Tetragonal trapezohedron

Last updated March 31, 2024

Tetragonal trapezohedron
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Type	trapezohedra
Conway	dA4
Coxeter diagram
Faces	8 kites
Edges	16
Vertices	10
Face configuration	V4.3.3.3
Symmetry group	D_4d, [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

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 spherical tiling, with 2 vertices on the poles, and alternating vertices equally spaced above and below the equator.

Related polyhedra

Family of n-gonal trapezohedra
Trapezohedron name	Digonal trapezohedron (Tetrahedron)	Trigonal trapezohedron	Tetragonal trapezohedron	Pentagonal trapezohedron	Hexagonal trapezohedron	Heptagonal trapezohedron	Octagonal trapezohedron	Decagonal trapezohedron	Dodecagonal trapezohedron	...	Apeirogonal trapezohedron
Polyhedron image										...
Spherical tiling image										Plane tiling image
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: 3.3.4.3.n v t e
Symmetry 4n2	Spherical		Euclidean	Compact hyperbolic				Paracomp.
Symmetry 4n2	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.∞

Related Research Articles

In geometry, an $n$ -gonal antiprism or $n$ -antiprism is a polyhedron composed of two parallel direct copies of an $n$ -sided polygon, connected by an alternating band of $2 n$ triangles. They are represented by the Conway notation $A n$ .

<span class="mw-page-title-main">Cube</span> Solid object with six equal square faces

In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets, or sides, with three meeting at each vertex. Viewed from a corner, it is a hexagon and its net is usually depicted as a cross.

<span class="mw-page-title-main">Octahedron</span> Polyhedron with eight triangular faces

In geometry, an octahedron is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex.

<span class="mw-page-title-main">Polyhedron</span> 3D shape with flat faces, straight edges and sharp corners

In geometry, a polyhedron is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices.

In geometry, a prismatoid is a polyhedron whose vertices all lie in two parallel planes. Its lateral faces can be trapezoids or triangles. If both planes have the same number of vertices, and the lateral faces are either parallelograms or trapezoids, it is called a prismoid.

In geometry, a cuboid is a hexahedron, a six-faced solid. Its faces are quadrilaterals. Cuboid means "like a cube", in the sense that by adjusting the lengths of the edges or the angles between faces, a cuboid can be transformed into a cube. In mathematical language a cuboid is a convex polyhedron whose polyhedral graph is the same as that of a cube.

In geometry, the gyroelongated square bipyramid is a polyhedron with 16 triangular faces. it can be constructed from a square antiprism by attaching two equilateral square pyramids to each of its square faces. The same shape is also called hexakaidecadeltahedron, heccaidecadeltahedron, or tetrakis square antiprism; these last names mean a polyhedron with 16 triangular faces. It is an example of deltahedron, and of a Johnson solid.

In geometry, an $n$ -gonaltrapezohedron, $n$ -trapezohedron, $n$ -antidipyramid, $n$ -antibipyramid, or $n$ -deltohedron is the dual polyhedron of an $n$ -gonal antiprism. The $2 n$ faces of an $n$ -trapezohedron are congruent and symmetrically staggered; they are called twisted kites. With a higher symmetry, its $2 n$ faces are kites.

<span class="mw-page-title-main">Triakis octahedron</span> Catalan solid with 24 faces

In geometry, a triakis octahedron is an Archimedean dual solid, or a Catalan solid. Its dual is the truncated cube.

<span class="mw-page-title-main">Gyrobifastigium</span> 26th Johnson solid (8 faces)

In geometry, the gyrobifastigium is the 26th Johnson solid. It can be constructed by joining two face-regular triangular prisms along corresponding square faces, giving a quarter-turn to one prism. It is the only Johnson solid that can tile three-dimensional space.

<span class="mw-page-title-main">Uniform polyhedron</span> Isogonal polyhedron with regular faces

In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive. It follows that all vertices are congruent.

In geometry, a pentagonal trapezohedron or deltohedron is the third in an infinite series of face-transitive polyhedra which are dual polyhedra to the antiprisms. It has ten faces which are congruent kites.

In geometry, a hexagonal trapezohedron or deltohedron is the fourth in an infinite series of trapezohedra which are dual polyhedra to the antiprisms. It has twelve faces which are congruent kites. It can be described by the Conway notation $dA6$ .

In geometry, an $n$ -gonaltruncated trapezohedron is a polyhedron formed by a $n$ -gonal trapezohedron with $n$ -gonal pyramids truncated from its two polar axis vertices. If the polar vertices are completely truncated (diminished), a trapezohedron becomes an antiprism.

<span class="mw-page-title-main">Midsphere</span> Sphere tangent to every edge of a polyhedron

In geometry, the midsphere or intersphere of a convex polyhedron is a sphere which is tangent to every edge of the polyhedron. Not every polyhedron has a midsphere, but the uniform polyhedra, including the regular, quasiregular and semiregular polyhedra and their duals all have midspheres. The radius of the midsphere is called the midradius. A polyhedron that has a midsphere is said to be midscribed about this sphere.

In geometry and topology, Conway polyhedron notation, invented by John Horton Conway and promoted by George W. Hart, is used to describe polyhedra based on a seed polyhedron modified by various prefix operations.

In geometry, a compound of two tetrahedra is constructed by two overlapping tetrahedra, usually implied as regular tetrahedra.

A tetradecahedron is a polyhedron with 14 faces. There are numerous topologically distinct forms of a tetradecahedron, with many constructible entirely with regular polygon faces.

In geometry, an enneahedron is a polyhedron with nine faces. There are 2606 types of convex enneahedron, each having a different pattern of vertex, edge, and face connections. None of them are regular.

In geometry, a diminished trapezohedron is a polyhedron in an infinite set of polyhedra, constructed by removing one of the polar vertices of a trapezohedron and replacing it by a new face (diminishment). It has one regular $n$ -gonal base face, $n$ triangle faces around the base, and $n$ kites meeting on top. The kites can also be replaced by rhombi with specific proportions.

References

1 2 3 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 .
1 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, S2CID 3236051 .
1 2 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, CiteSeerX 10.1.1.408.1550 , doi:10.1080/10586458.2004.10504548, MR 2118264, S2CID 1741871 .
1 2 3 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 .
1 2 3 4 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, archived from the original (PDF) on 2017-08-10, retrieved 2014-07-21.
↑ 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, vol. 1046, Berlin: Springer, pp. 465–476, doi:10.1007/3-540-60922-9_38, ISBN 978-3-540-60922-3, MR 1462118 .

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[e96-1] 1 2 3 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 .

[m99-2] 1 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, S2CID 3236051 .

[sz04-3] 1 2 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, CiteSeerX 10.1.1.408.1550 , doi:10.1080/10586458.2004.10504548, MR 2118264, S2CID 1741871 .

[css06-4] 1 2 3 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 .

[e13-5] 1 2 3 4 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, archived from the original (PDF) on 2017-08-10, retrieved 2014-07-21.

[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, vol. 1046, Berlin: Springer, pp. 465–476, doi:10.1007/3-540-60922-9_38, ISBN 978-3-540-60922-3, MR 1462118 .

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