Rhombic icosahedron

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Rhombic icosahedron
Rhombic icosahedron.png
Type Zonohedron
Faces 20 congruent golden rhombi
Edges 40
Vertices 22
Symmetry group D5d = D5v, [2+,10], (2*5)
Properties convex
A rhombic icosahedron Rhombic icosahedron.stl
A rhombic icosahedron

The rhombic icosahedron is a polyhedron shaped like an oblate sphere. Its 20 faces are congruent golden rhombi; [1] 3, 4, or 5 faces meet at each vertex. It has 5 faces (green on top figure) meeting at each of its 2 poles; these 2 vertices lie on its axis of 5-fold symmetry, which is perpendicular to 5 axes of 2-fold symmetry through the midpoints of opposite equatorial edges (example on top figure: most left-hand and most right-hand mid-edges). Its other 10 faces follow its equator, 5 above and 5 below it; each of these 10 rhombi has 2 of its 4 sides lying on this zig-zag skew decagon equator. The rhombic icosahedron has 22 vertices. It has D5d, [2+,10], (2*5) symmetry group, of order 20; thus it has a center of symmetry (since 5 is odd).

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Even though all its faces are congruent, the rhombic icosahedron is not face-transitive, since one can distinguish whether a particular face is near the equator or near a pole by examining the types of vertices surrounding this face.

Zonohedron

The rhombic icosahedron is a zonohedron.

The rhombic icosahedron has 5 sets of 8 parallel edges, described as 85 belts.

Rhombic icosahedron 5-color-paralleledges.png The edges of the rhombic icosahedron can be grouped in 5 parallel-sets, seen in this wireframe orthogonal projection.

The rhombic icosahedron forms the convex hull of the vertex-first[ clarification needed ] projection of a 5-cube to 3 dimensions[ citation needed ]. The 32 vertices of a 5-cube map into the 22 exterior vertices of the rhombic icosahedron, with the remaining 10 interior vertices forming a pentagonal antiprism.

In the same way, one can obtain a Rhombic dodecahedron from a 4-cube, and a rhombic triacontahedron from a 6-cube.

The rhombic icosahedron can be derived from the rhombic triacontahedron by removing a belt of 10 middle faces with parallel edges.

Rhombic triacontahedron middle colored.png
A rhombic triacontahedron can be seen as an elongated rhombic icosahedron.
Dual dodecahedron t1 H3.png
The rhombic icosahedron and the rhombic triacontahedron have the same 10-fold symmetric orthogonal projection. (*)

(*) (For example, on the left-hand figure):

The orthogonal projection of the (vertical) belt of 10 middle faces of the rhombic triacontahedron is just the (horizontal) exterior regular decagon of the common orthogonal projection.

Removal of a further belt of 8 faces with parallel edges from the icosahedron results in the Bilinski dodecahedron, which is topologically equivalent but not congruent to the regular rhombic dodecahedron.

Related Research Articles

In geometry, a dodecahedron or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three regular star dodecahedra, which are constructed as stellations of the convex form. All of these have icosahedral symmetry, order 120.

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<span class="mw-page-title-main">Stellation</span> Extending the elements of a polytope to form a new figure

In geometry, stellation is the process of extending a polygon in two dimensions, a polyhedron in three dimensions, or, in general, a polytope in n dimensions to form a new figure. Starting with an original figure, the process extends specific elements such as its edges or face planes, usually in a symmetrical way, until they meet each other again to form the closed boundary of a new figure. The new figure is a stellation of the original. The word stellation comes from the Latin stellātus, "starred", which in turn comes from the Latin stella, "star". Stellation is the reciprocal or dual process to faceting.

<span class="mw-page-title-main">Rhombus</span> Quadrilateral with sides of equal length

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<span class="mw-page-title-main">Rhombicosidodecahedron</span> Archimedean solid

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<span class="mw-page-title-main">Truncated icosidodecahedron</span> Archimedean solid

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<span class="mw-page-title-main">Decagon</span> Shape with ten sides

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<span class="mw-page-title-main">Rhombic dodecahedron</span> Catalan solid with 12 faces

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<span class="mw-page-title-main">Rhombic triacontahedron</span> Catalan solid with 30 faces

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<span class="mw-page-title-main">Trapezohedron</span> Polyhedron made of congruent kites arranged radially

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<span class="mw-page-title-main">Disdyakis triacontahedron</span> Catalan solid with 120 faces

In geometry, a disdyakis triacontahedron, hexakis icosahedron, decakis dodecahedron or kisrhombic triacontahedron is a Catalan solid with 120 faces and the dual to the Archimedean truncated icosidodecahedron. As such it is face-uniform but with irregular face polygons. It slightly resembles an inflated rhombic triacontahedron: if one replaces each face of the rhombic triacontahedron with a single vertex and four triangles in a regular fashion, one ends up with a disdyakis triacontahedron. That is, the disdyakis triacontahedron is the Kleetope of the rhombic triacontahedron. It is also the barycentric subdivision of the regular dodecahedron and icosahedron. It has the most faces among the Archimedean and Catalan solids, with the snub dodecahedron, with 92 faces, in second place.

<span class="mw-page-title-main">Trigonal trapezohedron</span> Polyhedron with 6 congruent rhombus faces

In geometry, a trigonal trapezohedron is a polyhedron with six congruent quadrilateral faces, which may be scalene or rhomboid. The variety with rhombus-shaped faces faces is a rhombohedron. An alternative name for the same shape is the trigonal deltohedron.

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<span class="mw-page-title-main">Regular dodecahedron</span> Convex polyhedron with 12 regular pentagonal faces

A regular dodecahedron or pentagonal dodecahedron is a dodecahedron composed of regular pentagonal faces, three meeting at each vertex. It is an example of Platonic solids, described as cosmic stellation by Plato in his dialogues, and it was used as part of Solar System proposed by Johannes Kepler. However, the regular dodecahedron, including the other Platonic solids, has already been described by other philosophers since antiquity.

In geometry, a quasiregular polyhedron is a uniform polyhedron that has exactly two kinds of regular faces, which alternate around each vertex. They are vertex-transitive and edge-transitive, hence a step closer to regular polyhedra than the semiregular, which are merely vertex-transitive.

<span class="mw-page-title-main">Medial rhombic triacontahedron</span> Polyhedron with 30 faces

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<span class="mw-page-title-main">Chamfer (geometry)</span> Geometric operation which truncates the edges of polyhedra

In geometry, chamfering or edge-truncation is a topological operator that modifies one polyhedron into another. It is similar to expansion: it moves the faces apart (outward), and adds a new face between each two adjacent faces; but contrary to expansion, it maintains the original vertices. For a polyhedron, this operation adds a new hexagonal face in place of each original edge.

<span class="mw-page-title-main">Icosahedron</span> Polyhedron with 20 faces

In geometry, an icosahedron is a polyhedron with 20 faces. The name comes from Ancient Greek εἴκοσι (eíkosi) 'twenty' and ἕδρα (hédra) 'seat'. The plural can be either "icosahedra" or "icosahedrons".

<span class="mw-page-title-main">Bilinski dodecahedron</span> Polyhedron with 12 congruent golden rhombus faces

In geometry, the Bilinski dodecahedron is a convex polyhedron with twelve congruent golden rhombus faces. It has the same topology but a different geometry than the face-transitive rhombic dodecahedron. It is a parallelohedron.

References

  1. Weisstein, Eric W. "Rhombic Icosahedron". mathworld.wolfram.com. Retrieved 2019-12-20.