Bilinski dodecahedron
In geometry, the Bilinski dodecahedron is a convex polyhedron with twelve congruent golden rhombus faces. It has the same topology as the face-transitive rhombic dodecahedron, but a different geometry. It is a parallelohedron, a polyhedron that can tile space with translated copies of itself.
History
This shape appears in a 1752 book by John Lodge Cowley, labeled as the dodecarhombus. It is named after Stanko Bilinski, who rediscovered it in 1960. Bilinski himself called it the rhombic dodecahedron of the second kind. Bilinski's discovery corrected a 75-year-old omission in Evgraf Fedorov's classification of convex polyhedra with congruent rhombic faces.Definition and properties
Definition
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 relative to the orientation of the corresponding rhombi in the rhombic dodecahedron.
Symmetry
Because of its reversal, the Bilinski dodecahedron has a lower order of symmetry; its symmetry group is that of a rectangular cuboid: [Dihedral symmetry in three dimensions|] of order 8. This is a subgroup of octahedral symmetry; its elements are three 2-fold symmetry axes, three symmetry planes, and a center of inversion symmetry. The rotation group of the Bilinski dodecahedron is of order 4.Vertices
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 and one rectangle, and its fourteen vertices in all are of four different kinds:- two degree-4 apices surrounded by four acute face angles ;
- four degree-4 vertices surrounded by three acute and one obtuse face angles ;
- four degree-3 vertices surrounded by three obtuse face angles ;
- four degree-3 vertices surrounded by two obtuse and one acute face angles.
Faces
The supplementary internal angles of a golden rhombus are:- acute angle:
- obtuse angle:
- eight apical faces with all four kinds of vertices,
- two side faces with alternate blue and red vertices,
- two side faces with alternating blue and green vertices.
Edges
The 24 edges of the Bilinski dodecahedron have the same length; but due to the reversal, they are of four different kinds:- four apical edges with black and red vertices,
- four apical edges with black and green vertices,
- eight side edges with blue and red vertices,
- eight side edges with blue and green vertices.
Cartesian coordinates
The vertices of a Bilinski dodecahedron with thickness 2 has the following Cartesian coordinates, where is the golden ratio:| other properties | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
The Bilinski dodecahedron of this size has:
In families of polyhedraThe Bilinski dodecahedron is a parallelohedron; thus it is also a space-filling polyhedron, and a zonohedron.Relation to rhombic dodecahedronIn 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.In the rhombic dodecahedron: every long body diagonal is parallel to the short diagonals of four faces. In the Bilinski dodecahedron: the longest body diagonal is parallel to the short diagonals of two faces, and to the long diagonals of two other faces; the shorter body diagonals are not parallel to the diagonal of any face. In any affine transformation of the rhombic dodecahedron: every long body diagonal remains parallel to four face diagonals, and these remain of the same length. Zonohedra with golden rhombic facesThe Bilinski dodecahedron can be formed from the rhombic triacontahedron 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. 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.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. A belt means a belt representing directional vectors, and containing coparallel edges with same length. The Bilinski dodecahedron has four belts of six coparallel edges. These zonohedra are projection envelopes of the hypercubes, with -dimensional projection basis, with golden ratio. For the specific basis is: For the basis is the same with the sixth column removed. For the fifth and sixth columns are removed.
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