Decagonal trapezohedron

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.

Decagonal trapezohedron
Typetrapezohedra
ConwaydA10
Coxeter diagram
Faces20 kites
Edges40
Vertices22
Face configurationV10.3.3.3
Symmetry groupD10d, [2+,20], (2*10), order 40
Rotation groupD10, [2,10]+, (2.2.10), order 20
Dual polyhedronDecagonal antiprism
Propertiesconvex, face-transitive

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 D10d of order 40. The rotation group is D10 of order 20.

Variations

One degree of freedom within symmetry from D10d (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 C10v (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.

See also
Family of n-gonal trapezohedra
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 ... Vāˆž.3.3.3

References
  1. McLean, K. Robin (1990), "Dungeons, dragons, and dice", The Mathematical Gazette, 74 (469): 243ā€“256, doi:10.2307/3619822, JSTOR 3619822.


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