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.
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.