The dodecahedron – 5

The divine proportion

The pentagon is a regular polygon (all of its sides are the same length, and all of its interior angles are the same).

It's unique because it contains within itself the golden mean or divine proportion, which can be simply expressed in this way: take a line of any length and divide it unequally into two parts so that the ratio of the whole line to the larger part is the same as the ratio of the larger part to the smaller.

Big deal? Well, yes, because it turns out that this proportion is one that we find in nature again and again, for example, in the spiral of a sea shell and the way in which plants unfold. It is also one of the most aesthetically pleasing proportions to look at, which is why the Parthenon at Athens is so admired.

When you draw lines between the vertices of a pentagon, the result is the pentacle, a five-pointed star, which encloses within itself another, smaller pentagon. It is in this pentacle that the golden mean reveals itself several times.

golden ratio between pentacle diagonal and pentagon edge    Golden ratio in acute triangle of pentacle    Golden ratio in obtuse triangle of pentacle


Pentagon formed by a folded strip of paperCuriously, 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.

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