geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, the tetrakis cuboctahedron is a convex polyhedron with 32 triangular faces, 48

edges Edge or EDGE may refer to: Technology Computing * Edge computing, a network load-balancing system * Edge device, an entry point to a computer network * Adobe Edge, a graphical development application * Microsoft Edge, a web browser developed by ...

, and 18 vertices. It is a dual of the

truncated rhombic dodecahedron In geometry, chamfering or edge-truncation is a topological operator that modifies one polyhedron into another. It is similar to Expansion (geometry), expansion, moving Face (geometry), faces apart and outward, but also maintains the original ...

. Its name comes from a topological construction from the cuboctahedron with the kis operator applied to the square faces. In this construction, all the vertices are assumed to be the same distance from the center, while in general octahedral symmetry can be maintain even with the 6 order-4 vertices at a different distance from the center as the other 12.

Related polyhedra

It can also be topologically constructed from the octahedron, dividing each triangular face into 4 triangles by adding mid-edge vertices (an ortho operation). From this construction, all 32 triangles will be equilateral. This polyhedron can be confused with a slightly smaller

Catalan solid In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. There are 13 Catalan solids. They are named for the Belgian mathematician Eugène Catalan, who first described them in 1865. The Catalan sol ...

, the tetrakis hexahedron, which has only 24 triangles, 32 edges, and 14 vertices. File:Tetrakis cuboctahedron on octahedron.png, Octahedron with edges bisected and faces divided into subtriangles of the tetrakis cuboctahedron File:Cuboctahedron.png, Cuboctahedron File:Tetrakishexahedron.jpg, Tetrakis hexahedron File:Octahemioctahedron.png, The nonconvex

octahemioctahedron In geometry, the octahemioctahedron or allelotetratetrahedron is a nonconvex uniform polyhedron, indexed as . It has 12 faces (8 triangles and 4 hexagons), 24 edges and 12 vertices. Its vertex figure is a crossed quadrilateral. It is one of n ...

looks like a concave tetrakis cuboctahedron with inverted

square pyramid In geometry, a square pyramid is a pyramid having a square base. If the apex is perpendicularly above the center of the square, it is a right square pyramid, and has symmetry. If all edge lengths are equal, it is an equilateral square pyramid, ...

s meeting at the polyhedron center.

References

John H. Conway John Horton Conway (26 December 1937 – 11 April 2020) was an English people, English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to ...

, Heidi Burgiel, Chaim Goodman-Strass, ''The Symmetries of Things'' 2008, ** Chapter 21: Naming the Archimedean and Catalan polyhedra and Tilings (p284)

External links

VTML polyhedral generator
Try "k4aC" ( Conway polyhedron notation) {{polyhedron-stub Polyhedra

Related polyhedra

See also

References

External links