Trapezohedron
From Free net encyclopedia
| Set of trapezohedra | |
|---|---|
| Image:Decagonal trapezohedron.png | |
| Faces | 2n kites |
| Edges | 4n |
| Vertices | 2n+2 |
| Face configuration | V3.3.3.n |
| Symmetry group | Dnd |
| Dual polyhedron | antiprism |
| Properties | convex, semi-regular (face-uniform) |
The n-agonal trapezohedron or deltohedron is the dual polyhedron of a regular n-agonal antiprism. Its 2n faces are congruent kites, also called deltoids (not trapezoids, so the name trapezohedron is misleading). The faces are symmetrically staggered.
The n-agon part of the name does not reference the faces here but arrangement of vertices around an axis of symmetry. The dual n-agonal antiprism has two actual n-agon faces.
The name deltohedron should not be confused with deltahedron.
An n-agonal trapezohedron can be decomposed into two equal n-gonal pyramids and an n-gonal antiprism.
Contents |
Forms
Image:Trigonal trapezohedron.png Image:Tetragonal trapezohedron.png Image:Pentagonal trapezohedron.png Image:Hexagonal trapezohedron.png Image:Octagonal trapezohedron.png Image:Decagonal trapezohedron.png
- Trigonal trapezohedron - 6 (rhombic) faces - dual octahedron
- A cube is a special case trigonal trapezohedron with square faces
- A trigonal trapezohedron is a special case rhombohedron with congruent rhombic faces
- Tetragonal trapezohedron - 8 kite faces - dual square antiprism
- Pentagonal trapezohedron - 10 kite faces - dual pentagonal antiprism
- Hexagonal trapezohedron - 12 kite faces - dual hexagonal antiprism
- Septagonal trapezohedron - 14 kite faces - dual septagonal antiprism
- Octagonal trapezohedron - 16 kite faces - dual octagonal antiprism
- Enneagonal trapezohedron - 18 kite faces - dual enneagonal antiprism
- Decagonal trapezohedron - 20 kite faces - dual decagonal antiprism
- ...n-agonal trapezohedron - 2n kite faces - dual n-agonal antiprism
In the case of the dual of a regular triangular antiprism the kites are rhombi, hence these trapezohedra are also zonohedra. They are called rhombohedron. They are cubes scaled in the direction of a body diagonal. Also they are the parallelepipeds with congruent rhombic faces.
A special case of a rhombohedron is one of the 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 trapezohedral cells.
- The pentagonal trapezohedron is the primary non-platonic solid polyhedron used as a die in roleplaying games such as Dungeons and Dragons. Having 10 sides, it can be used in repetition to generate any decimal-based uniform probability desired.
Symmetry
The symmetry group of an n-agonal 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.
See also
External links
- 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.it:Trapezoedro