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/