Each tessellation has a dual tessellation the cell centers in a tessellation are cell vertices in its dual tessellation. Therefore, all the interior elements which meet at the center of these polytopes have equilateral triangle inward faces, as in the dissection of the cuboctahedron into 6 square pyramids and 8 tetrahedra.Įach of these radially equilateral polytopes also occurs as cells of a characteristic space-filling tessellation: the tiling of regular hexagons, the rectified cubic honeycomb (of alternating cuboctahedra and octahedra), the 24-cell honeycomb and the tesseractic honeycomb, respectively. Radially equilateral polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge. This radial equilateral symmetry is a property of only a few uniform polytopes, including the two-dimensional hexagon, the three-dimensional cuboctahedron, and the four-dimensional 24-cell and 8-cell (tesseract).

(In the case of the cuboctahedron, the center is in fact the apex of 6 square and 8 triangular pyramids). Its center is like the apical vertex of a pyramid: one edge length away from all the other vertices. In a cuboctahedron, the long radius (center to vertex) is the same as the edge length thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Progression between a tetrahedron, expanded into a cuboctahedron, and reverse expanded into the dual tetrahedron Radial equilateral symmetry If these two triangular cupolas are twisted so triangles and squares line up, Johnson solid J 27, the triangular orthobicupola, is created. The cuboctahedron can be dissected into two triangular cupolas by a common hexagon passing through the center of the cuboctahedron. This dissection is expressed in the tetrahedral-octahedral honeycomb where pairs of square pyramids are combined into octahedra.

The cuboctahedron can be dissected into 6 square pyramids and 8 tetrahedra meeting at a central point. The area A and the volume V of the cuboctahedron of edge length a are:Ī = ( 6 + 2 3 ) a 2 ≈ 9.464 1016 a 2 V = 5 3 2 a 3 ≈ 2.357 0226 a 3. With the addition of 6 vertices of the octahedron, these vertices represent the 18 root vectors of the simple Lie group B 3. The cuboctahedron's 12 vertices can represent the root vectors of the simple Lie group A 3. This construction exists as one of 16 orthant facets of the cantellated 16-cell. The Cartesian coordinates for the vertices of a cuboctahedron (of edge length √ 2) centered at the origin are:Īn alternate set of coordinates can be made in 4-space, as 12 permutations of: Straight lines on the sphere are projected as circular arcs on the plane. This projection is conformal, preserving angles but not areas or lengths. The cuboctahedron can also be represented as a spherical tiling, and projected onto the plane via a stereographic projection. The skew projections show a square and hexagon passing through the center of the cuboctahedron. The last two correspond to the B 2 and A 2 Coxeter planes. The cuboctahedron has four special orthogonal projections, centered on a vertex, an edge, and the two types of faces, triangular and square. With D 3d symmetry, order 12, it is a triangular gyrobicupola.With T d symmetry, order 24, it is a cantellated tetrahedron or rhombitetratetrahedron.With O h symmetry, order 48, it is a rectified cube or rectified octahedron ( Norman Johnson).Fuller also called a cuboctahedron built of rigid struts and flexible vertices a jitterbug this object can be progressively transformed into an icosahedron, octahedron, and tetrahedron by folding along the diagonals of its square sides. Vector Equilibrium ( Buckminster Fuller) because its center-to-vertex radius equals its edge length (it has radial equilateral symmetry).The cuboctahedron was probably known to Plato: Heron's Definitiones quotes Archimedes as saying that Plato knew of a solid made of 8 triangles and 6 squares. Its dual polyhedron is the rhombic dodecahedron. an Archimedean solid that is not only vertex-transitive but also edge-transitive. As such, it is a quasiregular polyhedron, i.e. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces.

Ipicture of a cuboctahedron flattened