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Truncated dodecahedron
 
 
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In geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, the truncated dodecahedron is an Archimedean solid
Archimedean solid

In geometry an Archimedean solid is a highly symmetric, semi-regular convex set polyhedron composed of two or more types of regular polygons meeting in identical vertex ....
. It has 12 regular decagon
Decagon

In geometry, a decagon is any polygon with ten sides and ten angles, and usually refers to a regular polygon decagon, having all sides of equal length and all internal angles equal to 4π/5 ....
al faces, 20 regular triangular
Triangle

A triangle is one of the basic shapes of geometry: a polygon with three corners or wikt:vertex and three sides or edges which are line segments....
 faces, 60 vertices and 90 edges.








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In geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, the truncated dodecahedron is an Archimedean solid
Archimedean solid

In geometry an Archimedean solid is a highly symmetric, semi-regular convex set polyhedron composed of two or more types of regular polygons meeting in identical vertex ....
. It has 12 regular decagon
Decagon

In geometry, a decagon is any polygon with ten sides and ten angles, and usually refers to a regular polygon decagon, having all sides of equal length and all internal angles equal to 4π/5 ....
al faces, 20 regular triangular
Triangle

A triangle is one of the basic shapes of geometry: a polygon with three corners or wikt:vertex and three sides or edges which are line segments....
 faces, 60 vertices and 90 edges.

Geometric relations

This polyhedron
Polyhedron

|}A polyhedron is often defined as a geometry object with flat faces and straight edges .This definition of a polyhedron is not very precise, and to a modern mathematician is quite unsatisfactory....
 can be formed from a dodecahedron
Dodecahedron

A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex....
 by truncating
Truncation (geometry)

In geometry, a truncation is an operation in any dimension that cuts polytope vertices, creating a new Facet in place of each vertex....
 (cutting off) the corners so the pentagon
Pentagon

In geometry, a pentagon is any five-sided polygon. A pentagon may be simple or self-intersecting. The internal angles in a simple pentagon total 540?....
 faces become decagon
Decagon

In geometry, a decagon is any polygon with ten sides and ten angles, and usually refers to a regular polygon decagon, having all sides of equal length and all internal angles equal to 4π/5 ....
s and the corners become triangle
Triangle

A triangle is one of the basic shapes of geometry: a polygon with three corners or wikt:vertex and three sides or edges which are line segments....
s.

It is part of a truncation process between a dodecahedron and icosahedron:
Uniform Polyhedron 53 T0

Dodecahedron
Dodecahedron

A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex....
Uniform Polyhedron 53 T01

Truncated dodecahedron
Uniform Polyhedron 53 T1

Icosidodecahedron
Icosidodecahedron

An icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon....
Uniform Polyhedron 53 T12

Truncated icosahedron
Truncated icosahedron

The truncated icosahedron is an Archimedean solid. It comprises 12 regular pentagon faces, 20 regular hexagon faces, 60 vertices and 90 edges....
Uniform Polyhedron 53 T2

Icosahedron
Icosahedron

In geometry, an icosahedron isany polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangle s as faces....


It shares its vertex arrangement
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....
 with three uniform star polyhedra:
Great Ditrigonal Dodecicosidodecahedron

U42
Great ditrigonal dodecicosidodecahedron

In geometry, the great ditrigonal dodecicosidodecahedron is a nonconvex uniform polyhedron, indexed as U42.It shares its vertex arrangement with the truncated dodecahedron....
Great Icosicosidodecahedron

U48
Great icosicosidodecahedron

In geometry, the great icosicosidodecahedron is a nonconvex uniform polyhedron, indexed as U48.It shares its vertex arrangement with the truncated dodecahedron....
Great Dodecicosahedron

U63
Great dodecicosahedron

In geometry, the great dodecicosahedron is a nonconvex uniform polyhedron, indexed as U63.It shares its vertex arrangement with the truncated dodecahedron....


It is used in the cell-transitive hyperbolic space-filling tessellation, the bitruncated icosahedral honeycomb
Bitruncation

In geometry, a bitruncation is an operation on regular polytopes. It represents a Truncation beyond Rectification . The original edges are lost completely and the original faces remain as smaller copies of themselves....
.

Area and volume

The area A and the volume
Volume

The volume of any solid, liquid, plasma, vacuum or theoretical object is how much three-dimensional space it occupies, often quantified numerically....
 V of a truncated dodecahedron of edge length a are:

Cartesian coordinates

The following Cartesian coordinates define the vertices of a truncated
Truncation (geometry)

In geometry, a truncation is an operation in any dimension that cuts polytope vertices, creating a new Facet in place of each vertex....
 dodecahedron
Dodecahedron

A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex....
 with edge length 2(t-1), centered at the origin:

(0, ±1/t, ±(2+t))
(±(2+t), 0, ±1/t)
(±1/t, ±(2+t), 0)
(±1/t, ±t, ±2t)
(±2t, ±1/t, ±t)
(±t, ±2t, ±1/t)
(±t, ±2, ±t2)
(±t2, ±t, ±2)
(±2, ±t2, ±t)


where t = (1+√5)/2 is the golden ratio
Golden ratio

In mathematics and the arts, two quantities are in the golden ratio if the ratio between the sum of those quantities and the larger one is the same as the ratio between the larger one and the smaller....
 (also written f).

See also

  • Spinning truncated dodecahedron
  • icosahedron
    Icosahedron

    In geometry, an icosahedron isany polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangle s as faces....
  • icosidodecahedron
    Icosidodecahedron

    An icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon....
  • truncated icosahedron
    Truncated icosahedron

    The truncated icosahedron is an Archimedean solid. It comprises 12 regular pentagon faces, 20 regular hexagon faces, 60 vertices and 90 edges....


External links

  • The Encyclopedia of Polyhedra