Tridiminished icosahedron
In geometry, the tridiminished icosahedron is a Johnson solid that is constructed by removing three pentagonal pyramids from a regular icosahedron.
Construction
The tridiminished icosahedron can be constructed by removing three regular-faced pentagonal pyramid from a regular icosahedron. The aftereffect of such construction leaves five equilateral triangles and three regular pentagons. Since all of its faces are regular polygons and the resulting polyhedron remains convex, the tridiminished icosahedron is a Johnson solid, and it is enumerated as the sixty-third Johnson solid. This construction is similar to other Johnson solids as in gyroelongated pentagonal pyramid and metabidiminished icosahedron.One can construct the vertices of a tridiminished icosahedron with the following Cartesian coordinates:
where, obtained from the equation of a golden ratio.
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.
Properties
The surface area of a tridiminished icosahedron is the sum of all polygonal faces' area: five equilateral triangles and three regular pentagons. Its volume can be ascertained by subtracting the volume of a regular icosahedron from the volume of three pentagonal pyramids. Given that is the edge length of a tridiminished icosahedron, they are:A tridiminished icosahedron has three kinds of dihedral angles. These angles are between two triangles: 138.1°, triangle to pentagon: 100.8°, and two pentagons: 63.4°.