Sacred geometry – scepticism and open questions
Flower of life, merkaba and Platonic solids. What sacred geometry shares with the tetractys, and where scepticism is in order.
Anyone who studies the tetractys sooner or later comes across so-called sacred geometry. Both carry a reputation for fringe science and esotericism. A distinction is needed: the tetractys can be traced in the sources, whereas sacred geometry leaves many questions open.
In its popular form it goes back mainly to Drunvalo Melchizedek, whose civil name is Bernard Perona and who sees himself as a spiritual teacher. Naming oneself after the biblical priest-king Melchizedek is a matter of taste. At the centre of his teaching are two figures, the Flower of Life and the Merkaba, both presented as “patterns of creation” and ancient Egyptian mystery wisdom. Around them a market of books, seminars, amulets and printed merchandise has grown. Earning money is not wrong, but it may explain why the claims are so rarely checked. The aim of this page is not to tear the edifice down but to correct and complete it, as a suggestion also for those who teach it.
The Flower of Life
The pattern of overlapping circles in a hexagonal net is interpreted as the archetype of creation, from which all forms supposedly derive, and it is marketed on a large scale for “energetic purposes”.
The Flower of Life: 19 circles in a hexagonal net. Geometrically it is a section of the densest circle packing, six around one.
It is understandable that sixfold symmetries and the number 6 support the idea of a creation in six days, as in the Bible and the Kabbalah. But much speaks against the claimed origin and meaning:
- In no ancient culture is this pattern attested as the most important pattern of creation. Nowhere can it be shown to have been more than ornament.
- In Egypt there is not a single relief of it, only a faint, barely visible drawing in the Osireion at Abydos, in an inconspicuous place and of disputed age. Had the pattern been central, it would stand in central places.
- No alchemical and no kabbalistic text names this figure. The claim that the kabbalistic Tree of Life derives from the Flower of Life does not hold.
Kabbalistic depictions of hexagonal structures and circle packings do contain an important geometric statement, only one the flower pattern cannot have: space filling without gaps. The cubic crystal systems are connected both with these drawings and with the stella octangula (see The Kabbalah). As the sole pattern of creation the flower is far too little, and as regards the Kabbalah it misses the point.
The flower and the Platonic solids
That all Platonic solids can be projected onto the Flower of Life proves no special status for the pattern. That is selective perception. Other hexagonal tilings, sphere packings, triangular nets and honeycombs fit just as well. Moreover the five solids have further symmetry views that are not sixfold at all: besides vertex and face views there are edge views, and some views are four- or tenfold. The difference between 4 and 10 is 6. This brings one much closer to the many layers of the tetractys than a mere hexagonal structure (see The Platonic solids).
Flower of Life and sphere packing
Nassim Haramein also takes up the flower pattern to support his model of a “holographic unified field”, which it hardly needs. His basis is the Isotropic Vector Matrix of Buckminster Fuller, a truly remarkable spatial structure, identical with the face-centred cubic lattice of the densest sphere packing.
The crucial difference: the circles of the flower are not the spheres of this matrix. In the matrix the spheres only touch, they do not interpenetrate. Gaps remain between them, in space as in the plane, and in addition there are the octahedral holes with markedly larger circumspheres. It is precisely the touching spheres that give the structure its shape. “Gapless” applies only to the lattice of lines joining the centres, not to the packing itself. In nature it is the most space-saving arrangement of atoms.
The same centres, once as the Flower of Life, once as the densest packing. The flower circles are circumcircles, the packing circles incircles. 'Next stage' shows how a grid of half the diameter begins at the points of contact.
Laying both patterns on top of each other reveals an interesting duality. Joining all centres gives a net of triangles and hexagons, the honeycomb. The circles of the flower are the circumcircles of these hexagons, the circles of the packing the incircles. Whether Haramein’s model can do without the gaps is an open question, since it is meant to describe not matter but the vacuum. A contrast of vacuum and matter?
The spatial matrix has three different symmetry views, among them a hexagonal and a square one. The square view has nothing in common with the flower any more, and it shows that the large gaps cannot be closed with the flower either. Once again the spatial view leads to the tetractys, 4 and 6 (together 10), as with the Platonic solids.
An act of creation from the fractal
A far more coherent matrix of creation can be derived from the Isotropic Vector Matrix. In the plane it looks like this: at the points where the circles touch, their predetermined breaking points as it were, the structure halves. The breaking points form the centres of a new grid of circles with half the diameter, and so on. The same works with a square grid, and so also in the face-centred cubic lattice, which is itself a fractal (see From point to solid). This establishes a link to physics and chemistry, for here we are dealing with the order of matter.
The flower and the number π
One aspect of the overlapping circles has hardly been noticed. Joining the centres again gives the net of triangles and hexagons. Since Archimedes (around 250 BC) it has been known how the hexagon relates to π: for a hexagon in a circle, perimeter and diameter are exactly as 6 to 2, that is 3. As the number of vertices grows the polygon approaches the circle, and the ratio approaches π = 3.14159 …
Archimedes doubled the vertices from the hexagon up to the 96-gon. The coloured lenses between polygon and circle are the remainder after the decimal point.
On this reading the lens-shaped remainders between hexagon and circle correspond to the irregular distribution of the primes: the 3 before the decimal point stands for the regular rhythm of six of the twin primes, the irregular digits after it for the irregular distribution. And this irregularity corresponds to the principle of the pentagram, the golden ratio and so the principle of life (see The number π and Simplex – multidimensional tetrahedra). Seen this way, the Flower of Life deserves its name after all.
The Merkaba
There are interesting parallels, on the other hand, in the Merkaba. This is what sacred geometry calls the stella octangula, Kepler’s star tetrahedron, interpreted as the basis of a “rotating energy field” or “light body”. It consists of an octahedral core with eight tetrahedra set on it, and its eight tips form a cube. It is exactly the section of the densest-packing lattice to which the lattice owes its name “face-centred cubic”. The stella octangula has two important symmetry views. Almost only the hexagonal one is noticed, the square one hardly at all.
The stella octangula or Merkaba: two interpenetrating tetrahedra whose eight vertices form a cube.
That “Merkaba” is composed of the Egyptian words Mer, Ka and Ba is attested nowhere. The word is Hebrew and means the throne chariot of God from Ezekiel’s vision. Merkabah mysticism is a pre-kabbalistic current concerned with the vision of God, and the term lives on in the Kabbalah. Linking the stella octangula as a “light body” with the chariot of God is therefore not far-fetched, even if the details are wrong. More in The throne of God and the number 24.
Merkaba, Kaaba, Kabbalah
The Sefer Yetzirah describes how God makes room for himself: a tetrahedron, an octahedron and a cube become visible one after another (see The Kabbalah). Exactly these first three of the five Platonic solids are also the building blocks of Fuller’s matrix. There the cube stands for infinite space, the infinity of God, and magical-kabbalistic systems work with such a cube of space.
The cube appears again and again in religious tradition:
- The Kaaba in Mecca, a cuboid building in the courtyard of the Sacred Mosque, is regarded as the “House of God”.
- In Freemasonry the cubic stone is the image of the perfected human being who fits seamlessly into the building of Solomon’s Temple. Its right angle stands for right conduct and the law of the Architect of all worlds. This shows the Old Testament heritage of Freemasonry, whose roots lie in the cathedral builders’ lodges, the Kabbalah and Rosicrucianism.
- The Holy of Holies of Solomon’s Temple was a cube, twenty cubits long, wide and high (1 Kings 6:20), as was the innermost room of the Tabernacle, the portable sanctuary.
- The New Jerusalem of Revelation is equally long, wide and high (Rev 21:16).
That Merkaba, Kaaba and Kabbalah sound alike is linguistically pure speculation. In content, though, they are plainly related: the throne chariot of God, the House of God and the search for an immediate experience of God.
What is divine about the cube?
The question remains what is so special about the cube and the stella octangula. The cube is the only Platonic solid that fills space on its own, and the only one whose eight solid angles together make exactly one full sphere (see The Platonic solids). The cause lies in its right angles. That this seems unremarkable to us is because the right angle is everywhere in our world and therefore looks banal.
The religious-philosophical interpretation is simple:
- The full sphere stands for perfect wholeness.
- Gapless space filling stands for unity in multiplicity.
- From any centre out to infinity everything fits together, omnipresence. God dwells in every point of the fractal structure and pervades infinite space.
Whoever has truly understood the dualities between these solids also sees that this interlocking blueprint is to be understood as a unity.
Twelve and twenty-four
The cube and its related solids are twelvefold and twenty-fourfold throughout, like the 24 elders around the throne of God, the twelve disciples, the twelve tribes of Israel and the twelve gates of the heavenly Jerusalem:
| Solid | Twelve and twenty-four |
|---|---|
| Cube | 6 squares × 4 vertices = 24, 12 edges, 3 edges at each of 8 vertices = 24 |
| Octahedron | 8 triangles × 3 vertices = 24, 12 edges |
| Stella octangula | 24 faces, 24 outer edges, 12 inner edges |
| Cuboctahedron | 24 edges, 12 vertices |
| Rhombic dodecahedron | 12 faces, 24 edges, 4 edges at each of 6 vertices = 24, 3 edges at each of 8 vertices = 24 |
Around the centre of a sphere, too, the three axes x, y and z lay eight octants, 8 × 3 = 24. And the first subdivision steps of these solids show the same ratios again. For combinatorial reasons this is no surprise, since the solids consist only of triangles and squares. Up to 24 only 1, 2 and 3 play a part, and it is exactly they that order all remaining primes to infinity in a rhythm of six (see The number 24).
Add the three dimensions of space and one arrives at the divine unity in trinity in the sense of Platonism and Pythagoreanism: a point 1, a line 2, a surface 3, but only a solid 4 grasps the whole. Platonism and Pythagoreanism were quite evidently also the religious-philosophical foundation of Jewish and Christian mysticism.
What sacred geometry gets right
The conviction that circle, triangle, square, pentagram and hexagram are more than ornamental forms is justified. But the reason lies not in revelations but in verifiable properties: angle sums, tileability, divisibility. The motto: draw it yourself, calculate it and check it.