The Isometry Group of Chinese Checker Space

Abstract

In this article, we, firstly, find that the spheres in the Chinese Checkers space are deltoidal icositetrahedrons. Then we show that the group of isometries of the 3-dimensional space with respect to Chinese Checkers metric is the semi-direct product of deltoidal icositetrahedron group G(D) and T (3), where G(D) is the (Euclidean) symmetry group of the deltoidal icositetrahedron and T (3) is the group of all translations of the 3-dimensional space.

Isometry Groups of Chamfered Cube and Chamfered Octahedron Spaces

“…Those properties are invariant under the group of motions and geometry studies those properties. There are a lot of studies about group of isometries of a space (See [7,10,11])…”

“…So there are some metrics which unit spheres are convex polyhedrons. That is, convex polyhedrons are associated with some metrics (See [3,4,6,7,[9][10][11][12][13][14][15]). This influence us to the question "Are there some metrics of which unit spheres are the Catalan Solids?".…”

Inversions and Fractal Patterns in Alpha Plane

“…Then, some authors developed and studied on various aspect of these topics. For example, Gelişgen and Kaya [14,15] extended the α−distance to three and n dimensional spaces, respectively. Afterwards, Colako glu [8] extended the α−metric for α ∈ [0, π/2].…”

Truncated Truncated Dodecahedron ve Truncated Truncated Icosahedron Uzayları

“…In geometry, a polyhedron is simply a three-dimensional solid which consists of a collection of polygons, usually joined at their edges. The term "polyhedron" is used somewhat differently in algebraic topology, where it is defined as a space that can be built from such "building blocks" as line segments, triangles, tetrahedra, and their higher dimensional analogs by "gluing them together" along their faces [1]. The word derives from the Greek poly(many) plus the Indo-European hedron(seat).…”

“…Taxicab metric's unit sphere is an octahedron, another Platonic Solid. In [1,2,5,6,7,8,9,10,11,12] the authors give some metrics which the spheres of the 3-dimensional analytical space furnished by these metrics are some of Platonic solids, Archimedian solids and Catalan solids. So there are some metrics which unit spheres are convex polyhedrons.…”