Golden Section

4 5 Ratio between two numbers a and b Ratio between a and b a b or ab Inverse ratio b a or ba Means b, between a and c Arithmetic Mean b of a and c b Harmonic Mean b of a and c b Geometric Mean b of a and c b ac Proportion between two ratios Discontinuous 4 termed Continuous 3 termed a b c d a b b c a b c e.g., 4 8 5 10 note b is the geometric has invariant ratio 1 2 mean of a and c Platos World Soul Extended continuous geometric proportion 1 2 2 4 4 8 1 3 3 9 9 27 invar. ratio 1 2 invar. ratio 1 3 or 12 or 13 Lambda diagram a c 2 2ac a c 1 2 3 4 9 8 27 6 12 18 raTio, Means ProPorTion continuous geometric proportion Ratio logos is the relation of one number to another, for instance 48 4 is to 8. However, proportion analogia is a repeating ratio that typically involves four terms, so 48 510 4 is to 8 is as 5 is to 10. The Pythagoreans called this a four termed discontinuous proportion. The invariant ratio here is 12, repeated in both 48 and 510. An inverted ratio reverses the terms, so 84 is the inverse of 48, the invariant ratio now 21. Standing between the twotermed ratio and the fourtermed proportion is the threetermed mean in which the middle term is in the same ratio to the first as the last is to it. The geometric mean between two numbers is equal to the square root of their product. Thus, the geometric mean of, say, 1 and 9 is 1x9 3. This geometric mean relationship is written as 139, or, inverted, as 931. It can also be written more fully as a continuous geometric proportion where these two ratios repeat the same invariant ratio of 13. Thus, 13 39. The 3 is the geometric mean held in common by both ratios, binding, or interlacing them together in what the Pythagoreans called a threetermed continuous geometric proportion. Plato holds continuous geometric proportion to be the most profound cosmic bond. In his Timaeus the world soul binds together, into one harmonic resonance, the intelligible world of forms including pure mathematics above, and the visible world of material objects below, through the 1, 2, 4, 8 and 1, 3, 9, 27 series. This results in the extended continuous geometric proportions, 12 24 48, and 13 39 927 see opposite.