Angle sums and full circles
The square is the only polygon whose angles add up to exactly one full circle. The decagon with all its stars has exactly ten.
Everyone knows that the angles of a triangle add up to 180°. Hardly anyone notices another fact: the square is the only polygon whose interior angles add up to exactly one full circle, 4 × 90° = 360°. The pentagon already has 540°, one and a half circles, the hexagon two.
The number of degrees is a human convention and does not matter here. What counts is multiples of the full circle. In this measure a polygon with n vertices has the angle sum
(n − 2) / 2
Counting the star figures too
To arrive at 1 + 2 + 3 + 4 = 10 it is not enough to look at the polygons alone. Every star figure drawn in also has an angle sum. For the star {n/k} it is, in full circles,
(n − 2k) / 2
Each further figure therefore has exactly one full circle less than the one before.
| n | Figures | Full circles per figure | Sum |
|---|---|---|---|
| 3 | 1 | 0.5 | 0.5 |
| 4 | 1 | 1 | 1 |
| 5 | 2 | 1.5 + 0.5 | 2 |
| 6 | 2 | 2 + 1 | 3 |
| 7 | 3 | 2.5 + 1.5 + 0.5 | 4.5 |
| 8 | 3 | 3 + 2 + 1 | 6 |
| 9 | 4 | 3.5 + 2.5 + 1.5 + 0.5 | 8 |
| 10 | 4 | 4 + 3 + 2 + 1 | 10 |
| 11 | 5 | 4.5 + 3.5 + 2.5 + 1.5 + 0.5 | 12.5 |
| 12 | 5 | 5 + 4 + 3 + 2 + 1 | 15 |
The total of full circles of all figures in a simplex (red) compared with the number of vertices (dashed). The table shows the breakdown for n = 3 to 12.
The table shows two correspondences:
- Square = 1 full circle, decagon = 4 full circles. The decagon alone has four full circles, as many as the tetractys has rows.
- Decagon with all its stars = 10 full circles. The figures of the decagon have 4 + 3 + 2 + 1 full circles, ten in all. The decagon is the only simplex whose number of vertices equals the total of its full circles: 10 = 10.
Smaller simplexes have fewer full circles than vertices, larger ones more. At ten the scissors open, and the tetractys follows directly from the geometry of the circle.
Shot through with fours
Carry the series further and it turns out that the whole system is shot through with intervals of four, and that it is “decimally coded”: ten and its multiples keep appearing as turning points, without any need for the decimal system. How this shows up in number theory is described in The disorder of the primes.
A further finding from the division table: for the number 25 and the divisor 10, angle sum and quotient agree, both are 2.5. At the same time 25 is the first number at which a prime position in the rhythm of six stays empty.