- Source: 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. The aftereffect of such construction leaves five equilateral triangles and three regular pentagons. 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
{\displaystyle J_{63}}
. This construction is similar to other Johnson solids as in gyroelongated pentagonal pyramid and metabidiminished icosahedron.
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.
Properties
The surface area of a tridiminished icosahedron
A
{\displaystyle A}
is the sum of all polygonal faces' area: five equilateral triangles and three regular pentagons. Its volume
V
{\displaystyle V}
can be ascertained by subtracting the volume of a regular icosahedron with the volume of three pentagonal pyramids. Given that
a
{\displaystyle a}
is the edge length of a tridiminished icosahedron, they are:
A
=
5
3
+
3
5
(
5
+
2
5
)
4
a
2
≈
7.3265
a
2
,
V
=
15
+
7
5
24
a
3
≈
1.2772
a
3
.
{\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}}}
See also
Snub 24-cell, a 4-polytope whose vertex figure is a tridiminished icosahedron
References
External links
Weisstein, Eric W., "Tridiminished icosahedron" ("Johnson Solid") at MathWorld.
Kata Kunci Pencarian:
- Daftar bentuk matematika
- Tridiminished icosahedron
- Augmented tridiminished icosahedron
- Regular icosahedron
- Gyroelongated pentagonal pyramid
- Johnson solid
- Enneahedron
- List of polygons, polyhedra and polytopes
- List of mathematical shapes
- Pentagonal pyramid
- List of small polyhedra by vertex count
