Decagonal trapezohedron

Decagonal trapezohedron

Type	trapezohedra
Conway	dA10
Coxeter diagram
Faces	20 kites
Edges	40
Vertices	22
Face configuration	V10.3.3.3
Symmetry group	D_10d, [2⁺,20], (2*10), order 40
Rotation group	D₁₀, [2,10]⁺, (2.2.10), order 20
Dual polyhedron	Decagonal antiprism
Properties	convex, face-transitive

In geometry, a decagonal trapezohedron (or decagonal deltohedron) is the eighth in an infinite series of face-uniform polyhedra which are dual polyhedra to the antiprisms. It has twenty 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 decagonal trapezohedron is D_10d of order 40. The rotation group is D₁₀ of order 20.

Variations[]

One degree of freedom within symmetry from D_10d (order 40) to D₁₀ (order 20) 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_10v (cyclic) symmetry, order 20, and is called an unequal or asymmetric decagonal 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 C₁₀ (cyclic) symmetry, order 10, and is called an unequal twisted decagonal trapezohedron.

Spherical tiling[]

The decagonal 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 <10>

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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