What is a Cuboctahedron?Date: 01/02/2001 at 21:22:44 From: T.P. Subject: Cuboctahedron What is a cuboctahedron? Date: 01/03/2001 at 11:05:02 From: Doctor Rick Subject: Re: Cuboctahedron Hi, T.P. Start with a cube. Mark the center of each edge. Then slice off each corner right down to those marks. The original edges are completely gone, and instead you have edges that join the marks you made. Each face of the cube has been reduced to a smaller, tilted square. A new, triangular face has been added where each of the 8 vertices used to be. That's a cuboctahedron. It has 6 square faces (like a cube) and 8 equilateral triangular faces (like an octahedron). Each vertex is the same: two square faces and two triangular faces meet at the vertex, with the squares and triangular faces alternating. A polyhedron whose faces are all regular (like squares and equilateral triangles) and whose vertices are all the same is called an Archimedean solid. A cuboctahedron is an Archimedean solid. Now start with an octahedron. Mark the center of each edge. Slice off each corner of the octahedron, down to the marks. The original edges are completely gone, and instead you have edges that join the marks you made. Each face of the octahedron has been reduced to a smaller, inverted triangle. A new, square face has been added where each of the 6 vertices used to be. You just made a cuboctahedron again. In this sense a cuboctahedron is halfway between the cube and the octahedron: you can make it from either one, by doing exactly the same things to each. This works because the cube and the octahedron have a lot in common. They are called duals. A cube has as many faces as the octahedron has vertices, and as many vertices as the octahedron has faces, and they have the same number of edges. How does the cuboctahedron compare with them? For a rotating image of a cuboctahedron, see: The Uniform Polyhedra - MathConsult, Dr. R. Mader For a paper model, see: Paper Model Cuboctahedron - G. Korthals Altes http://www.geocities.com/SoHo/Exhibit/5901/cube_octahedron.html - Doctor Rick, The Math Forum http://mathforum.org/dr.math/ |
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