Truncated cuboctahedron

In geometry, the truncated cuboctahedron or great rhombicuboctahedron is a special shape known as an Archimedean solid. It was named by the astronomer Kepler as a kind of truncation of another shape called a cuboctahedron. This interesting solid has 12 flat square faces, 8 perfectly shaped hexagonal faces, and 6 regular octagonal faces. All together, it has 48 points where the faces meet, called vertices, and 72 lines where the faces connect, called edges.

One special feature of the truncated cuboctahedron is that each of its faces has a kind of balance called point symmetry. This means that if you turn the face halfway around, it still looks the same. Because of this, the shape is also called a 9-zonohedron.

The truncated cuboctahedron can fit together with other shapes in a pattern that covers space without gaps. It can tessellate with shapes known as octagonal prisms, making beautiful and repeating designs. This shape is important in the study of geometry and helps us understand how different faces can come together in three dimensions.

Names

There is a nonconvex uniform polyhedron with a similar name: the nonconvex great rhombicuboctahedron.

The name truncated cuboctahedron, given originally by Johannes Kepler, is misleading: an actual truncation of a cuboctahedron has rectangles instead of squares; however, this nonuniform polyhedron is topologically equivalent to the Archimedean solid unrigorously named truncated cuboctahedron. Alternate interchangeable names are: Truncated cuboctahedron (Johannes Kepler), Rhombitruncated cuboctahedron (Magnus Wenninger), Great rhombicuboctahedron (Robert Williams), Great rhombcuboctahedron (Peter Cromwell), Omnitruncated cube or cantitruncated cube (Norman Johnson), Beveled cube (Conway polyhedron notation).

Cuboctahedron and its truncation

Cartesian coordinates

The Cartesian coordinates for the vertices of a truncated cuboctahedron with edge length 2 and centered at the origin are all the permutations of (±1, ±(1 + √2), ±(1 + 2√2)). This means you can arrange these numbers in different orders and change their signs to find all the points where the vertices are located.

Area and volume

The area and volume of a truncated cuboctahedron can be found if we know the length of its edges, called a. The area A is about 61.76 times a squared, and the volume V is about 41.80 times a cubed. These formulas help us learn about the size of this special shape.

Dissection

The truncated cuboctahedron is like a rhombicuboctahedron with extra shapes on it. Picture putting small cubes on the 12 square sides of a rhombicuboctahedron. The areas around it can also hold 6 square cupolas and 8 triangular cupolas.

If we take away the central rhombicuboctahedron and some of these added shapes, we can make interesting ring-like shapes called Stewart toroids. For instance, taking away the 6 square cupolas makes one kind of toroid, and taking away the 8 triangular cupolas makes another kind. By deciding which shapes to take away, we can create many different kinds of these ring-like forms.

Uniform colorings

There is only one uniform coloring of the truncated cuboctahedron. In this coloring, each face type has its own color. There is also a 2-uniform coloring. This uses alternating colors on the hexagonal faces because of tetrahedral symmetry.

Orthogonal projections

The truncated cuboctahedron has two special orthogonal projections in the A₂ and B₂ Coxeter planes. These projections show the shape from certain angles. They help us see the shape's symmetry. There are also many other ways to look at the shape. These show different parts of its structure.

Orthogonal projections

Centered by	Vertex	Edge 4-6	Edge 4-8	Edge 6-8	Face normal 4-6
Image
Projective symmetry	+
Centered by	Face normal Square	Face normal Octagon	Face Square	Face Hexagon	Face Octagon
Image
Projective symmetry

Spherical tiling

The truncated cuboctahedron can be shown as a pattern on a sphere, called a spherical tiling. We can draw this pattern on a flat surface using a method called stereographic projection. This method keeps the angles the same but changes the sizes and lengths. On the flat drawing, lines that would be straight on the sphere look like curved arcs.

Orthogonal projection	square-centered	hexagon-centered	octagon-centered
	Stereographic projections

Full octahedral group

The truncated cuboctahedron has a special kind of symmetry called full octahedral symmetry. It has 48 vertices. These vertices match the elements of this symmetry group. Each face of its dual shape is a basic area of this group.

The edges of the solid match 9 reflections in the group. For example, edges between octagons and squares match 3 reflections between opposite octagons. Hexagon edges match 6 reflections between opposite squares. Some smaller symmetry groups match parts of the truncated cuboctahedron’s vertices.

Subgroups and corresponding solids
Truncated cuboctahedron tr{4,3}	Snub cube sr{4,3}	Rhombicuboctahedron s2{3,4}	Truncated octahedron h1,2{4,3}	Icosahedron
[4,3] Full octahedral	[4,3]+ Chiral octahedral	[4,3+] Pyritohedral	[1+,4,3] = [3,3] Full tetrahedral	[1+,4,3+] = [3,3]+ Chiral tetrahedral
all 48 vertices	24 vertices			12 vertices

Related polyhedra

The truncated cuboctahedron is related to the cube and regular octahedron. It is part of a family of shapes with special corner arrangements, shown in diagrams. For numbers smaller than six, these shapes can be drawn on a sphere or curved surfaces. The truncated cuboctahedron is the first in a series of shapes made from hypercubes.

Main article: Vertex configuration
Main articles: Omnitruncated polyhedra, Zonohedrons
Further information: Truncated triheptagonal tiling

Bowtie tetrahedron and cube contain two trapezoidal faces in place of each square.

Truncated cuboctahedral graph

In graph theory, the truncated cuboctahedral graph shows the points and lines of a special 3D shape called the truncated cuboctahedron. This shape is one of the Archimedean solids.

The graph has 48 points and 72 lines connecting these points. It is special because of its symmetry and its link to the study of 3D shapes.