The beauty of mathematics in nature

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The polar equation for a golden spiral is the same as for other logarithmic spirals, but with a special value of b:

r = ae^{b\theta}\,

or

\theta = \frac{1}{b} \ln(r/a),

with e being the base of natural logarithms, a being an arbitrary positive real constant, and b such that when theta  is a right angle (a quarter turn in either direction):

e^{b\theta_\mathrm{right}}\, = \phi

Therefore, b is given by

b = {\ln{\phi} \over \theta_\mathrm{right}}.

The numerical value of b depends on whether the right angle is measured as 90 degrees or as \textstyle\frac{\pi}{2} radians; and since the angle can be in either direction, it is easiest to write the formula for the absolute value of b (that is, b can also be the negative of this value):

|b| = {\ln{\phi} \over 90} = 0.0053468\, for in degrees;
|b| = {\ln{\phi} \over \pi/2} = 0.306349\, for in radians.

An alternate formula for a logarithmic and golden spiral is:

r = ac^{\theta}\,

where the constant c is given by:

c = e^b\,

which for the golden spiral gives c values of:

c = \phi ^ \frac{1}{90} \doteq 1.0053611

if is measured in degrees, and

c = \phi ^ \frac{2}{\pi} \doteq 1.358456.

if is measured in radians

See, mathematics is beautiful!

OK, so I stole the text from Wikipedia. I can’t really remember all this mathematics from high school, but I can still appreciate the beauty.

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