The Group of Transformations which Preserving Distance on Some Polyhedral Space

Abstract

$3$-dimensional analytical space which is covered by a metric is called a Minkowski geometry. In the Minkowski geometries, the unit balls are symmetric, convex closed sets. So there are Minkowski geometries which unit spheres are rhombic triacontahedron, icosidodecahedron and disdyakis triacontahedron. One of the fundamental problems in geometry for a space with a metric is to determine the group of isometries. In this article we show that the group of isometries of the $3-$dimensional space covered by $RT-metric$, $ID-metric$ and $DT-metric$ are the semi-direct product of $I_{h} $ and $T(3)$, where Icosahedral group $I_{h}$ is the (Euclidean) symmetry group of the icosahedron and $T(3)$ is the group of all translations of the $3-$ dimensional space.

Keywords

• Catalan solids
• Archimedean solids
• isometry group
• icosahedral symmetry
• rhombic triacontahedron
• icosidodecahedron
• disdyakis triacontahedron

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