Trapezohedron
| Set of trapezohedra | |
|---|---|
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| Conway notation | dAn |
| Schläfli symbol | { } ⨁ {n}[1] |
| Coxeter diagrams |    
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| Faces | 2n kites |
| Edges | 4n |
| Vertices | 2n + 2 |
| Face configuration | V3.3.3.n |
| Symmetry group | Dnd, [2+,2n], (2*n), order 4n |
| Rotation group | Dn, [2,n]+, (22n), order 2n |
| Dual polyhedron | antiprism |
| Properties | convex, face-transitive |
The n-gonal trapezohedron, antidipyramid, antibipyramid or deltohedron is the dual polyhedron of an n-gonal antiprism. With a highest symmetry, its 2n faces are congruent kites (also called deltoids). The faces are symmetrically staggered.
The n-gon part of the name does not reference the faces here but arrangement of vertices around an axis of symmetry. The dual n-gonal antiprism has two actual n-gon faces.
An n-gonal trapezohedron can be dissected into two equal n-gonal pyramids and an n-gonal antiprism.
Name
These figures, sometimes called deltohedra, must not be confused with deltahedra, whose faces are equilateral triangles.
In texts describing the crystal habits of minerals, the word trapezohedron is often used for the polyhedron properly known as a deltoidal icositetrahedron.
Symmetry
The symmetry group of an n-gonal trapezohedron is Dnd of order 4n, except in the case of a cube, which has the larger symmetry group Od of order 48, which has four versions of D3d as subgroups.
The rotation group is Dn of order 2n, except in the case of a cube, which has the larger rotation group O of order 24, which has four versions of D3 as subgroups.
One degree of freedom within Dn 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 Cnv symmetry, order 2n. These can be called unequal or asymmetric 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, Cn symmetry, order n.
| Type | Twisted trapezohedra | Unequal trapezohedra | Unequal and twisted | |
|---|---|---|---|---|
| Symmetry | Dn, (nn2), [n,2]+ | Cnv, (*nn), [n] | Cn, (nn), [n]+ | |
| Image (n=6) |
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Forms
A n-trapezohedron has 2n quadrilateral faces, with 2n+2 vertices. Two vertices are on the polar axis, and the others are in two regular n-gonal rings of vertices.
| Trapezohedron name | Digonal trapezohedron (Tetrahedron) |
Trigonal trapezohedron | Tetragonal trapezohedron | Pentagonal trapezohedron | Hexagonal trapezohedron | Heptagonal trapezohedron | Octagonal trapezohedron | Decagonal trapezohedron | Dodecagonal trapezohedron | ... | 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 | 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 |
Special cases:
- n=2: A degenerate form, form a geometric tetrahedron with 6 vertices, 8 edges, and 4 degenerate kite faces that are degenerated into triangles. Its dual is a degenerate form of antiprism, also a tetrahedron.
- n=3: In the case of the dual of a triangular antiprism the kites are rhombi (or squares), hence these trapezohedra are also zonohedra. They are called rhombohedra. They are cubes scaled in the direction of a body diagonal. Also they are the parallelepipeds with congruent rhombic faces.
A 60° rhombohedron, dissected into a central regular octahedron and two regular tetrahedra - A special case of a rhombohedron is one in which the rhombi which form the faces have angles of 60° and 120°. It can be decomposed into two equal regular tetrahedra and a regular octahedron. Since parallelepipeds can fill space, so can a combination of regular tetrahedra and regular octahedra.
Examples
- Crystal arrangements of atoms can repeat in space with trigonal and hexagonal trapezohedral cells.[2]
- The pentagonal trapezohedron is the only polyhedron other than the Platonic solids commonly used as a die in roleplaying games such as Dungeons & Dragons. Having 10 sides, it can be used in repetition to generate any decimal-based uniform probability desired. Two dice of different colors are typically used for the two digits to represent numbers from 00 to 99.
Star trapezohedra
Self-intersecting trapezohedron exist with a star polygon central figure, defined by kite faces connecting each polygon edge to these two points. A p/q-trapezohedron has Coxeter-Dynkin diagram 
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| 5/2 | 5/3 | 7/2 | 7/3 | 7/4 | 8/3 | 8/5 | 9/2 | 9/4 | 9/5 |
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| 10/3 | 11/2 | 11/3 | 11/4 | 11/5 | 11/6 | 11/7 | 12/5 | 12/7 | |
 
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See also
- Diminished trapezohedron
- Rhombic dodecahedron
- Rhombic triacontahedron
- Bipyramid
- Truncated trapezohedron
- Conway polyhedron notation
- The Haunter of the Dark, a short story by H.P. Lovecraft in which a fictional ancient artifact known as The Shining Trapezohedron plays a crucial role.
References
- ^ N.W. Johnson: Geometries and Transformations, (2018) ISBN 978-1-107-10340-5 Chapter 11: Finite symmetry groups, 11.3 Pyramids, Prisms, and Antiprisms, Figure 11.3c
- ^ Trigonal-trapezohedric Class, 3 2 and Hexagonal-trapezohedric Class, 6 2 2
- Anthony Pugh (1976). Polyhedra: A visual approach. California: University of California Press Berkeley. ISBN 0-520-03056-7. Chapter 4: Duals of the Archimedean polyhedra, prisma and antiprisms
External links
- Weisstein, Eric W. "Trapezohedron". MathWorld.
- Weisstein, Eric W. "Isohedron". MathWorld.
- Virtual Reality Polyhedra The Encyclopedia of Polyhedra
- VRML models (George Hart) <3> <4> <5> <6> <7> <8> [permanent dead link] <9> <10>
- Conway Notation for Polyhedra Try: "dAn", where n=3,4,5... example "dA5" is a pentagonal trapezohedron.
- Paper model tetragonal (square) trapezohedron