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.
What geometry could represent numbers better than polygons whose number of vertices is a natural number? Polygons with all their stars drawn in, the simplexes, are as it were numbers made flesh (see The simplex – the fixed form of number).
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).
How to read the figures
All the figures follow the same rule. Place n points evenly on a circle, start at one point and join it to the k-th point further on, from there again to the k-th, and so on until you return to the start. If you return before all points have been reached, you begin another figure of the same kind at the next free point. This is written n/k, and the number under each figure is the quotient n : k.
Try it yourself: choose n and k and press ‘Draw’. The lines appear in the order in which you would draw them with a pen. Each new colour is a fresh start, because the figure closed before all points were reached.
The four start figures are each the first example of their kind that encloses an area:
- Triangle, 3/1. Three points, join each neighbour. A point and a segment do not yet enclose an area, the triangle is the first polygon of all. Quotient 3.
- Double triangle, 6/2. Six points, join every second one. After three steps you are back at the start, although only three points have been reached. So you begin a second triangle at the next point: together the six-pointed star, because 6 is divisible by 2. The smaller 4/2 would give only two lines without area. Quotient 3.
- Pentagram, 5/2. Five points, join every second one. You reach all five points in a single stroke and go round the centre twice, because 5 and 2 have no common divisor. With fewer than five points no star is possible. Quotient 2.5.
- Double pentagram, 10/4. Ten points, join every fourth one. 10 is not divisible by 4, but both are divisible by 2. That is why the star closes after five points, and a second pentagram fills the remaining five. This is the smallest number at which this kind appears. Quotient 2.5.
So the figures are no arbitrary selection. They follow necessarily from asking for the smallest number at which each of the four possible relations between number and divisor first becomes visible as a figure.
| 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.
It is worth noticing how often one goes round the centre while drawing. Joining every second point, one circles the centre twice and gets two pentagons. With every third point one circles it three times and draws the star in one stroke. With every fourth, four times, and the last possible star figure in the decagon appears, two pentagrams. With every fifth point only lines remain that pass exactly through the centre, the “zero point”. These line stars have a zero status in more than one respect. Numerically: 10 : 1 = 10, 10 : 2 = 5, 10 : 3 = 3.33…, 10 : 4 = 2.5 and 10 : 5 = 2, which again belongs to the second kind.
Bear in mind: the three relations are the only ones possible between a natural number and a divisor.
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. But only the key: the tetractys is not finished with it, this is where it begins. From this geometry alone ten plays the leading role in the system of prime distribution, quite independently of the decimal system. After ten a system of grids and symmetric structures begins in which the numbers 10 and 4 assign the primes their places.
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.
An important addendum: why only ten?
Strictly speaking, ten is not the first number at which the third relation occurs, “not divisible, but with a common divisor”. Taking in each case the smallest suitable divisor, the series looks like this:
| Number | smallest suitable divisor | Quotient | Position |
|---|---|---|---|
| 6 | 4 (common divisor 2) | 1.5 | below the bound 2 |
| 8 | 6 (common divisor 2) | 1.33 | below the bound 2 |
| 9 | 6 (common divisor 3) | 1.5 | below the bound 2 |
| 10 | 4 (common divisor 2) | 2.5 | above the bound 2 |
The numbers 1, 2 and 3 naturally cannot qualify, and 5 and 7 are primes. What decides is the bound 2. Below it, with quotients smaller than 2, a number can no longer be split into at least two whole parts greater than 1 anyway. For the question of the primes these cases therefore play no role. The first constellation above the bound is 10 : 4 = 2.5, and 2.5 × 10 = 25 is the first number on a twin position 6n ± 1 that is not prime.
The geometry fits: the simplexes end at the divisor n/2, at the lines through the centre of the circle. Anything after that would only be repetition. So the line through the centre is not merely a geometric limit, it is the same bound 2 as in number theory. In the division table the quotients above this bound take up exactly a quarter of all cells, 1 : 4 (see Division table). That is how closely geometry and number theory agree here.
Does all this have anything to do with what the Pythagoreans called the tetractys 2500 years ago? A remarkably clear answer is given by Speusippus’ sentences on the Pythagorean numbers (see The ancient sources).
A striking parallel
The start figures repeat a pattern of the triangular numbers when they are ordered by the size of their number, 3, 5, 6, 10. 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.