Today is π-day. Investigating the properties of a circle is one of my favorite hobbies, so this is an appropriate event for some discussion on the circle. I love drawing circles and lines on a piece of paper. It's fascinating to discover natural laws in the circle. At each of my drawings, I gain some new insghts. Recently I came across a Cut The Knot post presenting an inventive way to divide a circular area into any number of equal parts, with compass and ruler. I googled a little more over it and curiously, it seems that the presented yin and yang-like solution is the only one that has been presented (if you know of other published references, I would be eager to hear about). It appeared in a Mathematical Association book in 1995 but I didn't find any other reference to solutions of this problem. This is surprising, because area division is a fundamental problem with broad applications in this world where sharing physical resources equally is a necessary guarantee for peace. Playing a bit with the elementary components of the yin and yang pattern gives related solutions, of which a random one is sketched in Figure 1.
I convinced myself that the most general solution is in fact a quite natural and intuitive one, because it figures the way how circular patterns often grow in nature, like the annual rings in wood. Other patterns also give some insight in various ways to divide a circular area into parts, like the beautiful quintuple emulsions formed with microfluidics of the figured image (courtesy of A. Abate/ Harvard University).
The general rule to have a circular area divided equally into N of such nested circles is that, for a starting circle of radius 1, the surrounding Nth circle radius is the square root of N (noted √N).
The first circle has radius 1, area π.
The second circle has radius √2, area 2π.
The third circle has radius √3, area 3π.
The fourth circle has radius √4 (= 2), area 4π.
And so on...
Square roots can be easily drawn with compass and ruler, so the same for nested circles with radius equal to √N, where N is incremented from 1 to any positive integer value. Figure 4 shows a convenient procedure for N=1 to 9:
Step 1. Draw nested circles with diameter 1 to 9 whose left diameter endpoints coincide.
Step 2. Draw a perpendicular to the diameter direction at the intersection of the right endpoint of the circle with diameter 1.
Step 3. Draw circles of radius equal to √N thanks to intersections of first circles with perpendicular at N=1.
Step 4. Erase lines and circles of intermediate steps. The obtained outer circle is divided into N equal parts each of area π.
Now, from this pattern, it is possible to start a set of variations on circular area division, because the nested circles are whole, contrarily to the yin and yang pattern. We may for example start to divide the circle of area 2π into two equal half-circles and shift those components randomly (see Figure 5). Or we may start to divide the circle of area 3π into equal parts, etc. I'm sure anyone can find original new ways to arrange and transform all those equal parts, in order to gain some new intuitions on the circle.