Tridiminished icosahedron

In geometry, the tridiminished icosahedron is a Johnson solid that is constructed by removing three pentagonal pyramids from a regular icosahedron. File:J63 tridiminished icosahedron.stl

Construction

The tridiminished icosahedron can be constructed by removing three regular-faced pentagonal pyramid from a regular icosahedron.Template:R The aftereffect of such construction leaves five equilateral triangles and three regular pentagons.Template:R 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 $J_{63}$.Template:R This construction is similar to other Johnson solids as in gyroelongated pentagonal pyramid and metabidiminished icosahedron.Template:R

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

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.Template:R

Properties

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

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

• 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.

As a vertex figure

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.Template:Sfnp

See also

• Augmented tridiminished icosahedron, a Johnson solid by attaching a tetrahedron from a tridiminished icosahedron

External links

• Template:MathWorld2