Octagonal trapezohedron

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.

Octagonal trapezohedron
Typetrapezohedron
ConwaydA8
Coxeter diagram
Faces16 kites
Edges32
Vertices18
Face configurationV8.3.3.3
Symmetry groupD8d, [2+,16], (2*8), order 32
Rotation groupD8, [2,8]+, (228), order 16
Dual polyhedronoctagonal 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 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
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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