The key to the tetractys
A number can stand to a divisor in only three ways. Together with the number itself they give four basic figures, and ten is the first number that contains all four.
Join every k-th vertex of a regular n-gon and you get a figure written {n/k}. Which kind of figure it becomes depends only on how n and k relate to each other. Every figure in the simplex therefore describes how the number n divides.
An example: 6 is divisible by 2 and by 3. In the hexagon, “join every second point” gives two triangles (2 × 3), and “join every third point” gives three lines through the centre (3 × 2).
Three kinds of divisibility, four figures
A natural number and a possible divisor can stand in exactly three relations:
- The number is divisible by the divisor.
- The number and the divisor are coprime, sharing no factor except 1.
- The number is not divisible, but shares a common factor greater than 1 with the divisor.
Add the number itself, the plain polygon with no divisor, and you get four basic figures. Each of them appears for the first time at a particular number. These first appearances are called the start figures here:
The four start figures: the triangle (3), the double triangle or hexagram (6 : 2), the pentagram (5 : 2) and the double pentagram (10 : 4).
| Figure | first appearance | divisibility |
|---|---|---|
| Polygon | 3 | the number itself, no divisor |
| Compound polygon | 6 : 2 = 3 | divisible |
| Star polygon | 5 : 2 = 2.5 | coprime |
| Compound star | 10 : 4 = 2.5 | not divisible, common factor 2 |
For even numbers there are also the lines through the centre (k = n/2). They have no area and belong to the second group, since n is divisible by k.
Why ten
The decagon is the first number whose simplex contains all four basic figures at once:
- 10 : 1 gives the decagon itself, the polygon.
- 10 : 2 gives two pentagons, a compound polygon.
- 10 : 3 gives a star that can be drawn in one stroke, a star polygon.
- 10 : 4 gives two pentagrams, a compound star.
- 10 : 5 gives five lines through the centre, the line star.
Four basic figures, united for the first time in the number ten: this is the real key to the tetractys 1 + 2 + 3 + 4 = 10.
The simplex of any n-gon, broken down into its figures. Under each figure is the quotient n : k. From n = 10 on, every even simplex contains all four start figures.
A striking parallel
The start figures repeat a pattern of the triangular numbers. 6 is the third triangular number, and the hexagram (the double triangle) is the third start figure. 10 is the fourth triangular number, and the double pentagram is the fourth start figure.
On this reading, polygons and simple star polygons are a kind of background noise. The compound figures, by contrast, form an ordering grid within the numbers: polygon and compound polygon lie row by row in the division table, the star figures as a grid over the whole area. How the order and disorder of the primes follow from this is shown in the section Number theory.