## Rabbit holes

Da Vinci

Divine proportion

Dodecahedron

Dymaxion

Euclid

Golden mean

Hexagon

Icosahedron

Icosidodecahedron

Making a lamp

Motivation

Octagon

Pentagon

Pacioli

Paper

Plato

Rhombicosido-

decahedron

Rhombicub-

octahedron

Symmetry

Truncation

Wood

Zometool

# The dodecahedron – 7

## The symmetries

As within the pentagon itself, the golden mean can be found within the dodecahedron. If you take three intersecting golden rectangles (one standing vertically on one of its shorter edges, one standing vertically on one of its longer edges and running through the center of and perpendicular to the first, and the third lying horizontally and running through the centers of the first two), the center of each face of a dodecahedron will touch one of the twelve corners of those three golden rectangles.

You can see three symmetries in the dodecahedron. From one angle, it has 2-fold symmetry. In other words, if you were to take one-half of the shape you see and fold it over on top of the other half, the two halves exactly mirror each other.

From another angle, it has 3-fold symmetry. Again, if you take one-third of the shape you see and fold it over on top of another third, they mirror each other.

Finally, from a third angle, you can view a 5-fold symmetry. In the following illustrations, the 5-fold symmetry of the dodecahedron (in red) is revealed within the pentagon itself.

Symmetry is significant because it is an attribute that naturally causes us to find pleasure in something we are viewing. Certainly this has roots in the 2-fold symmetry of our own bodies, as well as the symmetry we see in the natural world surrounding us.

The beauty of 5-fold symmetry can be found in most flowers and the fruits that are born from them (think of the cross section of an apple), some crystals (it is thought that the crystal of *fool's gold* or Pyrite might have provided the insight into the dodecahedron shape), and in the star fish of the sea.