Properties of Coxeter Andreev’s Tetrahedrons

Complete Set of CHC Tetrahedrons

“…n n n  . By Roeder's 2 nd condition (of Theorem 3.5): The following Theorem 4.6 and corollary 4.7 have been taken from [1] which will be used in this present work. Proof: Let T be a CHC tetrahedron with at least one vertex is of order   2,3,4 .…”

“…In this article, it is found that there are exactly 3 CHC (compact hyperbolic coxeter) tetrahedrons upto symmetry in real projective space. These 3 tetrahedrons can be realized uniquely [10] in Hyperbolic space and these are nothing but the 3 coxeter Andreev's tetrahedrons found in theorems 3.20 and 3.21 in [1]. This research can be extended to other compact as well as non-compact hyperbolic polytopes in spaces of different dimensions.…”

“…A simple polytope P in ndimensional hyperbolic space n H is said to be coxeter, if the dihedral angles of P are of the form n  where, n is a positive integer 2  . P. Kalita and B. Kalita [1] found that there are exactly one, four and thirty coxeter Andreev's tetrahedrons having respectively two edges of order 6 n  , one edge of order 6 n  and no edge of order 6 n  , n  upto symmetry. There is no complete classification of hyperbolic coxeter polytopes.…”

“…Hence, there is no such T of this type.From theorems 4.8 and 4.9, the total number of CHC tetrahedrons upto symmetry is: 21 3  . These are namely can be realized uniquely[10] in Hyperbolic space and these are nothing but the 3 coxeter Andreev's tetrahedrons found in theorems 3.20 and 3.21 in[1].…”