This state is known as TURBULENT TAYLOR VORTICES. Out of the chaotic system, a new order emerges. Form once again appears out of emptiness. Using our Buddhist analogy, there is a return to the 'mundane order', as it were. Of particular interest to us is how Stewart and Golubitsky are inclined to see this re-appearance of the 'Taylor Vortices', which occurs at very high speeds, as a SUPERIMPOSITION of pattern on the underlying turbulence or chaos.

This is not unlike the way in which the mandala, using the metaphor of 'superimposition', achieves the reconciliation of the two 'incommensurable' orders of existence (of form and emptiness, or appearance and essence). Buddhists (and other mystics) talk about a third, advanced stage of realization - in addition to the realization that 'form is emptiness', and 'emptiness is form'. This happens when one realizes that 'form is NOTHING OTHER than emptiness, and emptiness is NOTHING OTHER than form'! So form and emptiness are no longer conceived as exclusive polar opposites that need to ALTERNATE - they are simultaneously occuring, although this may seem quite paradoxical. The paradox is, however, the same one that intrigues and attracts and is embraced by contemporary chaos scientists - who now seem willing to speak of systems that are ESSENTIALLY chaotic manifesting APPARENT order, whether this proposition be paradoxical or not.

The contemporary chaos scientist is doing something, psychologically speaking, very similar to what the mystics who utilize the mandala as an explanatory device have been doing all along. They are conceiving of these 'post turbulent' patterns as 'superimposed, as it were' on the turbulence - a kind of layered phenomenon or epiphenonmenon resulting.

Pattern is construed as superimposed on an underlying 'chaotic' organization, in the same way that 'form', for the mystic, is conceived as an 'outer mandala' superimposed on the FORMLESS underlying substratum of existence which is the 'innermost' or 'secret' mandala (secret in the sense that it is not privy to us as an 'object' in everday 'dualistic' consciousness).

Both the mystic and the scientist are struggling to reconcile two incommensurable orders of existence. We say 'struggling', because both are operating within the context of languages that insist on logical consistency (ie, non-contradictory, non-paradoxical formulations) while self-contradiction is in reality unavoidable, at least at this rather profound level of description.

In the Couette System experiments describing above the outer cylinder was held fixed. In the 1960s and 70s, however, DiPrima and his students at Rensselaer Polytechnic Institute studied what happened in such a system when the outer cylinder is rotated in the OPPOSITE direction to the inner one - in which case the initial pattern that appears in the fluid is not a Taylor vortex, but a spiral.

There's a critical speed. If the outer cylinder is counterrotated at less than this critical speed, then the first pattern is Taylor vortices. Otherwise, the first pattern is spirals. If the speed of counterrotation is exactly equal to the critical speed, then both patterns occur simultaneously, and compete with each other to determine the actual pattern. We say that a MODE INTERACTION occurs [between the two modes of appearance - vortices and spirals] at the critical speed. Although you can't see it in the physics, both modes are there all the time. But on one side of the critical speed, the spirals are more stable and the vortices unstable; on the other side of the critical speed, the spirals are stable and vortices unstable. When spirals are stable, the mathematical model predicts a second bifurcation to Taylor vortices - but this state bifurcates from the now unstable Couette flow! So you won't observe it in the physical system. Similarly, when vortices are stable, there is a second bifurcation to spirals - but again starting from an unstable state. In the mathematical model, the two types of bifurcation switch orders at the critical speed. In the physical system, one disappears and the other suddenly shows up instead. (Stewart and Golubitsky, page 115)

The helical spirals that appear when the outer cylinder is turned in an opposite direction look and behave like the ones on a barber's pole. An illusion is created - as the stripes appear to move up the pole, while in fact they're actually simply rotating. 'It's quite disturbing to watch the spiral flow', Stewart and Golubitsky admit, 'you sit there waiting for the fluid to come out of the top of the apparatus, and feel somewhat foolish'. ³ Although the illusion created by the Shri Yantra was undoubtedly intentional and the illusion created by helical flow in a Couette system occurs naturally, the parallel is intriguing. Such 'illusions', as it turns out, seems to occur at comparatively advanced states in both systems.

Finally, when the speed of BOTH cylinders in the Couette system are made to VARY,

Pandora's box falls open, and almost as many flow patterns escape as did noxious insects in the Greek myth... There are wavy spirals and interpenetrating spirals, modulated wavy spirals, wavy outflow and wavy inflow boundaries, even spiral turbulence. (Stewart and Golubitsky, page 109

What is interesting here is that these patterns are conceived as emerging OUT of turbulent (ie, chaotic) flow. How do they arise? Why do they arise? Stewart and Golubitsky propose that as the speed of the inner cylinder in the initial series of experiments is increased, there is a progressive 'loss of degrees of symmetry' in the Couette system, resulting in an increase in pattern.

The Couette system - the apparatus - was designed to be a system with many degrees of symmetry, for the sake of simplicity and ease of calculation, as we've already mentioned.

The 'boring' Couette flow [that occured in Couette's original experiment] may be featureless, but that's not due to its lack of symmetry. On the contrary, it's due to its excessive degree of symmetry: it's unchanged by every single one of the symmetries listed. Couette flow has the same symmetry as the entire apparatus: there's no symmetry-breaking (and no pattern!). Rotate it, reflect it, translate it: it still looks uniform and featureless. Human psychology strikes again: TOO MUCH symmetry isn't perceived as pattern. (Stewart and Golubitsky, page 112)

But as one progresses through all of the forms of flow that arise as the inner cylinder is progressively speeded up (as described in the above) fewer and fewer of the symmetries in the original system are maintained.

Couette flow has full symmetry. Taylor vortex flow breaks some of it (most of the vertical translations). Wavy vortices break more. Modulated wavy vortices break most of the mixed spatio-temporal symmetries of wavy vortices: they become time-dependent in a much more complicated manner. Finally, the turbulent states have no obvious pattern, so presumably no symmetry at all.

Presumably ... but what about turbulent Taylor vortices? They may not have any symmetry, but they do have a sort-of pattern. We'll see in chapter 9 that even the turbulent states may have their own kind of symmetry. But for now, the essential point is this: the main sequence is a series of steps, at each of which symmetry is lost, until by the end it has all disappeared. It's a symmetry breaking cascade. (Stewart and Golubitsky, page 113)

A 'symmetry breaking cascade' - that is a phrase that could have been aptly used to describe the relationship between the nine superimposed mandalas in the Shri Yantra, starting with the symmetrical outermost mandala, and ending up with the asymmetrical innermost figure (where, by virtue of the illusion, symmetry is expected, but in fact ABSENT).

In the Enneagram we see something similar. In the hands-on PUZZLE that appears in Issue Four of the Journal, we explored some of the symmetry-related features of the Enneagram. In the next section of the present paper we will expand upon what was said there.

Degrees of Symmetry and The Enneagram

Imagine that you are given a circle with nine points on it, equally spaced. Using 9 straight lines, connect the points, but in such a way that each point has two and only two lines extending from it. Here are six solutions to that problem, six figures with varying degrees of 'reflective symmetry' -

Figure 1

1) By drawing lines from each point to its closest neighbors on the circle, one creates an equilateral nine-sided figure (shown in black). Nine axes (in pink) can be drawn through the center of the figure, each of which cuts the figure into identical mirror-image halves. This demonstrates that there are nine degrees of 'reflective symmetry' in this figure. Because there are so many degrees of symmetry in this figure, so much repetition, it is relatively uninteresting.

Figure 2

2) You can also draw three equilateral triangles (3-6-9, 1-4-7, 2-5-8 in the Enneagram), and the composite figure will, like figure 1, meet the specified conditions and also have nine degrees of reflective symmetry. This figure has the same 9 axes (in pink) as the diagram above - they are the 9 lines that pass through each enneagram point AND the center of the circle. There is a high degree of symmetry, and although a little more interesting, the pattern that is created is very easy to comprehend and also easy to describe in words ('connect every point on the circle with points that are 3 steps away on the circle').

Figure 3

3) This figure also meets the conditions, and has nine degrees of symmetry. We have chosen not to draw in the axes in this case, so as not to clutter the diagram. Notice how, as a byproduct of the way the lines are drawn, a little nine-pointed figure appears at the center of the diagram (which we have filled in with gray color to make it more pronounced). The diagram thus exhibits what we have been calling 'liminocentricity' (the innermost figure is structurally identical with the outermost). It is also a diagram that has a high degree of symmetry, however, and is just as easy to describe in words as the above figures ('connect ever point with points that are a distance of four steps away from it').

Figure 4

4) Here is another figure that meets all of the specified conditions. Like the others, it has 9 degrees of reflective symmetry. This figure may at first appear to be two superimposed pentagons, one slightly rotated. But the lines actually describe one continuous path that goes twice around the circle and wraps back on itself to make one closed circuit. Again, there is a high degree of symmetry and an easily described pattern (ie, 'moving in a clockwise direction, use your straight lines to connect every other point on the circle'). There is, however, a quality to this diagram that is a bit more elusive, and thus more entertaining.

Figure 5

5) This figure also meets all the conditions. But it has only three degrees of reflective symmetry. And in the diagram TWO distinctly separate figures emerge (the inner triangle, and an outer six-pointed figure), not just the illusion of two (as in figure 3). Trace the lines, and one finds that the 'movement' is more COMPLEX. There is no one simple rule (as in the four figures above) for connecting all the points. Two rules are required - one for the equilateral triangle and one for the other figure. The former utilizes the same 'skip two points on the circle' rule that we saw in Figure 2, but only if one starts at certain points (3, 6, or 9). And the latter uses a comparatively complex rule that requires one to alternate between 'taking two steps forward' on the circle and 'taking four steps forward' - but only if one starts at points 8, 2, or 5.

Figure 6

6) This last figure, the Enneagram, also has 9 straight lines, each of which connects two and only two points on the circle. But it has only one degree of reflective symmetry - it has 'bilateral' symmetry. Like Figure 5, though, it is composed of two separate figures. The second one, however, a six-pointed figure like the one in Figure 5, lacks its symmetry. The movement in the six-pointed sub-figure in the Enneagram is even more irregular than its counterpart in Figure 5, requiring a rule that is relatively complex - 'connect point One with the point that is 3 steps forward, and then, moving in a clockwise direction, progressively with the points that are 2 steps back, 6 steps forward, 3 steps back, 2 steps forward, and 4 steps forward'!

In this series of figures, as we 'break symmetry' by moving from figures with 9 degrees of reflective symmetry to 3 or less, we gain irregularities and there is more complexity in the pattern that is exhibited. This is very interesting, in light of Stewart and Golubitsky's thesis - that pattern formation occurs through symmetry-breaking. As degrees of symmetry in a structure are lost, more complex patterns emerge. An example of this is locomotion in bipeds and quadrapeds, which, according to Stewart and Golubitsky comprise a naturally occuring biological analogue for what happens in the series of shapes that emerge in the Couette System experiments.

Movement and Symmetry-breaking in the Enneagram

Human beings are bipeds with bilateral body symmetry. If we are to locomote in a manner that conserves the bilateral symmetry in our bodies, we would have to hop - putting both our right and left feet forward at the same time. But it is more efficient to 'break' bilateral symmetry and alternate foward placement of the right foot with forward placement of the left feet. In this way we can move FASTER and more effectively than we can by hopping.

Indeed, in quadrapeds we see not two, but rather a number of 'gaits'. And as the animal speeds up he or she CHANGES gait, at critical junctures. Horses, for instance, can walk, trot, pace, bound, transverse gallop and rotary gallop, canter, and pronk. The differences between these may be found in the order in which their four hooves meet the ground. Imagine that you are walking forward, and behind you is walking another individual, with his hands on your shoulders. This arrangement simulates a quadraped. If you and he are walking in lock-step, both putting your right foot forward at approximately the same time, this is similar to what the horse does when 'pacing'. On the other hand, if when you lift your left leg to bring it forward he lifts his right leg, and when you lift your right leg he lifts his left, they you are simulating what the horse does when it 'trots'.

You can diagram all of this in the following way. In the square configuration below, LF means 'left front foot', RF means 'right front foot', LH means 'left hind foot' and RH means 'right hind foot'. Imagine that when you are looking at this simple diagram you are viewing the horse from above, the perspective a rider would have.

     LF  RF
     LH  RH

Now, if we replace the letters with numbers that show the order in which these four feet hit the ground, we could diagram how a horse 'paces' in the manner shown in the figure below. Left front and left hind foot hit the ground together, as we said above, and then right front and right hind foot hit the ground together.

     1  2
     1  2

The 'trot' can be diagrammed in the following way. You can try this out on your hands and knees. You may find that it's actually easier to DO than to visualize doing.

     1  2
     2  1

The 'rotary gallop' looks like -

     1  2
     4  3

or (with opposite orientation) -

     1  4
     2  3

The 'bound' can be diagrammed as -

     1  1
     2  2

The horse's 'walk' or 'amble' may be diagrammed as -

     1  3
     4  2

or -

     1  3
     2  4

And the 'pronk' (which is like our 'hop') looks like -

     1  1
     1  1

The 'canter' is a rather strange gait that is difficult to diagram and does not come naturally to horses, if one is to believe Stewart and Golubitsky. They need to be trained to do it, and some never learn. The movement of the left front and right hind legs are presumably 'a rather arbitrary amount out of phase with the strongly coupled diagonal pair'. As there is a 'left footed' and a 'right footed' canter, it is possible to diagram the canter in two ways -

     2  1          1  2
     1  3          3  1

When horses pick up speed they shift from a walk to a trot to a canter and into a gallop. Wildebeasts shift directly from a walk to a canter. And crocodiles, by the way, can do both the rotary and transverse gallop.

The most symmetric gait is the 'pronk', and one doesn't get very far doing it. The trot has fewer symmetries than the pronk, and the pace and bound have similar symmetries to the trot. And the rotary gallop has even fewer symmetries than the trot - since no leg does exactly the same thing as any other.

We've now established two important general features of animal gaits. The first is that (as for the periodic motions of symmetric systems) their symmetries are MIXTURES of space and time. The second is that some gaits are less symmetric than others - transitions between gaits break symmetry. (Stewart and Golubitsky, page )

And, although there are evolutionary advantages to biological symmetry⁴, there are also advantages to breaking symmetry. For, in general, 'the faster the animal moves, the less symmetry its gait has'.⁵

But what does this have to do with the Enneagram? As the reader may recall, in our analysis of the movements that are associated with the mandala (and the mandala offering in particular) we discovered some that were quite counter-intuitive - and these were associated with the 'inner' figure of the mandala, representing the 'temporal' order. In the Enneagram it is the inner 6-pointed figure that moves according to the odd 1-4-2-8-5-7 sequence, 'breaking symmetry' in this way.

If the Enneagram were a fat, circular, bug with six feet it would probably walk with a gait that looks something like -

      4  1
     6    3
      5  2

The Enneagram, however, isn't a catepillar. And it is not physical locomotion that is its primary concern. But the question that Stewart and Golubitsky's analysis of locomotion and symmetry raises nevertheless applies to the Enneagram. For it DOES makes sense to ask 'What kind of advantage is born from this strange pattern in the Enneagram, which, as we've seen, results in an seven-fold LOSS of symmetry as compared to other ways in which the diagram could be drawn?'

It is probably safe to say that if there is an advantage to the loss of symmetry that results from the presence of the 6-pointed figure in the Enneagram, we will find it in the psychological PROCESSES of individuals - the equivalent in the mental life of individuals to locomotion in the physical sphere. As we've shown in our series on mandalas, an analysis of this 6-sided figure, the central tool associated with the 'Enneagram of Process' teachings, leads one to the concept of a NON-LINEAR sequencing order (ie, a circle) which encourages feedback and feedforward moves (leaps forward in the sequence, and leaps back). It does this in a way that is responsive (at point 6) to forces that break into the system from the outside (at point 3), or from higher 'levels' of the system, and produce something 'new' OUTSIDE of the system, at that higher level (at point 6). The Enneagram is thus, in its function, not unlike that other mandala that we've studied in our series, the Shri Yantra. The loss of symmetry deliberately designed into that figure, the reader may recall, results in the creation of OPPORTUNITY, an opportunity for the birth of something new - something that is indeed 'superfluous' from the point of view of the prevailing paradigm, which is thereby 'transcended'.

In this way the mandala in general, and the Enneagram in particular, reveals a basic truth about psychological process. At its most fundamental level, psychological process in the human being is a function of mental shifts that mirror the phases in a war between incommensurable rival paradigms as they are simultaneously held in mind by the individual.

To put it more simply - psychological 'process' emerges from how we deal with reconciling the incommensurable orders of existence that we cannot avoid as beings with a consciousness that is liminocentrically structured and ESSENTIALLY paradoxical. The bifurcation of consciousness that results from the paradoxical nature of its structure is the happy 'fault', the grand asymmetry in its design, which permits the entire world to come into existence for us. This is the asymmetry that is reflected in the design of the Shri Yantra. And the ramifications of this asymmetry with respect to psychological process is the aspect that the Enneagram, in particular, seeks to follow up.

Iterated Equations and the Enneagram

In general, structure is gained as symmetry is lost, say Stewart and Golubitsky. When a system is put under stress, 'the observed states lose symmetry and gain in complexity (both spatial and temporal)'. 'But what about turbulent Taylor vortices?', they ask. For, as the reader might recall, AFTER the speed of the inner cylinder in the Couette System is increased to the point at which all symmetries break down and a chaotic state appears, new forms arise. And THESE exhibit

... a state with more symmetry AND more structure ...

This seems quite contradictory to the proposition that only when symmetry breaks will structure appear.

What we find is a state that in some sense seems to exhibit more symmetry - at least on the average - than did the state from which it emerged. At present there's no proven explanation of what's happening in this form of symmetry-creation. We can't even throw it on a supercomputer and tell it to analyse the inner workings of the turbulent Taylor vortices, because the computation of turbulent vortices from the Navier-Stokes equations is well beyond the limits of current supercomputer power. So we're on our own: human imagination versus the infinite. (Stewart and Golubitsky, page 223)

As it turns out, symmetry-creation - which we liken here to the appearance of the 'superflous ninth' in the psychology of the mandala -can be witnessed when certain types of equations are 'iterated' by computer, and the results are plotted on screen. An initial point is (arbitarily) picked, and the x and y values of that point are fed into a fairly simple equation that is chosen for this purpose. This generates a second point, with new x and y values. That point is plotted on the screen, and the values are fed back into the equation, resulting in a third point. The same process is repeated to get the coordinates of a fourth point, and so on, until a very large number of points are graphed on the screen. They are rapidly thrown up onto the screen in a seemingly chaotic, random fashion. As one watches the screen it fills and eventually a pattern emerges.

The person who chooses the equation and the coordinates has no way of knowing, unless they've seen the iteration using this particular equation demonstrated before, what the pattern will look like. So one isn't in a position to DESIGN the results before-hand. The above pattern appeared when Stewart and Golubitsky iterated one particular equation, of the form z=x+iy. It resembles the Enneagram, with its central equilateral triangle (9-3-6), which breaks the nine points on the outer rim into three pairs (1-2, 4-5, and 7-8).

The figure to the right of this paragraph was NOT generated in the same way, but is somewhat similar. It is the central figure in Escher's CIRCLE 4, which shows three 'devils' surrounded by six 'angels'.

Escher was a master of illusion. Utilizing figure-ground reversals, he created patterns that in reality could not possibly exist. CIRCLE 4 (below, left) is a good example of that. What we are calling the 'central figure' in CIRCLE 4 (above right) is indisputably symmetrical (it has the same 3 degrees of symmetry as Figure Four, above). It creates the expectation that we could, if we wanted, expand it into a 'wallpaper' paper. So we are not surprised to see Escher do that, and then use the wallpaper to cover the surface of a sphere. So what we see in his drawing is, presumably, a rendering of what such a thing would look like if a painting were made of it, adjusting the figures in the pattern to compensate for perspective.

But if, in reality, one were to try and make such a wallpaper design, the interlocking pieces would not 'pack' in the way that Escher's drawing suggests. Only if the shapes within the figure were stretched in an odd way would it work, a fact that Escher hides from the viewer by mapping the projected wallpaper design onto a sphere, where one makes a mental adjustment for the apparent 'stretches', seeing them as necessary compensations for perspective. At the outer edges of the figure, where cumulative discrepencies would be revealed in a flat wallpaper design, the shapes become smaller and harder to see, and any distortion is rationalized as a consequence of contour and perspective.

In the SHRI YANTRA an expectation of symmetry is created at the outer fringes of the diagram - an expectation which, paradoxically, the diagram fails to fulfill at the center. In Escher's painting, the converse happens. The strong and clear inner figure, which promises three degrees of symmetry, AND the possibility of a repeating design, creates the ILLUSION of fulfillment, though real fulfillment of these expectations is in fact impossible.

Both the SHRI YANTRA and Escher's painting can be taken as suggesting the possibility that symmetry and asymmetry may not be mutually exclusive features. One can thus see them as percursors to the kind of investigations, into 'symmetry breaking' and 'symmetry creation', in which Chaoticians like Stewart and Golubitsky are involved.

Combining the features of both figures, we might conceptually imagine a third (liminocentric) figure that is asymmetric at BOTH extremes (innermost and outermost), and ONLY at the extremes, yet symmetric in the 'in between' regions. Such a figure would be an apt symbol for consciousness itself, and might arguable be called the quintessential 'mandala'.

If consciousness is indeed structured in this way, it would not be surprising if the (psychological) qualities that it manifests would frequently appear as POLARITIES that are subject to what Jung described as 'enantiodromic shifts' - i.e., shifts in which a thing, or quality, in its purist form, suddenly turns into its polar opposite. In such a system, those qualities that will appear as 'perfections' at one level, may manifest as 'defects' at another. The Enneagram provides us with a framework that invites looking at psychological qualities, and the personality types that are constructed out of such qualities, in precisely this way.