The central visual image representing the art & science of Alchemy is the Ouroboros: a dragon or snake swallowing its own tail. This symbol deftly reflects the totality of the alchemical process: the great circulation of Nature, by which, as described by the hermetic alchemist Peter Murien, “what was above as the very subtle spirit descends and becomes … the below, the earthy and fixed; then by reversing the process, what is below - fixed, heavy & earthy - ascends and gradually becomes volatile and subtle as the above.” Hence, the circle is closed: the snake/dragon has indeed swallowed its tail.
The Golden Chain: Alchemical Joining Of Heaven & Earth
The Greek philosopher Homer referred to this process as the “golden chain.” What is represented, along with the individual internal process of the yogi or yogini, is also the chain of alchemical adepts (those able to link Heaven and Earth), beginning - in the classical Alchemy lineage - with the Egyptian Hermes Trismegistus, who wrote: “Make what is above become like what is below, then what is below returns to what is above, thus creating the miracles of one thing.”
This same process - the descent of consciousness into matter, and the subsequent ascent of matter back into its more rarefied state (and, in nondual traditions, the linking of these two processes in a continuous loop) - is described in various other mystical traditions: most notably in Taoist Inner Alchemy texts, but also in, for instance, Patanjali’s Yoga Sutras.
Topology, Knots & The Ouroboros
In the language of Topology - the branch of mathematics that looks at what happens when certain geometric figures are stretched, twisted & folded (or, more precisely, when the space around these figures is distorted in these ways) - the Ouroboros is what’s called a knot: a closed curve in three-dimensional space. A salient characteristic of knots is that they can indeed be stretched, twisted & folded into all sorts of superficially different shapes, while retaining, mathematically, their sameness. The central question explored within Topology is how to know when one shape is simply a distorted version of another? The short answer is that this depends upon identifying what are called invariants: characteristics that remain, across variations in a given shape/knot.
Topology Meets Quantum Computing
Topology has recently been employed in the service of the newly-emerging science of quantum computing, as a resource for addressing the problem of decoherence. In quantum computers (which are at this point mostly still theoretical), information is stored on sub-atomic particles. The major problem with these so-called “qubits” is their susceptibility to decoherence: disturbance of their quantum space/structure by the Newtonian environment. Each time such a disturbance happens, and the quantum state of the particles is affected, the information stored on them is lost or distorted.
This is where Topology, potentially, is able to come to the rescue. For information stored on topological structures is impervious to local error, since small pieces of lost information can easily be reconstructed, given what is known of the overall pattern/shape of which it was a part. (Imagine a donut, and a small piece on the surface of the donut being lost … Since we can still see what the overall shape of the donut is, we can easily replace that missing link.) So what is needed then, in terms of quantum computing, are sub-atomic particles which can be mapped topologically.
Braids & Dancing Anyons
And it just so happens ... that when an electron liquid on a two-dimensional crystalline surface is subjected to extremely cold temperatures and a strong magnetic force, sub-atomic particles called anyons are created. These anyons move about the surface - they “dance” - in patterns which physicists call braids, and which can be mapped topologically. As a result, these anyon braids (defining what’s called a Quantum Hall fluid) are very good candidates for storing information in a quantum computer, in a way which side-steps the problem of decoherence.
Another interesting thing about anyons is that they carry a fractional charge. Whereas protons carry a single positive charge, and neutrons a single negative charge, anyons carry charges corresponding to any real number. Different fractional states, i.e. charges, define different “flavors” of Quantum Hall fluid, and determine how complex a computation is possible, using that fluid as a carrier of information in a quantum computer.
Assuming this all works out, and we’re able to use these anyon braids to store information: What would be the advantage of such a topological quantum computer? For one thing - speed. A classical/Newtonian computer operates via bits (housed in circuits or silicon chips) that are either on or off. In line with the so-called weirdness of the quantum world, qubits (say, in the form of anyon braids housed in Quantum Hall fluid) exist in combinations of on and off, simultaneously (a phenomenon known as the superposition of states). This property allows a quantum computer to perform multiple computations, and get multiple answers, all at the same time. So a calculation that might take a classical computer millions or billions of years to complete, would be completed via a quantum computer in a matter of minutes!
