Use of Block Toeplitz Matrix in the Study of Möbius Pairs of Simplexes in Higher-Dimensional Projective Space

A simplex in a projective space of dimension n is expressed by a matrix of order n  + 1, where each row represents the homogeneous coordinates of a vertex of the simplex with respect to a reference frame. In the present study, a block Toeplitz matrix is used to express a simplex which forms a Möbius...

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Bibliographic Details
Published in:Arnold mathematical journal 2020-06, Vol.6 (2), p.189-197
Main Authors: Panda, Golak Bihari, Misra, Saroj Kanta
Format: Article
Language:English
Subjects:
Apexes
Chains
Mathematics
Mathematics and Statistics
Research Contribution
Tetrahedra
Online Access:Get full text
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Summary:A simplex in a projective space of dimension n is expressed by a matrix of order n  + 1, where each row represents the homogeneous coordinates of a vertex of the simplex with respect to a reference frame. In the present study, a block Toeplitz matrix is used to express a simplex which forms a Möbius pair along with the reference simplex. A pair of mutually inscribed, circumscribed tetrahedrons is called a Möbius pair. The existence of such pairs of simplexes in higher-dimensional (odd) projective spaces is already established. In the present study an existence of an infinite chain of simplexes in a five-dimensional projective space is established where any two simplexes from the chain form a Möbius pair in some order of their vertices. This is studied with the help of powers of a block Toeplitz matrix. Also, attempt has been made to generalize this result to 2 n  + 1-dimensional projective space.
ISSN:2199-6792
2199-6806
DOI:10.1007/s40598-020-00142-y