Volumetric relationships of polyhedra possessing cubic symmetry

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The striking symmetrical interrelationships of the Platonic and Archimedean

solids are paralleled by their fortunate volumetric interrelationships.  The unique

multi-panel connection feature of the Polymorf system facilitates a straightforward

demonstration by construction of this dynamic.

For example the octahedron can be sectioned into eight octants.

.  Volume. = 1 octahedron = 4 tetrahedrons . Fig. 19 - Octahedron sectioned into octants (12 right triangles, 6 pinges)
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Then four of these octants can be combined with a tetrahedron to make a cube.

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. Volume = 1/2 octahedron + 1 tetrahedron = 3 tetrahedra (12 right triangles, 4 large triangles, 18 pinges) Fig. 20 - Assembling the cube from four octants and a tetrahedron
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It can be demonstrated that the octahedron is equal to four tetrahedral volumes.

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Therefore the volume of the assembled cube is equal to three tetrahedral volumes.

. Exercise:  Section the octahedron as shown.  Disassemble it into eight octants. Assemble the cube from these octants and a tetrahedron.

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