
The
symmetry of the tetrahedron 


The tetrahedron is the least symmetrical of the Platonic solids. However it does 


share
some of the cube's symmetry elements. As shown previously in Fig.
4a it can be 


inscribed
in the cube to reveal its symmetry in a straightforward manner. As Fig. 4a 



demonstrated,
the tetrahedron shares the cube's four [111] 
rotational axes
and (111) planes of 3fold symmetry. And, 
like the cube,
it has six (110) mirror planes. But it has only 
three [110] axes and (110) planes of 2fold rotational
symmetry 
and no
(100) planes  thirteen symmetry elements in all. 




Likewise, the octahedron can also be inscribed in the tetrahedron to reveal its sym 


metry
by association. In fact a Polymorf model can be built of an octahedron inscribed 


in
a tetrahedron inscribed in a cube to associate the symmetry of all three polyhedra. 







Fig.
8a  Octahedron inscribed 
Fig.
8b  Octahedron inscribed in 
in
a tetrahedron 
a
tetrahedron inscribed in a cube 

Required
parts 

20
triangles 
20
large triangles, 12 small squares 
24
pinges 
24
right triangles, 72 pinges 

. 



Fig. 8 
Associating the symmetry of the tetrahedron by inscription 
. 




Exercise: The (110) mirror
planes of the cube and tetrahedron have the same orienta 

tion but the [110] axes and (110)
planes of rotational symmetry do not. Explain why. 

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Geometry rules!  Symmetry association by inscription 