Last Wednesday night, and going on into the early hours of Thursday morning, I started to give some serious thought to writing this talk. An early question to settle is how to begin. A historical or classical introduction is often a favourite fall-back, as some of you may have noticed; but in a talk which seeks to explore some recent ideas about the nature of the Cosmos and the Mathematical structures pressed into service for modelling it's processes, the scope for a classical theme which I had not used before seemed a bit limited. 'Knots' appeared to hold out more promise than 'Superstrings', but the only reference I could call to mind was the tale of Alexander the Great being confronted with a particularly ugly specimen at Gordium while campaigning in Asia.

Getting out Encyclopaedia, I made a few notes, but it wasn't particularly inspiring. The bland assessment was that the event was a folksy myth to the effect that this intricately knotted rope could only be untied by the person who was to rule Asia.

I decided to sleep on it. A further few hours into Thursday morning I bestirred myself, switched the television on, and cast round the channels for something to decorate that corner of the room, settling on something which looked like a fifties Roman epic. After making a cup of coffee, I returned to my room and peered through the morning blur at a youthful Richard Burton, gladiatorially attired, in discourse with some older folk in more secular garb around a large black rope mass tied around what appeared to be the hafts of a chariot. At this point I turned the TV sound on. Indeed the scene was Gordium, and Alexander of Macedon was in the process of drawing his sword to sunder the fabled Knot.

One of those weird co-incidences that seems to crop up, and for which the mathematical odds would defy calculation.

DOUGHNUTS, TEA-CUPS AND PRETZELS

"A Topologist is someone who cannot tell the difference between a torus and a teacup" - Corny Mathematicians Joke.

Topology is sometimes referred to as 'rubber sheet' geometry. It classifies objects and structures on the basis of whether their forms can be stretched or compressed to resemble each other, while preserving certain invariant properties. Breaks and tears are excluded.

A circle and a square are topologically equivalent, because, if they were made of rubber, the one can be stretched and squeezed into the shape of the other. In contrast, there is no equivalence between a circular ring and a round disk such as a 10 pence coin. The essential difference lies in the existence of a hole in the ring - that feature is referred to as a topological invariant. The circular ring or torus, though, could be stretched and squeezed so that the bulk of the material of which it is made is contorted into a cup shape, with the invariant hole forming the handle.

If a bicycle inner tube or some other toroidal object such as a loop of rope or wire is twisted such that it appears like a figure eight or a symbol of infinity, then, quite clearly, it can be untwisted again to resume it's original shape. Provided the object is perceived as being in three dimensional space, there is no topological difference between a torus and a loop twisted so that it has a single crossing. The situation is quite different for example from that of two interlocking rings, although in a two dimensional representation the appearance might be closely similar. There is also a clear difference between two simple loops juxtaposed so that they overlap and two interlocking loops - that configuration which mathematicians sometimes call a Hopf Link.

The simplest knots also consist of a single loop of wire or rope, they are known as Trefoil Knots, but unlike the simple twisted figure-eight structure above, they are not capable of being constructed by a simple topological transformation from a plain torus. It is this essential difference which distinguishes them as knots, and a topologist would certainly perceive that a pretzel was different from a ring- doughnut. Mathematicians classify knotted structures by analysing, in tedious detail, the nature of each and every crossing and intersection in any presented knot formation; from this procedure, by applying simple rules, it can be determined whether two structures are equivalent. That is, whether one knot-form can be translated into another by simple moves which do not involve making breaks in the material from which the knot is constructed, or modifying the Topological Invariants.

To a topologist who practises magic there is a very significant difference, not only between a Unicursal Hexagram and a 'Star of David' Hexagram, but also between a six-pointed star drawn by superimposing two triangles and one formed by interlocking the triangles. The same sort of mathematical distinctions can be made between Pentagrams which are constructed by simply overlaying each stroke on top of the pre-existing ones and one made by interweaving the component straight lines. A topologist would be much more concerned about these differences than by the orientation of the finished figure. Indeed the corny joke with which I opened this section might be rephrased to define a topologist as a magical practitioner who cannot tell the difference between an aright and an inverted Pentagram - so-called 'white' magicians might find the whole subject upsetting on that basis!

Like most mathematics when reduced to the simplest axiomatic level, much of the foregoing discussion is little more than a laboured restatement of the Bloody Obvious; but for all that it yields a precise mathematical system by which all knots, tangled loops and cat's cradles, no matter how complicated, can be described in a compact way. It would be possible to use the same rules to describe examples of celtic interlacing or even an Aran Sweater in mathematical terms; though the latter task might be tedious without a computer.

Different techniques of analysing knots mathematically have been developed during this century, but one of the most succinct is that of Von Neumann Algebra, which is discussed in an article by Dr Vaughan F R Jones of University of California, Berkeley, in the November 1990 issue of Scientific American magazine. Jones, the recipient of this years Fields Medal (a mathematical equivalent of the Nobel Prize), has made the remarkable discovery that the same Von Neumann Algebras can also be employed to model phenomena in the domain of Statistical Mechanics.

THE ARROW OF TIME

Classical mechanics is that field of study which allows us to model the behaviour of simple assemblages of objects and their interactions in response to forces acting upon them. A familiar example is that of snooker balls on a table. If two balls collide with each other, or if a single ball experiences an elastic collision with the edge of the table, the resultant motion can be quite accurately predicted. Moreover, if a recording is made of any single collision event in isolation and then replayed, the behaviour of the objects involved will appear similar regardless of whether the recording is run forwards or backwards. These simple interactions are not time dependent.

Statistical mechanics, in contrast, deals with interactions in systems with large numbers of particles. Picture a situation where a large number of snooker balls are in motion on a frictionless table, where a central dividing wall has been placed on the table, such that all the balls are confined on one side. If the central barrier is removed the balls will spread out across the whole of the table, and the circumstance will not arise such that they all wind up back on the same side of the table again.

It might be said that a state of relative order, where the balls are confined in a relatively small space, will become more disordered in that the passage of time leads to a lessening of constraint upon the balls and, left to their own devices, they will not regain their former relatively ordered condition. The same concept is succinctly illustrated by Stephen Hawking, who uses the example of a porcelain tea-cup as a relatively ordered structure; if the tea-cup is dropped on the ground and it breaks, then that process occurs in synchronisation with the perceived direction of time passing. If we see re-played recordings of that particular event, there will be no difficulty in discerning the difference between a re-play in the normal direction of time, and a re-play in the reverse direction where the pieces of the cup collect themselves back together.

Phenomena such as those described conform with a well established Law of Physics known as the Second Law of Thermodynamics which states that in any closed system, Entropy (or disorder) will increase with the passage of time. From this law, the direction of time can be defined as being that of increasing entropy.

In carrying the Von Neumann Algebras of Knot Theory across to the field of Statistical Mechanics, the topological invariants of Knot Theory have their equivalence in the invariants of Thermodynamics such as the direction or Arrow of Time.

Many people think of Chaos and Entropy as being synonymous terms for 'Disorder' - my view of these entities is quite different. In earlier papers I have given numerous illustrations of the way in which relatively ordered situations can arise from the processes of Chaos Mathematics operating on physical reality. In these instances it would seem as if some fundamental breach of the Second Law of Thermodynamics is taking place. Stunning co-incidences can occur, such as that concerning the Gordian Knot, with which I opened this paper, or that by which the Moon's axial rotation period exactly equals it's Earth orbital period. These co-incidences have a magical or cyber-morphic quality which arises out of Chaos - in the one case as a result of some weird synchronicity between the evolution and timing of my programme of lectures and the organisation of the 'BBC 2' television programme schedules; and in the latter case because of an orbital resonance which develops over millions of years in a multi-body system where movements are constrained by the influence of gravity.

I would therefore argue that the essential difference between Chaos and Entropy is that Chaos incorporates, and even gives rise to, cyber-morphic system structure/control phenomena which can emerge and operate in a fashion which is independent of time and it's direction, while Entropy is nothing more than bog-standard thermodynamic disorder. Moreover, the cyber-morphic component in Chaos is that which can be modelled by reference to the Mandlebrot Set, which, since it does not require a time axis for it's construction, can account for instances of information transfer and emergence that occur independently of time.

The cyber-morphic content of the Second Law of Thermodynamics itself, and the Statistical Mechanics which models it's processes, are echoed in the system structure/control embedded in the mathematics of Knot Theory; that is in the topological invariants - crossings, links and holes -which determine whether one knot structure is the same as another.

Some may feel that the foregoing analysis is nothing more than a shining example of the sort of pseudo-intellectual bullshit which Chaos Magicians use as a cheap attempt to distract more conventional practitioners from sound traditional techniques. To such doubters I would make the following proposal: Next time you trace a pentagram in the course of a banishing or some other ritual procedure, apply a little topology to your visualisation. A pentagram (whether aright or inverted) is likely to be much more effective and to exhibit greater persistence if it is visualised as being knotted or interlaced in the process of construction. Try laying one out with wire or string - if you simply overlay each stroke on top of the previous ones, the form can easily reduce to a simple loop; if, on the other hand, you interweave the sides, a knotted structure will result which retains its topological invariants. This is likely to prove significantly more effective as a protective device, unless, of course, you find yourself in magical combat with some latter day Alexander the Great.

A PROTO-UNIVERSE LASHED BY LIGHT-SPEED COSMIC WHIPS?

Besides the processes at work in enforcing the Second Law of Thermodynamics, Statistical Mechanics has also been found to be useful in modelling events which physicists describe as Phase Changes. The most familiar examples of phase changes are those which occur when water freezes and when it boils. In chemical terms there is no difference between molecules of ice, water or steam - they all consist of two hydrogen atoms and one oxygen atom, but their physical properties bear little similarity. The determining factor in the phase adopted by water, or any other substance, is the amount of energy associated with it's constituent molecules; this in turn is related to the thermodynamic properties of temperature and pressure.

It has been postulated by theoretical physicists that irregularities which arose in the course of the initial chao-inflationary period of the expansion of the universe may have resulted in phase changes within the primal super-hot cosmic material. One of the results of such phase changes may have been that the primal matter configured into Cosmic Strings (or 'super-strings') from which the early galaxies are said to have coalesced. The theoretical image is one of spaghetti-like tendrils of matter spanning the breadth of the universe, and lashing back and forth at velocities approaching the speed of light. Although the observed distribution of galaxies is in accord with this theory, and although mathematical models exist to explain these super-strings, there has always been some difficulty in visualising what they may have been like.

Recent work by Bernard Yurke of AT & T Bell Laboratories on the behaviour of liquid crystal, the substance used as a display medium in many digital watches and calculators may be providing some new insights. Liquid crystal consists of rod-like molecules. These tiny rods ordinarily have a random orientation, like molecules in a liquid, but they line up parallel to each other either when a small electric charge is applied, or when they are placed under some physical pressure. This process of lining up represents a phase change in the liquid crystal material, and Yurke noticed that the equations he derived to model the process appeared very similar to those proposed by the theoreticians to describe what had happened in the early universe.

Yurke has built a 'cosmic simulator' in a cell of liquid crystal some three millimetres wide, and by using a microscope and a video-recorder he has managed to capture remarkable visual representations of the sort of processes which may have given birth to the galaxies in the period after the Big Bang. There is a write-up, under the headline 'Liquid Sky' in the Nov 1990 issue of Scientific American, together with a photograph. This shows miniature 'cosmic strings' in the liquid crystal undergoing a phase change, together with tangled blobs which are said to represent 'textures', or knots in space which are postulated to have formed galaxies and clusters of galaxies. It is noted that some of the thicker strings pinch down to points which exhibit behaviour resembling that predicted for the 'magnetic monopoles' which are said to have existed in the early universe. According to Neil Turok, Yurke's co-worker and a cosmologist at Princeton University, the correspondence between liquid crystal and the cosmos is not perfect; because of friction, the superstrings in the simulator do not lash about at light speed, and the actual expansion process has not been incorporated.

So, next time you look at the changing digits on your liquid crystal watch, be aware that what you are watching is nothing less than a miniature representation of the processes which gave birth to the visible universe. No chemignosis necessary, just applied pressure, a microscope and a video camera!

MATHE-MAGICAL MODELS

In my paper on 'Chaos Invocation', presented to an Oxford Golden Dawn Society meeting earlier this year, and in other recently published work, I have explored a hypothesis which seeks to explain some phenomena within the domain of magic(k) in terms of toroidal structures resembling smoke-rings which are postulated to have some stability and persistence in the Earth's magnetic field.

The appeal of that concept lies partly in the notion that it may be possible for a human nervous system to generate such a structure by a process akin to Faraday induction. It is suggested that this may occur as a result of a surge or pulse of electrochemical activity such as might accompany a martial arts style Kiai or at the onset of orgasm; both of which types of event are traditionally associated with successful magical activity.

I have never suggested that that particular mechanism might be held up to account for all types of magical occurrence, and I would like now to examine some other mathematical concepts which may be of interest to anyone seeking to reach some rational understanding of phenomena which are sometimes thought of as magical in character.

It has been said, if not by the Swiss mathematician Leonhard Euler then by an Open University Tutor in a seminar on his work, that any mathematical system having a 'rich internal structure' has some analogue in reality. That statement is more of a proposition or 'Lemma' than a 'Theorem', in that there do exist mathematical systems with an unarguably rich internal structure for which no immediately identifiable analogue in reality has been identified by orthodox mathematicians. One such example would be the 'Magic Squares' of Agrippa reproduced as the Appendix to my paper Pythagoras and the Mathesis of Chaos (included in 'Liber Cyber').

In the context of the foregoing examination of the mathematics of Knots, the Lemma might be restated as follows: Any system of mathematics exhibiting a rich internal structure will eventually be shown to have one or more analogues either in physical reality or in the 'meta-reality' of magical phenomena. (The Choronzon Lemma)

Where mathematical systems with more than one analogue in reality are concerned, Knot Theory presents a particularly fine example. Further cyber-morphic associations are to be found in applications which model the behaviour of Spin Fields associated with phase changes. In my paper on Chaos and Cosmos I identified the particle spin parameter as being the most fundamental example of the cyber-morph class of information, in that it does not affect a particle's energy or momentum, but merely the way in which it behaves in association with other particles.

Perhaps most intriguing of all, Knot Theory has applications in the modelling of peculiar topological structures known as 'Manifolds'. The simplest example of a Manifold, having only one dimension, is a wrap-around shift register. That is a data storage and/or processing structure of, say, sixteen bits, which might hold the following data:

1001001000010011

When the bits are shifted one position either to the left or right, the endmost bit is appended to the other end of the register, thus after a left shift the register example above would contain:

0010010000100111

The '1' in the most significant position wraps round to the least significant position, and the others all shift along one place.

Outside a computer's Central Processing Unit, the best illustration I can think of which approximates a one-dimensional manifold is that where a cheeky kid in a wide-format school photograph has run round the back while the camera pans along, so that s/he appears at both ends of the picture; there may be better examples.

Some computer game screen displays are configured as a two-dimensional manifold. These are of the type where some 'sprite' can exit screen right and re-appear at screen left (and vice versa) or exit at screen top and re-appear at screen bottom (and vice versa).

Topologically, a three-dimensional manifold can be thought of as a triple torus constructed on a cube such that a connection exists between the top and bottom faces, between the left and right faces, and between the back and front faces.

I am unaware of any orthodox physical example of either a two or three dimensional manifold outside a computer data structure, but the mathematical properties of such creatures are well defined and certainly possess a 'rich internal structure'. Any postulated process of teleportation might be modelled using a three-dimensional manifold, as might a 'worm-hole' through which matter was able to vanish from one location and reappear in another.

It has been suggested (Weeks & Thurston; Scientific American, July 1984) that such manifolds can be constructed on any of the regular solids of classical geometry. Besides those already discussed, this would allow manifolds of 4, 6 and 10 dimensions to be constructed on an octahedron (8 sides), dodecahedron (12 sides) and icosahedron (20 sides) respectively.

It would be possible to postulate that Dr Who's 'Tardis' might function in the domain of a four-dimensional space-time manifold, in that it is presented as being able to de-materialise and re-materialise in locations which are separated not only in distance but in time as well. The Kabbalist's 'Tree of Life' might bear interpretation as a model of the universe based on a ten-dimensional manifold. This perception would be consistent with the concept that all ten Sephiroth exist at all points in the cosmos.

Attempts have been made (unsuccessful as far as I am aware) to use some of these topological constructs as the basis of unified field theories which incorporate the four forces recognised by orthodox physics.

To conclude I would suggest that the Celts and other traditional cultures who give prominence to knots and interlacings in their artwork may, perhaps unwittingly, have given representation to a cyber-morphic concept which underpins some of the most fundamental structural characteristics of the universe. The importance of these structures has only quite recently come to be assimilated into the body of scientific knowledge.

REFERENCES

Choronzon, Fra                 Liber Cyber (Ecliptica) 1990

                                        The Wishing Well (Talking Stick Mag #2) 1990

Encyclopaedia Britannica   Gordian Knot; Topology; Manifolds;

Hawking, S W                   A Brief History of Time (Bantam) 1988

Horgan, J                          Liquid Sky (Scientific American) Nov 1990

Jones, V F R                     Knot Theory & Statistical Mechanics (Scientific American) Nov 1990

Weeks & Thurston           The Mathematics of Three-Dimensional Manifolds

                                       (Scientific American) Jul 1984