# Golden Section

6 7 PlaTos DiviDeD line knowing precisely where to cut So, returning to our puzzle, why does Plato ask us to make an uneven cut An even cut would result in a whole segment ratio of 21, and the ratio of the two equal segments would be 11. These ratios are not equal and so no proportion is present There is only one way to form a proportion from a simple ratio, and that is through the golden section. Plato wants you to discover a special ratio such that the whole to the longer equals the longer to the shorter. He knows this would result in his favourite bond of nature, a continuous geometric proportion. The inverse also applies, the shorter to the longer equals the longer to the whole. And why a line, rather than simply numbers Plato realised the answer is an irrational number that can be geometrically derived in a line, but cannot be expressed as a simple fraction see page 54. Solving this problem mathematically, and assuming the mean longer segment is 1, we find the greater golden value of 1.6180339... for the whole, and the lesser golden value 0.6180339... for the shorter. We term these fye the Greater and f fee the Lesser respectively. Notice that both their product and their difference is Unity. Furthermore, the square of the Greater is 2.6180339, or 1. Notice also that each is the others reciprocal, so that f is 1. In this book we will generally speak of the Greater as , the mean as Unity 1, and the Lesser as 1. Notice below left to right that Unity can act as the Greater whole, Mean longer segment or as the Lesser short segment.