Compounding the cube and the octahedron is another way to show that they are

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symmetrically identical.  In doing so the vertex of one polyhedron is juxtapositioned

.  Fig. 7a - Compound cube/ Fig. 7b - 2, 3, and 4-fold octahedron symmetry of cube/octahedron Required parts 32 large triangles, 6 large squares, 24 right triangles, 72 pinges .
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Fig. 7 - Compounding the cube and octahedron

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with the face center of the other so that their rotational axes and mirror planes

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coincide.   As a result the octahedron is said to be the dual of the cube and v.v.  See

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that the compounded polyhedra are created by raising the center point of their faces
until they can be connected by a straight line that bisects the intervening edge.

Note that the faces of the cube and octahedron modeled in Fig. 7 interpenetrate

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each other completely so that there is no ambiguity in the fact that the model was
compounded from two distinct and individual polyhedra.
. Exercise:  Demonstrate that the edges of the polyhedra are bisected and that straight lines connect the raised center points of their faces.

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