Tetragonal trapezohedron
Tetragonal trapezohedron | |
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Click on picture for large version. | |
Type | trapezohedra |
Coxeter diagram | |
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 |
The tetragonal trapezohedron or deltohedron is the second in an infinite series of face-uniform polyhedra which are dual to the antiprisms. It has eight faces which are congruent kites and is dual to the square antiprism.
Application
This shape has been used as a test case for hexahedral mesh generation,[1][2][3][4][5] simplifying an earlier test case of Rob Schneider 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 Schneider's 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]
Related polyhedra
2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | ... |
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As spherical polyhedra | |||||||||||
The tetragonal trapezohedron is first in a series of dual snub polyhedra and tilings with face configuration V3.3.4.3.n.
Symmetry 4n2 [n,4]+ |
Spherical | Euclidean | Compact hyperbolic | Paracompact | ||||
---|---|---|---|---|---|---|---|---|
242 [2,4]+ |
342 [3,4]+ |
442 [4,4]+ |
542 [5,4]+ |
642 [6,4]+ |
742 [7,4]+ |
842 [8,4]+... |
∞42 [∞,4]+ | |
Snub figure |
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.∞ |
Coxeter Schläfli |
sr{2,4} |
sr{3,4} |
sr{4,4} |
sr{5,4} |
sr{6,4} |
sr{7,4} |
sr{8,4} |
sr{∞,4} |
Snub dual figure |
V3.3.4.3.2 |
V3.3.4.3.3 |
V3.3.4.3.4 |
V3.3.4.3.5 |
V3.3.4.3.6 | V3.3.4.3.7 | V3.3.4.3.8 | V3.3.4.3.∞ |
Coxeter |
References
- ↑ 1.0 1.1 1.2 Eppstein, David (1996), "Linear complexity hexahedral mesh generation", Proceedings of the Twelfth Annual Symposium on Computational Geometry (SCG '96), New York, NY, USA: ACM, pp. 58–67, arXiv:cs/9809109, doi:10.1145/237218.237237, MR 1677595.
- ↑ 2.0 2.1 Mitchell, S. A. (1999), "The all-hex geode-template for conforming a diced tetrahedral mesh to any diced hexahedral mesh", Engineering with Computers 15 (3): 228–235, doi:10.1007/s003660050018.
- ↑ 3.0 3.1 3.2 Schwartz, Alexander; Ziegler, Günter M. (2004), "Construction techniques for cubical complexes, odd cubical 4-polytopes, and prescribed dual manifolds", Experimental Mathematics 13 (4): 385–413, MR 2118264.
- ↑ 4.0 4.1 4.2 Carbonera, Carlos D.; Shepherd, Jason F.; Shepherd, Jason F. (2006), "A constructive approach to constrained hexahedral mesh generation", Proceedings of the 15th International Meshing Roundtable, Berlin: Springer, pp. 435–452, doi:10.1007/978-3-540-34958-7_25.
- ↑ 5.0 5.1 5.2 5.3 Erickson, Jeff (2013), "Efficiently hex-meshing things with topology", Proceedings of the Twenty-ninth Annual Symposium on Computational Geometry (SoCG '13), New York, NY, USA: ACM, pp. 37–46, doi:10.1145/2462356.2462403.
- ↑ Mitchell, Scott A. (1996), "A characterization of the quadrilateral meshes of a surface which admit a compatible hexahedral mesh of the enclosed volume", STACS 96: 13th Annual Symposium on Theoretical Aspects of Computer Science Grenoble, France, February 22–24, 1996, Proceedings, Lecture Notes in Computer Science 1046, Berlin: Springer, pp. 465–476, doi:10.1007/3-540-60922-9_38, MR 1462118.
External links
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