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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 |
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 |
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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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|
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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. |
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Exercise: Demonstrate that the
edges of the polyhedra are bisected |
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and that straight lines connect the
raised center points of their faces. |
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Geometry rules! - Symmetry association by inscription |