Octagonal trapezohedron

Octagonal trapezohedron

Type	trapezohedron
Conway	dA8
Coxeter diagram
Faces	16 kites
Edges	32
Vertices	18
Face configuration	V8.3.3.3
Symmetry group	D_8d, [2⁺,16], (2*8), order 32
Rotation group	D₈, [2,8]⁺, (228), order 16
Dual polyhedron	octagonal antiprism
Properties	convex, face-transitive

In geometry, a octagonal trapezohedron' or deltohedron is the sixth in an infinite series trapezohedra which are dual polyhedron to the antiprisms. It has sixteen faces which are congruent kites.

It is a 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 a octagonal trapezohedron is D_8d of order 32. The rotation group is D₈ of order 16.

Variations[]

One degree of freedom within symmetry from D_8d (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 C_8v (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.

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
- VRML model <8>

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[1] McLean, K. Robin (1990), "Dungeons, dragons, and dice", The Mathematical Gazette, 74 (469): 243–256, doi:10.2307/3619822, JSTOR 3619822.

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Trapezohedron name	Digonal trapezohedron (Tetrahedron)	Trigonal trapezohedron	Tetragonal trapezohedron	Pentagonal trapezohedron	Hexagonal trapezohedron	Heptagonal trapezohedron	Octagonal trapezohedron	Decagonal trapezohedron	Dodecagonal trapezohedron	...	Apeirogonal trapezohedron
Polyhedron image										...
Spherical tiling image										Plane tiling image
Face configuration	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