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Peter Richter - Institut für Theoretische Physik The Golden Section in Nature 2007 3 14

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Peter Richter - Institut für Theoretische Physik Ratios of numbers: a mathematical construct and a vote 1:11:1.414…1:1.618…1:2

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Peter Richter - Institut für Theoretische Physik Some people claim that Nautilus embodies the golden section – does it?

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Peter Richter - Institut für Theoretische Physik Does Apollo?

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Peter Richter - Institut für Theoretische Physik May be the Parthenon und Walhalla?

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Peter Richter - Institut für Theoretische Physik 0 g 1 g 1 1- g g = g 2 = 1- g g (1+ g) = 1 g = 1 1+ g = 1 1 + 1 1+ g = 1 1 + 1 1 1+ g = … Calculating the golden number

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Peter Richter - Institut für Theoretische Physik Harmonic approximations: Fibonacci numbers c 1:1 c'c' 1:2 g 2:3 a 3:5 g#g# 5:8 ? 8:13 G=1/g 1:1,618 … Tonica Octave sm 3rd lg 6th Quart Quint lg 3rd sm 6th

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Peter Richter - Institut für Theoretische Physik The golden number g is the most irrational of all numbers: even though its continued fraction approximation p n /q n converges quadratically fast with the denominator q n, this convergence is slower than for any other irrational number! Therefore, g embodies the principle of irrationality, it is as far as can be from any resonances like 1:1, 1:2, 2:3, … g may not only be considered on a line but also on a circle

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Peter Richter - Institut für Theoretische Physik The golden section on disk or cylinder

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Peter Richter - Institut für Theoretische Physik Sunflowers, daisies, thistles, … 138.0 o 137.5 o 136.5 o 34:55

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Peter Richter - Institut für Theoretische Physik Real or fake?

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Peter Richter - Institut für Theoretische Physik

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Double pendulum and roses …

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Peter Richter - Institut für Theoretische Physik Do we understand? Double pendulum: yes, from –Poincaré ~1890 –Kolmogorov, Arnold, Moser 1963 –numerical work in the 1980s Sunflowers, pine cones, and roses? partly, from –Leonardo da Vinci ~1510 –Kepler ~1610 –mathematical modelling of morphogenesis … but not really: closer analysis of the principles of regulation is needed, starting from a molecular level, identifying the important agents, finding appropriate equations and investigating their dynamics. The challenge: explain the universality of g!

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Peter Richter - Institut für Theoretische Physik

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The Golden Section The Divine Proportion The Golden Mean

The Golden Section The Divine Proportion The Golden Mean

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