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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