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Golden Section

16 17 lucas nuMber Magic integers perfectly formed from irrationals In addition to the Fibonacci numbers, nature occasionally uses another series, named after Edouard Lucas. Lucas numbers 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199. are similar to Fibonacci numbers in that they are additive each new number is the sum of the previous two numbers, and multiplicative each new number approximates the previous number multiplied by the modular . In fact any additive series will converge on the golden ratio, the Fibonacci and Lucas series just do it the quickest. Note that the first four integers the basis of the Tetraktys see page 55 are all Lucas numbers. What is fascinating about the Lucas numbers is that they are formed by alternately adding and subtracting the golden powers of and its reciprocal 1 , the two irrational parts either zipping together or peeling apart to form the integers opposite top. These are not approximations, but absolutely exact This extraordinary feature may be extended to the construction of Fibonacci numbers lower opposite. Incredibly, it turns out that all integers can be constructed out of golden section powers, providing us with a tantalising new way of constructing mathematics integers are secretly hiding their component golden powers. Together with Fibonaccis, Lucas numbers though more rare are sometimes found in the phyllotactic patterns of sunflowers at times as much as 1 in 10 in some species, and in certain cedars, sequoias, balsam trees, and other species. In general, the Lucas divergence angle of 99.5o 360o1 2 occurs in 1.5 of observed phyllotactic plant patterns, as compared to 92 for the Fibonacci driven divergence angle see title page. 2 F 1F2 1.61803398 0.38196601 1 F 1F 1.61803398 0.61803398 3 F2 1F2 2.61803398 0.38196601 4 F3 1F3 4.23606797 0.23606797 7 F4 1F4 6.85410196 0.14589803 11 F5 1F5 11.09016994 0.09016994 18 F6 1F6 17.94427191 0.05572808 29 F7 1F7 29.03444185 0.03444185 47 F8 1F8 46.97871376 0.02128623 76 F9 1F9 76.01315561 0.01315561 123 F10 1F10 122.9918693 0.0081306 199 F11 1F11 199.00502499 0.00502499 7 G4 L4 6 0 . . 8 1 5 4 4 5 1 8 0 9 1 8 9 0 Lucas no. 0 1 2 3 4 5 6 7 8 9 10 11 1 F 2 0 F0 0F2 G0 1 F 2 2 F 3 1 F1 1F2 G1 L2 1.61803398 0.38196601 F 2 3 F 4 1 F2 1F2 G2 L2 2.61803398 0.38196601 F 2 5 F 5 2 F3 2F2 G3 2 L2 4.23606797 0.76393202 F 2 8 F 6 3 F4 3F2 G4 3 L2 6.85410196 1.14589803 F 2 13 F 7 5 F5 5F2 G5 5 L2 11.09016994 1.90983005 F 2 21 F 8 8 F6 8F2 G6 8 L2 17.94427191 3.05572808 F 2 Fib. no. 2 3 4 5 6 7 8 The Lucas Series Eventermed members are formed by the addition of the greater and lesser powers of the golden section, oddtermed members by subtraction. Notice how the decimals in the odd terms are perfectly sliced off. The number 7 is formed by zipping together the fourth powers of F and 1F. Notice how the decimals sum to 9. Like the Lucas series, Fibonacci numbers can be expressed in terms of powers of the golden section. Notice the Fibonacci numbers reappearing in the equations these can be further collapsed into golden power terms by repeatedly using the same technique.
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