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  Orthogonal projections that look like golden rhombohedra | |||
  Other orthogonal projections | |||
  Pairs of golden rhombohedra (Animations) | |||
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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.
This shape appears in a 1752 book by John Lodge Cowley, labeled as the dodecarhombus. [1] [2] It is named after Stanko Bilinski, who rediscovered it in 1960. [3] Bilinski himself called it the rhombic dodecahedron of the second kind. [4] Bilinski's discovery corrected a 75-year-old omission in Evgraf Fedorov's classification of convex polyhedra with congruent rhombic faces. [5]
The Bilinski dodecahedron is formed by gluing together twelve congruent golden rhombi. These are rhombi whose diagonals are in the golden ratio:
The graph of the resulting polyhedron is isomorphic to the graph of the rhombic dodecahedron, but the faces are oriented differently: one pair of opposite rhombi has their long and short diagonals reversed, relatively to the orientation of the corresponding rhombi in the rhombic dodecahedron.
Because of its reversal, the Bilinski dodecahedron has a lower order of symmetry; its symmetry group is that of a rectangular cuboid: D2h, [2,2], (*222), of order 8. This is a subgroup of octahedral symmetry; its elements are three 2-fold symmetry axes, three symmetry planes (which are also the axial planes of this solid), and a center of inversion symmetry. The rotation group of the Bilinski dodecahedron is D2, [2,2]+, (222), of order 4.
Like the rhombic dodecahedron, the Bilinski dodecahedron has eight vertices of degree 3 and six of degree 4. It has two apices on the vertical axis, and four vertices on each axial plane. But due to the reversal, its non-apical vertices form two squares (red and green) and one rectangle (blue), and its fourteen vertices in all are of four different kinds:
The supplementary internal angles of a golden rhombus are: [6]
The faces of the Bilinski dodecahedron are twelve congruent golden rhombi; but due to the reversal, they are of three different kinds:
(See also the figure with edges and front faces colored.)
The 24 edges of the Bilinski dodecahedron have the same length; but due to the reversal, they are of four different kinds:
(See also the figure with edges and front faces colored.)
The vertices of a Bilinski dodecahedron with thickness 2 has the following Cartesian coordinates, where φ is the golden ratio:
| degree | color | coordinates |  |  |
|---|---|---|---|---|
| 3 | red | (0, ±1, ±1) | ||
| green | (±φ, 0, ±φ) | |||
| 4 | blue | (±φ, ±1, 0) | ||
| black | (0, 0, ±φ2) | |||
| Red/green/blue vertices are in the plane perpendicular to the axis of the same color. | ||||
| other properties |
|---|
The Bilinski dodecahedron of this size has:
|
The Bilinski dodecahedron is a parallelohedron; thus it is also a space-filling polyhedron, and a zonohedron.
In a 1962 paper, H. S. M. Coxeter claimed that the Bilinski dodecahedron could be obtained by an affine transformation from the rhombic dodecahedron, but this is false. [7]
In the rhombic dodecahedron: every long body diagonal (i.e. lying on opposite degree-4 vertices) is parallel to the short diagonals of four faces.
In the Bilinski dodecahedron: the longest body diagonal (i.e. lying on opposite black degree-4 vertices) is parallel to the short diagonals of two faces, and to the long diagonals of two other faces; the shorter body diagonals (i.e. lying on opposite blue degree-4 vertices) are not parallel to the diagonal of any face. [5]
In any affine transformation of the rhombic dodecahedron: every long body diagonal (i.e. lying on opposite degree-4 vertices) remains parallel to four face diagonals, and these remain of the same (new) length.
The Bilinski dodecahedron can be formed from the rhombic triacontahedron (another zonohedron, with thirty congruent golden rhombic faces) by removing or collapsing two zones or belts of ten and eight golden rhombic faces with parallel edges. Removing only one zone of ten faces produces the rhombic icosahedron. Removing three zones of ten, eight, and six faces produces a golden rhombohedron. [4] [5] Thus removing a zone of six faces from the Bilinski dodecahedron produces a golden rhombohedron. The Bilinski dodecahedron can be dissected into four golden rhombohedra, two of each type. [8]
The vertices of the zonohedra with golden rhombic faces can be computed by linear combinations of two to six generating edge vectors with coefficients 0 or 1. [9] A belt mn means a belt representing n directional vectors, and containing m coparallel edges with same length. The Bilinski dodecahedron has four belts of six coparallel edges.
These zonohedra are projection envelopes of the hypercubes, with n-dimensional projection basis, with golden ratio (φ). For n = 6, the specific basis is:
For n = 5, the basis is the same with the sixth column removed. For n = 4, the fifth and sixth columns are removed.
| Solid name | Triacontahedron | Icosahedron | Dodecahedron | Hexahedron (acute/obtuse) | Rhombus (2-faced) |
|---|---|---|---|---|---|
| Full symmetry | Ih (order 120) | D5d (order 20) | D2h (order 8) | D3d (order 12) | D2h (order 8) |
| n Belts of (2(n−1))n// edges [10] | 6 belts of 106 // edges | 5 belts of 85 // edges | 4 belts of 64 // edges | 3 belts of 43 // edges | 2 belts of 22 // edges |
| n(n−1) Faces [11] | 30 | 20 (−10) | 12 (−8) | 6 (−6) | 2 (−4) |
| 2n(n−1) Edges [12] | 60 | 40 (−20) | 24 (−16) | 12 (−12) | 4 (−8) |
| n(n−1)+2 Vertices [13] | 32 | 22 (−10) | 14 (−8) | 8 (−6) | 4 (−4) |
| Solid image |  |  |  |   |  |
| Parallel edges image |  |  |  | ||
| Dissection | 10  + 10  | 5 + 5 | 2 + 2 | ||
| Projective n-cube | 6-cube | 5-cube | 4-cube | 3-cube | 2-cube |
| Projective n-cube image |  |  |  |
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets, or sides, with three meeting at each vertex. Viewed from a corner, it is a hexagon and its net is usually depicted as a cross.
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.
In geometry, a regular icosahedron is a convex polyhedron with 20 faces, 30 edges and 12 vertices. It is one of the five Platonic solids, and the one with the most faces.
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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.
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.
In geometry, a zonohedron is a convex polyhedron that is centrally symmetric, every face of which is a polygon that is centrally symmetric. Any zonohedron may equivalently be described as the Minkowski sum of a set of line segments in three-dimensional space, or as a three-dimensional projection of a hypercube. Zonohedra were originally defined and studied by E. S. Fedorov, a Russian crystallographer. More generally, in any dimension, the Minkowski sum of line segments forms a polytope known as a zonotope.
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In geometry, a golden rhombus is a rhombus whose diagonals are in the golden ratio:
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