Tetragonal trapezohedron

Tetragonal trapezohedron
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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.

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 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

Family of trapezohedra V.n.3.3.3

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

The tetragonal trapezohedron is first in a series of dual snub polyhedra and tilings with face configuration V3.3.4.3.n.

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
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.∞
Gyro figures
Config.	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.∞

References

1. 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. 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. 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. 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. Erickson, Jeff (2013), "Efficiently hex-meshing things with topology", Proceedings of the Twenty-ninth Annual Symposium on Computational Geometry (SoCG '13) (PDF), New York, NY, USA: ACM, pp. 37–46, doi:10.1145/2462356.2462403.
6. 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

• Paper model tetragonal (square) trapezohedron
• Weisstein, Eric W., "Trapezohedron", MathWorld.