Tridiminished icosahedron

Template:Short description {{#invoke:Infobox|infobox}}Template:Template other </math>

| vertex_config = <math> 2 \times 3 \times (3 \times 5^2) + 3 \times (3^3 \times 5) </math>
| properties = convex,
non-composite
| net = Johnson solid 63 net.png

}}

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 <math> J_{63} </math>.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 <math> A </math> is the sum of all polygonal faces' area: five equilateral triangles and three regular pentagons. Its volume <math> V </math> can be ascertained by subtracting the volume of a regular icosahedron with the volume of three pentagonal pyramids. Given that <math> a </math> is the edge length of a tridiminished icosahedron, they are:Template:R <math display="block"> \begin{align}

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{align} </math>

See also

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

External links

• Template:MathWorld2