The truncated rhombicuboctahedron is a polyhedron, constructed as a

truncation In mathematics and computer science, truncation is limiting the number of digits right of the decimal point. Truncation and floor function Truncation of positive real numbers can be done using the floor function. Given a number x \in \mathbb ...

of the rhombicuboctahedron. It has 50 faces consisting of 18 octagons, 8 hexagons, and 24 squares. It can fill space with the truncated cube, truncated tetrahedron and triangular prism as a truncated

runcic cubic honeycomb The tetrahedral-octahedral honeycomb, alternated cubic honeycomb is a quasiregular space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of alternating regular octahedra and tetrahedra in a ratio of 1:2. Other names ...

Other names

*Truncated small rhombicuboctahedron *Beveled cuboctahedron

Zonohedron

As a zonohedron, it can be constructed with all but 12 octagons as regular polygons. It has two sets of 48 vertices existing on two distances from its center. It represents the Minkowski sum of a

cube 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. The cube is the only r ...

, a

truncated octahedron In geometry, the truncated octahedron is the Archimedean solid that arises from a regular octahedron by removing six pyramids, one at each of the octahedron's vertices. The truncated octahedron has 14 faces (8 regular hexagon, hexagons and 6 Squa ...

, and a rhombic dodecahedron.

Excavated truncated rhombicuboctahedron

The excavated truncated rhombicuboctahedron is a toroidal polyhedron, constructed from a truncated rhombicuboctahedron with its 12 irregular octagonal faces removed. It comprises a network of 6 square cupolae, 8 triangular cupolae, and 24 triangular prisms. It has 148 faces (8 triangles, 126 squares, 8 hexagons, and 6 octagons), 312 edges, and 144 vertices. With

Euler characteristic In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space ...

χ = f + v - e = -20, its genus (g = (2-χ)/2) is 11. Without the triangular prisms, the toroidal polyhedron becomes a truncated cuboctahedron.

Related polyhedra

The

truncated cuboctahedron In geometry, the truncated cuboctahedron is an Archimedean solid, named by Kepler as a truncation of a cuboctahedron. It has 12 square faces, 8 regular hexagonal faces, 6 regular octagonal faces, 48 vertices, and 72 edges. Since each of its fac ...

is similar, with all regular faces, and 4.6.8 vertex figure. : The triangle and squares of the rhombicuboctahedron can be independently rectified or truncated, creating four permutations of polyhedra. The partially truncated forms can be seen as edge contractions of the truncated form. The ''truncated rhombicuboctahedron'' can be seen in sequence of

rectification Rectification has the following technical meanings: Mathematics * Rectification (geometry), truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points * Rectifiable curve, in mathematics * Recti ...

and

operations from the cuboctahedron. A further alternation step leads to the '' snub rhombicuboctahedron''.

References

* * Coxeter '' Regular Polytopes'', Third edition, (1973), Dover edition, (pp. 145–154 Chapter 8: Truncation) *

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-Strauss, ''The Symmetries of Things'' 2008, {{ISBN, 978-1-56881-220-5

External links