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 cross
The cross is one of the oldest symbols. Taken apart, its meaning explains itself: the horizontal stands for the earth, the vertical for heaven, the cross for their union. It divides the circle, the sign of completeness, into four equal parts and stands for the four cardinal directions. In ancient Egypt four pillars supported the vault of heaven. The cross was probably also a tool for observing the sky, evidence of astronomical and nautical knowledge, and the sun wheel as a sign of the yearly cycle belongs in this line. Think of the two bars as axes in space and it becomes a spatial cross, a coordinate system of all directions, an image of the universe.
A pyramid is nothing but the spatial version of the cross: the crossing point is drawn upwards and points to heaven. Mirror a second pyramid at the base towards the earth and an octahedron arises, whose axes and edges again form spatial crosses. Because of its symmetry each of its six vertices is the apex of a pyramid. It has 6 vertices, 12 edges and 8 faces, and it can be seen in two ways: as three interlocking squares with 12 sides together, or as eight triangles with 24. And it has a cross view and a six-pointed star view. Cross and six-pointed star, geometric laws as the basis of religious symbols. Whether the builders of the Great Pyramid knew the octahedron, the inverted pyramid, because of this cross symbolism remains conjecture.
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).
Pyramid and double pyramid in magic
Human orientation in space has always been the chief meaning of the cross. In magical rituals the four cardinal directions correspond to the four elements fire, air, water and earth. Added to them is the vertical, the link to “higher” and “lower” powers. Six directions from a centre again describe an octahedron, a double pyramid.
Many protective circles have a cross in the middle. Whoever stands in it wants to bring his four elemental forces, willing, feeling, thinking and being (see Freemasonry), into a cosmic order. He himself takes the central, raised position, the quinta essentia, the fifth element, the ordering and observing centre of his microcosm, the apex of the imagined pyramid: reason. The base is formed by the four forces of his mind, which he seeks to master.
There are also magic circles in the form of a hexagram. They correspond to the double pyramid, the octahedron. To the four elements and the quinta essentia is then added the unconscious, the “sixth sense” at the apex of the downward pyramid, and in the centre rests the divine self as the seventh. In this way the magician unites with the spatial order of the macrocosm:
| Form | Realm | Sign |
|---|---|---|
| pyramid | microcosm | pentagram |
| double pyramid, octahedron | macrocosm | hexagram |
The seventh day, seven planets, seven metals
The day of rest after creation in six days has many signs in hermeticism: seven planets, seven metals, the seventh heaven. The hexagram gathers them all in its seventh point, the centre. In the hexagram of the planets the sun sits in the middle, and the circle with a point is both the sign of creation and the astrological sign of the sun.
An engraving from Johann Daniel Mylius’ Opus medico-chymicum (1618), later also used in the Musaeum Hermeticum (Frankfurt 1625, enlarged 1678), shows the seven metals as the earthly counterpart of the seven planets. In a cave in the earth sit seven figures, in the middle Apollo with the lyre for gold, the counterpart of the sun. Above the cave sit three figures, the divine trinity: the left holds an upright triangle, the middle one a hexagram, the right an inverted triangle. The four elements and the planetary principles frame the scene. Below, on earth, the alchemical transformation from substance to medicine takes place. The upper trinity corresponds to the first three sefirot, separated by the abyss, here the earth’s crust, from the other seven. Together there are ten figures, like the ten sefirot and the Pythagorean tetractys.
How the Sefer Yetzirah seals the six directions of space, from 5 to 10, is described in The Kabbalah. The six directions can be shown as an octahedron with six vertices just as well as a cube with six faces, for the two solids are dual to each other. Space, the divine and the pyramid are connected here not only “mystically” but in a logically traceable way. A parallel in number space: the rhythm of six of the twin primes from 1 × 2 × 3 orders the primes to infinity and so links the spiritual world of numbers with experienced space of length, breadth and height (see The order of the primes).
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.
Among knightly and order crosses two basic forms prevail: the cross pattée with its spatula-like widening arms and a more or less pronounced swallowtail form. People of “noble mind” and institutions with a divine or charitable mission considered themselves worthy of this sign. Is the cross pattée perhaps an unfolded pyramid, seen from above?
A pyramid with equal edges, unfolded and seen from above: four triangles around a square. The outline gives the shape of a cross pattée, the notches between the triangles the swallowtail of many order crosses.
The swallowtails then mark the difference between the square base and the unfolded triangles, and the wearers of the cross would have identified with the pyramid, the sign of the divine. Number-theoretic diagrams show this cross shape too: in the division table as a coordinate system, the entered divisors give similar figures (see Tetractys and coordinate system). Whether there is a direct connection here is, however, more speculative than with the unfolded pyramid.
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 |
That this lattice consists of triangles, squares and hexagons has a simple reason: only these three polygons fill plane and space without gaps on their own (see Circle, triangle and square). Whoever really wants to grasp the structure should build it, with magnetic rods, toothpicks or tennis balls. Buckminster Fuller explained it in his lectures with infectious enthusiasm, and that enthusiasm is probably shared only by those who have built the lattice themselves. Nassim Haramein, too, declares this matrix the formula of the world, perhaps not entirely without reason, though no judgement on the soundness of his theories can be made here (see Sacred geometry).
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.