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Quasi-regular Polyhedra The Cuboctahedron and the Icosidodecahedron
A polyhedron is called quasi regular if it consists of two sets of regular polygons, lets say, m-sided and n-sided respectively, and it is constructed such that each polygon in one set is completely surrounded by members of the other set. For the cuboctahedron m = 3 n = 4 For the icosidodecahedron m = 3 n = 5 Observing both of the solids above it can clearly be seen that four faces meet at each vertex in the cyclic order m, n, m, n,. Because of this, these polyhedra have some special characteristics. One characteristic is that they and their duals are ‘edge regular’ meaning all the edges are equivalent. As a consequence they are like the platonics and have the same dihedral angle all around. Generally, a quasi regular is to be found inside the compound of (m, n) and (n, m). The vertices of the quasi regular solids can be obtained by truncating the platonic solids to the edge midpoints. Thus, the quasi regular solids can be obtained by truncating the platonic solids to the edge mid points.
The Cuboctahedron
3,4,3,4 The cubeoctahedron is a special type of polyhedron in that it can be derived by means of two different processes. The first of these processes is shown below and this involves the truncation of a cube. Note that this truncation is different to that involved in the formation of the original truncated hexahedron. The second method of formation is through the assemblage of the hexahedron and the icosahedron. The name of this polyhedron suggests a close relationship to the cube and the octahedron and indeed this is true. The six squares are on the facial planes of the cube and the eight triangles are on the facial planes of an octahedron. If you make a compound of the two platonics, the octahedron and the cube (which would mean an assemblage of two or more polyhedra, usually interpenetrating and having a common centre), the portion of space which is common to both polyhedra is the shape of the cuboctahedron.
The Formation of the Cuboctahedron by Truncation
The Compound of the Octahedron and the Hexahedron
If one visualises the area common to both of the solids shown above it will become evident that the resulting solid is also the cuboctahedron.
The Icosidodecahedron
The above image clearly satisfies the quasi-regular characteristic stating that, a polyhedron is called quasi regular if it consists of two sets of regular polygons, and it is constructed such that each polygon in one set is completely surrounded by members of the other set. As with the cuboctahedron the icosidodecahedron has a close relationship with two platonic solids, the icosahedron and the dodecahedron. If the icosahedron and the dodecahedron are assembled together with a common centre point, then the surfaces that are common to both will leave the shape of the icosidodecahedron ensuring of course that the vertices of one penetrates through the centres of the others faces, (ie. they have a common centre). The Compound of the Icosahedron and the Dodecahedron
If one visualises the area common to both of the solids shown above it will become evident that the resulting solid is the icosidodecahedron. These quasi regular solids are also called combinatorial solids as they are a combination of two other solids.
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Table of Contents
Polyhedra & Spherical Geometry Spherical Projection of the Cube
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