Bilinski dodecahedron

Bilinski dodecahedron with edges and front faces colored by their symmetry positions. It has D2h symmetry, order 8, containing 3 orthogonal mirrors, bisecting 4 of the 12 rhombic faces.

In geometry, the Bilinski dodecahedron is a 12-sided convex polyhedron with congruent rhombic faces. It has the same topology but different geometry from the face transitive rhombic dodecahedron, another 12-sided polyhedron with congruent rhombic faces.

History

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]

Properties

As a zonohedron, a Bilinski dodecahedron can be seen with 4 sets of 6 parallel edges. Contracting any set of 6 parallel edges to zero length produces golden rhombohedra.

In the rhombi of the Bilinski dodecahedron, the ratio of lengths of the two diagonals is the golden ratio; that is, the faces of this shape are golden rhombi. In contrast, for the standard rhombic dodecahedron the corresponding ratio is the square root of 2.[6]

This shape is a zonohedron. Like the rhombic dodecahedron, it can tile three-dimensional space by translation, making it a parallelohedron.[5]

Relation to rhombic dodecahedron

The Bilinski dodecahedron and rhombic dodecahedron have the same topology: their vertices, edges, and faces correspond one-for-one, in an adjacency-preserving way. However, their geometry is different. In a 1962 paper,[7] H. S. M. Coxeter claimed that the Bilinski dodecahedron could be obtained by an affine transformation from the rhombic dodecahedron, but this is false. For, in the Bilinski dodecahedron, the long body diagonal is parallel to the short diagonals of two faces, and to the long diagonals of two other faces. In the rhombic dodecahedron, the corresponding body diagonal is parallel to four short face diagonals, and in any affine transformation of the rhombic dodecahedron this body diagonal would remain parallel to four equal-length face diagonals. Another difference between the two dodecahedra is that, in the rhombic dodecahedron, all the body diagonals connecting opposite degree-4 vertices are parallel to face diagonals, while in the Bilinski dodecahedron the shorter body diagonals of this type have no parallel face diagonals.[5]

The Bilinski dodecahedron can be formed from the rhombic triacontahedron (another zonohedron with 30 golden rhombic faces) by removing or collapsing two zones or belts of faces with parallel edges. Removing only one of these two zones produces, instead, the rhombic icosahedron, and removing three produces the golden rhombohedra.[4][5] The Bilinski dodecahedron can be dissected into four golden rhombohedra, two of each type.[8]

The vertices of these zonohedra can be computed by linear combinations of 3 to 6 vectors. A belt mn means n directional vectors, each containing m coparallel congruent edges. The Bilinski dodecahedron has 4 belts of 6 coparallel edges.

These zonohedra are projection envelopes of the hypercubes, with n-dimensional projection basis, with golden ratio, φ. The specific basis for n=6 is:

x = (1, φ, 0, -1, φ, 0)
y = (φ, 0, 1, φ, 0, -1)
z = (0, 1, φ, 0, -1, φ)

For n=5 the basis is the same with the 6th column removed, and for n=4 the 5th and 6th column are removed.

Zonohedra with golden rhombic faces
Faces Triacontahedron Icosahedron Dodecahedron Hexahedron
Full
symmetry		 Ih
Order 120		 D5d
Order 20		 D2h
Order 8		 D3d
Order 12		 Dih2
Order 4
Belts 106 85 64 43 22
Faces 30 20
(−10)		 12
(−8)		 6
(−6)		 2
(−4)
Edges 60 40
(−20)		 24
(−16)		 12
(−12)		 4
(−8)
Vertices 32 22
(−10)		 14
(−8)		 8
(−6)		 4
(−4)
Image
Solid
Image
Parallels
Dissection 10 + 10 5 + 5 2 + 2
Projective
polytope		6-cube5-cube4-cube3-cube2-cube
Image
Projective
n-cube

References

  1. ↑ Hart, George W. (2000), "A color-matching dissection of the rhombic enneacontahedron", Symmetry: Culture and Science, 11 (1–4): 183–199, MR 2001417 .
  2. ↑ Cowley, John Lodge (1752), Geometry Made Easy; Or, a New and Methodical Explanation of the Elements of Geometry, London, Plate 5, Fig. 16 . As cited by Hart (2000).
  3. ↑ Bilinski, S. (1960), "Über die Rhombenisoeder", Glasnik Mat. Fiz. Astr., 15: 251–263, Zbl 0099.15506 .
  4. 1 2 Cromwell, Peter R. (1997), Polyhedra: One of the most charming chapters of geometry, Cambridge: Cambridge University Press, p. 156, ISBN 0-521-55432-2, MR 1458063 .
  5. 1 2 3 4 Grünbaum, Branko (2010), "The Bilinski dodecahedron and assorted parallelohedra, zonohedra, monohedra, isozonohedra, and otherhedra", The Mathematical Intelligencer, 32 (4): 5–15, doi:10.1007/s00283-010-9138-7, MR 2747698 .
  6. ↑ A new rhombic dodecahedron, Matt Parker, standupmaths
  7. ↑ Coxeter, H. S. M. (1962), "The classification of zonohedra by means of projective diagrams", Journal de Mathématiques Pures et Appliquées, 41: 137–156, MR 0141004 . Reprinted in Coxeter, H. S. M. (1968), Twelve geometric essays, Carbondale, Ill.: Southern Illinois University Press, MR 0310745 (The Beauty of Geometry. Twelve Essays, Dover, 1999, MR 1717154).
  8. ↑ "Golden Rhombohedra", CutOutFoldUp, retrieved 2016-05-26
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