Hexagonal Trapezohedron
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Hexagonal Trapezohedron
Hexagonal trapezohedron
Type trapezohedra
Conway dA6
Coxeter diagram
Faces 12 kites
Edges 24
Vertices 14
Face configuration V6.3.3.3
Symmetry group D6d, [2+,12], (2*6), order 24
Rotation group D6, [2,6]+, (66), order 12
Dual polyhedron hexagonal antiprism
Properties convex, face-transitive

The hexagonal trapezohedron or deltohedron is the fourth in an infinite series of face-uniform polyhedra which are dual polyhedron to the antiprisms. It has twelve faces which are congruent kites.

## Variations

One degree of freedom within D6 symmetry changes the kites into congruent quadrilaterals with 3 edges lengths. In the limit, one edge of each quadrilateral goes to zero length, and these become bipyramids.

Crystal arrangements of atoms can repeat in space with hexagonal trapezohedral cells.[1]

If the kites surrounding the two peaks are of different shapes, it can only have C6v symmetry, order 12. These can be called unequal trapezohedra. The dual is an unequal antiprism, with the top and bottom polygons of different radii. If it twisted and unequal its symmetry is reduced to cyclic symmetry, C6 symmetry, order 6.

Example variations
Type Twisted trapezohedra (isohedral) Unequal trapezohedra Unequal and twisted
Symmetry D6, (662), [6,2]+, order 12 C6v, (*66), [6], order 12 C6, (66), [6]+, order 6
Image
(n=6)
Net

## Related polyhedra

Family of n-gonal trapezohedra
Polyhedron image ... Apeirogonal trapezohedron
Spherical tiling image Plane tiling image
Face configuration Vn.3.3.3 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