geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, a tetragonal trapezohedron, or deltohedron, is the second in an infinite series of trapezohedra, which are dual to the

antiprism In geometry, an antiprism or is a polyhedron composed of two Parallel (geometry), parallel Euclidean group, direct copies (not mirror images) of an polygon, connected by an alternating band of triangles. They are represented by the Conway po ...

s. It has eight faces, which are

congruent Congruence may refer to: Mathematics * Congruence (geometry), being the same size and shape * Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure * In modu ...

kites A kite is a tethered heavier than air flight, heavier-than-air craft with wing surfaces that react against the air to create Lift (force), lift and Drag (physics), drag forces. A kite consists of wings, tethers and anchors. Kites often have ...

, and is dual to the

square antiprism In geometry, the square antiprism is the second in an infinite family of antiprisms formed by an even number, even-numbered sequence of triangle sides closed by two polygon caps. It is also known as an ''anticube''. If all its faces are regular ...

Applications

In mesh generation

This shape has been used as a test case for hexahedral

mesh generation Mesh generation is the practice of creating a polygon mesh, mesh, a subdivision of a continuous geometric space into discrete geometric and topological cells. Often these cells form a simplicial complex. Usually the cells partition the geometric ...

,..... simplifying an earlier test case posited by mathematician Robert Schneiders in the form of a

square pyramid In geometry, a square pyramid is a Pyramid (geometry), pyramid with a square base and four triangles, having a total of five faces. If the Apex (geometry), apex of the pyramid is directly above the center of the square, it is a ''right square p ...

with its boundary subdivided into 16 quadrilaterals. In this context the tetragonal trapezohedron has also been called the cubical octahedron, quadrilateral octahedron, or octagonal spindle, because it has eight quadrilateral faces and is uniquely defined as a combinatorial polyhedron by that property. Adding four cuboids to a mesh for the cubical octahedron would also give a mesh for Schneiders' pyramid. 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,. and an explicit mesh of this type has been constructed. However, it is unclear whether a decomposition of this type can be obtained in which all the cuboids are convex polyhedra with flat faces.

In art

A tetragonal trapezohedron appears in the upper left as one of the polyhedral "stars" in

M. C. Escher Maurits Cornelis Escher (; ; 17 June 1898 – 27 March 1972) was a Dutch graphic artist who made woodcuts, lithography, lithographs, and mezzotints, many of which were Mathematics and art, inspired by mathematics. Despite wide popular int ...

's 1948 wood engraving

Stars A star is a luminous spheroid of plasma held together by self-gravity. The nearest star to Earth is the Sun. Many other stars are visible to the naked eye at night; their immense distances from Earth make them appear as fixed points of ...

Spherical tiling

The tetragonal trapezohedron also exists as a

spherical tiling In geometry, a spherical polyhedron or spherical tiling is a tessellation, tiling of the sphere in which the surface is divided or partitioned by great arcs into bounded regions called ''spherical polygons''. A polyhedron whose vertices are equi ...

, with 2 vertices on the poles, and alternating vertices equally spaced above and below the equator. : Spherical tetragonal trapezohedron

Related polyhedra

The ''tetragonal trapezohedron'' is first in a series of dual snub polyhedra and tilings with

face configuration In geometry, a vertex configuration is a shorthand notation for representing a polyhedron or Tessellation, tiling as the sequence of Face (geometry), faces around a Vertex (geometry), vertex. It has variously been called a vertex description, vert ...

V3.3.4.3.''n''.

References

External links

Paper model tetragonal (square) trapezohedron
* Polyhedra {{Polyhedron-stub