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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. |
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 |
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Volume. |
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= 1
octahedron |
=
4 tetrahedrons |
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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. |
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Exercise: Section the octahedron
as shown. Disassemble it into eight octants. |
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Assemble the cube from these octants
and a tetrahedron. |
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Geometry rules! - Volumetric relationships of polyhedra |