A Maths Problem You WILL Understand

Think of a whole number. If it’s even, halve it. If it’s odd, multiply it by three and add one. Keep doing the same. You will find you have a series of numbers which go up and down in value, and if you plotted them on a graph they’d often seem to bounce up and down like hailstones in a hailstorm. For this reason, they’re called “hailstone numbers”.

Here’s an example. 5, 16, 8, 4, 2, 1, 4, 2, 1 . . .

Here’s another: 42, 21, 64, 32, 16, 8, 4, 2, 1 . . .

At first glance you might expect the situation to be as follows; the larger the number, the more steps it takes to reach the final cycle of 4, 2, 1 . . . It seems quite simple, but it isn’t. The number 27, for example, takes a hundred and eleven steps to get to the cycle and goes as high as 9 232. There seems to be no way to predict exactly how long the sequence will last before it gets to the cycle, although it is unsurprisingly true that larger numbers, particularly odd ones, tend to take longer. For instance, 27 is an odd number and the number 2596148429267413814265248164610048 also takes a hundred and eleven steps to reach the loop, and is considerably larger than 27. In case you’re interested, 27’s sequence is:

27, 82, 41, 124, 62, 31, 94, 47, 142, 71, 214, 107, 322, 161, 484, 242, 121, 364, 182,
91, 274, 137, 412, 206, 103, 310, 155, 466, 233, 700, 350, 175, 526, 263, 790, 395,
1186, 593, 1780, 890, 445, 1336, 668, 334, 167, 502, 251, 754, 377, 1132, 566,
283, 850, 425, 1276, 638, 319, 958, 479, 1438, 719, 2158, 1079, 3238, 1619, 4858,
2429, 7288, 3644, 1822, 911, 2734, 1367, 4102, 2051, 6154, 3077, 9232, 4616,
2308, 1154, 577, 1732, 866, 433, 1300, 650, 325, 976, 488, 244, 122, 61, 184, 92,
46, 23, 70, 35, 106, 53, 160, 80, 40, 20, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1, …

All of this is easy to understand. Any numerate person with a late primary school level of maths could make these calculations and test the sequences. We the not particularly mathematically-skilled public have no problem getting this and doing the work on it to a certain extent. It also seems to be pure mathematics so far as I can tell: there seem to be no applications for it at all, except perhaps for that frondy thing at the top of this post which looks nice. Much of my own ability to do maths is blocked by my failure to get to grips with calculus, as I suspect it is for many other people. I don’t consider the obstacle to be entirely insurmountable but I’ve never actually succeeded in vaulting over it. But there’s another aspect to this set of sequences which seems to stymie everyone, no matter how adept at mathematics they might be. This is known as Collatz’s Conjecture.

In maths, a conjecture is a statement which seems to be true on the basis of preliminary evidence but has never been proven or disproven. A famous example is Goldbach’s Conjecture: every even whole number from four upwards is the sum of two prime numbers. This has been demonstrated for every such number up to at least 1000000000000000000, but there’s still no known proof or disproof.

Collatz’s Conjecture is that every whole number treated in this way will eventually collapse into the sequence “4, 2, 1, . . .” rather than either going on forever or being part of a different sequence. It’s clearly true for any power of two, but they get further and further apart the higher you go, and in their case there’s a simple relationship between their size and the number of steps before it happens. You might therefore think that prime numbers which are one less than a power of two, known as Mersenne Primes, would have some kind of relationship but they don’t seem to have. 127, for example, takes forty-six steps to reach the cycle.

If the distributions of the steps for each number are plotted on a graph with a logarithmic scale to the Y-axis, before they collapse into the cycle the movements are pretty close to looking random, although there’s an algorithm to them so they aren’t. It occurs to me that the paths might be similar to the movements of atoms and molecules in an atmosphere thin enough at the surface of a moon or planet to be a collisionless gas, but maybe not. It might also be that it would work as a pseudorandom number generator with that logarithmic step and the omission of the final falls to the cycle. It’s also remarkably easy to write a program to do this. For instance, in BASIC:

10 LET A=<value>

20 IF A DIV 2 * 2=A THEN LET A=A/2: GOTO 20

30 LET A=3*A+1

40 GOTO 20

It can also be implemented easily in assembler using shifting and adding: the above code is actually unnecessarily complex, and this may not be coincidence. John Conway of “Life” fame managed to generalise this sequence to produce a Turing Machine, i.e. a general computer which can, given time, do anything any digital computer can, and this opens it up to comparison with the Halting Problem. The Halting Problem is whether an arbitrary computer program, given an input, will finish running or continue forever. Alan Turing proved that there is no way to show that this will happen. If these sequences are shown to be sufficiently similar to computer programs, the Collatz Conjecture would therefore be shown to be unprovable. Conway came up with an esolang (esoteric programming language) called FRACTRAN which was Turing-complete in 1987, based on this sequence.

The largest number tested for this is 268, which is 295147905179352825856. Riho Terras was able to prove that almost no number reaches a point below its original value, and limits to the values were arrived at in 1979 and 1994 which showed that the function can rise as slowly as possible.

Although every number tested does gravitate to 4,2,1…, that isn’t enough to prove that it always happens. For instance, it could be that high enough numbers wouldn’t do this, and since there are infinity whole numbers, simply testing every step is impossible. There are two not mutually-exclusive possibilities. One is that there is a number out there somewhere which will never start to fall towards the cycle. The other is that there is a set of numbers which is part of a completely different loop. Both of these could be true, or only one. If there is such a cycle, it’s been proven that it must have at least 17 087 915 members, meaning that it can’t be practically proven by one person doing pen-and-paper calculations. It’s also true that if the calculation is changed to 3x-1, two other cycles appear:

5, 14, 7, 20, 10, 5,…


17, 50, 25, 74, 37, 110, 55, 164, 82, 41, 122, 61,
192, 91, 272, 136, 68, 34, 17,…

There might also be more of them.

There is something of an argument which suggests the Collatz Conjecture is true. It goes like this: one more than thrice an odd number must be even, so it will divide by two. There is then a 50-50 chance that the result will also be even, and therefore also divisible by two. The longer the sequence is, the more likely this is to happen. This is a statistical argument, but some numbers in the sequence are omitted completely in these calculations and others crop up a lot more than they “should”, so the neat bell-curve that might be expected might not be forthcoming.

The numbers do, however, obey Benford’s Law. Benford’s Law was first noticed when books of logarithms in university libraries started to get dirtier towards the front of the books than the back. This is because numbers which begin with smaller digits are much more common than those which begin with larger ones. This applies, for example, to the lengths of rivers, electricity bills, stock and house prices. More than thirty percent of such numbers begin with a one, slightly more than a sixth begin with a two, an eighth begin with a three and so on, until only 4.6% begin with a nine. Benford’s Law does approximately apply to Collatz sequences, and it gets closer the more numbers are included. It’s also true of numbers in any base other than binary. It works best when considering data which span several orders of magnitude and is used to detect fraud in elections and accounting. This presumably means there are computer programs out there on the Dark Web or something which use Benford’s Law to befuddle forensic accountancy. Collatz sequences might find a use there.

Although the Collatz Conjecture seems useless as far as direct applications are concerned, it does have educational value. It shows, for example, how simple formulæ can lead to highly complicated systems, and that there are attractors as mentioned in Happy Catastrophe.

If the same thing is done with negative integers, there are three loops. These are:

-1, -2, -1. . .

-5, -14, -7, -20, -10, -5 . . .

-17, -50, -25, -74, -37, -110, -55, -164, -82, -41, -122, -61, -182, -91, -272, -136, -68, -34, -17 . . .

The image at the top of the post is produced using Collatz sequences. One way of thinking of them is as a tree. All known positive integers tested eventually settle down to the sequence 4, 2, 1 . . . , so this can be seen as something like a trunk or a final confluence of tributaries to a river. There are also other confluences further up the tree. If the transition to an odd number is drawn on this directed graph at a 20° angle in one direction and that to an even one as 8° in the other, you end up with what’s shown above. These angles can be adjusted and you end up with various shapes which look like living organisms such as corals, bryozoa or shrubs.

The oddity about this problem is that it jumps so rapidly from a simple issue which can be understood by anyone who knows basic arithmetic into a problem which has never been solved by the most skilled and advanced mathematicians who have applied themselves to it. This raises the question of why we are able to understand almost anything. Even though there are a huge number of mathematical problems which can and have been solved, it isn’t clear how we are able to do that. It seems that it could easily have turned out that we wouldn’t be able to do maths at all because it all turned out to be too hard, and where would that leave us? Why are we able to solve any mathematical problems? Also, most mathematicians consider this problem to be significant but also one which is likely to absorb all someone’s time without them coming up with a useful result, so it’s also the ultimate time-waster, or at least seems to be. Hence this entire post may or may not have been a waste of time. What do you think?

Catastrophe Theory

By Salix alba – Own work, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=26446257

I’ve already covered the topic of fractals and Chaos Theory, but the arrival and popularity of these two obscures a slightly earlier and rather similar mathematical topic which has a number of things in common with them, although it’s a lot “smoother”. This was Catastrophe Theory.

On 28th July 1975, BBC-2 broadcast a ‘Horizon’ documentary entitled ‘Happy Catastrophe’ which got a larger response from its viewership than any other ‘Horizon’ episode. It clearly captured the public’s imagination, attracting more correspondence than any other ‘Horizon’ up until that point, and in fact stuck in my own mind more than most other programmes at the time. Looking back at it, I found a number of other episodes in the mid-’70s quite memorable, such as the one on epilepsy and another on Erich von Däniken, which I mention here, but certainly this is one of them, and in fact epileptic seizures themselves could be modelled using catastrophe theory (CT) itself. To an extent, I want to blog about CT today, but I’m also interested in why it was so popular, and why it seems to be largely ignored today.

CT deals with discontinuities, which are moments of sudden change. For example, if you take a thin card and press it at its sides, it will do very little for quite a while, then suddenly crumple or flip into a different shape, and letting go of the card will not lead to its return to anything like the flat form it had before, although it will tend to spring back a little. The same applies to a snapping rubber band under tension and a host of other situations, such as the epileptic seizures I mentioned just now, although one would hope in this last case that the brain can in fact fairly quickly return to a more organised state. Unfortunately this is rarely not so, in which case it becomes a medical emergency.

The programme’s title, ‘Happy Catastrophe’, is interesting. When we use the word in English, and it is of course a Greek word, we generally mean something negative. The Greek word, “καταστροφη”, consists of the words “κατα”, meaning “down”, and “στρεφειν” – to turn, in other words a “downturn”, and with the usual connotations of falling does indeed have negative connotations. The word was prominently used in drama, where it referred to the fourth and final part of a play, after protasis, epitasis and katastasis. We’re familiar with it today through tragedy, but in fact it also applied to comedy, and in that setting it referred to a happy ending such as a wedding. Hence our own usage has become predominantly negative, but for some time I attempted to use it with a more neutral connotation, which in fact makes the word a lot more useful, although it can be confusing and we don’t really have control over the meaning of words, particularly when we lack something like L’Académie Française. There were two types of catastrophe, whether happy or otherwise, in Greek drama. In a simple catastrophe, there’s simply a transition from dramatic events to a quieter set of circumstances without any change in character, unravelling or revelation. Complex catastrophes involve sudden discoveries by the character or sudden changes in fortune which are feasible and upon which the plot depends, rather than being a deus ex machina. In a way, simple catastrophes occupy one side of the graph whereas complex ones occupy the other. This is what I mean:

Taken from here. Will be removed on request.

A simple catastrophe can be thought of as a movement across the steady slope on the left hand side of this graph. It descends into repose without anything huge happening. I don’t know what examples there are of this but to be honest they sound a bit boring. Complex catastrophes, on the other hand, are movements along the right hand side of the graph and involve events “falling off a cliff” in such a way that they permanently change things. This graph is of course the “cusp catastrophe”. It makes me wonder what the variable labelled as “u” is in drama. ‘Œdipus Rex‘ definitely occupies the right hand side – it has a low value of u, whatever that might be. It’s also important to remember that if you turn this graph upside down, you more or less have the same graph, and that therefore comedies are also catastrophic in nature. ‘Much Ado About Nothing’ is just as catastrophic as ‘Œdipus Rex‘, but in a positive way.

Incidentally, in what I’ve just said I can’t help but be reminded of this:

Can you usefully take a quantitative approach to literature? In a way the answer is a definite “yes”, because for instance you could look at repetition of certain words and phrases or the prosody or rhyme scheme of a particular poem, but in general it does have a bad rap. But I can’t help noticing that when John Keating gets the pupils to rip out the introduction to ‘Understanding Poetry’, it is a catastrophic event, and of course later in the film there are other incidents more deserving of the word, but there’s no going back once the introduction has been ripped out, as the end of the film illustrates.

The cusp catastrophe graph looks like the kind of shape you’d get if you held a thin sheet of metal horizontally and bent it towards or away from you. This is because that situation is in fact a catastrophe with two control dimensions and one behaviour dimension. The buckling which occurs on one side of the sheet is dramatically greater than on the other. This now sounds like an engineering or metallurgy issue, but can be used for drama, as with the 1951 film ‘No Highway In The Sky’, which involves the catastrophic failure of aircraft in this way. In this case the behaviour axis involves the plane falling out of the sky and killing everyone, although there’s another catastrophe where Theodore Honey deliberately damages a plane to prevent it taking off and killing the occupants:

I’ve mentioned control and behaviour dimensions, or axes, without really explaining what they are. To elaborate, it makes sense to consider the simplest possible models, including non-catastrophic ones, which have two dimensions. A section of a two-dimensional line graph can have a number of shapes relevant to CT. It can be a slope, a trough, a peak or a fold. Except for the slope, these are all the same basic shape. With a fold, the shape is like a C rather than a U or an “n”. This means that as the control variable increases, the behaviour of the system can either become more dramatic or less so, to choose one possible label for a variable, but will be stuck in that trend unless the other variable reduces considerably. Or, it can be reflected along the Y axis and will be stuck in a trend unless that variable increases a lot. This is the “zone of inaccessibility” and can be shown in several other examples.

There are substances whose melting points are not the same as their freezing points. That is, if a solid of this nature is heated, it will melt at a particular point, but if the resultant liquid is then cooled, it may need to be made colder than the temperature at which it melted to solidify. I seem to remember that cocoa butter does this, but there are many examples. Similarly, when tuning in an analogue radio with a manual tuner, one can find a station, then tune up past it and then find that it seems to be on a lower frequency than one previously found it when twisting the knob back again. These are examples of the kind of behaviour which is modelled in the overhang found in the cusp catastrophe. A value can increase smoothly until it leaps to a higher value if another value is high, but can also stay on the lower surface, and likewise can stay on the higher surface until it is lower than when it initially leapt up. I have a feeling that tidiness is like this. It takes more effort to tidy something up in one big go than it appears to when one does it bit by bit, and then it slips down into untidiness more easily.

Adding a dimension clearly results in three-dimensional graphs, and again there are a certain number of these. Incidentally, before I go on I want to point out that CT graphs only focus on a narrow range of variables where something interesting is occurring, and are therefore small portions of potentially infinite graphs. The two-dimensional “fold” catastrophe could easily diverge to an ever-increasing but smooth extent along its control axis, even to infinity. Also, in illustrating these graphs the section can be a small map of a much larger landscape, such as one including peaks and basins or mountains and valleys. It’s just that the distinctive shapes can be broken down in this way.

Three-dimensional graphs could just be extensions of two-dimensional ones, so for example a valley could just be long and not do much interesting in the Z-axis, so all the types still exist in three and more dimensions and are not cancelled out by the new ones, but each added dimension does introduce additional graphs. In the three-dimensional case, X and Z can be the controls and Y the behaviour, which makes the surfaces more relatable as they’re more like topographical features. There’s the slope which rises diagonally to the axes, the peak, what I’m going to call the “crater”, which is a dent in a surface, and two less familiar shapes, the col and the cusp. I want to mention the col even though it isn’t catastrophic, because it’s less well-known or easy to relate to than the others.

A col is a gap between two peaks. These are often nameless locations, although passes are cols. They occur also in air pressure patterns, where there’s a low-pressure point between two high pressure weather systems. There’s also the saddle:

Saddles differ from cols in continuing to curve away in both directions, concave on one side and convex on the other. A col is the central point of a saddle according to one definition.

The cusp is crucially different from all of these because it has a kind of asymmetry to it along one axis, although it also is rotationally symmetrical in that turning it 180° around the axis labelled u in the earlier graph, assuming it’s aligned correctly, will lead the same shape. This mixture of asymmetry and symmetry doesn’t apply to the other shapes and the cusp is the only discontinuous shape involved.

These shapes appeal to the eye, and it’s been said that CT is particularly visual. It shares this feature with many fractals and the Mandelbrot Set, and in this respect serves as a kind of herald to those later, particularly visually appealing, mathematical excursions. It also has a kind of universalising tendency, which despite its name has been described as modelling rather than a theory. Calling it a theory is a bit like counting two legs on a person and seeing that there are two stars in a binary star system and calling that “integer theory”. It’s more that this kind of model can be applied to natural phenomena, and as seen above with the illustration of catastrophes as a dramatic device, also in the social sciences and humanities. The issue of their beauty may be similar to the beauty of regular fractals and the Mandelbrot Set, in that certain features echo the characteristics of being a product of the Universe, which is who we are in one respect.

There are a total of seven graphs, according to CT, which can between them be used to model all discontinuities. These are: the fold, cusp, butterfly, swallowtail, hyperbolic umbilic, parabolic umbilic and elliptic umbilic. The hyperbolic umbilic is illustrated at the start of this post, where it comprises the upper part of the image. Because it’s a five-dimensional shape, the illustration isn’t exactly what it “looks” like, but is in fact what’s known as the bifurcation set of the hyperbolic umbilic. This is a projection of the shapes which are discontinuous in the graph. In the case of the cusp, this is a kind of curved V-shape extending to infinity or the edge of the graph, like a kind of shadow cast by illumination on a transparent model, or alternatively, and this is more important than it might seem, the kind of light reflected by illuminating a smooth metallic version. The bifurcation set of an hyperbolic umbilic is like two superimposed half-pipes at a shallow angle to each other semicircular in cross-section at opposite ends smoothly becoming curved V-shapes at the other. That probably isn’t very clear. It has two behaviour dimensions rather than one, and three control dimensions. Umbilics are points on locally spherical surfaces, and hyperbolic ones have just one ridge line passing through the point in question, which if I’ve described the above clearly means the point of intersection between the two half-pipes. It’s interesting to contemplate what it would be like to skateboard around the bifurcation set of an hyperbolic umbilic.

The other two umbilics are the parabolic and elliptic. Elliptic umbilics have three control and two behaviour dimensions and the bifurcation set looks like a cross-sectionally curved triangular prism pinched smoothly to a point at the centre, which is the three ridge points passing through the umbilic point. Finally, the parabolic umbilic is six-dimensional, with four control and two behavioural dimensions, making it particularly hard to visualise as even the bifurcation set has four dimensions, but are transitional between hyperbolic and elliptic umbilics, with two ridges, one of which is singular. Visualised using the fourth dimension as time, running in one direction the bifurcation of a parabolic umbilic looks like a shrinking paper plane crashing through the fold in a sheet of paper folded into a V-shape while another V-folded paper shape at the bottom is flattening out and bowing outward.

The other two are the rather less awkwardly-named butterfly and swallowtail. The former is interestingly named because of the butterfly effect, but is not more closely linked to that than the others. It’s five-dimensional, with four control dimensions and one behaviour dimension, and has been used to model eating disorders. It looks odd, even reduced to three dimensions, which effectively destroys its usefulness but enables one to work out what it’s doing, as it looks like a cusp catastrophe with three cusps linked in a kind of triangle. That is, a triangle can be drawn between the three points where the cusps split off from the smooth side, but that triangle isn’t oriented in three-dimensional space unless the butterfly is rotated in such a way that most of it is in hyperspace.

The swallowtail catastrophe is so named because a mathematician was trying to describe it to a blind person, who responded that it sounded like a swallowtail, which it does. It’s merely four-dimensional and its bifurcation set looks like a swallowtail at one end with a U-shape above it with the tail diminishing into the U halfway along. This has one behaviour dimension and three of control. Salvador Dalí’s last painting, if it was his, in 1983, was based on this graph, and was entitled “The Swallow’s Tail”:

This is a cross-section of the bifurcation set with some extra bits added. The monoline S shape is a cross-section of the cusp catastrophe. Dalí described CT as “the most beautiful æsthetic theory in the world”. The artist used to kind of “riff” on scientific theories in an artistic way, using them as inspiration without necessarily understanding them in an analytical way. He also included a formula describing the swallowtail in his 1983 painting linked here entitled ‘El rapte topològic d’Europa. Homenatge a René Thom’. The last few years of his life are controversial because it’s alleged that he was made to sign canvases by his carers which would later be used to paint forgeries, and the above painting may not be his because his hands were said to be too shaky for him to draw such a line, which brings Britney Spears to my mind. After completing this painting, if he did, Dalí tried to enter a state of suspended animation through fasting and died five years later, soon after giving the visiting Juan Carlos a drawing entitled ‘The Head Of Europa’.

One way of looking at these graphs is to see the compartments as representing different stable states. Hence the six “cells” of the parabolic umbilic plus the seventh open space nearby are each conditions some systems can enter if there are four main factors determining their behaviour, which can in turn be described in terms of two factors. The same can be applied to the others.

I mentioned Dalí’s tremor making his creation of ‘The Swallowtail’ questionable, but in fact tremor and noise are not likely to disturb the behaviour of catastrophes. They’re quite stable in this respect, which calls into question the often-quoted explanation as to why they’re now so seldom modelled in this way being that not many systems can be adequately described with so few variables. This property is accompanied by what are called “attractors”, which CT has in common with Chaos Theory. An attractor is a set of states a system tends to drift towards, or in this case jump towards. Each one of the cells I mentioned just now is an attractor. After having got there, the system will tend to continue to be at least somewhat like that. It occurs to me in fact that limerence could be modelled in this way. It’s easy to get fixated on someone but it can be a lot harder to get over them. That, then, would be literally an attractor: a person one finds attractive. This suggests it would be fruitful to work out which control variables are involved, since in certain crucial circumstances, people do end up suffering from long-term limerence. However, discussing it and other psychological models in this way raises the question of positivism, which can be criticised on the grounds of reductivism.

You may or may not have heard of Gartree Prison, which was well-known for its helicopter escape in 1987. I have two personal connections to Gartree. One is that it ended up housing the bloke who abducted me in 1989 and the other is that one of my tutors on the herbalism course was married to a Gartree prison guard. Rather startlingly, Gartree prison disturbances were modelled using CT, more specifically the cusp catastrophe. This makes for a significant case study of the application of CT to social phenomena. When this was done, CT was riding on a wave of popularity triggered by the ‘Horizon’ broadcast and was possibly quite immature in its development, although as a modelling method it dates back to Edwardian times, the modelling having been published in 1976. The control variables seem to have been tension and alienation, which were assessed quantitatively, an approach which seems quite vague. They were based on governor applications, inmates requesting segregation, staff absenteeism, welfare visits and inmates in the punishment cells, and the shape of the graph seems to have been derived using a method which, it’s said, could have been made to fit almost any data set. There may have been an issue in the dominant connotations of the word “catastrophe” here, because it tends to be interpreted as negative and would perhaps consequently tend to lead to applications of the theory to model negatively-perceived events such as prison riots. It might also have been used by the prison service to make its operations and management appear more scientific than it actually was. And in any case, scientific management is widely regarded as a bad thing, at least for workers, as it’s seen as leading to redundancy, monotonous work, exploitation of workers, and from the management side expensive to implement, time-consuming and leading to a deterioration in quality. This could have implications for the situation inside prisons, as they are also workplaces for the staff and sometimes also for prisoners, so simply making the measures required might impair the function of the institution.

This could be applied more widely to other institutions such as mental hospitals and schools. For instance, if it successfully predicted grades in a school and also ways of manipulating variables in order to get those grades onto a higher tier of the graph, it wouldn’t necessarily improve less quantifiable measures of school performance. Likewise, a similar approach might lead to higher “cure” rates in a mental hospital, but that would only be in terms of particular paradigms of “abnormal” behaviour. Could it be applied to increase the quality of poetry? Maybe it could. Maybe J Evans-Pritchard would be able to measure the greatness of the poetry output by all these “cured” psychotics and high-achieving school-leavers with his scale. Or, maybe we just like to imagine that we aren’t reducible in such a way to a few variables and graphs, but maybe we’re wrong about that.

The modelling here, and in the other two as far as I know fictional examples I gave (the mental health one is less fictional than one might think), is applied to systems which depend on many assumptions about how society should be. For instance, assuming the prison study was valid, it might still fail to show anything because prisons of that kind are constrained by social factors always to be on one side of the cusp, and whereas manipulating the variables beyond that range is theoretically possible, doing so would not be possible given factors like level of public funding, policy regarding responses to crime and the nature of the buildings used. Then again, maybe we do want an entirely evidence-based set of policies. I would personally prefer that. It’s called socialism.

In a realm entirely outside the question of social policy, meditation, states of consciousness or mental illness, catastrophe graphs turn up in another rather surprising place: caustics. Caustics are projections of light rays reflected or refracted by a reflective or transparent medium onto a surface. I mentioned previously that a model of a cusp catastrophe could be made of mirror-like reflective material and be illuminated, and such a situation could lead to the projection of a caustic onto a flat screen. Caustics are the kind of light pattern you see when you look down into a clean, empty mug into which sunlight is shining, and they alter their shape and size according to the angle of incidence. They can also be seen in the dappling effect on a sandy seabed of waves on a sunny day. They can also have a kind of three-dimensional appearance, and in the teacup case they seem to look rather like a swallowtail bifurcation set, but in three dimensions in each case. Moving the cup leads to a different section of the graph. Caustics are odd because they’re always sharp and it isn’t clear what’s so special about the area they illuminate as opposed to its surroundings. They’ve also historically been problematic in computer graphics because depicting them accurately is computationally intensive, so in CGI they tend to be more decorative than realistic. It would be interesting to know whether catastrophe theory could simplify or has ever been used to generate caustics in computer images. Moreover, it would also be interesting to know if images of three-dimensional slices of higher-dimensional CT graphs could be accurately generated using three-dimensional reflective surfaces to generate their caustics.

A major question remains. Why don’t we hear so much about CT nowadays when it was so popular forty-odd years ago? An answer might be found in an illustration from herbalism, and at this point I shall intrepidly venture onto the territory of one of my other blogs. It’s been noted that herbal prescriptions with an odd number of remedies tend to be more successful than those with an even number. This needs to be restrained in various ways. For instance, it doesn’t mean that an even-numbered ℞ can be made more effective by omitting one of the herbs or adding one which is not relevant to the patient’s needs. I hypothesised that the reason for this was that an odd-numbered prescription could be modelled in terms of relative doses using catastrophe theory, whereas an even-numbered ℞ couldn’t. However, there are a number of problems with this which can be extended to other situations. The herbs here are presumed to be the control dimensions of the graph. A fold catastrophe has one control dimension, a cusp two, a swallowtail three, a butterfly four, a hyperbolic umbilic three, parabolic four and elliptic three, so the number of remedies would seem to have to be three or one if this is to hold true. In fact ℞s tend to have five or seven remedies, if one is in the low number of remedies in high doses as am I, because I feel the high number of remedies in low doses is beginning to look like homeopathy. Hence it can’t be applied to most herbal prescriptions other than simples, and there would have to be something which makes the fold, swallowtail and hyperbolic and elliptic umbilics distinctive in terms of their efficacy, which may be true but I’m not sure about that. But there’s a bigger problem which applies more widely. Herbs are not single remedies. They generally include a large number of different compounds with various effects on each other and physiology. Thus it seems implausible to apply catastrophe theory to herbalism, and this can be broadened out into biology more generally, since in most biological situations the number of control dimensions would be too high for CT to be relevant.

CT is still applicable to engineering and physics, but its intended target, the inexact sciences such as sociology, psychology and ecology, is rather more slippery. It does still happen, for instance in modelling the population dynamics of aphids via the butterfly catastrophe (it would have to be named after an insect – presumably the swallowtail is useful for modelling bird migration), but there really do seem to be too many variables and the smoothing effect initially claimed doesn’t seem to hold. That said, the formulæ used to generate the graphs are quite simple, and this could lend them to use in computer games, both in generating caustics on the graphics side and the likes of political and social interactions in games like Sim City.