Tridiminished icosahedron

The tridiminished icosahedron can be constructed by removing three regular-faced pentagonal pyramid from a regular icosahedron.^[1] The aftereffect of such construction leaves five equilateral triangles and three regular pentagons.^[2] Since all of its faces are regular polygons and the resulting polyhedron remains convex, the tridiminished icosahedron is a Johnson solid, after American mathematician Norman W. Johnson who listed the 92 such polyhedra. It is enumerated as the sixty-third Johnson solid ${\displaystyle J_{63}}$.^[3] This construction is similar to other Johnson solids as in gyroelongated pentagonal pyramid and metabidiminished icosahedron.^[1]

One can construct the vertices of a tridiminished icosahedron with the following Cartesian coordinates: $${\displaystyle (\pm 1,0,\varphi ),(1,0,-\varphi ),(\varphi ,\pm 1,0),(0,\varphi ,1),(-\varphi ,-1,0),(0,-\varphi ,\pm 1),}$$ where ${\displaystyle \varphi =(1-{\sqrt {5}})/2}$ is a golden ratio, obtained from the equation ${\displaystyle \varphi ^{2}=\varphi +1}$.^[4]

The tridiminished icosahedron is a non-composite polyhedron. That is, no plane intersects its surface only in edges, so that it cannot be thereby divided into two or more convex, regular-faced polyhedra.^[5]

The surface area of a tridiminished icosahedron ${\displaystyle A}$ is the sum of all polygonal faces' area: five equilateral triangles and three regular pentagons. Its volume ${\displaystyle V}$ can be ascertained by subtracting the volume of a regular icosahedron from the volume of three pentagonal pyramids. Given that ${\displaystyle a}$ is the edge length of a tridiminished icosahedron, they are:^[2] $${\displaystyle {\begin{aligned}A&={\frac {5{\sqrt {3}}+3{\sqrt {5\left(5+2{\sqrt {5}}\right)}}}{4}}a^{2}&\approx 7.3265a^{2},\\V&={\frac {15+7{\sqrt {5}}}{24}}a^{3}&\approx 1.2772a^{3}.\end{aligned}}}$$

A tridiminished icosahedron has a three-dimensional symmetry group ${\displaystyle C_{3\mathrm {v} }}$ of order six. It has three kinds of dihedral angles. These angles are yielded in the following calculations.^[6]

• An angle between two adjacent triangles is around 138.1°, equal to that of a regular icosahedron and that of a pentagonal pyramid.
• A triangle-to-pentagon angle is around 100.8°. The result is obtained by subtracting the pentagon-to-triangle angle of a pentagonal pyramid from the triangle-to-triangle angle of a regular icosahedron.
• An angle between two adjacent pentagons is around 63.4°. The result is obtained by subtracting the pentagon-to-triangle angle of a pentagonal pyramid from the tridiminished icosahedron's triangle-to-pentagon angle.

The tridiminished icosahedron is a vertex figure of a snub 24-cell, a four-dimensional polytope consisting of 120 regular tetrahedra and 24 icosahedra as the cells.^[7]