Hexagonal trapezohedron

Summary

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
Typetrapezohedron
Faces12 kites
Edges24
Vertices14
Vertex configurationV6.3.3.3
Coxeter diagram
Symmetry groupD6d, [2+,12], (2*6), order 24
Rotation groupD6, [2,6]+, (66), order 12
Dual polyhedronhexagonal antiprism
Propertiesconvex, 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

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The symmetry a hexagonal trapezohedron is D6d of order 24. The rotation group is D6 of order 12.

Variations

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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 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        

Spherical tiling

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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.

 
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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 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 ... V∞.3.3.3

References

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  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.
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