Rhombic hexecontahedron

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Rhombic hexecontahedron
Rhombic hexecontahedron.png
Type Stellation of rhombic triacontahedron
Vertices62 (12+20+30)
Edges120 (60+60)
Faces60 golden rhombi
Symmetry Ih, [5,3], (*532)
Properties non-convex, zonohedron
3D model of a rhombic hexecontahedron Rhombic hexecontahedron.stl
3D model of a rhombic hexecontahedron

In geometry, a rhombic hexecontahedron is a stellation of the rhombic triacontahedron. It is nonconvex with 60 golden rhombic faces with icosahedral symmetry. It was described mathematically in 1940 by Helmut Unkelbach. [1]

Contents

It is topologically identical to the convex deltoidal hexecontahedron which has kite faces.

Dissection

The rhombic hexecontahedron can be dissected into 20 acute golden rhombohedra meeting at a central point. This gives the volume of a hexecontahedron of side length a to be and the area to be .

Acute golden rhombohedron.png

Construction

A rhombic hexecontahedron can be constructed from a rhombic triacontahedron. Construction of Rhombic hexecontahedron from Rhombic Triacontahedron.gif
A rhombic hexecontahedron can be constructed from a rhombic triacontahedron.

A rhombic hexecontahedron can be constructed from a regular dodecahedron, by taking its vertices, its face centers and its edge centers and scaling them in or out from the body center to different extents. Thus, if the 20 vertices of a dodecahedron are pulled out to increase the circumradius by a factor of φ + 1/2 ≈ 1.309, the 12 face centers are pushed in to decrease the inradius to 3 − φ/2 ≈ 0.691 of its original value, and the 30 edge centers are left unchanged, then a rhombic hexecontahedron is formed. (The circumradius is increased by 30.9% and the inradius is decreased by the same 30.9%.) Scaling the points by different amounts results in hexecontahedra with kite-shaped faces or other polyhedra.

Every golden rhombic face has a face center, a vertex, and two edge centers of the original dodecahedron, with the edge centers forming the short diagonal. Each edge center is connected to two vertices and two face centers. Each face center is connected to five edge centers, and each vertex is connected to three edge centers.

Stellation

The rhombic hexecontahedron is one of 227 self-supporting stellations of the rhombic triacontahedron. Its stellation diagram looks like this, with the original rhombic triacontahedron faces as the central rhombus.

Rhombic hexecontahedron stellation diagram.png

The great rhombic triacontahedron contains the 30 larger intersecting rhombic faces:

DU54 great rhombic triacontahedron.png DU54 facets.png

In Brazilian culture, handcrafted rhombic hexecontahedra used to be made from colored fabric and cardboard, called giramundos ("world turners" in Portuguese) or happiness stars, sewn by mothers and given as wedding gifts to their daughters. The custom got lost with the urbanization of Brazil, though the technique for producing the handicrafts was still taught in Brazilian rural schools up until the first half of the twentieth century. [2]

The logo of the WolframAlpha website is a red rhombic hexecontahedron and was inspired by the logo of the related Mathematica software. [3]

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.

<span class="mw-page-title-main">Icosidodecahedron</span> Archimedean solid with 32 faces

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<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">Catalan solid</span> 13 polyhedra; duals of the Archimedean solids

In mathematics, a Catalan solid, or Archimedean dual, is a polyhedron that is dual to an Archimedean solid. There are 13 Catalan solids. They are named after the Belgian mathematician Eugène Catalan, who first described them in 1865.

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

In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of 2 types. It is a Catalan solid, and the dual polyhedron of the cuboctahedron.

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

The rhombic triacontahedron, sometimes simply called the triacontahedron as it is the most common thirty-faced polyhedron, is a convex polyhedron with 30 rhombic faces. It has 60 edges and 32 vertices of two types. It is a Catalan solid, and the dual polyhedron of the icosidodecahedron. It is a zonohedron.

<span class="mw-page-title-main">Deltoidal icositetrahedron</span> Catalan solid with 24 kite faces

In geometry, the deltoidal icositetrahedron is a Catalan solid. Its 24 faces are congruent kites. The deltoidal icositetrahedron, whose dual is the (uniform) rhombicuboctahedron, is tightly related to the pseudo-deltoidal icositetrahedron, whose dual is the pseudorhombicuboctahedron; but the actual and pseudo-d.i. are not to be confused with each other.

<span class="mw-page-title-main">Disdyakis dodecahedron</span> Geometric shape with 48 faces

In geometry, a disdyakis dodecahedron,, is a Catalan solid with 48 faces and the dual to the Archimedean truncated cuboctahedron. As such it is face-transitive but with irregular face polygons. It resembles an augmented rhombic dodecahedron. Replacing each face of the rhombic dodecahedron with a flat pyramid creates a polyhedron that looks almost like the disdyakis dodecahedron, and is topologically equivalent to it.

<span class="mw-page-title-main">Deltoidal hexecontahedron</span> Catalan polyhedron

In geometry, a deltoidal hexecontahedron is a Catalan solid which is the dual polyhedron of the rhombicosidodecahedron, an Archimedean solid. It is one of six Catalan solids to not have a Hamiltonian path among its vertices.

<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">Great stellated dodecahedron</span> Kepler–Poinsot polyhedron

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<span class="mw-page-title-main">Rhombic icosahedron</span>

The rhombic icosahedron is a polyhedron shaped like an oblate sphere. Its 20 faces are congruent golden rhombi; 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).

<span class="mw-page-title-main">Great icosidodecahedron</span> Polyhedron with 32 faces

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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.

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

In geometry, the medial rhombic triacontahedron is a nonconvex isohedral polyhedron. It is a stellation of the rhombic triacontahedron, and can also be called small stellated triacontahedron. Its dual is the dodecadodecahedron.

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

In geometry, the great rhombic triacontahedron is a nonconvex isohedral, isotoxal polyhedron. It is the dual of the great icosidodecahedron (U54). Like the convex rhombic triacontahedron it has 30 rhombic faces, 60 edges and 32 vertices.

<span class="mw-page-title-main">Golden rhombus</span> Rhombus with diagonals in the golden ratio

In geometry, a golden rhombus is a rhombus whose diagonals are in the golden ratio:

<span class="mw-page-title-main">First stellation of the rhombic dodecahedron</span> Self I intersecting polyhedron with 12 faces

In geometry, the first stellation of the rhombic dodecahedron is a self-intersecting polyhedron with 12 faces, each of which is a non-convex hexagon. It is a stellation of the rhombic dodecahedron and has the same outer shell and the same visual appearance as two other shapes: a solid, Escher's solid, with 48 triangular faces, and a polyhedral compound of three flattened octahedra with 24 overlapping triangular faces.

<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. Grünbaum (1996b)
  2. Artesanato se antecipou à descoberta de poliedro [Handicraft anticipated the discovery of a polyhedron] (in Portuguese), IMPA , retrieved 2019-01-08
  3. "What's in the Logo? That Which We Call a Rhombic Hexecontahedron—Wolfram|Alpha Blog".

Bibliography