Dodecagonal trapezohedron
| Dodecagonal trapezohedron | |
|---|---|
| |
| Type | trapezohedron |
| Conway | dA12 |
| Coxeter diagram |     
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| Faces | 24 kites |
| Edges | 48 |
| Vertices | 26 |
| Face configuration | V12.3.3.3 |
| Symmetry group | D12d, [2+,24], (2*12), order 48 |
| Rotation group | D12, [2,12]+, (2.2.12), order 24 |
| Dual polyhedron | Dodecagonal antiprism |
| Properties | convex, face-transitive |
In geometry, a dodecagonal trapezohedron or deltohedron is one in an infinite series of trapezohedra, duals to the antiprisms. It has 24 faces which are congruent kites.
It is an isohedral figure, (face-transitive), having all its faces the same. More specifically, all faces must be not merely congruent but must be transitive, i.e. must lie within the same symmetry orbit. Convex isohedral polyhedra are the shapes that will make fair dice.[1]
Symmetry
The symmetry of a dodecagonal trapezohedron is D12d of order 48. The rotation group is D12 of order 24.
Variations
One degree of freedom within symmetry from D12d (order 48) to D12 (order 24) changes the congruent kites into congruent quadrilaterals with three edge lengths, called twisted kites, and the trapezohedron is called a twisted trapezohedron.
If the kites surrounding the two peaks are not twisted but are of two different shapes, the trapezohedron can only have C12v (cyclic) symmetry, order 24, and is called an unequal or asymmetric dodecagonal trapezohedron. Its dual is an unequal antiprism, with the top and bottom polygons of different radii. These are still isohedral.
If the kites are twisted and of two different shapes, the trapezohedron can only have C12 (cyclic) symmetry, order 12, and is called an unequal twisted dodecagonal trapezohedron.
Quasicrystals
The dodecagonal trapezohedron is within the highest symmetry forms of quasicrystal identified.[2][3]
Spherical tiling
The dodecagonal trapezohedron also exists as a spherical tiling, with 2 vertices on the poles, and alternating vertices equally spaced above and below the equator.
See also
| Family of n-gonal trapezohedra | |||||||||||
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| Polyhedron image | 
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Error creating thumbnail: | ... | Apeirogonal trapezohedron |
| Spherical tiling image | File:Spherical digonal antiprism.png | File:Spherical trigonal trapezohedron.png | File:Spherical tetragonal trapezohedron.png | File:Spherical pentagonal trapezohedron.png | File:Spherical hexagonal trapezohedron.png | File:Spherical heptagonal trapezohedron.png | File:Spherical octagonal trapezohedron.png | File:Spherical decagonal trapezohedron.png | Error creating thumbnail: | Plane tiling image | File:Apeirogonal trapezohedron.svg |
| Face configuration Vn.3.3.3 | V2.3.3.3 | V3.3.3.3 | V4.3.3.3 | V5.3.3.3 | V6.3.3.3 | V7.3.3.3 | V8.3.3.3 | V10.3.3.3 | V12.3.3.3 | ... | V∞.3.3.3 |
References
- ↑ McLean, K. Robin (1990), "Dungeons, dragons, and dice", The Mathematical Gazette, 74 (469): 243–256, doi:10.2307/3619822, JSTOR 3619822.
- ↑ Point Groups and Single Forms of Quasicrystals with Eightfold and Twelvefold Symmetry 1989 Shi Nicheng, Liao Libing
- ↑ POINT GROUPS AND SINGLE FORMS OF QUASICRYSTAL Zhao Wenxia Chen Jingzhong Wan Anwa
External links
- Weisstein, Eric W. "Trapezohedron". MathWorld.
- Virtual Reality Polyhedra www.georgehart.com: The Encyclopedia of Polyhedra
- Conway Notation for Polyhedra Try: "A12"
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