From point to solid
The four rows of the tetractys as the four steps of space, and how stacked triangles become a tetrahedron.
One of the oldest readings of the tetractys takes its four rows as steps of dimension:
- One point has no extent.
- Two points define a line, the first dimension.
- Three points not on one line span a surface, a triangle.
- Four points not in one plane enclose a solid, the tetrahedron.
With four points, then, three-dimensional space is complete. The tetrahedron is the simplest solid there is: four vertices, six edges and four triangular faces, with every vertex joined to every other.
An old teaching
This reading is old and well attested. Speusippus, Plato’s nephew and his successor at the Academy, wrote a book On the Pythagorean Numbers, of which a longer passage survives. In it he says that one is the point, two the line, three the triangle and four the pyramid, and that each is the first of its kind. Therefore, says Speusippus, ten is perfect, for in 1 + 2 + 3 + 4 it contains all the steps from point to solid.
Euclid begins his Elements around 300 BC with the same steps: “A point is that which has no part. A line is breadthless length.” The surface follows, the solid only in the eleventh book. And as late as 1730 the apprentice in Samuel Prichard’s Masonic catechism answers the question about the four principles of masonry: “Point, Line, Superficies, and Solid”, the line being “Length without Breadth”, almost word for word as in Euclid (see Polygons and star polygons in Freemasonry).
Why does it stop at four? Aristotle gives the answer at the beginning of his work On the Heavens: a magnitude can extend in one, two or three directions, and there are no more. The solid is therefore complete, “for three is all”. With the point as beginning there are four steps. A fourth direction had been considered earlier, but only the mathematics of the nineteenth century worked out a geometry with any number of dimensions (see below).
The tetractys in space
The figure can also be lifted literally into space. Lay spheres in a triangle of 10, put a layer of 6 on top, then 3, and a single sphere at the top. The result is a pyramid with a triangular base, a tetrahedron of spheres. Every layer is itself a triangular number.
Each layer is a triangle of points. Together 1 + 3 + 6 + 10 = 20 spheres form a tetrahedron. Drag to rotate the model.
The totals are called tetrahedral numbers: 1, 4, 10, 20, 35, 56 … Notice that ten appears twice here: as the fourth triangular number and as the third tetrahedral number.
The arrangement in the model is no accident. It is how a greengrocer stacks oranges and how the atoms of many metals arrange themselves: the densest way to pack equal spheres. Johannes Kepler conjectured this in 1611. It was proved only by the American mathematician Thomas Hales, who in 1998 presented a proof resting on extensive computer calculations. After years the referees could confirm it only “99 per cent”. So Hales and a team had it checked completely in the Flyspeck project by computer programs that verify every logical step. This succeeded in 2014 and was published in 2017. A good four hundred years after Kepler the conjecture is thus finally a theorem.
And beyond?
The pattern does not stop at the tetrahedron. Add a fifth point and join it to all four vertices, and you get the four-dimensional counterpart of the tetrahedron. Such figures are called simplexes. They can no longer be built, but they can be drawn, and then they reveal a surprising inner order. That is the subject of the next page.
Sources and further reading
Sources
- Speusippus, fragment from On the Pythagorean Numbers, preserved in the Theologumena arithmeticae
- Euclid, Elements I, definitions 1 and 2
- Aristotle, On the Heavens I 1, 268a
- Samuel Prichard, Masonry Dissected, London 1730
- Thomas C. Hales et al., A formal proof of the Kepler conjecture, Forum of Mathematics, Pi 5, 2017