CTK Insights

08 Aug

Leonardo's Petals

Leonardo da Vinci's Notebooks contain many examples of geometric designs with circles and squares. Here's two design patterns obtained as a combination of a circle and four circles half its size. If r is the radius of the big circle, the small ones have radius \frac{1}{2}r.

The two dark patterns both consists of the same number of petals or lenses and so have the same area. What is it?

Let G and W be the respective areas of the gray flower and the remaining white region. Observe that

\frac{1}{2}G+\frac{1}{4}W=\pi \left(\frac{1}{2}r\right)^{2}.

It follows that G+G+W is equal to the area of four small (of radius \frac{1}{2}r circles.) But now turn to another diagram:

Let D be the area of the dark region. By the way we defined the three areas, G+W+D equals the area of the big circle (that of radius r). In the Notebooks, da Vinci combines these conditions:

G+G+W=\pi \left(\frac{1}{2}r\right)^{2}=\pi r^{2}=G+W+D

which shows that G=D. For the next step, Leonardo relocates the eight half-petals forming the gray flower to fill the pieces outside the square:

The resulting darkened area equals G+D=2G, on the one hand, and (\pi -2)r^{2}, on the other. This is because the square of diagonal 2r has the side \sqrt{2}r and the area of 2r^{2}. For the area P of a single petal he got

P=\frac{1}{8}(\pi - 2)r^{2}.

(Another of Leonardo's mathematical exploits can be found elsewhere.)

References

  1. Alexander J. Hahn, Mathematical Excursions to the World's Great Buildings, Princeton University Press, 2012

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