Number theory

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 $\pi$ and the primes are connected. He showed that the sum of the reciprocals of all square numbers equals $\pi^{2}/6$, and that this sum can be written as a product over all primes. This has long been proven, and the $\pi$ comes from the sum over all numbers. Many still find it astonishing that through the product it is also linked with the primes. The thesis: the simplexes, figures in a circle that show how numbers divide, are the key. In essence and meaning they correspond to the circle number: as the circumference relates to the diameter, so the natural number, the n-gon, relates to its possible divisors, the lines through the circle. Through Pascal’s triangle the simplexes are also directly linked to number theory.

The hexagon and the three

It has been known since Archimedes that the hexagon and $\pi$ 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 $\pi = 3.14159$ …

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.

Vertices

Squares on the sides of an n-gon, folded inwards, with alternating fill. The readout shows the ratio of perimeter to diameter of the inscribed polygon approaching π from below. The table beside it shows the overlap area, which approaches π from above.

According to Andreas Weiss’s calculation, the total intersection area of the folded squares, measured in unit squares, approaches $\pi$ as the number of vertices grows: 3.14159 … for very many vertices.

With each vertex a square is added, while the circle grows and the individual intersections shrink. That is why the total changes only slightly and becomes stable only at infinity. That is exactly the area of a circle of radius 1. His idea: in the formula for the area of a circle, $\pi$ 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.

That the total tends to $\pi$ can be proven. From the 12-gon on the area of overlap is exactly $n \tan\tfrac{\pi}{n}$, the area of the polygon enclosing the unit circle, and that this area tends to $\pi$ is the theorem of Archimedes. So the folded squares rebuild his method in a surprising way (see Approaching π).

Further observations on these figures:

• In the pentagon all five squares overlap in places, in the hexagon all six, in the heptagon up to four, from the octagon to the hendecagon up to three.
• 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:

Only up to 10/4 are all four principles of the tetractys contained. After that the positions repeat at intervals of four.

n-gon

PolygonCompound polygonStar polygonCompound starLine star

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 $\pi /4 \approx 78.5$ %. And $\pi \times 25 = 78.54$, at 25, the first number at which the order of the twin primes breaks.

Sources and further reading

Sources

• Holger Ullmann, tetraktys.de, since 2010
• Archimedes, Measurement of a Circle, 3rd century BC
• Leonhard Euler, Introductio in analysin infinitorum, Lausanne 1748