Thinking about pentagons and dodecahedrons – 6

Curiously, if you tie an overhand knot in a strip of paper, carefully flattening it so that all of the edges meet, you can see the pentagon revealed in the knot itself. The golden mean is revealed in the ratio between the longest crossing edge of the strip on top and any of the edges of the pentagon that is formed. In this example, the paper is blue on one side and green on the other:
 

Plato's enchantment with the dodechadron

As the ancient Greek mathematicians began to explore three-dimensional space, they discovered that there are five regular solids (or polyhedrons) that can be formed from regular polygons. For example, the cube is the regular solid that is formed from squares. Of these five, three are based on equilateral triangles, one is based on the square, and the remaining one is based on the pentagon.

Plato attributed significant philosophical significance to these five solids, so much so that they came to be known as the Platonic solids. He associated the four elements—air, earth, water, and fire—with the four solids made from triangles and squares.

The fifth polyhedron, the pentagon-based dodecahedron, symbolized the heavens to Plato: "There still remained a fifth construction, which the gods used for embroidering the constellations on the whole heaven."

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