Octagonal trapezohedron
| Octagonal trapezohedron | |
|---|---|
| Octagonal trapezohedron | |
| Type | trapezohedron |
| Conway | dA8 |
| Coxeter diagram | File:CDel node fh.png File:CDel node fh.pngFile:CDel 16.png File:CDel node fh.png File:CDel node fh.pngFile:CDel 8.pngFile:CDel node fh.png
|
| Faces | 16 kites |
| Edges | 32 |
| Vertices | 18 |
| Face configuration | V8.3.3.3 |
| Symmetry group | D8d, [2+,16], (2*8), order 32 |
| Rotation group | D8, [2,8]+, (228), order 16 |
| Dual polyhedron | octagonal antiprism |
| Properties | convex, face-transitive |
In geometry, an octagonal trapezohedron or deltohedron is the sixth in an infinite series of trapezohedra which are dual polyhedron to the antiprisms. It has sixteen 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 an octagonal trapezohedron is D8d of order 32. The rotation group is D8 of order 16.
Variations
One degree of freedom within symmetry from D8d (order 32) to D8 (order 16) 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 C8v (cyclic) symmetry, order 16, and is called an unequal or asymmetric octagonal 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 C8 (cyclic) symmetry, order 8, and is called an unequal twisted octagonal trapezohedron.
Spherical tiling
The octagonal 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
References
- ↑ McLean, K. Robin (1990), "Dungeons, dragons, and dice", The Mathematical Gazette, 74 (469): 243–256, doi:10.2307/3619822, JSTOR 3619822.
External links
- Weisstein, Eric W. "Trapezohedron". MathWorld.
- Virtual Reality Polyhedra The Encyclopedia of Polyhedra
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