Set of trapezohedra | |
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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-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.
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.
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 D_{3} 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.
Type | Twisted trapezohedra | Unequal trapezohedra | Unequal and twisted | |
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Symmetry | D_{n}, (nn2), [n,2]^{+} | C_{nv}, (*nn), [n] | C_{n}, (nn), [n]^{+} | |
Image (n=6) |
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Net |
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 | |||||||||||
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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:
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 .
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 | |
Wikimedia Commons has media related to Trapezohedra. |