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

Use of Block Toeplitz Matrix in the Study of Möbius Pairs of Simplexes in Higher-Dimensional... 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 2n + 1-dimensional projective space. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Arnold Mathematical Journal Springer Journals

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

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Publisher
Springer Journals
Copyright
Copyright © Institute for Mathematical Sciences (IMS), Stony Brook University, NY 2020
ISSN
2199-6792
eISSN
2199-6806
DOI
10.1007/s40598-020-00142-y
Publisher site
See Article on Publisher Site

Abstract

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 2n + 1-dimensional projective space.

Journal

Arnold Mathematical Journal – Springer Journals

Published: Jun 12, 2020

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