Thinking about pentagons and dodecahedrons – 1

The book Islamic Patterns by Keith Critchlow revealed the wonder-full essence of the pentagon to me. The pentagon contains within it the golden mean in the ratio between the length of one of its edges and one of its diagonals. And within the polyhedron that is derived from pentagons, the dodecahedron, the golden mean shows up in the measurements I needed to make to determine the angles of the dodecahedron.

The dodecahedron - frolicking pentagons

The dodechahedron is one of the Platonic solids. Pictured here with an added base, the dodecahedron is made up of twelve pentagons.

From a woodworking perspective, the trick is to figure out the angle between two of the pentagon faces. This dihedral angle is repeated throughout the shape. Once you know this angle, you can create one edge of a pentagon with one side angled at 1/2 of the dihedral angle. The you can simply repeat that edge five times to form a pentagon, and repeat that pentagon twelve times to form the dodecahedron.

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