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.
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.
Te(n) = n · (n + 1) · (n + 2) / 6
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 in 1998.
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.