a -simplex [9] if ad oly if P has exactly + sides Specifically, a -simplex is a -dimesioal polytope which is the covex hull of its + vertices I particular, a lie is a -dimesioal, a triagle is a -dimesioal ad a tetrahedro is a -dimesioal simplex A Reflectio group is a group geerated by reflectios o the sides of polyhedra A triagle group [, 5] is a ifiite reflectio group which ca be realized geometrically by sequeces of reflectio across the sides of a triagle There are three types of triagle groups-euclidea, Spherical ad Hyperbolic These groups arise i arithmetic geometry A tessellatio is a family of tiles with a give shape or shapes that cover the surface (D regio) without gaps or overlaps The idea of tessellatig a D space with D shape or shapes ca be geeralized from the idea of fillig a D regio with D shape or shapes Triagle groups preserve a tilig by triagles I this paper, the coxeter diagrams, ad their associated groups of the triagle groups-euclidea, Spherical ad Hyperbolic have bee studied The tessellatios of triagles with the triagle groups have also bee discussed, ad fially foud the gram spectrums of triagles with triagle groups usig Maple Software Defiitio : A triagle ( lm,, ) across the sides,, i X E S H = / / with agles,, αβγ, is defied [5] as ( l m ) a b c a b c ( ab) ( bc) ( ca) π π π, ad respective reflectios abc,, l m l m { },, =,, = = = = = = Defiitio : Let lm,, be the positive itegers ad defie [5] λ to be λ = + + () l m π π π A triagle group ( lm,, ) which is geerated by the reflectios of all sides of the triagle with agles,, is l m said to be Euclidea triagle group if λ =, Spherical triagle group if λ >, (c) Hyperbolic triagle group if λ < Lemma : The triagle groups ( lm,, ), ( ml,, ), ad ( ml,, ) ( lm,, ) ( ml,, ) ( ml,, ) are isomorphic [5] That is, Correspodig author: Praab Kalita* Departmet of Mathematics, Gauhati Uiversity, Guwahati-784, Assam, Idia Iteratioal Joural of Mathematical Archive- 5(6), Jue 4 64

2 Praab Kalita*/ Gram Spectrums of Triagles with Triagle Groups / IJMA- 5(6), Jue-4 Defiitio 4: A coxeter diagram [4] is a graph Γ= ( V, E) with edges labeled by a elemet m {, 4,5, } { } For simplicity, the label m = is suppressed Let Γ= ( V, E) be a coxeter diagram Γ For all v, v Vv, v k ( vi vj) if, E = m if lebel of v, v is m ( i j) Now the group G ( Γ ) associated with Γ is the group geerated by the symbols v i k ( ) vv i j vi =, = for all vi, vj Vv, i vj If Γ is the discoected uio of two subgraphs Γ ad G Γ G Γ the direct product ( ) ( ), defie i j i j V subject to the relatios G Γ is Γ, the ( ) TRIANGLE GROUPS, COXETER DIAGRAMS, THEIR ASSOCIATED GROUPS AND TESSELLATIONS Euclidea Triagle Group Substitutig λ = i (), we have + + = l m Upto permutatios, the oly values for the triples (,, ) group [] Clearly (,, 6) is a equilateral triagle The coxeter diagrams for (,, 6), (, 4, 4) ad (,,),, 6,, 4, 4,,,, is called Euclidea triagle is a lm are ( ) ( ) ( ) -6 right triagle, (, 4, 4) is a isosceles right triagle, ad (,,) are i figure : (c) Figure - : Coxeter diagrams of (,, 6), (, 4, 4), (c) (,,) The coxeter groups of (,, 6), (, 4, 4) ad (,,) are respectively C, G, ad A The tessellatios [] of figure E geerated by reflectig i the sides of (,, 6), (, 4, 4) ad (,,) are illustrated i 4, IJMA All Rights Reserved 65

3 Praab Kalita*/ Gram Spectrums of Triagles with Triagle Groups / IJMA- 5(6), Jue-4 igure - : Tessellatios of E geerated by reflectig i the sides of (,, 6) (c), (, 4, 4) ad (c) (,,) Spherical Triagle Group Substitutig λ > i (), we have + + > + + > l m l m Upto permutatios, the oly values for the triples ( lm,, ) are (,, ), (,, ), ( ) ( ) triagle group The coxeter diagrams for (,, ), (,, ), (,, 4 ), ad (,,5) are i figure :,,4,,,5, is called spherical 4 5 (c) (d) Figure - : Coxeter diagrams of (,, ), (,, ), (c) (,, 4), (d) (,,5) The respective coxeter groups [5] of (,, ), (,, ), (,, 4 ), ad (,,5) are A I( ), A, BC, H The tessellatios [, 5] of S geerated by reflectig i the sides of (,, ), (,, ), (,, 4 ), ad (,,5) are illustrated i figure 4 (c) (d) (e) 4, IJMA All Rights Reserved 66

4 Figure - 4: Tessellatios of Praab Kalita*/ Gram Spectrums of Triagles with Triagle Groups / IJMA- 5(6), Jue-4 (d) (,, 4), (e) (,,5) Theorem : If,, S geerated by reflectig i the sides of (,,5), (,,6), (c) ( ) Area T = α + β + γ π αβγ are the agles of a spherical triagle T, the ( ) ( ) Corollary : I tessellatio of a spherical D surface agles αβγ,, is obtaied by N( T) = α + β + γ π,,, S, the total umber N( T ) of spherical triagles T with ( ) Proof: The area of a spherical triagle T with agles αβγ,, is ( ) ( ) is Therefore, the total umber N( T ) of spherical triagles T is ( ) N T The total umber of triagles [] i tessellatio of = = Area α + β + γ π ( T ) ( ) Area T = α + β + γ π ad the area [] of S with spherical triagle group is listed i table Table - Triagle Group Area ( T ) = ( α + β + γ) π Total No of Triagles i π π π π (,, ) + + π = (,,) (,, 4) (,,5) Hyperbolic Triagle Group π π π π + + π = 6 π π π π + + π = 4 π π π π + + π = 5 Substitutig λ < i (), we have + + < + + < l m l m 4 π = 4 π = 6 48 π = π = This iequality has a ifiite umber of solutios for ( lm,, ), havig at most oe of lm,, is The solutios for ( lm,, ) havig exactly oe of lm,, is, say l = upto symmetry are: (, m, ) = (,, 7 ),(, 4, 5 ),(,5, 5 ),(, m 6, 6; m ) The coxeter diagrams of ( l=, m, ) ad ( l, m, ) > are: S S m l m Figure - 5: Coxeter diagram of ( l=, m, ), ( l>, m, ) 4, IJMA All Rights Reserved 67

5 Praab Kalita*/ Gram Spectrums of Triagles with Triagle Groups / IJMA- 5(6), Jue-4 Theorem : If αβγ,, are the agles of a geeralized [] hyperbolic triagle T, the Area ( T ) = π ( α + β + γ) π The triagle (,, 7) has a special importace as it has least area The tessellatio [, 5] of B geerated by 4 reflectig i the sides of (,, 7) is illustrated i figure 6 Figure - 6: Tessellatios of B geerated by reflectig i the sides of (,, 7) SPECTRUM OF GRAM MATRICES OF TRIANGLES ASSOCIATED THE TRIANGLE GROUPS The Gram matrix is the most essetial ad atural tool associated to a simplex The geometric properties of a simplex are eclosed i the eigevalues of a Gram matrix The Gram matrix G of a k -simplex i X whose sides are s, s,, s k matrix with th etry is cosθ, θ is the agle betwee the sides s i ad s j A simplex is Euclidea, spherical or hyperbolic accordig to the determiat of the gram matrix is, positive or egative respectively Gram matrix takes a importat role i scietific computig, statistical mechaics ad radom matrix theory [] I our study, we fid the gram spectrums of triagles with triagle groups usig Maple Software The gram is the ( k ) ( k ) matrix G is symmetric (real), the eigevalues of G are real ad hece ca be ordered, say λ λ λ λ The spectrum of a gram matrix is said to be gram spectrum Let G be a gram matrix with eigevalues λ, λ, λ,, λr havig respective multiplicities m, m, m,, mr The the gram spectrum of G is writte as λ λ λ λr m m m mr σ ( G) = or σ ( G) = ( λ, λ, λ,, λr ) m m m mr Defiitio : The Gram matrix of a triagle with agles (,, ) θ θ θ is defied as cosθ cosθ G = cosθ cosθ cosθ cosθ The gram eergy of the Gram Matrix G havig eige values λ, λ, ad defied as E( G) = λ i= i The mai results of this study have bee established here as follows: λ of a triagle with agles (,, ) Theorem : The characteristic equatio of the Gram matrix of a triagle with agles (,, ) λ λ cos θi λ+ cos θi + cosθi = i= i= i= Proof: The characteristic equatio of the Gram matrix of a triagle with agles (,, ) λ cosθ cosθ cosθ λ cosθ = cosθ cosθ λ θ θ θ is ( λ ) + cosθcosθcosθ + cosθcosθcosθ ( ) ( ) ( ) θ θ θ is λ cos θ λ cos θ λ cos θ = θ θ θ θ θ θ ( cos + cos + cos ) + ( ) λ λ θ θ θ λ cos + cos + cos + cos cos cos = θ θ θ is λ λ cos θi λ+ cos θi + cosθi = i= i= i= 4, IJMA All Rights Reserved 68

6 Praab Kalita*/ Gram Spectrums of Triagles with Triagle Groups / IJMA- 5(6), Jue-4 Corollary : The characteristic equatio of the Gram matrix of a triagle with agles ( θθθ,, ) is ad the gram spectrum is ( ) ( ) λ λ θ λ θ θ cos + cos + cos = ( cos θ, cos θ, cosθ) + + Corollary 4: The gram eergy of the Gram matrix of a k -simplex i X is ( k + ) Proof: The Gram matrix G of a k -simplex i X whose sides are s, s,, s k + th etry is cosθ, θ is the agle betwee the sides s i ad s j is the ( k ) ( k ) + + matrix with Therefore, the diagoal etries are all ad the gram eergy (sum of the eigevalues) is the trace of the matrix ad k + hece ( ) Result 5: The gram eergy of the Gram matrix of a triagle is Result 6: Gram spectrums for Euclidea Triagle Group are listed i table ( lm,, ) (,, 6) (, 4, 4) Table - Characteristic Polyomial Gram spectrum + = (,, ) + = (,, ) λ λ λ λ λ λ 9 (,,) λ λ + λ = 4 Result 7: Gram spectrums for Spherical Triagle Group are listed i table ( lm,, ) Table -,, Characteristic Polyomial Gram spectrum (,, ) (,, ) + = (,, ) λ λ λ π π λ λ + cos λ + cos = (,,) 5 λ λ + λ = π π, + cos, cos, +, (,, 4) 9 λ λ + λ = 4 4, +, (,,5) π π λ λ + cos λ + cos = , π + + cos, 5 π + cos 5 Result 8: Gram spectrums for Hyperbolic Triagle Group are listed i table 4 4, IJMA All Rights Reserved 69

7 ( lm,, ) (,, 7) 4 CONCLUSIONS Praab Kalita*/ Gram Spectrums of Triagles with Triagle Groups / IJMA- 5(6), Jue-4 Table - 4 The coxeter diagrams ad their associated groups of the triagle groups-euclidea, Spherical ad Hyperbolic have bee studied here Moreover, we have studied about the tessellatios of triagles with the triagle groups, ad fially foud the gram spectrums of triagles with triagle groups usig Maple Software REFERENCES Characteristic Polyomial Gram spectrum π π λ λ + cos λ + cos = 4 4 (, 4, 5) 5 π π λ λ + cos λ + cos = (,5, 5) π π λ λ + cos + cos λ 5 π π + cos + cos = 5 (, m 6, 6; m ) π π λ λ + cos + cos λ m π π + cos + cos = m [] Joh G Ratcliffe, Foudatios of Hyperbolic Maifolds, 994 by Spriger-Verlag, New York, Ic [] [] P Kalita ad B Kalita, Properties of Coxeter Adreev s Tetrahedros, IOSR Joural of Mathematics (IOSR-JM) e- ISSN: 78-8, p-issn: Volume 9, Issue 6, 4, PP 8-5 [4] M Hazewikel, at el, The Ubiquity of Coxeter-Dyki Diagrams, Nieuw Archief Voor Wiskude (), XXV (977), 57-7 [5] Dipakar Modal, Itroductio to Reflectio Groups, Triagle Group (Course Project), April 6, [6] H S M Coxeter, Discrete groups geerated by reflectios, A Of Math 5 (94), 588-6, π + + 4cos, π + 4cos, π + + 4cos, π + 4cos, π π + + 4cos + cos, 5 π π + 4cos + cos 5, π π + cos + cos, m π π cos + cos m [7] Wilhelm Magus, Noeuclidea Tesselatios ad Their Groups Academic Press New York ad Lodo, 974 4, IJMA All Rights Reserved 7

8 Praab Kalita*/ Gram Spectrums of Triagles with Triagle Groups / IJMA- 5(6), Jue-4 [8] Alexadre V Borovik, Aa Borovik, Mirror ad Reflectio: The Geometry of Fiite Reflectio Groups Spriger- Uiversitext, [9] ewikipediaorg/wiki/simplex [] J Mcleod, Hyperbolic Coxeter Pyramids, Advaces i Pure Mathematics, Scietific Research,,, 78-8 [] Rolad K W Roeder, Compact hyperbolic tetrahedra with o-obtuse dihedral agles, August,, arxivorg/pdf/math/648 [] Aleksadr Kolpakov, O extremal properties of hyperbolic coxeter polytopes ad their reflectio groups, Thesis No: 766, e-publide, [] Aa Felikso, Pavel Tumaarki, Coxeter polytopes with a uique pair of o itersectig facets, Joural of Combiatorial Theory, Series A 6 (9) [4] Pavel Tumarki, Compact Hyperbolic Coxeter polytopes with + facets, The Electroic Joural of Combiatorics 4 (7) [5] Rolad KW Roeder, Joh H Hubbard ad William D Dubar, Adreev s Theorem o Hyperbolic Polyhedra, A Ist Fourier, Greoble 57, (7), [6] D Cooper, D Log ad M Thistlethwaite, Computig varieties of represetatios of hyperbolic - maifolds ito SL 4,, Experimet Math 5, 9-5 (6) ( ) [7] D A Derevi, A D Medykh ad M G Pashkevich, O the volume of symmetric tetrahedro, Siberia Mathematical Joural, Vol 45, No 5, pp , 4 [8] Yuhi Cho ad Hyuk Kim O the volume formula for hyperbolic tetrahedral Discrete Comput Geom, (): 47-66, 999 [9] E B Viberg, Hyperbolic Reflectio Groups, Uspekhi Mat Nauk 4, 9-66 (985) [] R Guo ad Y Wag, Eigevalues of Gram Matrices [] E B Viberg, The absece of crystallographic groups of reflectios i Lobachevsk spaces of large dimesios, Tras Moscow Math Soc 47 (985), 75- Source of support: Nil, Coflict of iterest: Noe Declared [Copy right 4 This is a Ope Access article distributed uder the terms of the Iteratioal Joural of Mathematical Archive (IJMA), which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited] 4, IJMA All Rights Reserved 7