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

Peter Richter - Institut für Theoretische Physik

The Golden Section in Nature

中国科学院上海生命科学研究院计算生物研究所 2007 年 3 月 14 日


Ratios of numbers: a mathematical construct and a vote

1:1 1:1.414… 1:1.618… 1:2


Some people claim that Nautilus embodies the golden section – does it?


Does Apollo?


May be the Parthenon und Walhalla?


0 g 1

g

1

1- g

g=

g2 = 1- g

g (1+ g) = 1

g =1

1+ g=

1

1 +1

1+ g

=1

1 +1

1+1

1+ g

= …

Calculating the golden number


Harmonic approximations: Fibonacci numbers

c

1:1

c'

1:2

g

2:3

a

3:5

g#

5:8

?

8:13

G=1/g

1:1,618 …

Toni

ca

Oct

ave

sm 3

rdlg

6th

Qua

rtQ

uint

lg 3

rdsm

6th


The golden number g is the most irrational of all numbers:

even though its continued fraction approximation pn/qn converges quadratically fast with the denominator qn, 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


The golden section on disk or cylinder


Sunflowers, daisies, thistles, …

138.0o

137.5o

136.5o

34:55


Real or fake?


Double pendulum and roses …


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”!

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

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