Tridiminished icosahedron

Tridiminished icosahedron
Type	Johnson J62 – J63 – J64
Faces	5 triangles 3 pentagons
Edges	15
Vertices	9
Vertex configuration	${\displaystyle 2\times 3\times (3\times 5^{2})+3\times (3^{3}\times 5)}$
Symmetry group	${\displaystyle C_{3\mathrm {v} }}$
Properties	convex, non-composite
Net

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.^[1] The aftereffect of such construction leaves five equilateral triangles and three regular pentagons.^[2] 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 ${\displaystyle J_{63}}$.^[3] This construction is similar to other Johnson solids as in gyroelongated pentagonal pyramid and metabidiminished icosahedron.^[1]

One can construct the vertices of a tridiminished icosahedron with the following Cartesian coordinates: $${\displaystyle (\pm 1,0,\sigma ),(1,0,-\sigma ),(\sigma ,\pm 1,0),(0,\sigma ,1),(-\sigma ,-1,0),(0,-\sigma ,\pm 1),}$$ where ${\displaystyle \sigma =(1-{\sqrt {5}})/2}$, obtained from the equation of a golden ratio ${\displaystyle \sigma ^{2}=\sigma +1}$.^[4]

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.^[5]

Properties

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

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°.^[6]

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.^[7]

See also

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

References

1. Gailiunas, Paul (2001), "A Polyhedral Byway" (PDF), in Sarhangi, Reza; Jablan, Slavik (eds.), Bridges: Mathematical Connections in Art, Music, and Science, Bridges Conference, pp. 115–122.
2. Berman, Martin (1971), "Regular-faced convex polyhedra", Journal of the Franklin Institute, 291 (5): 329–352, doi:10.1016/0016-0032(71)90071-8, MR 0290245.
3. Francis, Darryl (August 2013), "Johnson solids & their acronyms", Word Ways, 46 (3): 177
4. Koca, Mehmet; Al-Ajmi, Mudhahir; Koca, Nazife Ozdes (2011). "Quaternionic representation of snub 24-cell and its dual polytope derived from ${\displaystyle \mathrm {E} _{8}}$ root system". Linear Algebra and Its Applications. 434 (4): 977–989. arXiv:0906.2109. doi:10.1016/j.laa.2010.10.005. ISSN 0024-3795. S2CID 18278359.
5. Timofeenko, A. V. (2009), "Convex Polyhedra with Parquet Faces" (PDF), Doklady Mathematics, 80 (2): 720–723, doi:10.1134/S1064562409050238.
6. Johnson, Norman W. (1966), "Convex polyhedra with regular faces", Canadian Journal of Mathematics, 18: 169–200, doi:10.4153/CJM-1966-021-8, MR 0185507, S2CID 122006114; see Table III, line 63.
7. Johnson (1966), p. 174.

External links

• Weisstein, Eric W., "Tridiminished icosahedron" ("Johnson Solid") at MathWorld.