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. This is only the tip of the iceberg: continue the series of numbers and it turns out that the whole system is pervaded by intervals of four and structured in steps of ten.
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. This happens for only four figures at all (16/4, 18/3, 18/6, 25/10), and only here with a fractional value. At the same time 25 is the first number at which a prime position in the rhythm of six stays empty.
Half and full circles
Nature counts in whole circles, one circle is the factor 1. To grasp the essence of 1 + 2 + 3 + 4 = 10, number-theoretical peculiarities alone are not enough. One has to see a fact that is hardly noticed: the series of numbers and all its divisors contain a kind of rotation connected with the circle and thus with the number π. Like a wave, whose circular motion divides into four sections, the series of numbers runs in quarters (see The tetractys at a glance).
The triangle has half a circle, one more corner makes it a whole one. Every odd quantity begins a new half circle, the following even quantity closes it. The “closing” number is thus always even, and a full circle always takes two points. That is why Pythagoreans and Platonists ascribed male-creative qualities to odd numbers and female-preserving ones to even numbers. The German word for odd, ungerade, “not straight”, even carries a note of incompleteness.
A triangle of full circles
Take only the even polygons 4, 6, 8 and 10, and of them only the figures whose angle sums are whole full circles, that is, all of them up to the lines through the centre, and arrange them in rows: a triangle appears.
The figures of the even polygons 4, 6, 8 and 10 with their angle sums in full circles. The rows give 1, 3, 6 and 10, the triangular numbers of the triangle of points. The colours show the four figure types.
The rows contain 1, 2, 3 and 4 figures, and their angle sums together give 1, 3, 6 and 10 full circles, exactly the triangular numbers of the Pythagorean triangle of points. The decagon is the only simplex whose angle sum in full circles equals the number of its vertices. At the same time, in 10/4, it shows for the first time a star made of stars, two pentagrams, and so is the first to unite all four kinds of divisibility.
Sum up what comes together here:
- the strict correspondence of odd and even numbers to half and whole circles
- the quantities 4 and 10 and their fusion into the double pentagram at position 10/4
- the link to the triangular numbers
- the direct correspondence to the kinds of divisibility in number theory
- the start of the series with point, line, triangle and tetrahedron, that is vertex, edge, face and solid
- and the exact agreement of this “geometric division table” with the lambdoma of music theory
All this leaves little doubt that precisely this complex of themes is meant when the Pythagoreans spoke of the tetractys. Clear pointers are given by Speusippus’ sentences on the Pythagorean numbers (see The ancient sources). And since the beginning of an order shapes its further course, the larger numbers in the division table show this structure too.