Gram Matrix of Coxeter Symmetric Tetrahedrons with Dihedral Spherical Triangle Group

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International Journal of Advanced Research in Engineering and Technology (IJARET) Volume 10, Issue 6, November-December 2019, pp. 453-461, Article ID: IJARET_10_06_050 Available online at http://iaeme.com/Home/issue/IJARET?Volume=10&Issue=6 ISSN Print: 0976-6480 and ISSN Online: 0976-6499 DOI: 10.34218/IJARET.10.6.2019.050

GRAM MATRIX OF COXETER SYMMETRIC TETRAHEDRONS WITH DIHEDRAL SPHERICAL TRIANGLE GROUP

Pranab Kalita Department of Mathematics, Pandit Deendayal Upadhyaya Adarsha Mahavidyalaya, Dalgaon, Darrang, Assam, India

ABSTRACT Simplex is the building block of space in geometry. It is basically a polytope which is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The gram matrix is the most essential and natural tool associated to a simplex. The geometric properties of a simplex are enclosed in the eigenvalues of a gram matrix. In this paper, we have classified the coxeter symmetric tetrahedrons with dihedral spherical triangle group, and it has been found that there are four such coxeter symmetric tetrahedrons with dihedral spherical triangle group upto symmetry. We also found the determinants and eigenvalues of the gram matrices of those four tetrahedrons using Maple Software. Keywords: Triangle Groups, Tetrahedron, Gram Matrix, Spectrum Cite this Article: Pranab Kalita, Gram Matrix of Coxeter Symmetric Tetrahedrons with Dihedral Spherical Triangle Group, International Journal of Advanced Research in Engineering and Technology, 10(6), 2019, pp. 453-461. http://iaeme.com/Home/issue/IJARET?Volume=10&Issue=6

1. INTRODUCTION AND PRELIMINARIES Simplex (in plural, simplexes or simplices) is the building block of space in geometry. It is basically a polytope which is a generalization [8] of the notion of a triangle or tetrahedron to arbitrary dimensions. A n-dimensional polytope P in X = E n / S n / H n , with n  0 , is a n-simplex [20, 21] if and only if P has exactly n+1 sides. Specifically, a n-simplex is a n-dimensional polytope which is the convex hull of its n+1 vertices. In particular, a line is a 1-dimensional, a triangle is a 2-dimensional and a tetrahedron is a 3-dimensional simplex. A Reflection group is a group generated by reflections on the sides of polyhedra. A triangle group [12, 16] is an infinite reflection group which can be realized geometrically by sequences of reflection across the sides of a triangle. There are three types of triangle groups-Euclidean, Spherical and Hyperbolic. These groups arise in arithmetic geometry. A tessellation is a family of tiles with a given shape or shapes that cover the surface (2D region) without gaps or overlaps. The idea of

http://iaeme.com/Home/journal/IJARET 453 [email protected] Gram Matrix of Coxeter Symmetric Tetrahedrons with Dihedral Spherical Triangle Group tessellating a 3D space with 3D shape or shapes can be generalized from the idea of filling a 2D region with 2D shape or shapes. Triangle groups preserve a tiling by triangles.    Definition 1.1: A triangle ( l ,m, n) in X = E 2 / S 2 / H 2 with angles , , , and respective l m n reflections a,b,c across the sides  , , , is defined [12] as

n l m  ( l ,m ,n ) = a ,b ,c a 2 = b 2 = c 2 = ( ab ) = (bc ) = ( ca ) = 1 Definition 1.2: Let l ,m, n be the positive integers  2 and define [12]  to be 1 1 1  = + + −1 (1) l m n A triangle group ( l ,m, n) which is generated by the reflections of all sides of the triangle    with angles , , is said to be (a) Euclidean triangle group if  = 0 , (b) Spherical triangle l m n group if   0 , (c) Hyperbolic triangle group if   0 . Lemma 1.3: The triangle groups ( l , m, n) , ( m, n, l) , and ( n, m, l) are isomorphic [12]. That is,

 ( l , m, n)   ( m, n, l )  ( n, m, l).

Definition 1.4: A coxeter diagram [7] is a graph  = (V , E) with edges labeled by an element m3 ,4,5,   . For simplicity, the label m = 3 is suppressed. Let  = (V , E) be a coxeter diagram  . For all v i ,v j V ,v i  v j , define

2 if v ,v  E  ( i j ) kij =  m if lebel of ( vi , v j ) is m

Now the group G ( ) associated with  is the group generated by the symbols vi V subject

kij 2 to the relations ( vi v j ) =1, vi =1 for all v i ,v j V ,v i  v j . If  is the disconnected union of two subgraphs 1 and  2 , then G ( ) is the direct product G( 1 )  G( 2 ) .

2. SPHERICAL TRIANGLE GROUP, COXETER DIAGRAMS AND TESSELLATIONS Substituting   0 in (1), we have 1 1 1 1 1 1 + + − 1  0  + +  1 l m n l m n Upto permutations, the only values for the triples (l ,m, n) are ( 2,2, n  2) , ( 2,3,3) , ( 2,3,4) ,( 2,3,5) , is called spherical triangle group. The coxeter diagrams for  (2 ,2, n  2 ),  (2 ,3,3 ),  (2 ,3,4 ), and ( 2,3,5) are in fig. 1:

(a) (b)

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5 4

Figure 1 Coxeter diagrams of (a)  (2 ,2, n  2) , (b) ( 2,3,3) , (c) ( 2,3,4) , (d) ( 2,3,5)

The respective coxeter groups [12] of  ( 2,2, n  2 ) ,  ( 2,3,3 ) , ( 2,3,4 ), and ( 2,3,5) are

A1  I 2 ( n) , A3 , BC3 , H3 . The tessellations [12] of S 2 generated by reflecting in the sides of  ( 2,2, n  2 ) , ( 2,3,3 ),  (2 ,3,4 ), and ( 2,3,5) are illustrated in figure 2.

(a) (b)

Figure 2: Tessellations of S 2 generated by reflecting in the sides of (a)  (2 ,2,5) , (b)  (2 ,2,6) , (c)  (2 ,3,3) , (d)  (2 ,3,4) , (e) ( 2,3,5) .

Theorem 2.1: If  , , are the angles of a spherical triangle T , then Area ( T ) = (  +  +  ) − Corollary 2.2: In tessellation of a spherical 2D surface S 2 , the total number N ( T) of spherical triangles T with angles  ,  , is obtained by

4 N (T ) = (  +  +  ) −

Proof: The area of a spherical triangle T with angles  , , is Area (T ) = (  +  +  ) − and the area [16] of S 2 is 4 . Therefore, the total number N ( T) of spherical triangles T is

4 4 N ( T ) = = Area( T) (  +  +  )− 

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The total number of triangles [16] in tessellation of S 2 with spherical triangle group is listed in table 1.

Table 1

Triangle Group Area ( T ) = (  +  +  ) − Total No. of Triangles in S 2  (2 ,2, n  2)       4  + +  −  = = 4n  2 2 n  n  n ( 2,3,3)       4  + +  −  = = 24  2 3 3  6  6 ( 2,3,4)       4  + +  −  = = 48  2 3 4  12  12 ( 2,3,5)       4  + +  −  = = 120  2 3 5  30  30

3. COXETER SYMMETRIC TETRAHEDRONS WITH DIHEDRAL SPHERICAL TRIANGLE GROUP Definition 3.1: The angle between two faces of a polytope, measured from perpendiculars to the edge created by the intersection of the planes is called a dihedral angle. A coxeter dihedral  angle is a dihedral angle of the form where, n is a positive integer  2 . n

Dihedral Angle

Figure 3 A dihedral angle of a cube  Definition 3.2: If the dihedral angle of an edge of a polytope is ,n is a positive number, then n n is said to be the order of the edge. We define a trivalent vertex to be of order ( l ,m, n) if the three edges at that vertex are of order l ,m, n . Definition 3.3: We define a tetrahedron T as CST-DST (Coxeter Symmetric Tetrahedron with Dihedral Spherical Triangle Group) if it satisfies the following properties: • Dihedral angles at the edges adjacent to one vertex are coxeter, • It is symmetric, that is, pair of disjoint edges are having same dihedral angles,

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• Dihedral angles at the edges adjacent to one vertex satisfy the property of spherical triangle group.    We denote a CST-DST T with the dihedral angles A = , B = ,C = at the edges adjacent l m n to one vertex (figure 4) by CST-DST = T (l , m, n ).

l m n

n m l

Figure 4 A CST-DST = T (l , m, n )

4. GRAM MATRIX OF CST-DSTS

The Gram matrix G of a k -simplex in X whose sides are s 1, s 2, ,sk +1 is the ( k + 1)  ( k +1) matrix with ijth entry is − cosij ,  ij is the angle between the sides si and s j . The gram matrix G is symmetric (real), the eigenvalues of G are real and hence can be ordered, say 1  2  3  n

. The spectrum of a gram matrix G with eigenvalues 1 , 2 , 3 , , r having respective multiplicities m1 , m 2 , m 3, ,m r is written as       1 2 3 r m1 m2 m3 mr  ( G ) =   or  ( G) = ( 1 , 2 , 3 , , r ) .  m1 m2 m3 mr  Definition 4.1: Consider a hyperbolic or a spherical tetrahedron T ( A, B, C , D, E ,F ) with dihedral angles A, B ,C at the edges adjacent to one vertex and D, F , F are the dihedral angles are opposite to them respectively (figure 5).

A B C

F D E

Figure 5 A T ( A, B, C , D, E ,F) Then the gram matrix of a tetrahedron T ( A, B, C , D, E ,F )

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is defined as  1 − cos A − cos B −cos F   − cos A 1 − cos C −cos E G =    − cos B − cos C 1 −cos D    − cos F − cosE −cos D 1 

Definition 4.2: A tetrahedron T ( A, B, C , D, E ,F) is said to be symmetric if A = D, B = E ,C = F and the gram matrix of a symmetric tetrahedron T ( A = D, B = E ,C = F )

is defined as:  1 − cos A − cos B −cos C   − cos A 1 − cos C − cosB G =    − cos B − cosC 1 − cosA    − cos C − cos B −cos A 1 

The Gram matrix is the most essential and natural tool associated to a simplex. The geometric properties of a simplex are enclosed in the eigenvalues of a Gram matrix. Gram matrix takes an important role in scientific computing, statistical mechanics and random matrix theory. In our study, we classify all the CST-DSTs, and also find the determinants and eigenvalues of the gram matrices of all CST-DSTs using Maple Software. Corollary 4.3: In a CST-DST tetrahedron T , the order of one vertex is one of the forms: ( 2,2, n  2) , ( 2,3,3) , ( 2,3,4) , ( 2,3,5). Proof: Consider the dihedral angles at the edges adjacent to one vertex of a CST-DST    tetrahedron T are A = , B = ,C = . Therefore, the order of that vertex is ( l ,m, n) , l ,m, n are l m n positive integers  2 such that:    1 1 1 + +    + +  1 l m n l m n

Upto permutations, the only values for the triples (l ,m, n) are:

( 2,2, n  2) , ( 2,3,3) ,( 2,3,4) , ( 2,3,5)

(Note: For different values of n , there will be infinite numbers of tetrahedrons, but we treat these as single category) Corollary 4.4: In a symmetric tetrahedron T , if any vertex is of order ( l ,m, n) , then the other three vertices are also of order (l ,m, n).

Proof: Let T be a symmetric tetrahedron with a vertex v 1 is of order (l ,m, n) . Since T is symmetric, therefore, each pair of disjoint edges are having same dihedral angles and hence we have the following figure 6.

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l m n

v 4 n l v2 m v3

Figure 6: A Symmetric Tetrahedron

From figure, it is clear that the other three vertices v 2,v3 and v4 are also of order ( l ,m, n) . Corollary 4.5: There are exactly 4 CST-DSTs upto symmetry Proof: Using corollary 4.3 and 4.4, it can be easily proved that there are exactly 4 CST-DSTs, namely

2 2 n 2 3 3

n 2 3 2 2 3

CST-DST-1 = T ( 2,2, n  2) CST-DST-2 = T ( 2,3,3)

2 3 4 2 3 5

4 3 2 5 3 2

CST-DST-3 = T (2, 3,4) CST-DST-4 = T (2 ,3,5)

Figure 7 The 4 CST-DSTs Theorem 4.6: The spectrum of the Gram matrix G of a symmetric tetrahedron T ( A = D, B = E ,C = F ) is

1 − cos A −cos B −cos C,   1 + cos A +cos B −cos C,  ( G) =   1 + cos A− cos B +cos C,   1 − cos A + cos B +cos C,

Proof: The characteristic equation of the Gram matrix of a symmetric tetrahedron T ( A = D, B = E ,C = F ) is

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−1 cos A cos B cosC cos A  −1 cos C cos B = 0 cos B cos C  −1 cos A cos C cos B cos A  −1

 4 − 4 3 + ( 6 − 2cos 2 A − 2cos 2 B −2cos2 C) 2 + ( 4cos 2 A + 4cos 2 B + 4cos 2 C + 8cos A cos B cos C − 4) +1 − 2cos 2 A − 2cos 2 B − 2cos 2 C − 2cos 2 A cos 2 B − 2cos 2 B cos 2 C −2 cos2 C cos 2 A + cos4 A + cos 4 B + cos4 C − 8cos A cos B cos C = 0 On solving 1 − cos A −cos B −cos C,   1 + cos A +cos B −cos C,  ( G ) =   1 + cos A− cos B +cos C,   1 − cos A + cos B +cos C,

Result 4.7: Determinants and Spectrums of Gram matrices of the 4 CST-DSTs are listed in table 2.

Table 2 CST-DST determinant Spectrum

CST-DST-1 = T ( 2,2, n  2) 2    4      1 − 2cos + cos 2    2        1 + 1− sin   ,1 − 1− sin  ,  n   n    n   n      2    2     1 + 1− sin   ,1 − 1− sin      n   n   CST-DST-2 = T ( 2,3,3) 0 ( 0,1,1,2) CST-DST-3 = T (2, 3,4) 7  1 2 1 2  −  + , − ,  16  2 2 2 2   3 2 3 2   + , −   2 2 2 2 

CST-DST-4 = T ( 2,3,5) 9 5 2    4     1    1     − cos   + cos    +cos   , −cos  ,  16 2  5   5  2  5  2  5     3    3      +cos   , −cos     2  5  2  5  

5. CONCLUSIONS In this paper, we have classified the CST-DSTs, and found that there are 4 such tetrahedrons upto to symmetry. We also found the determinants and eigenvalues of the gram matrices of the 4 CST-DSTs using Maple Software. This research can be extended to higher dimensional simplexes.

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