Polyhedra cuboctahedron

A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cube has 12 identical vertices and 24 identical edges where 2 triangles and 2 squares meet each, each of which separates a triangle from a square. As such, it is a quasi-regular polyhedron, that is, an Archimedean polyhedron that is an edge transition as well as a vertex transition. It is the only polyhedron that is radially equilateral convex.

Area and Volume
Area & Volume

A double polyhedron is a rhombic dodecahedron.


The cube was probably known to Plato: The definition of Heron cites Archimedes that Plato knew a solid made up of eight triangles and six squares.
A cuboctahedron  is a unique convex polyhedron whose long radius (from center to vertex) is equal to the length of its edges. So the long diameter (from one vertex to the opposite vertex) is the length of two edges. This radial equilateral symmetry is a property of several uniform polytopes, including two-dimensional hexagons, three-dimensional hexahedrons, and four-dimensional 24-cells and 8-cells (orthotropic). A radially equilateral polytope can consist of long radii in an equilateral triangle meeting at the center of the polytope, each contributing two radii and an edge. Thus, all internal elements that meet at the center of these polytopes have equilateral triangular inner faces, equivalent to dissecting a cube into 6 equilateral triangles and 8 tetrahedra. Each of these radially equilateral polytopes arises from cells in characteristic space-filling tessellations, such as regular hexagonal tilings, hexahedral honeycombs (alternating cubes and octahedrons), 24-cell honeycombs, and tesseractic honeycombs, respectively. Each tessellation has a double tessellation. The cell center of a tessellation is the cell vertex of a double tessellation. Regular sphere packing, which is known to be the densest in 2nd, 3rd, and 4th dimensions, uses the cell centroid of one of these tessellations as the sphere centroid.

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A cube has octahedral symmetry. The first asterisk is a composite of a cube and a double octahedron, the vertices of the cubes being at the midpoints of both edges.

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