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Truncated icosidodecahedron
 
 
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The truncated icosidodecahedron 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 30 square
Square (geometry)

In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....
 faces, 20 regular hexagon
Hexagon

In geometry, a hexagon is a polygon with six edges and six Vertex . A regular hexagon has Schl?fli symbol ....
al faces, 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, 120 vertices and 180 edges. Since each of its faces has point symmetry (equivalently, 180° rotation
Rotation

A rotation is a movement of an object in a circular motion. A two-dimensional object rotates around a center of rotation. A Three-dimensional space object rotates around a line called an axis....
al symmetry), the truncated icosidodecahedron is a zonohedron
Zonohedron

A zonohedron is a convex set polyhedron where every face is a polygon with point symmetry or, equivalently, symmetry under rotations through 180?....
.


The name truncated icosidodecahedron, originally given by Johannes Kepler
Johannes Kepler

Johannes Kepler was a Germans mathematician, astronomer and astrologer, and key figure in the 17th century Scientific revolution. He is best known for his eponymous Kepler's laws of planetary motion, codified by later astronomers based on his works Astronomia nova, Harmonices Mundi, and Epitome of Copernican Astrononomy....
, is somewhat misleading.






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The truncated icosidodecahedron 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 30 square
Square (geometry)

In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....
 faces, 20 regular hexagon
Hexagon

In geometry, a hexagon is a polygon with six edges and six Vertex . A regular hexagon has Schl?fli symbol ....
al faces, 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, 120 vertices and 180 edges. Since each of its faces has point symmetry (equivalently, 180° rotation
Rotation

A rotation is a movement of an object in a circular motion. A two-dimensional object rotates around a center of rotation. A Three-dimensional space object rotates around a line called an axis....
al symmetry), the truncated icosidodecahedron is a zonohedron
Zonohedron

A zonohedron is a convex set polyhedron where every face is a polygon with point symmetry or, equivalently, symmetry under rotations through 180?....
.

Other names


Alternate interchangeable names include:
  • Great rhombicosidodecahedron
  • Rhombitruncated icosidodecahedron
  • Omnitruncated
    Omnitruncation (geometry)

    In geometry, an omnitruncation is an operation applied to a regular polytope in a Wythoff construction that creates a maximum number of facets....
     icosidodecahedron


The name truncated icosidodecahedron, originally given by Johannes Kepler
Johannes Kepler

Johannes Kepler was a Germans mathematician, astronomer and astrologer, and key figure in the 17th century Scientific revolution. He is best known for his eponymous Kepler's laws of planetary motion, codified by later astronomers based on his works Astronomia nova, Harmonices Mundi, and Epitome of Copernican Astrononomy....
, is somewhat misleading. If you truncate
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....
 an 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....
 by cutting the corners off, you do not get this uniform figure: instead of square
Square (geometry)

In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....
s you get golden rectangle
Golden rectangle

A golden rectangle is a rectangle whose side lengths are in the golden ratio, 1: , that is, or approximately 1:1.618.A distinctive feature of this shape is that when a square section is removed, the remainder is another golden rectangle; that is, with the same proportionality s as the first....
s. However, the resulting figure is topologically
Topology

Topology is a major area of mathematics that has emerged through the development of concepts from geometry and set theory, such as those of space, dimension, shape, transformation and others....
 equivalent to this and can always be deformed until the faces are regular.
Icosidodecahedron

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....

A literal geometric truncation
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....
 of the icosidodecahedron produces rectangular
Rectangle

In geometry, a rectangle is a Closed set planar quadrilateral with four right angles. A rectangle with vertices ABCD would be denoted as .A rectangle with adjacent sides of lengths a and b has area ab and diagonals of equal length ....
 faces rather than squares
Square (geometry)

In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....
.


The alternative name great rhombicosidodecahedron (as well as rhombitruncated icosidodecahedron) refers to the fact that the 30 square faces lie in the same planes as the 30 faces of the rhombic triacontahedron
Rhombic triacontahedron

In geometry, the rhombic triacontahedron is a convex set polyhedron with 30 rhombus faces. It is an Archimedean solid solid, or a Catalan solid....
 which is dual to the 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....
. Compare to small rhombicosidodecahedron.

One unfortunate point of confusion is that there is a nonconvex uniform polyhedron of the same name. See uniform great rhombicosidodecahedron
Uniform great rhombicosidodecahedron

In geometry, the uniform great rhombicosidodecahedron is a nonconvex uniform polyhedron, indexed as U67. It is also called the quasirhombicosidodecahedron....
.

Area and volume

The surface area A and the volume V of the truncated icosidodecahedron of edge length a are:

Cartesian coordinates

Cartesian coordinates for the vertices of a truncated icosidodecahedron with edge length 2t − 2, centered at the origin, are all the even permutations of
(±1/t, ±1/t, ±(3+t)),
(±2/t, ±t, ±(1+2t)),
(±1/t, ±t2, ±(−1+3t)),
(±(-1+2t), ±2, ±(2+t)) and
(±t, ±3, ±2t),
where t = (1 + v5)/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....
.

Spherical tiling


The truncated icosidodecahedron
Truncated icosidodecahedron

The truncated icosidodecahedron is an Archimedean solid. It has 30 square faces, 20 regular hexagonal faces, 12 regular decagonal faces, 120 vertices and 180 edges....
 can also be represented as a spherical tiling, and projected onto the plane via a stereographic projection
Stereographic projection

In geometry, the stereographic projection is a particular mapping that projects a sphere onto a plane . The projection is defined on the entire sphere, except at one point — the projection point....
. This projection is conformal
Conformal map

In mathematics, a conformal map is a function which preserves angles. In the most common case the function is between domains in the complex plane....
, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane.


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 ....
-centered
Hexagon
Hexagon

In geometry, a hexagon is a polygon with six edges and six Vertex . A regular hexagon has Schl?fli symbol ....
-centered
square
Square (geometry)

In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....
-centered
Spherical tilingStereographic projections (face-centered)


See also

  • Spinning great rhombicosidodecahedron
  • 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....
  • great truncated icosidodecahedron
    Great truncated icosidodecahedron

    In geometry, the great truncated icosidodecahedron is a nonconvex uniform polyhedron, indexed as U68....
  • 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....
  • truncated cuboctahedron
    Truncated cuboctahedron

    The truncated cuboctahedron is an Archimedean solid. It has 12 Square faces, 8 regular hexagonal faces, 6 regular octagonal faces, 48 vertices and 72 edges....


External links

  • The Encyclopedia of Polyhedra