Hexagonal trapezohedron

In geometry, a hexagonal trapezohedron or deltohedron is the fourth in an infinite series of trapezohedra which are dual polyhedra to the antiprisms. It has twelve faces which are congruent kites. It can be described by the Conway notation dA6.

Hexagonal trapezohedron
Type	trapezohedron
Faces	12 kites
Edges	24
Vertices	14
Vertex configuration	V6.3.3.3
Coxeter diagram
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

It is an isohedral (face-transitive) figure, meaning that all its faces are the same. More specifically, all faces are not merely congruent but also transitive, i.e. lie within the same symmetry orbit. Convex isohedral polyhedra are the shapes that will make fair dice.^[1]

Symmetry

The symmetry a hexagonal trapezohedron is D_6d of order 24. The rotation group is D₆ of order 12.

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 a hexagonal trapezohedral configuration around one atom, which is always enantiomorphous,^[2] and comprises space groups 177–182.^[3] Beta-quartz is the only common mineral with this crystal system.^[4]

If the kites surrounding the two peaks are of different shapes, it can only have C_6v 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

Spherical tiling

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

Related polyhedra

Uniform hexagonal dihedral spherical polyhedra
Symmetry: [6,2], (*622)							[6,2]+, (622)	[6,2+], (2*3)
{6,2}	t{6,2}	r{6,2}	t{2,6}	{2,6}	rr{6,2}	tr{6,2}	sr{6,2}	s{2,6}
Duals to uniforms
V62	V122	V62	V4.4.6	V26	V4.4.6	V4.4.12	V3.3.3.6	V3.3.3.3

Family of n-gonal trapezohedra

Trapezohedron name	Digonal trapezohedron (Tetrahedron)	Trigonal trapezohedron	Tetragonal trapezohedron	Pentagonal trapezohedron	Hexagonal 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	...	V∞.3.3.3

References

1. McLean, K. Robin (1990), "Dungeons, dragons, and dice", The Mathematical Gazette, 74 (469): 243–256, doi:10.2307/3619822, JSTOR 3619822.
2. 3 2 and Hexagonal-trapezohedric Class, 6 2 2
3. Hahn, Theo, ed. (2005). International tables for crystallography (5th ed.). Dordrecht, Netherlands: Published for the International Union of Crystallography by Springer. ISBN 978-0-7923-6590-7.
4. "Crystallography: The Hexagonal System". www.mindat.org. Retrieved 6 January 2023.

External links

• Weisstein, Eric W. "Trapezohedron". MathWorld.
• Virtual Reality Polyhedra The Encyclopedia of Polyhedra
  • VRML model <6> Archived 2020-09-15 at the Wayback Machine