Английская Википедия:Bilinski dodecahedron
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.
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]
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:
- <math>\varphi = { {1 + \sqrt{5} } \over 2} \approx 1.618~034 .</math>
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.
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|Шаблон:Math]] 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 Шаблон:Math 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 (red and green) and one rectangle (blue), and its fourteen vertices in all are of four different kinds:
- two degree-4 apices surrounded by four acute face angles (vertical-axis vertices, black in first figure);
- four degree-4 vertices surrounded by three acute and one obtuse face angles (horizontal-axial-plane vertices, blue in first figure);
- four degree-3 vertices surrounded by three obtuse face angles (one vertical-axial-plane vertices, red in first figure);
- four degree-3 vertices surrounded by two obtuse and one acute face angles (other vertical-axial-plane vertices, green in first figure).
Faces
The supplementary internal angles of a golden rhombus are:[6]
- acute angle:
- <math>\alpha = \arctan 2 \approx 63.434~949 ^ \circ ;</math>
- obtuse angle:
- <math>\beta = \pi - \arctan 2 \approx 116.565~051 ^ \circ .</math>
The faces of the Bilinski dodecahedron are twelve congruent golden rhombi; but due to the reversal, they are of three different kinds:
- eight apical faces with all four kinds of vertices,
- two side faces with alternate blue and red vertices (front and back in first figure),
- two side faces with alternating blue and green vertices (left and right in first figure).
(See also the figure with edges and front faces colored.)
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 (in first figure),
- four apical edges with black and green vertices (in first figure),
- eight side edges with blue and red vertices (in first figure),
- eight side edges with blue and green vertices (in first figure).
(See also the figure with edges and front faces colored.)
Cartesian coordinates
The vertices of a Bilinski dodecahedron with thickness 2 has the following Cartesian coordinates, where Шаблон:Mvar is the golden ratio:
| degree | color | coordinates | Файл:Right-handed coordinate system (y to back).png | Файл:Bilinski dodecahedron (gray).png |
|---|---|---|---|---|
| 3 | red | (Шаблон:Math) | ||
| green | (Шаблон:Math) | |||
| 4 | blue | (Шаблон:Math) | ||
| black | (Шаблон:Math) | |||
| 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:
|
In families of polyhedra
The Bilinski dodecahedron is a parallelohedron; thus it is also a space-filling polyhedron, and a zonohedron.
Relation to rhombic dodecahedron
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.
Zonohedra with golden rhombic faces
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 Шаблон:Mvar means a belt representing Шаблон:Mvar directional vectors, and containing Шаблон:Mvar coparallel edges with same length. The Bilinski dodecahedron has four belts of six coparallel edges.
These zonohedra are projection envelopes of the hypercubes, with Шаблон:Mvar-dimensional projection basis, with golden ratio (Шаблон:Mvar). For Шаблон:Math the specific basis is:
For Шаблон:Math the basis is the same with the sixth column removed. For Шаблон:Math 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 Шаблон:Nowrap | 6 belts of 106 // edges |
5 belts of 85 // edges |
4 belts of 64 // edges |
3 belts of 43 // edges |
2 belts of Шаблон:Nowrap |
| n(n−1) Faces[10] | 30 | 20 (−10) |
12 (−8) |
6 (−6) |
2 (−4) |
| 2n(n−1) Edges[11] | 60 | 40 (−20) |
24 (−16) |
12 (−12) |
4 (−8) |
| n(n−1)+2 Vertices[12] | 32 | 22 (−10) |
14 (−8) |
8 (−6) |
4 (−4) |
| Solid image | Файл:Rhombic triacontahedron middle colored.png | Файл:Rhombic icosahedron colored as expanded Bilinski dodecahedron.png | Файл:Bilinski dodecahedron as expanded golden rhombohedron.png | Файл:Acute golden rhombohedron.pngФайл:Flat golden rhombohedron.png | Файл:GoldenRhombus.svg |
| Parallel edges image | Файл:Rhombic tricontahedron 6x10 parallels.png | Файл:Rhombic icosahedron 5-color-paralleledges.png | Файл:Bilinski dodecahedron parallelohedron.png | ||
| Dissection | 10Файл:Acute golden rhombohedron.png + 10Файл:Flat golden rhombohedron.png | 5Файл:Acute golden rhombohedron.png + 5Файл:Flat golden rhombohedron.png | 2Файл:Acute golden rhombohedron.png + 2Файл:Flat golden rhombohedron.png | ||
| Projective n-cube |
6-cube | 5-cube | 4-cube | 3-cube | 2-cube |
| Projective n-cube image |
Файл:6Cube-QuasiCrystal.png | Файл:5-cube-Phi-projection.png | Файл:4-cube-Phi-projection.png |
References
External links
- VRML model, George W. Hart: Шаблон:URL
- animation and coordinates, David I. McCooey: Шаблон:URL
- A new Rhombic Dodecahedron from Croatia!, YouTube video by Matt Parker
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- ↑ Шаблон:Citation.
- ↑ Шаблон:Citation. As cited by Шаблон:Harvtxt.
- ↑ Шаблон:Citation.
- ↑ 4,0 4,1 Шаблон:Citation.
- ↑ 5,0 5,1 5,2 Шаблон:Citation.
- ↑ Шаблон:Citation. See in particular table 1, p. 188.
- ↑ Шаблон:Citation. Reprinted in Шаблон:Citation (The Beauty of Geometry. Twelve Essays, Dover, 1999, Шаблон:MR).
- ↑ Шаблон:Citation
- ↑ Let V denote the number of vertices and ek denote the k-th generating edge vector, where 1 ≤ k ≤ n;
for 2 ≤ n ≤ 3, V = card(𝒫 {e1,...,en}) = 2n;
for 4 ≤ n ≤ 6, V < 2n, because some of the linear combinations of four to six generating edge vectors with coefficients 0 or 1 end strictly inside the golden rhombic zonohedron. - ↑ A golden rhombic zonohedron has each pair of opposite rhombi corresponding to two among n generating edge vectors, so it has:
F = 2×(n2) = n(n−1) faces. - ↑ A golden rhombic zonohedron has each face lying on four edges, and each edge lying in two faces, so it has:
E = 4F/2 = 2F = 2n(n−1) edges. - ↑ A golden rhombic zonohedron has:
V = E−F+2 = 2n(n−1)−n(n−1)+2 = n(n−1)+2 vertices.
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