What is the tetractys?

Ten points in four rows, one of the oldest figures in mathematics, and why it is more than a sum.

The Greek word tetraktys simply means “fourness”. It names a figure of ten points arranged in four rows: one point at the top, two below it, then three, and four at the bottom. Together they form an equilateral triangle.

Rows

The tetractys: 1 + 2 + 3 + 4 = 10. Use the slider to build the other triangular numbers.

A number you can see

The Greeks did not write numbers with digits the way we do. They often thought of them as arrangements of pebbles or points, so a number had a shape. Some quantities can be laid out as a square (4, 9, 16 …), others as a rectangle, still others as a triangle. Numbers of the last kind are still called triangular numbers:

1, 3, 6, 10, 15, 21, 28, 36, 45, 55 …

The ten of the tetractys is the fourth of them. In general, the n-th triangular number is

T(n) = 1 + 2 + … + n = n · (n + 1) / 2

The formula can be read straight off the picture: put two identical triangles of points against each other and you get a rectangle of n by (n + 1) points. The triangle is half of it.

More than a sum

That 1 + 2 + 3 + 4 makes ten is no deep insight. Yet the Pythagoreans saw a basic pattern of the world in this figure, for several reasons that are followed up one by one here:

Geometry: The four rows can be read as point, line, surface and solid, the steps in which space comes into being. More under From point to solid.
Music: The ratios of the first four numbers, 1 : 2, 2 : 3 and 3 : 4, are exactly the string lengths of the three purest intervals: octave, fifth and fourth. More under Number and sound.
The ten: For the Pythagoreans the series of numbers closed with ten. Everything beyond was repetition. What the ancient sources say about this is under The ancient sources.

The figure is taken seriously here, without claiming more for it than can be shown. Where tradition is uncertain, that is said. Where something can be checked, you can try it yourself in the graphics.

Why ten of all numbers?

Anyone searching the internet for the Pythagorean tetractys mostly lands on esoteric sites or on interpretations from music theory. The musical explanations have their value, but none of them can say what the first four numbers have to do with their sum, ten. Instead, various groups of four from harmony are declared to be the tetractys. Nowhere does one read that the tetractys represents something concrete and meaningful in mathematics. That is contradictory, since Pythagoras brought very solid mathematics into the world.

This site sets out to show that behind the harmless 1 + 2 + 3 + 4 = 10 there is more than esotericism, vague conjecture or ancient philosophy. In the geometry of circles and spheres, of regular polygons, polyhedra and their higher-dimensional relatives, one keeps finding that ten plays an ordering, limiting and determining role. Three examples:

The triangle of points itself. The points of all triangles before the ten-triangle add up to ten again. At ten, so to speak, “the scissors open”. The same pattern appears in space, in sphere packings and their crystal lattices. See The triangle of points.
The decagon simplex. Only for the decagon is the number of vertices equal to the number of full circles that all its figures together contain as angles. Smaller simplexes have fewer, larger ones more. See Angle sums and full circles. The same geometry also reveals the order of the primes, in grids that start in steps of ten and grow in steps of four, again the numbers of the tetractys. See The disorder of the primes.
The fractal decagon. Carry the Sierpiński triangle over to the decagon and it breaks into ten decagons that overlap and so form quadrilaterals. Each of the ten decagons contains exactly ten quadrilaterals, arranged from the edge to the centre as 1 + 2 + 3 + 4. Smaller even polygons contain fewer quadrilaterals than vertices, larger ones more. See The tetractys in the fractal polygons.

In all three cases the opening scissors rest on the sequence of triangular numbers, 1, 3, 6, 10, 15, 21 …, together with a grid of fours or the square numbers. This only becomes visible if one looks not just at the figure with the ten but at the figures before and after it in a row. Besides ten, only six has the same property: in the six-triangle, three new points join three. Six and ten lie four apart, and with their divisors 2, 3 and 5 they form the grids that also order the distribution of the primes. Perhaps that is what Pythagoras wanted to show his pupils with the counting pebbles.

Number, geometry and philosophy

There are good reasons to look at number theory from the point of view of geometry, more precisely the geometry of circles and spheres with their perfect symmetry. Some of these figures reproduce the divisibility of the natural numbers exactly. And this geometry is free of the accidents that the decimal system, or any other counting system, brings with it. The questions that follow are philosophical: what are numbers, and is there an intelligence behind their order? They are raised on the page Number theory + geometry = philosophy.

A natural science like mathematics is a soulless affair without philosophy. And it is a pity that the beautiful geometry of stars has hardly found its way into schools to this day. It would enliven a subject that many people find bone-dry. Readers of these pages are invited not simply to believe but to check for themselves and to question established doctrine afresh. Perhaps in this way, with the help of fragments of old knowledge, the mathematical foundations of the old Hermeticists can be brought back to life.

Open questions

Some of what has come down as mysticism may be an unrecognised or forgotten connection with a scientific background. A few questions this site pursues:

What is really meant by the four elements fire, air, water and earth? In religions and in the Hermetic tradition alike they form the philosophical basis. One possible assignment: fire = energy, air = time, water = space, earth = matter (see The tetractys in mysticism, Hermeticism and religion).
Do the four fundamental forces of physics, gravity and the electromagnetic, weak and strong interactions, have to do with the fourness of the tetractys? Describing them in one theory is the greatest unsolved problem of physics. Put simply: mechanics and quantum physics do not fit under one roof.
Quantum physicists describe forces with highly symmetric geometric models, the Lie groups. Some of them look confusingly like the geometry of the simplexes. The exceptional group E8 can be drawn as eight nested 30-gon figures, the 240 points of an eight-dimensional root system (the group itself has 248 dimensions). The numbers 24 and 30 also play a role in the order of the primes (see Quantum physics, magic and the unconscious).
What is the deeper meaning of 1 + 2 + 3 + 4 = 10? (see The key to the tetractys)
How is the ten of the triangle of points related to the lambdoma of harmonics?
How is the pentagram, the Pythagoreans’ sign of recognition, related to the lambdoma and the ten-point triangle?
And how do these three fragments of tradition flow into the all-embracing tetractys? If everything springs from the One, everything must also be traceable back to the One.

This site offers neither a doctrine of salvation nor the final theory of everything. It wants to give independent-minded readers hints about where digging is worthwhile.