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.


4octantsleft109x100res100transgif256.gif (5815 bytes)

sectocta109x100res100trangif256.gif (4971 bytes)


4octantsright109x100res100transgif256.gif (5815 bytes)

= 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.


seccube12an1trans153x125res72gif256.gif (12588 bytes)


     = 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.


page 15 of lesson

Exercise:  Section the octahedron as shown.  Disassemble it into eight octants.

page 17 of lesson

Assemble the cube from these octants and a tetrahedron.

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