“You believe this floor exists because you feel it resist you… But do you think your ideas do not resist you? Find me then two unequal diameters in a circle, or three equal ones in an ellipse. Find me the square root of eight and the cube root of nine… It resists your mind. Do not, therefore, doubt its reality,” thus argued the French philosopher Nicolas Malebranche (1638-1715), expounding upon an ancient variety of Platonism (or mathematical realism).
A more extreme viewpoint would be that the only realities that can exist are mathematical in nature. Somehow it would also imply that human thought — and, in fact, consciousness too — has an underlying geometric structure. While speaking of this, Jacques Lacan invoked the Möbius strip, a two-dimensional surface that has only one side, because (writes Owen Hewitson of LacanOnline) “We don’t have a divide between an ‘above’ and ‘below’, or between a surface and a depth. The unconscious is not hidden in the deep and the surface is not the superficial.”
Certain Hindu schools acknowledge three foreground states of consciousness (instead of Lacan’s two) such as waking, dreaming, and dreamless sleeping. All three are a dance of the senses upon the background of turiya — a space which transcends and contains them all. Tibetan Buddhists acknowledge six distinct, mezzanine states of transformation called the bardo . In the Sufi meditative state of adraak , a person is able to perceive things without being able to see or hear them, as if from a higher dimension.
Viewing lower objects from a higher dimension is a clever technique in mathematics. For example, “to end up with a 3-dimensional Penrose tiling, one needs to start with a stack of six-dimensional cubes.” The vertices of this lattice are subjected to a filter, and those that pass are projected at a particular angle onto three-dimensional space, like the shadow of a sundial. “Astoundingly, when connected by lines of equal length, these vertices become the nodes of a perfect quasicrystal...” writes Tony Robbin in Shadows of Reality . Poised upon a delicate fulcrum between order and chaos, the pattern will repeat itself into infinity, with imperceptible changes — like an irrational number, which has no fixed periodicity.
A more fanciful version of the projection-slicing method caught the imagination of the Theosophists — occultists who were trying to create a heady cocktail of Eastern mysticism while keeping a watchful eye on Western science (the poet William Butler Yeats was one of their illustrious members). Like the trend of Cubism in painting, they too were fascinated by the fourth dimension, but considered it a spiritual domain. If one could trick the mind into seeing through a hypercube or tesseract , they reasoned, one could gain access to hermetic knowledge which even scientists did not possess.
Perhaps their most daring and extraordinary act, Occult Chemistry is a book by Annie Besant, Charles Webster Leadbeater and Curuppumullage Jinarajadasa. By what they call a “clairvoyant investigation” into matter, the 1908 book explains (prior to Bohr’s atomic model) the internal structure of atoms with images that suggest the existence of quarks. While utterly unscientific and ridiculous, the close agreement with a future discovery to be made only in 1968 is somewhat startling.
Perhaps the difference between a mystic and mathematician is only the rigour of hallucination. In another prophetic coincidence, the girih tiles on the Darb-e-Imam shrine in Isfahan, which go back to 1453, possess “properties consistent with self-similar fractal” tilings invented by Sir Roger Penrose in the 1970s. These Islamic mathematicians (banned from depicting Allah in any human form) had begun to worship geometry itself as the very image of God.
In a horror story called The Centaur (1911) by Algernon Blackwood, we find the idea that our mythical gods may just be forms of ancient knowledge: “Of such great powers or beings there may be conceivably a survival... a survival of a hugely remote period when... consciousness was manifested, perhaps, in shapes and forms long since withdrawn before the tide of advancing humanity... forms of which poetry and legend alone have caught a flying memory and called them gods, monsters, mythical beings of all sorts and kinds...”
The fifth avatar of the Hindu god Vishnu — the first of which was human — is said to be a dwarf known as Vamana or Trivikrama, who “stepped from heaven to earth with the first step, from earth to the netherworld with the second” and then measured the universe by placing his third step on the head of a king. This elastic, enlargeable god is an ancient memory of the Pythagorean theorem, a 90-degree right-triangle (the primal ‘atom’ of geometry) without which it would be impossible to measure two-dimensional space, depth in human vision, or the distance to stars and the farthest galaxies.
Rohit Gupta explores the history of science as Compasswallah; @fadesingh