Article 4MKWF Angles in the spiral of Theodorus

Angles in the spiral of Theodorus

by
John
from John D. Cook on (#4MKWF)

The previous post looked at how to plot the spiral of Theodorus shown below.

theodorus.svgWe stopped the construction where we did because the next triangle to be added would overlap the first triangle, which would clutter the image. But we could certainly have kept going.

If we do keep going, then the set of hypotenuse angles will be dense in the circle, with no repeats.

The nth triangle has sides of length 1 and an, and so the tangent of nth triangle's acute angle is 1/an. The angle formed by the nth hypotenuse is thus

arctan(1) + arctan(1/a2) + arctan(1/a3) + " + arctan(1/an).

Here's a plot of the first 99 hypotenuse angles.

theodorus_angles.svg

Here's the code that produced the plot.

 from numpy import * import matplotlib.pyplot as plt plt.axes().set_aspect(1) N = 100 theta = cumsum(arctan(arange(1,N)**-0.5)) plt.scatter(cos(theta), sin(theta)) plt.savefig("theodorus_angles.svg")

If we change N to 500 we get a solid ring because the angles are closer together than the default thickness of dots in a scatterplot.

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