Rhombicosidodecahedron

Rhombicosidodecahedron
(Click here for rotating model)
Type	Archimedean solid
Elements	F = 62, E = 120, V = 60 (χ = 2)
Faces by sides	20{3}+30{4}+12{5}
Schläfli symbol	$r\begin{Bmatrix} 3 \\ 5 \end{Bmatrix}$
Wythoff symbol	3 5 \| 2
Coxeter-Dynkin
Symmetry	I_h
References	U₂₇, C₃₀, W₁₄
Properties	Semiregular convex
Colored faces	3. In Geometry an Archimedean solid is a highly symmetric semi-regular convex Polyhedron composed of two or more types of Regular polygons meeting In Mathematics, and more specifically in Algebraic topology and Polyhedral combinatorics, the Euler characteristic is a Topological invariant In Mathematics, the Schläfli symbol is a notation of the form {pqr In Geometry, a Wythoff symbol is a short-hand notation created by mathematician Willem Abraham Wythoff, for naming the regular and semiregular polyhedra using a List of Symmetry groups on the sphere Spherical symmetry groups are also called Point groups in three dimensions. A regular Icosahedron has 60 rotational (or orientation-preserving symmetries and a total of 120 symmetries including transformations that combine a reflection and a rotation A uniform polyhedron is a Polyhedron which has Regular polygons as faces and is Transitive on its vertices (i A uniform polyhedron is a Polyhedron which has Regular polygons as faces and is Transitive on its vertices (i Harold Scott MacDonald "Donald" Coxeter CC ( February 9, 1907 – March 31, 2003) is regarded as one of the great This table contains an indexed list of the Uniform and stellated polyhedra from the book Polyhedron Models, by Magnus Wenninger 4. 5. 4 (Vertex figure)
Deltoidal hexecontahedron (dual polyhedron)	Net

The rhombicosidodecahedron, or small rhombicosidodecahedron, is an Archimedean solid. In Geometry a vertex figure is broadly speaking the figure exposed when a corner of a Polyhedron or Polytope is sliced off See also Deltoidal icositetrahedron In Geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the In Geometry the net of a Polyhedron is an arrangement of edge-joined Polygons in the plane which can be folded (along edges to become the faces of the polyhedron In Geometry an Archimedean solid is a highly symmetric semi-regular convex Polyhedron composed of two or more types of Regular polygons meeting It has 20 regular triangular faces, 30 regular square faces, 12 regular pentagonal faces, 60 vertices and 120 edges. A triangle is one of the basic Shapes of Geometry: a Polygon with three corners or vertices and three sides or edges which are Line Classification A square (regular Quadrilateral) is a special case of a Rectangle as it has four right angles and equal parallel sides Regular pentagons The term pentagon is commonly used to mean a regular convex pentagon, where all sides are equal and all interior angles are equal (to

The name rhombicosidodecahedron refers to the fact that the 30 square faces lie in the same planes as the 30 faces of the rhombic triacontahedron which is dual to the icosidodecahedron. In Geometry, the rhombic triacontahedron is a convex Polyhedron with 30 rhombic faces An icosidodecahedron is a Polyhedron with twenty triangular faces and twelve pentagonal faces

It can also called a cantellated dodecahedron or a cantellated icosahedron from truncation operations of the uniform polyhedron. In Geometry, a cantellation is an operation in any dimension that cuts a Regular polytope edges and vertices creating a new facet in place of each edge and vertex A uniform polyhedron is a Polyhedron which has Regular polygons as faces and is Transitive on its vertices (i

1 Geometric relations
2 Cartesian coordinates
3 Vertex arrangement
4 See also
5 References
6 External links

Geometric relations

If you blow up an icosahedron by moving the faces away from the origin the right amount, without changing the orientation or size of the faces, and do the same to its dual dodecahedron, and patch the square holes in the result, you get a rhombicosadodecahedron. In Geometry, expansion is a Polytope operation where facets are separated and moved radially apart and new facets are formed at separated elements (vertices In Geometry, an icosahedron ( Greek: eikosaedron, from eikosi twenty + hedron seat /ˌaɪ In Mathematics, the origin of a Euclidean space is a special point, usually denoted by the letter O, used as a fixed point of reference A dodecahedron is any Polyhedron with twelve faces but usually a regular dodecahedron is meant a Platonic solid composed of twelve regular Pentagonal Therefore, it has the same number of triangles as an icosahedron and the same number of pentagons as a dodecahedron, with a square for each edge of either.

The rhombicosidodecahedron shares the vertex arrangement with the small stellated truncated dodecahedron, and with the uniform compounds of 6 or 12 pentagrammic prisms. In Geometry, the small stellated truncated dodecahedron is a nonconvex Uniform polyhedron, indexed as U58 In Geometry, the pentagrammic prism is one in an infinite set of nonconvex prisms formed by square sides and two regular Star polygon caps in this case

The Zometool kits for making geodesic domes and other polyhedra use slotted balls as connectors. The term zome is used in several related senses A zome in the originalsense is a building using unusual geometries(different from the standard house or other building which A geodesic dome is an almost spherical shell structure based on a network of Great circles ( Geodesics lying approximately on the surface of a Sphere The balls are "expanded" small rhombicosidodecahedra, with the squares replaced by rectangles. The expansion is chosen so that the resulting rectangles are golden rectangles.

Cartesian coordinates

Cartesian coordinates for the vertices of a rhombicosidodecahedron centered at the origin are

(±1, ±1, ±τ³),

(±τ³, ±1, ±1),

(±1, ±τ³, ±1),

(±τ², ±τ, ±2τ),

(±2τ, ±τ², ±τ),

(±τ, ±2τ, ±τ²),

(±(2+τ), 0, ±τ²),

(±τ², ±(2+τ), 0),

(0, ±τ², ±(2+τ)),

where τ = (1+√5)/2 is the golden ratio (also written φ). In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane 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

Vertex arrangement

The rhombicosidodecahedron shares its vertex arrangement with 3 nonconvex uniform polyhedrons:

Small dodecicosidodecahedron

Small rhombidodecahedron

Small stellated truncated dodecahedron

References

Williams, Robert (1979). See Vertex figure for the local description of faces around a vertex of a polyhedron or tiling A uniform polyhedron is a Polyhedron which has Regular polygons as faces and is Transitive on its vertices (i In Geometry, the small dodecicosidodecahedron is a nonconvex Uniform polyhedron, indexed as U33 In Geometry, the small rhombidodecahedron is a nonconvex Uniform polyhedron, indexed as U39 In Geometry, the small stellated truncated dodecahedron is a nonconvex Uniform polyhedron, indexed as U58 A dodecahedron is any Polyhedron with twelve faces but usually a regular dodecahedron is meant a Platonic solid composed of twelve regular Pentagonal In Geometry, an icosahedron ( Greek: eikosaedron, from eikosi twenty + hedron seat /ˌaɪ An icosidodecahedron is a Polyhedron with twenty triangular faces and twelve pentagonal faces The rhombicuboctahedron, or small rhombicuboctahedron, is an Archimedean solid with eight triangular and eighteen square faces The truncated icosidodecahedron is an Archimedean solid. It has 30 regular square faces 20 regular Hexagonal faces 12 regular Decagonal faces Robert Williams, Rob Williams or Bob Williams may refer to United Kingdom Sir Robert Williams 2nd Baronet (c The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X. (Section 3-9)

External links

Eric W. Weisstein, Small Rhombicosidodecahedron (Archimedean solid) at MathWorld. Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W
The Uniform Polyhedra
Virtual Reality Polyhedra The Encyclopedia of Polyhedra
The Rhombi-Cosi-Dodecahedron Website

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