World Library  
Flag as Inappropriate
Email this Article

Trapezohedron

Article Id: WHEBN0000719845
Reproduction Date:

Title: Trapezohedron  
Author: World Heritage Encyclopedia
Language: English
Subject: Cube, Hexagonal trapezohedron, Octagonal trapezohedron, Decagonal trapezohedron, Dice
Collection:
Publisher: World Heritage Encyclopedia
Publication
Date:
 

Trapezohedron

Set of trapezohedra
Decagonal trapezohedron.
Schläfli symbol { } ⨁ {n}
Coxeter diagram
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 or deltohedron is the dual polyhedron of an n-gonal antiprism. Its 2n faces are congruent kites (also called trapezia or 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 decomposed 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.

Forms

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 special case of a rhombohedron is one in 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.

A degenerate form, n=2, 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.

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.

If the kites surrounding the two peaks are of different shapes, it can only have Cnv symmetry, order 2n.

Examples

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 9/5










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









See also

References

  • 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
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
 
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
 
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.
 


Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.