Set of dual-uniform n-gonal trapezohedra | |
---|---|

Type | dual-uniform in the sense of dual-semiregular polyhedron |

Faces | 2n congruent kites |

Edges | 4n |

Vertices | 2n + 2 |

Vertex configuration | V3.3.3.n |

Schläfli symbol | { } ⨁ {n}^{ [1] } |

Conway notation | dA_{n} |

Coxeter diagram | |

Symmetry group | D_{nd}, [2^{+},2n], (2*n), order 4n |

Rotation group | D_{n}, [2,n]^{+}, (22n), order 2n |

Dual polyhedron | (convex) uniform n-gonal antiprism |

Properties | convex, face-transitive, regular vertices^{ [2] } |

In geometry, an **n-gonal****trapezohedron**, **n-trapezohedron**, **n-antidipyramid**, **n-antibipyramid**, or **n-deltohedron** is the dual polyhedron of an n-gonal antiprism. The **2***n* faces of an n-trapezohedron are congruent and symmetrically staggered; they are called *twisted kites*. With a higher symmetry, its 2*n* faces are *kites* (also called *deltoids*).^{ [3] }

The "n-gonal" part of the name does not refer to faces here, but to two arrangements of each n vertices around an axis of n-fold 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.

These figures, sometimes called delt**o**hedra, must not be confused with delt**a**hedra, whose faces are equilateral triangles.

*Twisted* *trigonal*, *tetragonal*, and *hexagonal trapezohedra* (with six, eight, and twelve *twisted* congruent kite faces) exist as crystals; in crystallography (describing the crystal habits of minerals), they are just called *trigonal*, *tetragonal*, and *hexagonal trapezohedra*. They have no plane of symmetry, and no center of inversion symmetry;^{ [4] }^{,}^{ [5] } but they have a center of symmetry: the intersection point of their symmetry axes. The trigonal trapezohedron has one 3-fold symmetry axis, perpendicular to three 2-fold symmetry axes.^{ [4] } The tetragonal trapezohedron has one 4-fold symmetry axis, perpendicular to four 2-fold symmetry axes of two kinds. The hexagonal trapezohedron has one 6-fold symmetry axis, perpendicular to six 2-fold symmetry axes of two kinds.^{ [6] }

Crystal arrangements of atoms can repeat in space with trigonal and hexagonal trapezohedron cells.^{ [7] }

Also in crystallography, the word *trapezohedron* is often used for the polyhedron with 24 congruent non-twisted kite faces properly known as a * deltoidal icositetrahedron *,^{ [8] } which has eighteen order-4 vertices and eight order-3 vertices. This is not to be confused with the *dodecagonal trapezohedron*, which also has 24 congruent kite faces, but two order-12 apices (i.e. poles) and two rings of twelve order-3 vertices each.

Still in crystallography, the *deltoid dodecahedron*^{ [9] } has 12 congruent non-twisted kite faces, six order-4 vertices and eight order-3 vertices (the * rhombic dodecahedron * is a special case). This is not to be confused with the * hexagonal trapezohedron *, which also has 12 congruent kite faces,^{ [6] } but two order-6 apices (i.e. poles) and two rings of six order-3 vertices each.

An **n-trapezohedron** is defined by a regular zig-zag skew **2***n*-gon base, two symmetric apices with no degree of freedom right above and right below the base, and quadrilateral faces connecting each pair of adjacent basal edges to one apex.

An n-trapezohedron has two apical vertices on its polar axis, and 2*n* basal vertices in two regular n-gonal rings. It has **2***n* congruent kite faces, and it is isohedral.

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 |

Special cases:

*n*= 2. A degenerate form of trapezohedron: 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. 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. They are also the parallelepipeds with congruent rhombic faces.- A special case of a rhombohedron is one in which the rhombi forming 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.

*n*= 5. The pentagonal trapezohedron is the only polyhedron other than the Platonic solids commonly used as a die in roleplaying games such as*Dungeons & Dragons*. Being convex and face-transitive, it makes fair dice. Having 10 sides, it can be used in repetition to generate any decimal-based uniform probability desired. Typically, two dice of different colors are used for the two digits to represent numbers from 00 to 99.

The symmetry group of an n-gonal trapezohedron is D_{nd} = D_{nv}, of order 4*n*, except in the case of *n* = 3: a cube has the larger symmetry group O_{d} of order 48 = 4×(4×3), which has four versions of D_{3d} as subgroups.

The rotation group of an n-trapezohedron is D_{n}, of order 2*n*, except in the case of *n* = 3: a cube has the larger rotation group O of order 24 = 4×(2×3), which has four versions of D_{3} as subgroups.

Note: Every n-trapezohedron with a regular zig-zag skew **2***n*-gon base and 2*n* congruent non-twisted kite faces has the same (dihedral) symmetry group as the dual-uniform n-trapezohedron, for *n* ≥ 4.

One degree of freedom within symmetry from D_{nd} (order 4*n*) to D_{n} (order 2*n*) changes the congruent kites into congruent quadrilaterals with three edge lengths, called *twisted kites*, and the *n*-trapezohedron is called a *twisted trapezohedron*. (In the limit, one edge of each quadrilateral goes to zero length, and the *n*-trapezohedron becomes an *n*-bipyramid.)

If the kites surrounding the two peaks are not twisted but are of two different shapes, the *n*-trapezohedron can only have C_{nv} (cyclic with vertical mirrors) symmetry, order 2*n*, and is called an *unequal* or *asymmetric trapezohedron*. Its dual is an *unequal n-antiprism *, with the top and bottom *n*-gons of different radii.

If the kites are twisted and are of two different shapes, the *n*-trapezohedron can only have C_{n} (cyclic) symmetry, order *n*, and is called an *unequal twisted trapezohedron*.

Trapezohedron type | Twisted trapezohedron | Unequal trapezohedron | Unequal twisted trapezohedron | |
---|---|---|---|---|

Symmetry group | D_{6}, (662), [6,2]^{+} | C_{6v}, (*66), [6] | C_{6}, (66), [6]^{+} | |

Polyhedron image | ||||

Net |

A **star p/q-trapezohedron** (where 2 ≤

A star *p*/*q*-trapezohedron has two apical vertices on its polar axis, and 2*p* basal vertices in two regular p-gonal rings. It has **2***p* congruent kite faces, and it is isohedral.

Such a star *p*/*q*-trapezohedron is a *self-intersecting*, *crossed*, or *non-convex* form. It exists for any regular zig-zag skew star **2***p*/*q*-gon base (where 2 ≤ *q* < **1***p*).

But if *p*/*q* < 3/2, then (*p* − *q*)360°/*p* < *q*/2360°/*p*, so the dual star antiprism (of the star trapezohedron) cannot be uniform (i.e. cannot have equal edge lengths); and if *p*/*q* = 3/2, then (*p* − *q*)360°/*p* = *q*/2360°/*p*, so the dual star antiprism must be flat, thus degenerate, to be uniform.

A dual-uniform star *p*/*q*-trapezohedron has Coxeter-Dynkin diagram .

5/2 | 5/3 | 7/2 | 7/3 | 7/4 | 8/3 | 8/5 | 9/2 | 9/4 | 9/5 |
---|---|---|---|---|---|---|---|---|---|

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

Wikimedia Commons has media related to Trapezohedra .

- 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.

In geometry, an **n-gonal antiprism** or **n-antiprism** is a polyhedron composed of two parallel direct copies of an n-sided polygon, connected by an alternating band of 2*n* triangles. They are represented by the Conway notation A*n*.

A (symmetric) n-gonal **bipyramid** or **dipyramid** is a polyhedron formed by joining an n-gonal pyramid and its mirror image base-to-base. An n-gonal bipyramid has 2*n* triangle faces, 3*n* edges, and 2 + *n* vertices.

In geometry, a **cube** is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross.

In geometry, a **dodecahedron** or **duodecahedron** is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three regular star dodecahedra, which are constructed as stellations of the convex form. All of these have icosahedral symmetry, order 120.

In geometry, an **octahedron** is a polyhedron with eight faces. The term is most commonly used to refer to the **regular** octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex.

In geometry, a **prism** is a polyhedron comprising an n-sided polygon base, a second base which is a translated copy of the first, and n other faces, necessarily all parallelograms, joining corresponding sides of the two bases. All cross-sections parallel to the bases are translations of the bases. Prisms are named after their bases, e.g. a prism with a pentagonal base is called a pentagonal prism. Prisms are a subclass of prismatoids.

In mathematics, a **Catalan solid**, or **Archimedean dual**, is a dual polyhedron to an Archimedean solid. There are 13 Catalan solids. They are named for the Belgian mathematician Eugène Catalan, who first described them in 1865.

In geometry, the **rhombic dodecahedron** is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of 2 types. It is a Catalan solid, and the dual polyhedron of the cuboctahedron.

In geometry, the **snub disphenoid**, **Siamese dodecahedron**, **triangular dodecahedron**, **trigonal dodecahedron**, or **dodecadeltahedron** is a convex polyhedron with twelve equilateral triangles as its faces. It is not a regular polyhedron because some vertices have four faces and others have five. It is a dodecahedron, one of the eight deltahedra, and is the 84th Johnson solid. It can be thought of as a square antiprism where both squares are replaced with two equilateral triangles.

In geometry, the term **semiregular polyhedron** is used variously by different authors.

A **hexagonal bipyramid** is a polyhedron formed from two hexagonal pyramids joined at their bases. The resulting solid has 12 triangular faces, 8 vertices and 18 edges. The 12 faces are identical isosceles triangles.

In geometry of 4 dimensions or higher, a **double prism** or **duoprism** is a polytope resulting from the Cartesian product of two polytopes, each of two dimensions or higher. The Cartesian product of an n-polytope and an m-polytope is an (*n*+*m*)-polytope, where n and m are dimensions of 2 (polygon) or higher.

In geometry, a **trigonal trapezohedron** is a rhombohedron in which, additionally, all six faces are congruent. Alternative names for the same shape are the *trigonal deltohedron* or *isohedral rhombohedron*. Some sources just call them *rhombohedra*.

In geometry, a **tetragonal trapezohedron**, or **deltohedron**, is the second in an infinite series of trapezohedra, which are dual to the antiprisms. It has eight faces, which are congruent kites, and is dual to the square antiprism.

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.

In geometry, an n-gonal**truncated trapezohedron** is a polyhedron formed by a n-gonal trapezohedron with n-gonal pyramids truncated from its two polar axis vertices. If the polar vertices are completely truncated (diminished), a trapezohedron becomes an antiprism.

In geometry, a tessellation of dimension 2 or higher, or a polytope of dimension 3 or higher, is **isohedral** or **face-transitive** if all its faces are the same. More specifically, all faces must be not merely congruent but must be *transitive*, i.e. must lie within the same *symmetry orbit*. In other words, for any two faces A and B, there must be a symmetry of the *entire* figure by translations, rotations, and/or reflections that maps A onto B. For this reason, convex isohedral polyhedra are the shapes that will make fair dice.

In geometry, a **compound of two tetrahedra** is constructed by two overlapping tetrahedra, usually implied as regular tetrahedra.

In geometry, a **diminished trapezohedron** is a polyhedron in an infinite set of polyhedra, constructed by removing one of the polar vertices of a trapezohedron and replacing it by a new face (diminishment). It has one regular n-gonal base face, n triangle faces around the base, and n kites meeting on top. The kites can also be replaced by rhombi with specific proportions.

- ↑ 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 - ↑ "duality".
*maths.ac-noumea.nc*. Retrieved 2020-10-19. - ↑ Spencer 1911, p. 575, or p. 597 on Wikisource, CRYSTALLOGRAPHY, 1. CUBIC SYSTEM, TETRAHEDRAL CLASS, footnote: « [Deltoid]: From the Greek letter δ, Δ; in general, a triangular-shaped object; also an alternative name for a trapezoid ». Remark: a twisted kite can look like and even be a trapezoid.
- 1 2 Spencer 1911, p. 581, or p. 603 on Wikisource, CRYSTALLOGRAPHY, 6. HEXAGONAL SYSTEM,
*Rhombohedral Division*, TRAPEZOHEDRAL CLASS, FIG. 74. - ↑ Spencer 1911, p. 577, or p. 599 on Wikisource, CRYSTALLOGRAPHY, 2. TETRAGONAL SYSTEM, TRAPEZOHEDRAL CLASS.
- 1 2 Spencer 1911, p. 582, or p. 604 on Wikisource, CRYSTALLOGRAPHY, 6. HEXAGONAL SYSTEM,
*Hexagonal Division*, TRAPEZOHEDRAL CLASS. - ↑ Trigonal-trapezohedric Class, 3 2 and Hexagonal-trapezohedric Class, 6 2 2
- ↑ Spencer 1911, p. 574, or p. 596 on Wikisource, CRYSTALLOGRAPHY, 1. CUBIC SYSTEM, HOLOSYMMETRIC CLASS, FIG. 17.
- ↑ Spencer 1911, p. 575, or p. 597 on Wikisource, CRYSTALLOGRAPHY, 1. CUBIC SYSTEM, TETRAHEDRAL CLASS, FIG. 27.

- 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 - Spencer, Leonard James (1911). . In Chisholm, Hugh (ed.).
*Encyclopædia Britannica*. Vol. 07 (11th ed.). Cambridge University Press. pp. 569–591.

- Weisstein, Eric W. "Trapezohedron".
*MathWorld*. - Weisstein, Eric W. "Isohedron".
*MathWorld*. - Virtual Reality Polyhedra The Encyclopedia of Polyhedra
- Paper model tetragonal (square) trapezohedron

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