### 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 is the radius of the big circle, the small ones have radius .

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

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

It follows that is equal to the area of four small (of radius circles.) But now turn to another diagram:

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

which shows that . 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 , on the one hand, and , on the other. This is because the square of diagonal has the side and the area of . For the area of a single petal he got

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

### References

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