Gap-free filling of space
Tetrahedra and octahedra fill space together. Their smallest common fractal has 10 vertices and 24 edges: 1 + 2 + 3 + 4 and 1 × 2 × 3 × 4.
In 1975 the American engineer and visionary Richard Buckminster Fuller (1895–1983) presented his own cosmology and theory of design in Synergetics. Its basic assumption: the tetrahedron and its relation to the sphere provide the mathematical template of the universe. His spatial lattice, the Isotropic Vector Matrix, is exactly the structure that had occupied Johannes Kepler 400 years earlier. Studying the sixfold symmetry of snowflakes, Kepler had come to the question of how densely circles and spheres can be packed. The densest packing is therefore also called Kepler’s packing, or properly the face-centred cubic lattice.
The structure of many metals follows this lattice, because crystals keep their volume as small as possible. Turned around, one could say that matter follows the “anatomy of space”. Energy such as light, heat and sound spreads spherically from a source, and from atoms to stars the sphere is nature’s basic form. Perhaps this lattice is far more than a question of crystallography. It probably also lies behind the creation account of the Sefer Yetzirah, the oldest book of the Kabbalah (see The Kabbalah).
Tetrahedron and octahedron
Place a tetrahedron of the same edge length on every face of an octahedron, cover its free faces with octahedra again, and so on, and you fill the whole of space without gaps. On every face sits a solid of the other kind. Stack tetrahedra vertex to vertex and octahedral gaps appear, giving a tetrahedral fractal. Stack octahedra and tetrahedral gaps appear, giving an octahedral fractal. Both pyramids, the triangular and the square one, are parts of the same structure, only seen from different directions.
A section of the spatial lattice. Every edge belongs to tetrahedra and octahedra at the same time.
The angular solids turn into spheres if you inflate every vertex into a sphere until they all touch. That is Kepler’s packing, the densest of all. Only Thomas Hales proved it, in 1998. The vertices of the crystal solids always lie at the centres of the spheres, and equal spheres produce equal distances, hence regular solids.
The stella octangula
The stella octangula, also called Kepler’s star or star tetrahedron, is the section of this lattice to which the lattice owes its name. The centres of the eight outer spheres form a cube. They enclose six inner spheres that form the octahedral core, each sitting exactly at the centre of a face of the cube. Hence the name: face-centred cubic.
The stella octangula can be understood in three ways:
- A tetrahedron sits on each of the eight faces of an octahedron.
- Two large tetrahedra interpenetrate. Their intersection is a small octahedron.
- Wedges are cut out of a cube along its twelve edges down to the face centres.
The stella octangula, on its own, inside its enclosing cube, and as a packing of spheres (8 cube corners + 6 face centres).
Ratios of volume
At equal edge length an octahedron has exactly four times the volume of a tetrahedron. The cube around a stella octangula contains 8 tetrahedra, plus 12 quarter-octahedra along the edges and a whole octahedron at the centre, so 4 octahedra and 8 tetrahedra in all. That equals 24 tetrahedra or 6 octahedra. The stella octangula itself has exactly half the volume of the cube.
Double the edge length, one step in the fractal, and every volume grows eightfold: the octahedral parts multiply by six, the tetrahedral parts by four. In the whole lattice there are twice as many tetrahedra as octahedra, but because the octahedron has four times the volume, the octahedra take up twice as much space as the tetrahedra overall.
The primal fractal: 10 vertices, 24 edges
Point, line, triangle and tetrahedron show the fourness of vertex, edge, face and space, but not yet a meaningful ten. The tetrahedron also has a weakness: on its own it cannot fill space.
What happens when the tetrahedron, the representative of our third dimension, becomes a fractal? The first step of division halves all six edges. The six midpoints are the vertices of an octahedron. From 4 vertices and 6 edges come 10 vertices and 24 edges: four small tetrahedra and an octahedron at the centre.
The tetrahedron after the first step of division: four tetrahedra (red) around an octahedron (blue). Stacked as spheres they are 1 + 3 + 6 = 10.
Here the link to the tetractys appears:
- After this first step nothing new happens, because tetrahedra and octahedra fill space without end.
- This space-filling primal fractal has 10 vertices and 24 edges: 1 + 2 + 3 + 4 = 10 and 1 × 2 × 3 × 4 = 24. Both numbers also play a part in the distribution of the primes (see The number 24).
- It contains two kinds of cell in the ratio 1 : 2, with volumes in the ratio 1 : 4. The two smallest cells are in balance.
Stacked as spheres, the primal fractal has the layers 1, 3 and 6, ten spheres in all. That is again the balance from the triangle of points: the sum before the triangle of ten is ten.
Three layers of 1, 3 and 6 spheres make the ten spheres of the primal fractal.
The tetractys of solids
As it grows denser, the tetrahedron–octahedron lattice forms two further important solids, four in all:
- Tetrahedron
- Tetrahedron + octahedron
- Tetrahedron + octahedron + cuboctahedron
- Tetrahedron + octahedron + cuboctahedron + stella octangula (within the cube outline)
Cuboctahedron and stella octangula are linked by duality and interpenetration with two further solids: the rhombic dodecahedron and the cube. Neither is made directly of tetrahedra and octahedra, but through new connections between the sphere centres they span a lattice of their own, and both also fill space by themselves. In this view the cuboctahedron and the stella octangula mediate between the old tetrahedron–octahedron net and the new lattice.
The solid angle of a full sphere
In the plane, the angles around every vertex of a tiling must add up to a full circle. In space, the solid angles around every vertex must make a full sphere. The solids that manage this are again made only of triangles and squares and have only 3-, 4- and 6-fold symmetries.
Around the centre of a sphere two complementary forms appear: the outward-pointed stella octangula and the outward-curved cuboctahedron. In the cuboctahedron the corners of the tetrahedra and octahedra point inwards, towards the centre of the sphere. In the stella octangula they point outwards. Every node of the lattice is a vertex of 6 octahedra and 8 tetrahedra. Exactly this arrangement forms a cuboctahedron. Because threefold, fourfold and sixfold symmetry meet in both, 12 and 24 are the basic numbers of their symmetry. Their smallest cells, however, remain tetrahedra and octahedra.