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 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, and 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

The tridiminished icosahedron is non-composite polyhedron, meaning it is convex polyhedron that cannot be separated by a plane into two or more regular polyhedrons.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 with the volume of three pentagonal pyramids. Given that $a$ is the edge length of a tridiminished icosahedron, they are:Template:R $$\begin{array}{ccc}A& =\frac{5\sqrt{3}+3\sqrt{5(5+2\sqrt{5})}}{4}{a}^2& \approx 7.3265a^2,\\ V& =\frac{15+7\sqrt{5}}{24}{a}^3& \approx 1.2772a^3.\end{array}$$

See also

• Snub 24-cell, a 4-polytope whose vertex figure is a tridiminished icosahedron

External links

• Template:MathWorld2