Tetragonal trapezohedron

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Tetragonal trapezohedron
Tetragonal trapezohedron
Click on picture for large version.
Type trapezohedra
Conway dA4
Coxeter diagram CDel node fh.pngCDel 2x.pngCDel node fh.pngCDel 8.pngCDel node.png
CDel node fh.pngCDel 2x.pngCDel node fh.pngCDel 4.pngCDel node fh.png
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

In geometry, a tetragonal trapezohedron, or deltohedron, is the second in an infinite series of trapezohedra, which are dual to the antiprisms. It has eight faces, which are congruent kites, and is dual to the square antiprism.

In mesh generation[edit]

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[edit]

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[edit]

The tetragonal trapezohedron also exists as a spherical tiling, with 2 vertices on the poles, and alternating vertices equally spaced above and below the equator.

Spherical tetragonal trapezohedron.png

Related polyhedra[edit]

Family of n-gonal trapezohedra
Trapezohedron name Digonal trapezohedron
Trigonal trapezohedron Tetragonal trapezohedron Pentagonal trapezohedron Hexagonal trapezohedron Heptagonal trapezohedron Octagonal trapezohedron Decagonal trapezohedron Dodecagonal trapezohedron ... Apeirogonal trapezohedron
Polyhedron image Digonal trapezohedron.png TrigonalTrapezohedron.svg Tetragonal trapezohedron.png Pentagonal trapezohedron.svg Hexagonal trapezohedron.png Heptagonal trapezohedron.png Octagonal trapezohedron.png Decagonal trapezohedron.png Dodecagonal trapezohedron.png ...
Spherical tiling image Spherical digonal antiprism.png Spherical trigonal trapezohedron.png Spherical tetragonal trapezohedron.png Spherical pentagonal trapezohedron.png Spherical hexagonal trapezohedron.png Spherical heptagonal trapezohedron.png Spherical octagonal trapezohedron.png Spherical decagonal trapezohedron.png Spherical dodecagonal trapezohedron.png Plane tiling image Apeirogonal trapezohedron.svg
Face configuration 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:
Spherical Euclidean Compact hyperbolic Paracomp.
242 342 442 542 642 742 842 ∞42
Spherical square antiprism.png Spherical snub cube.png Uniform tiling 44-snub.png H2-5-4-snub.svg Uniform tiling 64-snub.png Uniform tiling 74-snub.png Uniform tiling 84-snub.png Uniform tiling i42-snub.png
Spherical tetragonal trapezohedron.png Spherical pentagonal icositetrahedron.png Tiling Dual Semiregular V3-3-4-3-4 Cairo Pentagonal.svg H2-5-4-floret.svg
Config. V3. V3. V3. V3. V3. V3. V3. V3.3.4.3.∞


  1. ^ a b c 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, S2CID 3266195.
  2. ^ a b 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, S2CID 3236051.
  3. ^ a b c 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, doi:10.1080/10586458.2004.10504548, MR 2118264, S2CID 1741871.
  4. ^ a b c 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. ^ a b c d 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, S2CID 10861924.
  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[edit]