Polymorf - Knowhere - Icosahedron, Dodecahedron symmetry

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 and 11 - Icosahedron and dodecahedron*

2-fold rotational symmetry, ten axes and planes of 3-fold symmetry, and six axes and

planes of 5-fold 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 3-fold 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.

Page 9 Geometry rules! - Symmetry association by inscription