# Golden Section

22 23 growTh DiMinuTion through the looking glass Nature pulses with cycles and rhythms of increase and decrease. Heraclitus, a Presocratic influence on Plato, noted The way up and the way down are one and the same. Observe the waxing and waning of the moon, the circle of the year, the interplay of day and night, the breath of the tides, the systole and diastole of the beat of the heart, and the expansion and contraction of the lungs. The explosive growth of a star is often followed by implosion, and the negative entropy in the ordered organisation of life is balanced by the positive entropy of disorder and death. In chaos theory, the golden section governs the chaos border, where order passes into and emerges out of disorder see appendix V. Demanding simplicity and economy, nature appears to require an accretion and diminution process that is simultaneously additive and multiplicative, subtractive and divisional. This demand is satisfied perfectly only by the golden section powers, and in practice by Fibonacci and Lucas approximations. In the table opposite top, notice how we can move upwards in growth by both addition and multiplication, and move down, diminishing, by subtraction and division. The fulcrum is Unity acting as the geometric mean in golden relationship to both the increase of the deficient Lesser and decrease of excessive Greater. Think of an oak tree. It shoots up as fast as it can from an acorn, only to slow, mature and fractalise its space towards a limit, becoming a new relative unity, what Aristotle called an entelechy, the form it grows into. Like Alice in Wonderland, nature simultaneously grows and diminishes to relative limits. 144 89 55 34 21 13 8 5 3 2 1 89 55 34 21 13 8 5 3 2 1 1 55 34 21 13 8 5 3 2 1 1 0 123 76 47 29 18 11 7 4 3 1 2 199 123 76 47 29 18 11 7 4 3 1 322 199 123 76 47 29 18 11 7 4 3 G M L G M L Fibonacci Lucas F7 F6 F5 F4 F3 F2 F 1 1F 1F2 1F3 1F4 1F5 1F6 1F7 F6 F5 F4 F3 F2 F 1 1F 1F2 1F3 1F4 1F5 1F6 1F7 1F8 F5 F4 F3 F2 F 1 1F 1F2 1F3 1F4 1F5 1F6 1F7 1F8 1F9 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 n Greater Mean Lesser d i m i n u t i o n t h e w a y d o w n g ro w th t h e w a y u p The Fibonacci approximate geometric mean is corrected alternately by 1 or 1 under the square root. So 3 is the approximate geometric mean of 2 and 5, as 2 x 51 9, and 5 is the approximate geometric mean of 3 and 8, 3 x 81 25. The Lucas approximate geometric mean is corrected alternately by 5 or 5 under the square root. So 4 is the approximate geometric mean of 3 and 7, 3 x 75 16, and 7 is the approximate geometric mean of 4 and 11, 4 x 115 49. The Golden Series shown opposite displays the unique simultaneous additive and multipicative qualities of the Golden Section. Multiplication Gn1 Gn x F Addition Gn1 Gn Mn Gn Gn1 Division Gn1 Gn F Subtraction Gn1 Mn Gn Ln Gn Gn2 These equations may be extended for Lesser and Mean values. Each term is simultaneously the sum of the preceding two and the product of the previous term multiplied by F. So F4 F2 F3 F2 x F2 F3 x F No other number behaves likes this, fusing addition and multiplication.