Truncated dodecahedron

Truncated dodecahedron
Type	Archimedean solid
Faces	32
Edges	90
Symmetry group	icosahedral symmetry ${\displaystyle \mathrm {I} _{\mathrm {h} }}$
Dihedral angle (degrees)	10-10: 116.57° 3-10: 142.62°
Dual polyhedron	Triakis icosahedron
Vertex figure
Net

In geometry, the truncated dodecahedron is an Archimedean solid. It has 12 regular decagonal faces, 20 regular triangular faces, 60 vertices and 90 edges.

Construction

The truncated dodecahedron is constructed from a regular dodecahedron by cutting all of its vertices off, a process known as truncation. This process of construction results in the pentagonal faces becoming decagonal faces, and the vertices become triangles. Therefore, it has 32 faces, 90 edges, and 60 vertices.^[1]

The truncated dodecahedron may also be constructed by using Cartesian coordinates. With an edge length ${\displaystyle 2\varphi -2}$ centered at the origin, they are all even permutations of $${\displaystyle \left(0,\pm {\frac {1}{\varphi }},\pm (2+\varphi )\right),\qquad \left(\pm {\frac {1}{\varphi }},\pm \varphi ,\pm 2\varphi \right),\qquad \left(\pm \varphi ,\pm 2,\pm (\varphi +1)\right),}$$ where ${\textstyle \varphi ={\frac {1+{\sqrt {5}}}{2}}}$ is the golden ratio.^[2]

Properties

The surface area ${\displaystyle A}$ and the volume ${\displaystyle V}$ of a truncated dodecahedron of edge length ${\displaystyle a}$ are:^[1] $${\displaystyle {\begin{aligned}A&=5\left({\sqrt {3}}+6{\sqrt {5+2{\sqrt {5}}}}\right)a^{2}&&\approx 100.991a^{2}\\V&={\frac {5}{12}}\left(99+47{\sqrt {5}}\right)a^{3}&&\approx 85.040a^{3}\end{aligned}}}$$

The dihedral angle of a truncated dodecahedron between two regular dodecahedral faces is 116.57°, and that between triangle-to-dodecahedron is 142.62°

The truncated dodecahedron is an Archimedean solid, meaning it is a highly symmetric and semi-regular polyhedron, and two or more different regular polygonal faces meet in a vertex.^[3] It has the same symmetry as the regular icosahedron, the icosahedral symmetry.^[4] The polygonal faces that meet for every vertex are one equilateral triangle and two regular decagon, and the vertex figure of a truncated dodecahedron is ${\displaystyle 3\cdot 10^{2}}$. The dual of a truncated dodecahedron is triakis icosahedron, a Catalan solid,^[5] which shares the same symmetry as the truncated dodecahedron.^[6]

The truncated dodecahedron is non-chiral, meaning it is congruent to its own mirror image.^[4]

Truncated dodecahedral graph

Truncated dodecahedral graph
5-fold symmetry Schlegel diagram
Vertices	60
Edges	90
Automorphisms	120
Chromatic number	3
Chromatic index	3
Properties	Cubic, Hamiltonian, regular, zero-symmetric
Table of graphs and parameters

In the mathematical field of graph theory, a truncated dodecahedral graph is the graph of vertices and edges of the truncated dodecahedron, one of the Archimedean solids. It has 60 vertices and 90 edges, and is a cubic Archimedean graph.^[7]

Circular

References

Further reading

• Cromwell, P. (1997). Polyhedra. United Kingdom: Cambridge. pp. 79–86 Archimedean solids. ISBN 0-521-55432-2.

External links

• Weisstein, Eric W., "Truncated dodecahedron" ("Archimedean solid") at MathWorld.
  • Weisstein, Eric W. "Truncated dodecahedral graph". MathWorld.
• Klitzing, Richard. "3D convex uniform polyhedra o3x5x - tid".
• Editable printable net of a truncated dodecahedron with interactive 3D view

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Archimedean solids

Truncated tetrahedron	Truncated cube	Truncated octahedron	Truncated dodecahedron	Truncated icosahedron	Cuboctahedron
Icosidodecahedron	Rhombicuboctahedron	Truncated cuboctahedron	Rhombicosidodecahedron	Truncated icosidodecahedron	Snub cube
Snub dodecahedron