
The tetrahedron is a "selfdual". The vertices and faces of one can be
juxtaposed 
. 
with another to
form two completely interpenetrating tetrahedra. This is called a 
. 
stellated
tetrahedron or stella octangula. The stella octangula itself can be inscribed in 
. 
the cube to
demonstrate its symmetry. Stellating the tetrahedron restores all of the 
. 


Fig. 9a  large
stella octangula 
Fig. 9b  small
stella octangula 
. 

Required
parts 
s.o.  32 large
triangles, 36 pinges 
s.o.  32 small
triangles, 36 pinges 
cube  48 right
triangles, 48 pinges 
cube  24 right
triangles, 12 pinges 
. 

. 
Fig. 9 
Alternative models of a stella octangula inscribed in a cube 
. 
. 
. 
cubic symmetry
elements that were lacking in the tetrahedron alone. Therefore the 
. 
stella
octangula possesses full cubic symmetry. 
. 

Exercise: Explain why the
tetrahedron is missing some cubic 

symmetry elements and why they are
restored when it is stellated. 

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Geometry rules!  Symmetry association by inscription 