Simplexes, number theory and π
The hexagon and the three before the decimal point, squares folded into an n-gon, and figures whose angles add up to exactly one full circle.
Leonhard Euler already knew that π and the primes are connected. He showed that the sum of the reciprocals of all square numbers equals π²/6, and that this sum can be written as a product over all primes. Why π, of all numbers, turns up in a statement about primes is still not really understood. The thesis: the simplexes, figures in a circle that show how numbers divide, are the key.
The hexagon and the three
It has been known since Archimedes that the hexagon and π are closely linked. In the hexagon the perimeter is exactly three times the diameter of the circumscribed circle. As the number of vertices grows, the polygon approaches the circle and the ratio approaches π = 3.14159 …
The lines through the centre of the hexagon correspond to 6 : 3, or 2 × 3. There is an image in this: the three before the decimal point stands for the ordered rhythm of six of the twin primes, the irregular decimal places for the irregular distribution of the primes.
Folded squares
Andreas Weiss raised a further connection in 2013. On each side of a square place a square of the same size and fold all four inwards: they cover one another completely, four overlaps, a consequence of the right angles. Add a vertex, making a pentagon, and five squares fold inwards and now overlap only partly. With more vertices the side length stays 1, the circle grows, and the overlaps change.
Squares on the sides of an n-gon, folded inwards, with alternating fill. The readout shows the ratio of perimeter to diameter approaching π.
According to Andreas Weiss’s calculation, the total intersection area of the folded squares, measured in unit squares, approaches π as the number of vertices grows: 4 for the square, about 3.63 for the pentagon, about 3.195 for the 14-gon, and 3.14159 … for very many vertices. That is exactly the area of a circle of radius 1. His idea: in the formula for the area of a circle, π need not be merely a factor. It can itself be understood as an area, multiplied by the square of the radius. The radius is then just a multiple of the unit circle.
Further observations on these figures:
- In the pentagon up to four squares overlap, in the hexagon three, in the heptagon two.
- From the heptagon on, a hole opens at the centre that grows with the number of vertices.
- From the dodecagon on, there are only single overlaps.
Fittingly, 7 is the sum of 3 and 4, and 12 their product. Triangle and square are exactly the two shapes that arise when the centres of touching circles are joined (see Circle, triangle and square).
Figures with exactly one full circle
All star figures that arise from the overlaps of the squares necessarily have the angle sum of one full circle. Star polygons with exactly these angles exist only in even simplexes, at these places in the division table:
| Cell | Figure |
|---|---|
| 4/1 | Polygon |
| 6/2 | Compound polygon |
| 8/3 | Star polygon |
| 10/4 | Compound star |
| 12/5 | Star polygon |
| 14/6 | Compound star |
| 16/7 | Star polygon |
| … | alternating without end |
Only up to 10/4 are all four principles of the tetractys contained. After that the positions repeat at intervals of four.
The simplex of the decagon. The figure 10/4, two pentagrams, has an angle sum of exactly one full circle.
25 and the area of the circle
One last observation: a circle fills its circumscribing square to π/4 ≈ 78.5 %. And π × 25 = 78.54, at 25, the first number at which the order of the twin primes breaks.