Quantum physics, magic and the unconscious
Symmetry as the language of physics, number as the archetype of order, and a 30-gon at the heart of the Lie group E8. Where matter, symbol and soul touch.
Magic, mysticism and modern physics seem to have nothing to do with one another. And yet all three work with the same raw material: symmetry. The magician draws a circle around a centre, the Kabbalist joins 22 letters into a circle of 231 gates, and the physicist describes the fundamental forces of nature with symmetry groups. The question of this chapter: is it the same order, once in the image, once in the soul, once in matter?
Symmetry is physics
In 1918 the mathematician Emmy Noether proved a theorem that became a foundation of modern physics: every symmetry of a physical system has a conserved quantity. That the laws of nature are the same tomorrow as today means conservation of energy. That they are the same here as there means conservation of momentum. That they hold in every direction means conservation of angular momentum. Symmetry is not nature’s ornament but its law.
Particle physics today is at its core a theory of symmetry groups. The forces between elementary particles are described by Lie groups, continuous symmetries that can be pictured as rotations in abstract spaces. Their structure can be represented by root systems: sets of vectors arranged symmetrically around a centre. Drawn in the plane, they produce figures that look like simplexes, stars and mandalas.
E8 and the 30-gon
The largest and most enigmatic of these structures is the exceptional group E8. It has 248 dimensions. Its root system consists of 240 vectors in eight-dimensional space. Projected onto the right plane, these vectors arrange themselves in eight rings of 30 points each: 8 × 30 = 240. Each ring is a regular 30-gon.
The 240 roots of E8, projected into the Coxeter plane and computed here from the coordinates. The colours distinguish the eight rings, the lines join neighbouring roots.
The 30-gon is no accident. 30 is the so-called Coxeter number of E8, and 30 = 2 × 3 × 5 is exactly the grid that orders the primes next after the rhythm of six (see The order of the primes). For comparison, the simplex of the 30-gon, every point joined to every other:
The 30-gon simplex: 30 points, 435 connections. Its figures show all divisors of 30 and all numbers coprime to it.
The figures of the E8 projection and of the 30-gon simplex look remarkably alike. Both show the same symmetry, both are structured by the divisors of 30. The structures with which quantum physicists describe working systems of physical forces are thus closely related to the geometry of the simplexes, which shows how the natural numbers divide. In 2007 the physicist Garrett Lisi even proposed fitting all particles and forces into E8. His theory is disputed, but in 2010 experiments on magnetic crystals measured excitations whose energies follow the symmetry of E8.
All is frequency
Quantum physics began with a discovery by Max Planck in 1900: energy is emitted in packets, and the energy of a packet is proportional to its frequency, E = h · f. Every particle has a wave, every wave a particle. With this the Pythagorean idea that the world is ordered by ratios of vibration arrived in physics (see All is number – all is frequency). Planck himself concluded that behind matter stands a mind that holds it together.
The magician and the unconscious
Magic, as handed down, works not with physics but with images. The magic circle (see The throne of God and the number 24) is a symbol meant to act on the magician’s unconscious: he stands at its centre, invokes the four elements and enters a trance. The Kabbalistic ritual of the 231 gates also works with visualisation: each letter is imagined, held and joined with all the others until the whole 22-gon stands before the inner eye.
The magic circle: an order around a centre, built of square, hexagrams and pentagrams.
Why should these figures in particular act on the soul? One answer was given by the psychologist Carl Gustav Jung. He observed that people in crisis spontaneously draw mandalas: circles around a centre, usually divided into four. He read them as images of the Self, the wholeness of the psyche, and fourness, the quaternity, as its basic pattern. Jung’s psychology knows four basic functions of consciousness, thinking, feeling, sensation and intuition, which recall the four elements and the four temperaments (see The tetractys in mysticism and religion).
For Jung, number was an archetype, a primal form of order that belongs to the soul as much as to the world. The natural numbers, in his view, are not invented but found. They have properties nobody knew of beforehand. This matches Philolaus’ statement that the order of number exists by nature and not by convention.
Pauli and Kepler
The bridge between physics and soul was built by none other than one of the founders of quantum physics. The Nobel laureate Wolfgang Pauli worked with Jung for years. In 1952 they published a book together. Pauli’s contribution was titled The Influence of Archetypal Ideas on the Scientific Theories of Kepler. In it Pauli showed how Kepler’s planetary laws grew out of an inner image: the sphere with its centre as an image of the Trinity, and the conviction that God created according to geometric archetypes. Exact science, Pauli concluded, draws on the same sources as mysticism and religion.
Three worlds, one order
Set the three realms side by side and a picture emerges:
| Physics | Magic and mysticism | Soul | |
|---|---|---|---|
| Basic form | symmetry group | magic circle | mandala |
| Centre | conserved quantity | the magician | the Self |
| Structure | root system, rings of 30 | hexagrams, pentagrams | quaternity |
| Substance | frequency | sound, sung letters | image, archetype |
Behind all three stands the geometry of circle and sphere, structured by polygons and star polygons. On this view the tetractys is not a metaphor linking physics, magic and soul, but the common order from which all three draw: the fourness in the circle, which appears alike in the root system of physics, in the magic circle and in the mandala of the soul.