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 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 | Triangular number T(n) | Sum of all T 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 |
Left: in the triangle of six, three new points join three old ones, a first balance. Right: the points of all triangles before the triangle of ten add up to ten again.
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.
Example 2: The angle sums of the simplexes
Draw all possible star polygons into an n-gon and you get the simplex. 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
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. From the pentagon on they overlap, and small quadrilaterals appear in the overlaps.
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.
Each of the ten sub-decagons contains exactly ten such quadrilaterals. Smaller 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:
- 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. 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. See The disorder of the primes.