Trapezohedron

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.

Set of trapezohedra

Conway notation	dA_n
Schläfli symbol	{ } ⨁ {n}[1]
Coxeter diagrams
Faces	2n kites
Edges	4n
Vertices	2n + 2
Face configuration	V3.3.3.n
Symmetry group	D_nd, [2⁺,2n], (2*n), order 4n
Rotation group	D_n, [2,n]⁺, (22n), order 2n
Dual polyhedron	antiprism
Properties	convex, face-transitive

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 D_nd of order 4n, except in the case of a cube, which has the larger symmetry group O_d of order 48, which has four versions of D_3d as subgroups.

The rotation group is D_n 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 D_n 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 C_nv 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, C_n symmetry, order n.

Example variations
Type	Twisted trapezohedra	Unequal trapezohedra	Unequal and twisted
Symmetry	D_n, (nn2), [n,2]⁺	C_nv, (*nn), [n]	C_n, (nn), [n]⁺
Image (n=6)
Net

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.

Family of trapezohedra Vn.3.3.3
Polyhedron
Tiling
Config.	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 .

Uniform dual p/q star trapezohedra up to p = 12
5/2	5/3	7/2	7/3	7/4	8/3	8/5	9/2	9/4

10/3	11/2	11/3	11/4	11/5	11/6	11/7	12/5	12/7

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> <9> <10>
- Conway Notation for Polyhedra Try: "dAn", where n=3,4,5... example "dA5" is a pentagonal trapezohedron.
Paper model tetragonal (square) trapezohedron

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[1] 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

[2] Trigonal-trapezohedric Class, 3 2 and Hexagonal-trapezohedric Class, 6 2 2