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

Abstract

This study presents the conditions of M_{T}(2)-equivalence for two systems of vectors {x₁,x₂,x₃} and {y₁,y₂,y₃} 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 {x₁,x₂,x₃} is obtained. Then, using the conditions of M_{T}(2)-equivalence, an answer is given to the open problem posed in sch. 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 col. 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.

Keywords

• Invariant,regular polygon,taxicab geometry

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