
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 
multipanel
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. 

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Geometry rules!  Volumetric relationships of polyhedra 