Pyramids, knightly crosses, spatial lattices
The pyramid is half an octahedron, the cannonball stack a piece of crystal lattice, and the cross of the knightly orders a fourfold symmetry. Three symbols, one geometry.
Pyramids, crosses and crystals are among humanity’s strongest symbols. They stand for eternity, for faith and for purity. Geometrically they are closely related: all three are sections of the same order of space, the lattice of the densest sphere packing (see Gap-free filling of space).
The pyramid
A pyramid with a square base and equal edges is exactly half an octahedron. Put two such pyramids together base to base and you get the octahedron, one of the five Platonic solids. The pyramid with a triangular base is the tetrahedron. Both are the smallest cells of the spatial lattice: where tetrahedra and octahedra alternate, they fill space without gaps.
Stack spheres into a pyramid with a square base, like cannonballs outside a fortress or oranges at a market, and the layers give the square numbers 1, 4, 9, 16 …, which add up to the square pyramidal numbers 1, 5, 14, 30, 55.
Spheres stacked into a square pyramid. Each sphere rests in the hollow of four spheres in the layer below. Drag to rotate the model.
The astonishing thing: this pyramid is the same lattice as the triangular stack of spheres, the face-centred cubic lattice of the densest packing. It is only seen from a different direction. Look at a side face of the square pyramid and you see triangles there, just as in the tetrahedral stack. Square and triangle, the two basic forms of tiling, are united in a single solid.
For comparison, the triangular stack. Both pyramids are sections of the same spatial lattice.
Many numerical claims surround the great pyramids of Egypt. What can be checked: in the pyramid of Khufu, the perimeter of the base is to the height almost exactly as 2π, with a deviation of less than one part in a thousand. Whether the builders intended this or whether it followed from their method of measurement (with a rolling drum, for instance) is disputed. Either way the pyramid, like the simplex, carries π in an angular form (see Simplexes and π).
The knightly cross
The cross pattée, the equal-armed cross with arms that widen outwards, became the emblem of the knightly orders. The Templars wore it in red on a white mantle, the Teutonic Order in black on white. The Order of St John bore the eight-pointed cross that, after its later seat, is called the Maltese cross.
Left, the cross pattée of the knightly orders. Right, the Maltese cross with its eight points.
Both crosses are fourfold symmetries around a centre, the same division as the circle with four segments in the Secret Symbols of the Rosicrucians. The square is the only polygon whose angles make exactly one full circle, and the cross divides the circle into four right angles. The eight points of the Maltese cross have been read as the eight Beatitudes of the Sermon on the Mount. Geometrically it is an eightfold figure, like the stella octangula in its square view and like the Ægishjálmur with its eight arms (see The throne of God and the number 24).
The spatial lattice
The lattice that contains pyramid, octahedron, tetrahedron and cube all at once is the face-centred cubic lattice. Buckminster Fuller called it the Isotropic Vector Matrix, Kepler described it as the densest packing of spheres, and the atoms of copper, gold and aluminium sit in it.
A section of the spatial lattice. It contains tetrahedra, octahedra, square pyramids, cubes, the cuboctahedron and the stella octangula.
On this view the symbols of mysticism are not arbitrary inventions but sections of this one lattice:
| Symbol | Section of the spatial lattice |
|---|---|
| Pyramid | half an octahedron |
| Tetrahedron, triangle | the smallest cell |
| Cube, the Masons’ “perfect ashlar” | outline around a stella octangula |
| Merkabah, Star of David | stella octangula, seen from a vertex as a hexagram |
| Cross | the fourfold view, along an axis of the cube |
| Flower of life | one layer of the packing, six spheres around one |
Whoever looks at a pyramid, a cross or a crystal thus sees a different side of the same order each time. It is the order into which equal spheres arrange themselves of their own accord when packed tightly together: the anatomy of space.