Rhombic Triacontahedron
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Rhombic Triacontahedron

Rhombic triacontahedron
Rhombictriacontahedron.svg
(Click here for rotating model)
Type Catalan solid
Coxeter diagram CDel node.pngCDel 5.pngCDel node f1.pngCDel 3.pngCDel node.png
Conway notation jD
Face type V3.5.3.5
DU24 facets.png

rhombus
Faces 30
Edges 60
Vertices 32
Vertices by type 20{3}+12{5}
Symmetry group Ih, H3, [5,3], (*532)
Rotation group I, [5,3]+, (532)
Dihedral angle 144°
Properties convex, face-transitive isohedral, isotoxal, zonohedron
Icosidodecahedron.svg
Icosidodecahedron
(dual polyhedron)
Rhombic triacontahedron Net
Net
3D model of a rhombic triacontahedron

In geometry, 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.

GoldenRhombus.svg
A face of the rhombic triacontahedron. The lengths
of the diagonals are in the golden ratio.
This animation shows a transformation from a cube to a rhombic triacontahedron by dividing the square faces into 4 squares and splitting middle edges into new rhombic faces.

The ratio of the long diagonal to the short diagonal of each face is exactly equal to the golden ratio, ?, so that the acute angles on each face measure , or approximately 63.43°. A rhombus so obtained is called a golden rhombus.

Being the dual of an Archimedean solid, the rhombic triacontahedron is face-transitive, meaning the symmetry group of the solid acts transitively on the set of faces. This means that for any two faces, A and B, there is a rotation or reflection of the solid that leaves it occupying the same region of space while moving face A to face B.

The rhombic triacontahedron is somewhat special in being one of the nine edge-transitive convex polyhedra, the others being the five Platonic solids, the cuboctahedron, the icosidodecahedron, and the rhombic dodecahedron.

The rhombic triacontahedron is also interesting in that its vertices include the arrangement of four Platonic solids. It contains ten tetrahedra, five cubes, an icosahedron and a dodecahedron. The centers of the faces contain five octahedra.

It can be made from a truncated octahedron by dividing the hexagonal faces into 3 rhombi:

A topological rhombic triacontahedron in truncated octahedron

Cartesian coordinates

Let be the golden ratio. The 12 points given by and cyclic permutations of these coordinates are the vertices of a regular icosahedron. Its dual regular dodecahedron, whose edges intersect those of the icosahedron at right angles, has as vertices the 8 points together with the 12 points and cyclic permutations of these coordinates. All 32 points together are the vertices of a rhombic triacontahedron centered at the origin. The length of its edges is . Its faces have diagonals with lengths and .

Dimensions

If the edge length of a rhombic triacontahedron is a, surface area, volume, the radius of an inscribed sphere (tangent to each of the rhombic triacontahedron's faces) and midradius, which touches the middle of each edge are:[1]

where ? is the golden ratio.

The insphere is tangent to the faces at their face centroids. Short diagonals belong only to the edges of the inscribed regular dodecahedron, while long diagonals are included only in edges of the inscribed icosahedron.

Dissection

The rhombic triacontahedron can be dissected into 20 golden rhombohedra: 10 acute ones and 10 obtuse ones.[2][3]

10 10
Acute golden rhombohedron.png
Acute form
Flat golden rhombohedron.png
Obtuse form

Orthogonal projections

The rhombic triacontahedron has four symmetry positions, two centered on vertices, one mid-face, and one mid-edge. Embedded in projection "10" are the "fat" rhombus and "skinny" rhombus which tile together to produce the non-periodic tessellation often referred to as Penrose tiling.

Orthogonal projections
Projective
symmetry
[2] [2] [6] [10]
Image Dual dodecahedron t1 v.png Dual dodecahedron t1 e.png Dual dodecahedron t1 A2.png Dual dodecahedron t1 H3.png
Dual
image
Dodecahedron t1 v.png Dodecahedron t1 e.png Dodecahedron t1 A2.png Dodecahedron t1 H3.png

Stellations

Rhombic hexecontahedron
An example of stellations of the rhombic triacontahedron.

The rhombic triacontahedron has 227 fully supported stellations.[4][5] Another stellation of the Rhombic triacontahedron is the compound of five octahedra. The total number of stellations of the rhombic triacontahedron is 358,833,097.

Related polyhedra

This polyhedron is a part of a sequence of rhombic polyhedra and tilings with [n,3] Coxeter group symmetry. The cube can be seen as a rhombic hexahedron where the rhombi are also rectangles.

6-cube

The rhombic triacontahedron forms a 32 vertex convex hull of one projection of a 6-cube to three dimensions.

6Cube-QuasiCrystal.png
The 3D basis vectors [u,v,w] are:
u = (1, ?, 0, -1, ?, 0)
v = (?, 0, 1, ?, 0, -1)
w = (0, 1, ?, 0, -1, ?)
RhombicTricontrahedron.png
Shown with inner edges hidden
20 of 32 interior vertices form a dodecahedron, and the remaining 12 form an icosahedron.

Uses

An example of the use of a rhombic triacontahedron in the design of a lamp

Danish designer Holger Strøm used the rhombic triacontahedron as a basis for the design of his buildable lamp IQ-light (IQ for "Interlocking Quadrilaterals").

STL model of a rhombic triacontahedral box made of six panels around a cubic hole – zoom into the model to see the hole from the inside

Woodworker Jane Kostick builds boxes in the shape of a rhombic triacontahedron.[6] The simple construction is based on the less than obvious relationship between the rhombic triacontahedron and the cube.

Roger von Oech's "Ball of Whacks" comes in the shape of a rhombic triacontahedron.

The rhombic triacontahedron is used as the "d30" thirty-sided die, sometimes useful in some roleplaying games or other places.

Christopher Bird, co-author of The Secret Life of Plants wrote an article for New Age Journal in May, 1975, popularizing the dual icosahedron and dodecahedron as "the crystalline structure of the Earth," a model of the "Earth (telluric) energy Grid." The EarthStar Globe by Bill Becker and Bethe A. Hagens purports to show "the natural geometry of the Earth, and the geometric relationship between sacred places such as the Great Pyramid, the Bermuda Triangle, and Easter Island." It is printed as a rhombic triacontahedron, on 30 diamonds, and folds up into a globe.[7]

See also

References

  1. ^ Stephen Wolfram, "[1]" from Wolfram Alpha. Retrieved 7 January 2013.
  2. ^ [2]
  3. ^ Dissection of the rhombic triacontahedron
  4. ^ Pawley, G. S. (1975). "The 227 triacontahedra". Geometriae Dedicata. Kluwer Academic Publishers. 4 (2-4): 221-232. doi:10.1007/BF00148756. ISSN 1572-9168.
  5. ^ Messer, P. W. (1995). "Stellations of the Rhombic Triacontahedron and Beyond". Structural Topology. 21: 25-46.
  6. ^ triacontahedron box - KO Sticks LLC
  7. ^ http://www.vortexmaps.com/grid-history.php

External links


  This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

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