Hexagonal trapezohedron
| Hexagonal trapezohedron | |
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face-transitive |
In
It is an
Symmetry
The symmetry a hexagonal trapezohedron is D6d 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.
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.
| Type | Twisted trapezohedra (isohedral) | Unequal trapezohedra | Unequal and twisted | |
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| Symmetry | D6, (662), [6,2]+, order 12 | C6v, (*66), [6], order 12 | C6, (66), [6]+, order 6 | |
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Spherical tiling
The hexagonal trapezohedron also exists as a
Related polyhedra
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[6,2] , (*622)
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[6,2]+, (622) | [6,2+], (2*3) | ||||||||||||
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| {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} | ||||||
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| V62 | V122 | V62 | V4.4.6 | V26 | V4.4.6 | V4.4.12 | V3.3.3.6 | V3.3.3.3 | ||||||
| Trapezohedron name | Digonal trapezohedron (Tetrahedron) |
Trigonal trapezohedron | Tetragonal trapezohedron | Pentagonal trapezohedron | Hexagonal trapezohedron | Heptagonal trapezohedron
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Octagonal trapezohedron
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Decagonal trapezohedron
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Dodecagonal trapezohedron
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... | Apeirogonal trapezohedron
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| Polyhedron image | 
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Spherical tiling image
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Plane tiling image
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Face configuration
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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
- JSTOR 3619822.
- ^ 3 2 and Hexagonal-trapezohedric Class, 6 2 2
- ISBN 978-0-7923-6590-7.
- ^ "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
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