Bilinski dodecahedron

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Bilinski dodecahedron (gray).png
Bilinski dodecahedron, ortho z.png
Bilinski dodecahedron, ortho y.png Bilinski dodecahedron, ortho x.png


Bilinski dodecahedron, ortho obtuse.png Bilinski dodecahedron, ortho acute.png
Orthogonal projections that look like golden rhombohedra
Bilinski dodecahedron, ortho matrix.png Bilinski dodecahedron, ortho slanted.png
Other orthogonal projections

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.


This shape appears in a 1752 book by John Lodge Cowley, labeled as the dodecarhombus.Cite error: Closing </ref> missing for <ref> tag Bilinski himself called it the rhombic dodecahedron of the second kind.[1] Bilinski's discovery corrected a 75-year-old omission in Evgraf Fedorov's classification of convex polyhedra with congruent rhombic faces.[2]


degree color coordinates
3 red (0, ±1, ±1) Right-handed coordinate system (y to back).png
green (±φ, 0, ±φ)
4 blue (±φ, ±1, 0)
black (0, 0, ±φ2)

Like its Catalan twin the Bilinski dodecahedron has eight vertices of degree 3 and six of degree 4. But due to its different symmetry it has four different kinds of vertices: The two on the vertical axis and four in each axial plane.

Its faces are 12 golden rhombi of three different kinds: 2 with alternating blue and red vertices (front and back), 2 with alternating blue and green vertices (left and right) and 8 with all four kinds of vertices.

The symmetry group of this solid is D2h. It is the same as that of a rectangular cuboid, has 8 elements, and is a subgroup of octahedral symmetry. The three axial planes are also the symmetry planes of this solid.

Relation to rhombic dodecahedron

In a 1962 paper,[3] 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.[2]

Related zonohedra

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.

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.[1][2] The Bilinski dodecahedron can be dissected into four golden rhombohedra, two of each type.[4]

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
Order 120
Order 20
Order 8
Order 12
Order 4
Belts 106 85 64 43 22
Faces 30 20
Edges 60 40
Vertices 32 22
Rhombic triacontahedron middle colored.png Rhombic icosahedron colored as expanded Bilinski dodecahedron.png Bilinski dodecahedron as expanded golden rhombohedron.png Acute golden rhombohedron.pngFlat golden rhombohedron.png GoldenRhombus.svg
Rhombic tricontahedron 6x10 parallels.png Rhombic icosahedron 5-color-paralleledges.png Bilinski dodecahedron parallelohedron.png
Dissection 10Acute golden rhombohedron.png + 10Flat golden rhombohedron.png 5Acute golden rhombohedron.png + 5Flat golden rhombohedron.png 2Acute golden rhombohedron.png + 2Flat golden rhombohedron.png
6-cube 5-cube 4-cube 3-cube 2-cube
6Cube-QuasiCrystal.png 5-cube-Phi-projection.png 4-cube-Phi-projection.png


  1. 1.0 1.1 Cromwell, Peter R. (1997), Polyhedra: One of the most charming chapters of geometry, Cambridge: Cambridge University Press, p. 156, ISBN 0-521-55432-2, .
  2. 2.0 2.1 2.2 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 .
  3. "The classification of zonohedra by means of projective diagrams", Journal de Mathématiques Pures et Appliquées 41: 137–156, 1962 . Reprinted in Coxeter, H. S. M. (1968), Twelve geometric essays, Carbondale, Ill.: Southern Illinois University Press  (The Beauty of Geometry. Twelve Essays, Dover, 1999, MR1717154).
  4. "Golden Rhombohedra", CutOutFoldUp,, retrieved 2016-05-26 

External links