Tetragonal trapezohedron
| Tetragonal trapezohedron | |
|---|---|
 Click on picture for large version. | |
| Type | trapezohedra |
| Conway | dA4 |
| Coxeter diagram |     
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| Faces | 8 kites |
| Edges | 16 |
| Vertices | 10 |
Face configuration |
V4.3.3.3 |
| Symmetry group | D4d, [2+,8], (2*4), order 16 |
| Rotation group | D4, [2,4]+, (224), order 8 |
| Dual polyhedron | Square antiprism |
| Properties | convex, face-transitive
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In
In mesh generation
This shape has been used as a test case for hexahedral mesh generation,[1][2][3][4][5] simplifying an earlier test case posited by mathematician Robert Schneiders in the form of a square pyramid with its boundary subdivided into 16 quadrilaterals. In this context the tetragonal trapezohedron has also been called the cubical octahedron,[3] quadrilateral octahedron,[4] or octagonal spindle,[5] because it has eight quadrilateral faces and is uniquely defined as a combinatorial polyhedron by that property.[3] Adding four cuboids to a mesh for the cubical octahedron would also give a mesh for Schneiders' pyramid.[2] As a simply-connected polyhedron with an even number of quadrilateral faces, the cubical octahedron can be decomposed into topological cuboids with curved faces that meet face-to-face without subdividing the boundary quadrilaterals,[1][5][6] and an explicit mesh of this type has been constructed.[4] However, it is unclear whether a decomposition of this type can be obtained in which all the cuboids are convex polyhedra with flat faces.[1][5]
In art
A tetragonal trapezohedron appears in the upper left as one of the polyhedral "stars" in M. C. Escher's 1948 wood engraving Stars.
Spherical tiling
The tetragonal trapezohedron also exists as a
Related polyhedra
| Trapezohedron name | Digonal trapezohedron (Tetrahedron) |
Trigonal trapezohedron | Tetragonal trapezohedron | Pentagonal trapezohedron | Hexagonal trapezohedron | Heptagonal trapezohedron
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Octagonal trapezohedron
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Decagonal trapezohedron
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Dodecagonal trapezohedron
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... | Apeirogonal trapezohedron
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| Polyhedron image | 
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Spherical tiling image
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Plane tiling image
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Face configuration
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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 |
The tetragonal trapezohedron is first in a series of dual snub polyhedra and tilings with
| 4n2 symmetry mutations of snub tilings: 3.3.4.3.n | ||||||||
|---|---|---|---|---|---|---|---|---|
| Symmetry 4n2 |
Spherical | Euclidean | Compact hyperbolic | Paracomp. | ||||
| 242 | 342 | 442 | 542 | 642 | 742 | 842 | ∞42 | |
| Snub figures |
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| Config. | 3.3.4.3.2 | 3.3.4.3.3 | 3.3.4.3.4 | 3.3.4.3.5 | 3.3.4.3.6 | 3.3.4.3.7 | 3.3.4.3.8 | 3.3.4.3.∞
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| Gyro figures |
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Config.
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V3.3.4.3.2 | V3.3.4.3.3 | V3.3.4.3.4 | V3.3.4.3.5
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V3.3.4.3.6 | V3.3.4.3.7 | V3.3.4.3.8 | V3.3.4.3.∞ |
References
External links
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