Chaos and Topology
by Tom Gilmore
All graphics by Tom Gilmore
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From the Greek, catastrophe means sudden dramatic unpredictable change. The nuance is that it may be expected but it may not be accurately predicted. In the case of a rainbow appearing, which is a catastrophic event, the cause is complex and relatively esoteric. A sudden chain reaction in the path of light rays is triggered by the surface anomalies of the spheroid water droplets.
In classic physics, in describing a catastrophic event, the fulcrum of change is called the "cusp". Sudden buckling, folding, exploding, combusting, boiling, breaking and so forth are all cusp catastrophes, where the event occurs at some specific threshold.
Some mathematicians, such as Rene Thom, have categorized cusps into types based on the parameters involved. Simple events such as the formation of a rainbow or a sonic boom from a jet "breaking the sound barrier" are called folds. It is like bending a stiff sheet of paper until it suddenly folds. A cusp is where two competing forms are in competition, such as in water changing back and forth from liquid and gaseous forms (boiling and condensing). Buffered events have compound cusps. Nudging a movable magnet on a table toward a stationary magnet, if the poles are opposite the movable magnet will suddenly jump to the stationary magnet. This jump occurs when the strength by proximity of the magnetic attraction overcomes the frictional resistance (due to gravity) from the surface of the table.
In nature the buffered event is used in animal behavior. When a predator crosses the buffer outer boundary the prey become nervous. If the predator crosses the inner buffer boundary the prey becomes alarmed and flees. This serves to keep the fleeing behavior from constantly interrupting the feeding behavior in the frequent instances when predators are only passing by and not a threat.
"Umbilic" events are typified by the buckling of a stiffened plate. For solids, umbilic events are hyperbolic, while for liquids they are elliptic and parabolic.
Bifurcation theory arose from the wobble catastrophe, which is circular umbilic. The Hopf bifurcation is when a steady state transforms to periodic oscillations, or the reverse. This process is utilized in neurons of the brain to fire signals off. The mystery of the will and how it causes the firing of neurons may have to do with bifurcation. Bifurcation appears to be an intermediate form of chaos. It is perhaps chaos which still contains the pattern of order that links it back to the ordered state it degraded from. This is in contrast to random chaos, which reverts to an indiscriminate order. There is a constant flux in the border between chaos and order, and the biology of life-processes depends entirely on this feature. Perfect order and perfect chaos are precluded, and everything is forced toward this buffered cusp where it is recycled in the catastrophe of spontaneous life.
Chaos and Free Will
The notion that catastrophic events can be triggered by minute causes is expressed in the popular axiom that it may be the disturbance of a butterfly's wings in the jungles of Argentina which cause a devastating April snowstorm in New York. This axiom also expresses the notion that nothing is totally chaotic, and that the appearance of chaos may be due to the seemingly trivial causes being overlooked.
The doctrine of determinism states that everything follows inevitably from the prior conditions. From this is deduced that there is no free will. However, mathematics reveal the inherent falseness of the doctrine. For example, differential equations commonly have multiple correct answers (eg. When y = x2, if y =4, x = 2 or -2}.
The stochastic doctrine is that free will consists of the choice from among the possible correct answers of the deterministic equation (in other words a choice from limited possibilities).
The chaotic doctrine is the commonly held view that free choice is capricious and unrestricted (as represented by the unreasonable actions of Man).
Order as an Attractor
When uranium decays to lead it takes one of many possible routes of alpha and beta decay through the intervening unstable elements (from element 92 to element 82). It is impossible to predict which route the decay will take, but regardless of the route, Uranium238 always ends up as Lead206. In the topology of probability, such a common endpoint is called an attractor. The complete set of decay paths describe a topological surface. (To see the routes, refer to abundance charts of the Isotopes.)
Following the line of Hindu thought expressed in the lotus flower, where the structure is composed of micro units of the identical structure (in turn composed of micro units of that structure), topologists envisioned the strange attractor (an attractor operating at multiple scales of expression). In mathematics, a set with common structure at all scales is called a fractal. These concepts relate to the forces of order acting on chaotic events. The chaos is restricted to a topological space and that defines the attractor function.
On the other direction, chaos is acting on order, and this is termed turbulence. In fluid dynamics, at a certain rate of flow, turbulence interrupts the regular predictable flow with chaotic flow. (Likewise with airfoil turbulence.) No mathematical description of turbulence has been discovered, but the region of chaotic turbulence can be handled as an attractor. Others (determinists) postulate that turbulence is actually quasiperiodic (composed of multiple periods), with so many contributing periods that the combined result appears random.
The issue of smoothness and continuousness is basic to the discrete vs. fluid models of physics. The fluid model is a mathematical invention which in effect extends the discrete reality by calculating theoretical intermediate positions between the discrete positions. That ‘continuousity’ is invalid is demonstrated by geometric paradox, such as that put forth by Helge von Koch in defining a curve of infinite length enclosing a finite area. The curve is made up of straight lines which approach infinite shortness. It is constructed from equilateral triangles where a triangle 1/3 as large is erected on all three sides, and repeating this process ad infinitum. The resulting object resembles a snowflake, has no tangents, and has no function defining it. This violates geometry because every curve must have tangents its entire length. More generally this fallacy of continuousness is expressed in the map paradox, where the shore lines are infinitely irregular and therefore the lines marking them, when fully straightened out, must be of infinite length.
Any natural fractal object, such as the branching system of a tree, has a finite number of scales of expression. Only imaginary mathematical fractals have infinite scales. Due to the practical limits of mathematical accuracy, in that an arbitrary number of decimal positions must be utilized, fractals fall into patterns of repeating rounding bifurcations. Mandelbrot sets explored such fractals in pictorial representation, and revealed the innate beauty of the discrete nature of number.
Iteration is a mathematical fractal. It takes a function (f) on a number (n) and repeatedly applies the function on the result. The series is f(n); f(f(n)); f(f(f(n))); etc. This imposes a nested structure.
An example which moves between order and chaos is the fractal f(n) = xn(1-x), where n>0, n<1, and x>0, x<4. Over long term iteration, regardless of the value of n, if x<3 the fractal application converges on one value. As x gets greater than 3 it oscillates between 2 values, and as x increases it oscillates between four values, then eight values, and so on, until at around 3.7 it goes chaotic. This results from the increasing pressure of precluded orders of precision in mathematical representation, and totally frustrates the proponents of determinism.
Topology has roots going back into the mists of time, beyond all recorded history. In Astrology, which also predates recorded history, the basic geometric concept is expressed in the symbol for the sun. The astrological symbol for the sun is written as a dot inside a circle. Pop up this 2-dimensional symbol in the 3rd dimension and it is a sphere inside a donut (torus). The spheroid and the torus are the two basic surfaces of solids in geometry. Topology deals with the surfaces of solids. Any blob of any shape (even a cube) is considered spheroid unless it has a hole in it, in which case it is torus.
The astronomical symbol of the sun (shown above in 3-D) is a dot inside a circle (a spheroid inside a Taurus).
The origin of modern topology can be traced to 1887 when King Oscar of Sweden offered an award of 2,500 crowns for the answer to the perplexing question regarding the ultimate stability of the orbital dynamics of the solar system. The question was whether the orbits were a closed path or unstable oscillations which might result in Earth being thrown out into space by the influence of the other planets. Newtonian gravitational dynamics can be calculated with accuracy for two bodies in space, but with 3 or more the mathematics become extraordinarily difficult, and perhaps insoluble. If it could be shown that all elements of the solar system ever repeated their relative positions at once, then it could be deduced that the bodies were following a closed path which would repeat forever. When rather quickly it was shown that the probability of position recurrence was diminishingly small, a different approach to solving Oscar's problem was needed.
The prominent mathematician of the time, Poincare, originated a model he called "analysis situs" to demonstrate the stability of the solar system. The concept was to take the orbital paths of the planets and construct a mathematical surface upon which it could be shown the path always occurred. Then these surfaces could serve as a closed path, proving the stability of the system. In 1889 King Oscar awarded the prize to Poincare, basically for inventing the field of topology, even though the proofs he derived were considered inconclusive. His work also introduced what in modern times is called "strange attractors" in chaos theory, which today is used in predicting weather patterns.
One property of the spheroid is that it is only closed on a dimensionless surface. With any thickness to the surface there is an inherent discontinuity somewhere. For example, the fur on an animal must have a part somewhere, meaning a place where the direction of the fur abruptly changes direction. The atmosphere of the Earth cannot be in motion on every spot at once, since there must be an abrupt change of direction somewhere as dictated by topological principles. A magnetic container must leak somewhere, where the magnetic force is perpendicular to the surface. A bubble must have a line of weakness in the tensile gradient where it can burst.
The Mobius Band
Evidence of the ancient knowledge of what is called the Mobius band can be seen in depictions on Tarot cards. In the Bohemian Tarot the character on the first card wears a hat with a floppy brim which has a twist in it. The Mobius band has such a twist in it. In modern versions of the Tarot the symbol of infinity (an "8" on its side) is shown over the head of the character on the first card. This clearly is a shorthand representation of the Mobius band, not disguised in the clothing as in the Bohemian version. Bohemian versions of the Tarot were attempting to hide knowledge from the prying pious eyes of the Inquisition.
One subject of Topology regards the properties of surfaces of solids. There are two basic solid forms; spheroid and torus. For the taurus, the number of holes in the solid is called its genus (g). The surface of solids can be completely smooth, or have all planar surfaces, or be a combination of both. Any polyhedron, no matter how complex, has a mathematical relationship between vertices (corners), edges, and faces. Tor the torus the sum of the vertices and faces always equals the number of edges. In the case of a spheroid, there are always two less edges than the sum of the vertices and faces. In the simplest example, the case of the cube, it has 6 faces, 8 vertices, and 12 edges (6+8 -12 = 2).
The surface of a regular torus can be constructed from plane surfaces. Taking a planar square and rolling it into a tube forms a cylinder. Taking opposite edges of the cylinder and rolling them inward to join, retains the inside hole and forms the torus surface. It seems anti-intuitive because there was no hole in the plane surface we began with. A hole was created by moving from 2-dimensions to 3-dimensions. If you take the square and roll it but twist at the same time so that the bottom joins to the top, the cylinder now is a Mobius band, but if you roll the edges inward to join, you find that 3-dimensional space precludes the full joining. This theoretical solid is called a Klein Bottle. You would need a 4th dimension of space to construct it in. The twist in the cylinder has inverted the edges of the original square.
What is happening is that the Mobius band is a flat-seeming surface, which being twisted, is actually already a 3-dimesnional solid. This is also anti-intuitive because of the flat surface of the Mobius band. The band is only a sub-section of the torus; a 2-dimensional cutout requiring 3-dimensions. It violates our sense of "sidedness" because we see the flat band and it looks to have 2 sides (and 2 edges). To see that the band is 3-dimensional refer to the graphic below, where the Taurus to the left (in red) is marked with one continuous line which crosses over from the outside to the inside on the first circuit, and then crosses back on the second circuit at the same point (blue lines on the red torus). By connecting the lines with a curved plane, and discarding the red Taurus surface, a Mobius band is cut out (blue band to the right). As can be seen on the torus, the edges are one line looped through the hole.
To create a Mobius band take a strip of paper and twist one end once, then glue or tape the ends together.
If the edge is given two twists then the upper side is again joined with upper side. In this case it has two edges, whereas with one twist it had only one edge. With 3 twists it again has one edge, with 4 twists it has two edges, with 5 twists one edge, and so on. Topologically it only matters whether the joined edges of the original square are aligned or inverted.
Cutting the Mobius Band
Although not localized, the edge of the Mobius band crosses the face of the band somewhere. When you poke a scissor through the center of the band face and cut down the middle, you cut across the edge which invisibly crosses the flat face, so the invisible edge is cut in two, leaving two twists in the band (which means it now has 2 faces and 2 edges (remember that there were two crossovers in the one edge-line on the Taurus, and when it is cut in half there are topologically still two crossovers).
When you cut the band with 2 twists in half, you simply have two bands with 2 twists, however, one band is looped twice through the other band. This is because there were two crossovers in the uncut 2-twist band.
A combination effect occurs when the Mobius band is cut in thirds. When you start the interior cut at about 1/3 way across the face, and just continue cutting, the cut will continue around the band twice. First one third is cut off, but it is attached by the crossover edges, then the remaining two thirds is cut in half. The first third is still a Mobius band attached by the uncut edge, and the rest of the cut then cuts the invisible crossing edge. The result is a band with a full twist, just as when you cut the Mobius band in half, but it is looped through the Mobius band that was cut in the first one-third of the total cut.