
Compounding the cube and the octahedron is another way to show that they are 
. 
symmetrically
identical. In doing so the vertex of one polyhedron is juxtapositioned 
. 


Fig. 7a 
Compound cube/ 
Fig. 7b  2,
3, and 4fold 
octahedron 
symmetry of
cube/octahedron 
Required
parts 
32
large triangles, 6 large squares, 24 right triangles, 72 pinges 
. 


. 
Fig. 7 
Compounding the cube and octahedron 
. 

. 
with the face
center of the other so that their rotational axes and mirror planes 
. 
coincide.
As a result the octahedron is said to be the dual of the cube and v.v. See 
. 
that the compounded polyhedra
are created by raising the center point of their faces 
until they can be connected by
a straight line that bisects the intervening edge. 
Note that the faces of the cube and octahedron modeled in Fig. 7 interpenetrate 
. 
each other completely so that
there is no ambiguity in the fact that the model was 
compounded from two distinct
and individual polyhedra. 
. 

Exercise: Demonstrate that the
edges of the polyhedra are bisected 

and that straight lines connect the
raised center points of their faces. 

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Geometry rules!  Symmetry association by inscription 