The triangle of points – just 1 + 2 + 3 + 4 = 10?
Why ten, of all numbers? Three observations in which one series of numbers catches up with another at exactly this point.
The triangular numbers 1, 3, 6, 10, 15 … are the only purely number-theoretical clue to the tetractys, and ten is the fourth of them. But what makes it so special? Why did the Pythagoreans all but worship four and ten? On a whim, out of religious conviction, or were they simply wrong? That would be odd, since Pythagoras brought very solid mathematics into the world.
The ten of the tetractys is often explained by our ten fingers and the decimal system. The thesis here is a different one: ten is not a human convention. It follows from the sequence of natural numbers and the geometry that belongs to them. Three examples show what is meant. In all three the same thing happens at ten: two series of numbers that have been apart until then meet, and afterwards the “scissors” open the other way.
Example 1: The triangle of points itself
Place the triangles of points side by side (1, 3, 6, 10, 15 …) and each time add up all the points of the preceding triangles. At first this sum stays smaller than the next triangle. At ten it is exactly equal: 1 + 3 + 6 = 10. From then on the sum is always larger.
| n | T(n) | sum before | |
|---|---|---|---|
| 1 | 1 | > | 0 |
| 2 | 3 | > | 1 |
| 3 | 6 | > | 4 |
| 4 | 10 | = | 10 |
| 5 | 15 | < | 20 |
| 6 | 21 | < | 35 |
| 7 | 28 | < | 56 |
| 8 | 36 | < | 84 |
Top: in the triangle of six, three new points join three old ones, a first balance. Bottom: the points of all triangles before the triangle of ten add up to ten again. The table continues the series.
This can also be written as a formula. The n-th triangular number is T(n) = n(n + 1) / 2. The sum of all triangles before it is the tetrahedral number
Te(n − 1) = (n − 1) · n · (n + 1) / 6
Divide one by the other and almost everything cancels:
Te(n − 1) : T(n) = (n − 1) : 3
So the ratio grows evenly with n. It is less than 1 as long as n is less than 4, exactly 1 at n = 4, that is at the fourth triangular number, ten, and larger ever after. The formula shows that the balance is no accident of a small number but has to lie at exactly this one point.
If the triangular number itself is counted into the sum, the picture shifts: then six faces the sum ten and ten faces the sum twenty, exactly double. This too holds for no other triangular number. The relation of six and ten will come up often on this site. One may object that this is just one way of looking at it. It becomes remarkable only because the same point reappears in quite different geometries. The triangle of points belongs to the hexagonal lattice of crystallography, and the same laws appear in space, in sphere packings and their crystal lattices.
Lay equal circles as densely as possible and the hexagonal lattice appears: six circles around each circle. Every triangle of points is a piece of it. With the slider the triangle grows row by row.
The sums in the right-hand column are the tetrahedral numbers. That the triangular and tetrahedral numbers coincide at ten can also be seen in space: a tetrahedron of three layers of spheres (1 + 3 + 6) contains ten spheres, exactly as many as the bottom layer of the next tetrahedron. More under From point to solid.
Left: three layers of 1, 3 and 6 spheres, together a tetrahedron of ten spheres. Right: the layer that would come next underneath, also ten spheres.
Example 2: The angle sums of the simplexes
The simplexes could be called another state of matter of numbers, especially when fanned out in a coordinate system and their star polygons examined one by one. With them one can truly step outside the decimal system in which we calculate, think and live. Hardly anyone can multiply in their head in another number system. Whoever works only with digits and ignores the matching geometry cannot see the wood for the trees. We owe the decimal system, it is said, to our ten fingers. That may be so. But does the geometry of the simplexes also count on ten fingers? The astonishing answer: yes, with two hands of five fingers each, the pentagram and the double pentagram (see The key to the tetractys). That the simplexes also make the order of the primes visible links number theory directly to the symmetry of circles and spheres.
Draw all possible star polygons into an n-gon and you get the simplex. This is what it looks like for the decagon:
The simplex of the decagon: each of the ten points joined to every other. The buttons show and hide the individual figures.
Take it apart into its figures and each has its own angle sum. The decagon itself has four full circles, the two pentagons together three, the star through every third point two, the two pentagrams one, and the lines through the centre have no area and no angle:
The figures of the decagon with their angle sums in full circles. Use the slider to compare other polygons: only for the decagon does the total equal the number of vertices.
Add up the interior angles of all these figures, counting in whole circles (360°), and the square has exactly one full circle. Smaller simplexes have fewer full circles than vertices, larger ones more. Only the decagon has exactly ten full circles.
| 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 |
Red line: total full circles of all figures in the simplex. Dashed: the number of vertices. The lines cross at the decagon.
The total of full circles itself grows in triangular numbers: for the decagon it is 4 + 3 + 2 + 1 = 10, the tetractys again. In detail under Angle sums and full circles.
Example 3: The fractal polygons
Anyone who has looked into fractals knows the Sierpiński triangle. For number theory it is especially important, since it is contained in Pascal’s triangle. Hardly known, by contrast, are polygons as fractals derived directly from it. Two conditions hold: as many new polygons arise as the figure has vertices, and their sides halve with each step. The first step is enough here, because everything essential already shows in it: whether the parts overlap and how many quadrilaterals arise. Further steps pile more and more overlaps on top of each other until the structure can hardly be made out. If you want to see them anyway, there is a slider on the page The tetractys in the fractal polygons. The graphic here therefore shows not a sequence of steps but how the first step changes as the number of vertices grows.
The Sierpiński triangle arises by replacing a triangle with three half-sized triangles at its corners. The same procedure can be applied to any n-gon: a half-sized copy goes at every corner. For the square, the four copies fill it without gaps, since squares tile the plane. The square with its right angles is the last fractal without overlaps. From the pentagon on the copies must overlap and group around a centre, and small quadrilaterals appear in the overlaps. With filled areas no structure can be seen any more, unless one fills alternately or draws only the outlines.
The first step of the Sierpiński procedure for n-gons. Alternating fill makes the overlaps visible. In the decagon, the quadrilaterals of one sub-decagon line up from the outside in rows of 1, 2, 3 and 4.
In all, the decagon produces 40 quadrilaterals. Each of the ten sub-decagons contains exactly ten of them, produced by its overlap with six of its neighbours. Smaller even polygons contain fewer, larger ones more, and again they are arranged in the rows of the tetractys. More under Fractal polygons.
What lies behind it
All three examples rest on the sequence of triangular numbers that the triangle of points shows directly. The thesis: at the ten counting stones Pythagoras had not merely found a pretty sum. He had before his eyes something that turns up again and again at the same place in geometry and number theory. Two further observations lead into number theory:
Six is actually the first place where the scissors open: in the triangle of six, three new points join three. This is exactly what can be laid out with counting stones, and it may be how Pythagoras showed it to his pupils:
3 + 3 = 6
1 + 3 + 6 = 10
Left: three new stones join three, the triangle of six. Right: the triangles of 1, 3 and 6 stones together have as many stones as the triangle of ten. The button lays the stones again.
But what lies behind it? Did the Pythagoreans merely take a childlike delight in triangular and square numbers? Or had they found in the ten-point triangle something truly significant, something that impressed them so deeply that it stirred religious feeling? Is the plain triangle of points perhaps the holy grail of the prime number question, which they called the tetractys? This is not a mere guess. Strictly speaking, the sieve of Eratosthenes is a sequence of regular, overlapping grids of points in the lambdoma. So it is not only about figurate numbers but about figurate fractions in a coordinate system in which the simplexes can also be fanned out (see The lambdoma and The sieve of Eratosthenes).
- Three, thought of as a grid of points, produces the rhythm of six in which the twin primes are arranged. Geometrically it corresponds to the step from the triangle to the six-pointed star, in the philosophy of religion to creation in six days. See The order of the primes.
- Five breaks this order open again from 25 = 5 × 5. Geometrically it corresponds to the pentagram, numerically to 5 : 2 = 2.5. In the philosophy of religion it corresponds to the Fall, the plunge into matter. See The disorder of the primes.