Coxeter Dihedral Symmetric Tetrahedrons with Triangle Groups: Euclidean, Spherical and Hyperbolic
Abstract
Abstract: In this article, we have classified the Coxeter Dihedral Symmetric (CDS) tetrahedrons with triangle groups: Euclidean, Spherical and Hyperbolic. We have calculated the gram spectrums of these CDS tetrahedrons with triangle groups: Euclidean, spherical and hyperbolic, and finally studied their existence in the spaces: Euclidean, spherical and hyperbolic.
MSC 2010 Codes: 51M05, 05C50, 15A45, 15A42, 05C69.
Keywords: Coxeter Dihedral Symmetric Tetrahedrons, Triangle groups, gram matrix, spectrum, Euclidean, eigen values
Cite this Article: Pranab Kalita. Coxeter Dihedral Symmetric Tetrahedrons with Triangle Groups: Euclidean, Spherical and Hyperbolic. Research & Reviews Discrete Mathematical Structures. 2019; 6(3): 1–12p.
Full Text:
PDFReferences
John G. Ratcliffe, Foundations of Hyperbolic Manifolds, ©1994 by Springer-Verlag, New York, Inc.
http://en.wikipedia.org/wiki/Triangle-group.
P. Kalita and B. Kalita, Properties of Coxeter Andreev’s Tetrahedrons, IOSR Journal of
Mathematics (IOSR-JM) e-ISSN: 2278-3008, p-ISSN: 2319-7676. Volume 9, Issue 6, 2014, PP
-105.
P. Kalita, “Gram Spectrums of Triangles with Triangle Groups”, IJMA-5(6), 2014, 64-71.
Dipankar Mondal, Introduction to Reflection Groups, Triangle Group (Course Project), April
, 2013.
H. S. M. Coxeter, Discrete groups generated by reflections, Ann. Of Math. 35 (1934), 588-621.
Wilhelm Magnus, Noneuclidean Tesselations and Their Groups. Academic Press New York
and London, 1974.
Alexandre V. Borovik, Anna Borovik, Mirror and Reflection: The Geometry of Finite
Reflection Groups. Springer-Universitext, 2010.
en.wikipedia.org/wiki/simplex.
J. Mcleod, Hyperbolic Coxeter Pyramids, Advances in Pure Mathematics, Scientific Research,
, 3, 78-82
Roland K. W. Roeder, Compact hyperbolic tetrahedra with non-obtuse dihedral angles,
August 10, 2013, arxiv.org/pdf/math/0601148.
Aleksandr Kolpakov, On extremal properties of hyperbolic coxeter polytopes and
their reflection groups, Thesis No: 1766, e-publi.de, 2012.
Anna Felikson, Pavel Tumaarkin, Coxeter polytopes with a unique pair of non intersecting
facets, Journal of Combinatorial Theory, Series A 116 (2009) 875-902.
Pavel Tumarkin, Compact Hyperbolic Coxeter polytopes with facets, The Electronic
Journal of Combinatorics 14 (2007).
Roland K.W. Roeder, John H. Hubbard and William D. Dunbar, Andreev’s Theorem on
Hyperbolic Polyhedra, Ann. Inst. Fourier, Grenoble 57, 3 (2007), 825-882.
D. Cooper, D. Long and M. Thistlethwaite, Computing varieties of representations of
hyperbolic 3- manifolds into , Experiment. Math. 15, 291-305 (2006)
D. A. Derevnin, A. D. Mednykh and M. G. Pashkevich, On the volume of symmetric
tetrahedron, Siberian Mathematical Journal, Vol. 45, No. 5, pp. 840-848, 2004
Yunhi Cho and Hyuk Kim. On the volume formula for hyperbolic tetrahedral. Discrete
Comput. Geom., 22(3): 347-366, 1999.
E. B. Vinberg, Hyperbolic Reflection Groups, Uspekhi Mat. Nauk 40, 29-66 (1985)
R. Guo and Y. Wang, Eigenvalues of Gram Matrices.
E. B. Vinberg, The absence of crystallographic groups of reflections in Lobachevskij spaces of
large dimensions, Trans. Moscow Math. Soc. 47 (1985), 75-112.
M. Hazewinkel, at .el, “The Ubiquity of Coxeter-Dynkin Diagrams”, Nieuw Archief Voor
Wiskunde (3), XXV(1977), 257-307.
P. Kalita and B. Kalita, Properties of Coxeter Andreev’s Tetrahedrons, ISOR Journal of
Mathematics, 2014, PP 81-105.
J. Mcleod, Hyperbolic Coxeter Pyramids, Advances in Pure Mathematics, Scientific Research,
, 3, 78-82
Roland K. W. Roeder, Compact hyperbolic tetrahedra with non-obtuse dihedral angles,
August 10, 2013, arxiv.org/pdf/math/0601148.
D. A. Derevnin, A. D. Mednykh and M. G. Pashkevich, On the volume of symmetric
tetrahedron, Siberian Mathematical Journal, Vol. 45, No. 5, pp. 840-848, 2004
E. B. Vinberg, The absence of crystallographic groups of reflections in Lobachevskij spaces of
Large dimensions, Trans. Moscow Math. Soc. 47 (1985), 75-112.
Refbacks
- There are currently no refbacks.
This site has been shifted to https://stmcomputers.stmjournals.com/