Equivalence conditions of two systems of vectors in the taxicab plane and its applications to taxicab polygons


ÖREN İ., ÇOBAN H. A.

COMMUNICATIONS FACULTY OF SCIENCES UNIVERSITY OF ANKARA-SERIES A1 MATHEMATICS AND STATISTICS, vol.69, pp.413-430, 2020 (ESCI) identifier

Abstract

This study presents the conditions of M-T (2)-equivalence for two systems of vectors {x(1), x(2,) x(3)} and {y(1), y(2), y(3)} in R-T(2), where M-T (2) is the group of all isometries of the 2-dimensional taxicab space R-T(2). Firstly a minimal complete system of M-T (2)-invariants of {x(1), x(2), x(3)} is obtained. Then, using the conditions of M-T (2)-equivalence, an answer is given to the open problem posed in [10, p.428]. Furthermore, an algorithm is given for constructing taxicab regular polygons in terms of M-T (2)-invariants. This algorithm is general and useful to construct the taxicab regular 2n-gons and gives a tool to solve special cases of the open problem posed in [2, p.32]. Besides, both the conditions of the taxicab regularity of Euclidean regular polygons and Euclidean regularity of taxicab regular polygons are given in terms of M-T (2)-invariants.