geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...

, the truncated dodecahedron is an Archimedean solid. It has 12 regular

decagon In geometry, a decagon (from the Greek δέκα ''déka'' and γωνία ''gonía,'' "ten angles") is a ten-sided polygon or 10-gon.. The total sum of the interior angles of a simple decagon is 1440°. A self-intersecting ''regular decagon'' i ...

al faces, 20 regular

triangular A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- collinea ...

faces, 60 vertices and 90 edges.

Geometric relations

This

polyhedron In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on ...

can be formed from a

regular dodecahedron A regular dodecahedron or pentagonal dodecahedron is a dodecahedron that is regular, which is composed of 12 regular pentagonal faces, three meeting at each vertex. It is one of the five Platonic solids. It has 12 faces, 20 vertices, 30 ed ...

by truncating (cutting off) the corners so the pentagon faces become

s and the corners become

triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colline ...

s. It is used in the

cell-transitive In geometry, a tessellation of dimension (a plane tiling) or higher, or a polytope of dimension (a polyhedron) or higher, is isohedral or face-transitive if all its faces are the same. More specifically, all faces must be not merely congrue ...

hyperbolic space-filling tessellation, the bitruncated icosahedral honeycomb.

Area and volume

The area ''A'' and the

volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). Th ...

''V'' of a truncated dodecahedron of edge length ''a'' are: :

\begin
A &= 5 \left(\sqrt+6\sqrt\right) a^2 &&\approx 100.990\,76a^2 \\
V &= \tfrac \left(99+47\sqrt\right) a^3 &&\approx 85.039\,6646a^3
\end

Cartesian coordinates

Cartesian coordinates for the vertices of a truncated

dodecahedron In geometry, a dodecahedron (Greek , from ''dōdeka'' "twelve" + ''hédra'' "base", "seat" or "face") or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagon ...

with edge length 2''φ'' − 2, centered at the origin, are all even permutations of: :(0, ±, ±(2 + ''φ'')) :(±, ±''φ'', ±2''φ'') :(±''φ'', ±2, ±(''φ'' + 1)) where ''φ'' = is the

golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ( ...

Orthogonal projections

The ''truncated dodecahedron'' has five special

orthogonal projection In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself (an endomorphism) such that P\circ P=P. That is, whenever P is applied twice to any vector, it gives the same result as if it wer ...

s, centered: on a vertex, on two types of edges, and two types of faces. The last two correspond to the A₂ and H₂

Coxeter plane In mathematics, the Coxeter number ''h'' is the order of a Coxeter element of an irreducible Coxeter group. It is named after H.S.M. Coxeter. Definitions Note that this article assumes a finite Coxeter group. For infinite Coxeter groups, there ar ...

Spherical tilings and Schlegel diagrams

The truncated dodecahedron can also be represented as a

spherical tiling In geometry, a spherical polyhedron or spherical tiling is a tiling of the sphere in which the surface is divided or partitioned by great arcs into bounded regions called spherical polygons. Much of the theory of symmetrical polyhedra is most c ...

, and projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane.

Schlegel diagram In geometry, a Schlegel diagram is a projection of a polytope from \mathbb^d into \mathbb^ through a point just outside one of its facets. The resulting entity is a polytopal subdivision of the facet in \mathbb^ that, together with the orig ...

s are similar, with a perspective projection and straight edges.

Vertex arrangement

It shares its

vertex arrangement In geometry, a vertex arrangement is a set of points in space described by their relative positions. They can be described by their use in polytopes. For example, a ''square vertex arrangement'' is understood to mean four points in a plane, equa ...

with three nonconvex uniform polyhedra:

Related polyhedra and tilings

It is part of a truncation process between a dodecahedron and icosahedron: This polyhedron is topologically related as a part of sequence of uniform truncated polyhedra with

vertex configuration In geometry, a vertex configurationCrystallography ...

s (3.2''n''.2''n''), and 'n'',3

Coxeter group In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean refle ...

symmetry.

Truncated dodecahedral graph

In the

mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

field of

graph theory In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...

, 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.

Notes

References

* (Section 3-9) *

External links

* ** *
Editable printable net of a truncated dodecahedron with interactive 3D viewThe Uniform Polyhedra
The Encyclopedia of Polyhedra {{Polyhedron navigator Uniform polyhedra Archimedean solids Truncated tilings