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
One can construct the vertices of a tridiminished icosahedron with the following Cartesian coordinates: <math display=block> (\pm 1, 0, \sigma), (1, 0, -\sigma), (\sigma, \pm 1, 0), (0, \sigma, 1), (-\sigma, -1, 0), (0, -\sigma, \pm 1), </math> where <math> \sigma = (1-\sqrt{5})/2 </math>, obtained from the equation of a golden ratio <math> \sigma^2 = \sigma + 1 </math>.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 regular or Johnson polyhedra.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 from 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>
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°.Template:R
As a 4-polytope cell
The tridiminished icosahedron is a cell of a snub 24-cell, a four-dimensional polytope consisting of 120 regular tetrahedra and 24 icosahedra vertex figures.Template:Sfnp
See also
- Augmented tridiminished icosahedron, a Johnson solid by attaching a tetrahedron from a tridiminished icosahedron