
Icosahedron
and dodecahedron symmetry 
. 
The icosahedron and the dodecahedron are duals of each other and therefore are 
. 
symmetrically
identical. Both possess fifteen mirror planes, fifteen axes and planes of 
. 



Fig. 10 
Icosahedron 
Fig.
11a  Dodecahedron  Fig. 11b 
Required
parts 
20 triangles 
60 triangles 
30 isosceles
triangles 
30 pinges 
90 pinges 
20 pinges, 4
rubber bands 

. 
. 
. 
Fig. 10 and
11  Icosahedron and dodecahedron* 
. 
. 
. 
2fold
rotational symmetry, ten axes and planes of 3fold symmetry, and six axes and 
. 
planes of
5fold symmetry for a total of 46 elements. In Fig. 11b rubber bands are 
used to outline
the shape of a cube inserted inside a dodecahedron to show that the 
dodecahedron
shares the four (100) mirror planes of the cube as well as its four [111] 
axes of 3fold
symmetry. 
*Note: Fig. 11a is a representation
of a dodecahedron that has concave pentagonal 
faces replacing
the classic pentagons. Fig. 11b is a dodecahedron nolid (the negative 
of a solid).
It is constructed from 30 isosceles triangles that converge on the center of 
the shape. Actually it is
mathematically impossible to construct with the dimensions 
of the isosceles triangles used
in Polymorf, but there is enough "give" in the model to 
make it work. (This is
why the Greeks did not rely on models to prove their theorems!) 
. 

Exercise: Build Fig. 11b
(including the rubber bands). Then compare 

the symmetry elements of the
dodecahedron to those of the cube. 

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Geometry rules!  Symmetry association by inscription 