Heptagonal trapezohedron

Heptagonal trapezohedron

Type	trapezohedron
Conway	dA7
Coxeter diagram
Faces	14 kites
Edges	28
Vertices	16
Face configuration	V7.3.3.3
Symmetry group	D_7d, [2⁺,14], (2*7), order 28
Rotation group	D₇, [2,7]⁺, (227), order 14
Dual polyhedron	heptagonal antiprism
Properties	convex, face-transitive

In geometry, a heptagonal trapezohedron or deltohedron is the fifth in an infinite series of trapezohedra which are dual polyhedron to the antiprisms. It has 14 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 heptagonal trapezohedron is D_7d of order 28. The rotation group is D₇ of order 14.

Variations

One degree of freedom within symmetry from D_7d (order 28) to D₇ (order 14) 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_7v (cyclic) symmetry, order 14, and is called an unequal or asymmetric heptagonal 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 C₇ (cyclic) symmetry, order 7, and is called an unequal twisted heptagonal trapezohedron.

Spherical tiling

The heptagonal trapezohedron also exists as a spherical tiling, with 2 vertices on the poles, and alternating vertices equally spaced above and below the equator.