Polymorf - Knowhere - Sectioning polyhedra

Volumetric relationships of polyhedra possessing cubic symmetry

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)

Then four of these octants can be combined with a tetrahedron to make a cube.

	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

It can be demonstrated that the octahedron is equal to four tetrahedral volumes.

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.

Page 16 Geometry rules! - Volumetric relationships of polyhedra