I don’t know how long this is going to be because it’s a bit of a thought dump, which actually everything on here is supposed to be, so this is actually more in the spirit of the intention of this blog than usual, to some extent. Then again, most of what I put on here is a bit like that – unstructured stream of consciousness stuff with a small readership.
But someone has said something which has made me think, and I was already thinking about this. They were concerned that they might be a figment of the imagination.
Let’s start from the beginning.
Everything used to be in the same place. Then something happened and things started being in different places. Supposèdly, anyway. This was the Big Bang. I’ve just come out of the other side of a phase of not believing in the Big Bang but now I think it probably did happen because it was preceded by a infinitely long period of time going backwards. The reason I didn’t believe in the Big Bang was that given that time is eternal and the conditions of the Big Bang can arise spontaneously by quantum events at any time, very improbably, the chances of being within measurable distance of the beginning of the Universe are zero. This is, incidentally, not the same as impossible. Infinitely improbable events happen all the time. If someone were to flip a coin forever the sequence of heads and tails they produce would have a probability of zero but it would have to be a particular sequence, and the Universe, in some places, does kind of consist of a figurative coin being flipped forever in the sense that there are random quantum fluctuations. However, there are infinitely more of these random fluctuations than the actual event they mimic, so you can be certain that we are not near the beginning of the Universe. However, the idea that we are in a habitable period of the Universe’s history is entirely different. In this case we’re not near the beginning, just near an event following the collapse (backwards expansion actually) of the Universe followed by a Big Bang, which probably happens a lot. This is fine because it eliminates the possibility of us being infinitely special. We’re just living in the period where it’s possible for us to exist instead, and so now I kind of believe in the Big Bang again.
Yesterday I mentioned cosmic strings. Since I’ve only just said what they are, or what I think they are, I’m going to go into that again just briefly this time. Cosmic strings are basically wrinkles or cracks in space which didn’t collapse down to the three dimensions we’re familiar with nowadays but stayed more like the early Universe, and as such are either very massive or very “anti-massive” and tend to warp space. This depends on string “theory” because of the extra dimensions, or rather it’s related to it. This made me wonder if it’s actually more than this. I should point out here that this is now just me, a philosopher and herbalist, thinking and not a physicist.
Up until I was thirteen, I used to believe that fermions were tiny vortices in space time, and that bosons, including light, were gravitational waves. Just to explain that briefly, bosons carry force and fermions are stuff, such as protons, neutrons and electrons. This happens to be similar to a nineteenth century theory of atoms that they were vortices in the æther, which was the medium believed to carry light. At this time, I seemed to have been assuming that space was a thing rather than a relationship, because presumably if something could swirl about it had to be something.
I was at some point disabused of this model by my physics teacher. Incidentally, to get a bit home-eddy (geddit?), this was the same physics teacher who inadvertently killed my interest in physics after an optics lesson about refractive indices, which by that point I’d known about for more than half my life, when I asked him why anything was transparent and he said “you don’t need to know that at O-level”, the problem being that he was of course constrained by time and resources to teach O-level physics and was therefore unable to help me pursue my highly motivated curiosity. Nowadays I still have no idea why some matter is transparent and some opaque. It would make sense to me if everything was opaque or everything transparent, and I can to some extent understand translucency, but I don’t understand why there’s a difference. Also, I only understand structural colour. I get that some atoms absorb or emit certain wavelengths of electromagnetic radiation when their electrons change energy level, but beyond that I get stuck. This has been going on for more than three dozen years now and it really bugs me sometimes.
It’s important to be aware of the patterns one’s mind tends to fall into because of how it’s predisposed to function. For instance, as a child I used to have a habit of trying to imagine two-dimensional phenomena in three or more dimensions. This initially gave me a frisson of intellectual superiority but after a short while I got worried that I was falling into a rigid mental tendency in this area. This, of course, was the mind of a nine year old child, so perhaps it reflects a developing brain rather than some ingrained issue. Nonetheless, it’s helpful to be suspicious of one’s own thought processes and try to step outside them sometimes, although literally speaking that’s impossible as one is still having thoughts about having thoughts. But it’s about self-awareness, and that’s definitely important. I bring this up because what I’m about to say seems quite like my childish vortex notion of matter.
Cosmic strings are supposed to be pinched-up wrinkles in space. They form into loops when they intersect with themselves, so presumably by now there are loads of looped strings around the Universe formed from former strings, and for all I know they also merge and become bigger strings or loops once again. But also, I couldn’t help thinking that these enormous strings, some gigaparsecs long apparently, really sound like superstrings, and that maybe at some other juncture in the early Universe after the Big Bang space was just really scruffed up and wrinkly, and this led to the formation of elementary particles, perhaps as different knots or maybe vibrating at different rates, which were nothing other than really tiny cosmic strings or loops. I would like to believe that this is what string theory is, but to be honest I have no idea at all other than the stuff in my own head, so I dunno, maybe. Little loops of vibrating multidimensional space?
I am purposely avoiding doing any research for this post. I’m as interested in the process giving rise to my thoughts, and therefore those of other people, as I am in the ideas themselves here, and I don’t want to set off the “someone else’s idea” alarm, so I’m not reading up on this right now.
What is the nature of space-time? Space stops everything from being in the same place and time stops everything from happening at once. There are other ways of measuring things based on space and time, such as temperature and weight. The absolute temperature scale begins at the lowest possible temperature, and it works like this. It was discovered that at the freezing point of water a gas had a particular volume, which was 1/273 smaller at -1°C, 2/273 smaller at -2 and so on, so the question arose, what happens at -273? It’s actually -273.15, but that would involve more accurate measurement than was possible at the time. The answer is quite simple. It always takes the same energy to reduce temperature by a certain proportion, or alternatively, the same energy is always lost when temperature goes down by the same proportion, so reducing temperature from 100°C to about -86°C, which is halving the temperature from 373 to 186.5 above absolute zero, takes as much energy as reducing it from -273.14°C to -273.145°C, so absolute zero can never be reached. Hence there are no negative absolute temperatures. Not every quantity measured can be negative. Mass might be an example of this. Nonetheless it makes sense to think of temperatures as located along a line, and the differences between them as measured along this line. After all, that’s what a thermometer is. Likewise with weight and a spring balance, the weight of an object is measured along a line. We abstract these quantities in terms of a dimension.
(Honest units on the left, pretend ones on the right)
(Honest units on neither side but left slightly more sensible than right)
Now a ruler could be the same kind of device, as could a protractor. They measure something, but that doesn’t make that thing any more of a “thing” than temperature, weight or pressure. Length, width and breadth are quantities along with direction, and from those we abstract the idea that there is a “thing” called space, and likewise from clocks, stopwatches and calendars we abstract the idea of the quantity we measure with those into a “thing” called time, and of course the two are related, and they do exist, but maybe they’re not things.
Why would it be a good idea not to think of space-time as a thing?
There’s a question which seems to betray a misunderstanding of the nature of space and time yet is constantly asked: what is the Universe expanding into? The reason this question gets asked is probably due to the idea of the expanding Universe being illustrated as an inflating balloon, which makes it sound like there is a larger, higher dimensional hyperspace into which the Universe is expanding. I’ve long maintained that this isn’t so. In fact the idea of the expanding Universe is that the maximum possible distance between two points is always increasing, and that beyond a certain distance the direction of an object reverses. This is true on a spherical surface if you think of it as flat, but for different reasons, so for example the maximum possible distance between two locations on Earth is (almost by definition) 20 000 km, and if you are on the equator and someone else near your antipodes is walking West, they will be East of you once they pass it. In the case of the Universe, however, it’s a property of space implied by geometry not being Euclidean and space is not a “thing” in the same way as Earth’s surface is. Consequently, although the Universe seems to have the topology of a hypersphere, there is no geometrical real hypersphere corresponding to that topology.
Or so I thought. I should point out that not everyone thinks this way. There is a thing called “‘brane theory”, where “‘brane” is short for “membrane”. According to this theory, there really is a hyperspace in which this universe and many others are expanding, and when they touch and cross each other new universes are made. Incidentally you should check that – I’m not looking anything up for the purposes of this exercise, but I think that’s what brane theory is. It’s also an amusingly similar word to “brain”, which is an intrinsically funny word. If brane theory is correct, that really is what the Universe is like and space is actually a thing.
Now to get back to superstring theory, which I admit I may have got completely wrong. If particles are superstrings, and superstrings are topological defects in the same way as cosmic strings are, then all matter is, is swirly bits of space, and the problem with that is that if space and time are not really “things”, there’s nothing to swirl and nothing to be anywhere or happen at a particular time, and there just is no time or space. So, huh? Does this mean that space-time is actually a thing, or just that I have misunderstood superstrings? Or, does it mean string theory is flawed? Probably not the last one.
Time and faster than light travel have for a long time been thought impossible. Before Einstein, nobody realised there was a cosmic speed limit so the issue of travelling at any speed would’ve been considered merely a problem of giving something enough energy to force it to do so. This was ultimately proven wrong due to a chain of reasoning beginning with the observation that light travels at the same speed in a vacuum regardless of how fast an observer is moving. As for time travel, this has existed as a literary trope for centuries, in the form of visions and dreams of the future or sleeping for a long time. Even the Bible has time travel in a sense, because it has prophecy and the resurrection. I’m personally inclined to regard dreams as not anchored to our own perception of the passage of time and am aware that they are sometimes precognitive. That is, I don’t just speculate that they might be: I assert that people have dreams which predict the future. I don’t know exactly how that works but a true sceptic will accept an incontrovertible fact and look for an explanation. K-skeptics will often deny facts if they don’t fit theories.
It took a long time for literature to get round to imagining time travel into the past rather than the future. H G Wells had his time traveller go into the future and report back, but although he is speculated to have gone into the past and disappeared permanently from the nineteenth Christian century, and the narrator speculates thus:
It may be that he swept back into the past, and fell among the blood-drinking, hairy savages of the Age of Unpolished Stone; into the abysses of the Cretaceous Sea; or among the grotesque saurians, the huge reptilian brutes of the Jurassic times. He may even now—if I may use the phrase—be wandering on some plesiosaurus-haunted Oolitic coral reef, or beside the lonely saline lakes of the Triassic Age.
H G Wells, ‘The Time Machine’, 1895
The very obvious big problem with upstream time travel is that it appears to cause paradoxes, that is, one can kill one’s own ancestor. There is a related paradox that one can take an item from the present day and leave it in the past, so that it becomes token-identical with it and has no origin, but these are really the same problem. However, there is a startling oddity regarding upstream time travel and physics which is not present with faster than light travel: nothing seems to rule it out in principle. There are practical difficulties in building time machines but they don’t appear to rely on problems related to time travel itself. It’s as if the problem with travelling faster than light were to do with sufficiently streamlining a spacecraft because space was filled with a tenuous gas rather than it being a fundamental issue with the nature of reality, but at first glance the idea of travelling faster than light seems less problematic than going backwards in time.
There may also be a close connection between the two problems. I’ve also failed to state the exact issue with travelling faster than light, because in fact there is nothing stopping an object from moving at the speed of light or even faster than it provided certain properties of an unusual nature are physically possible. What is impossible is for any object currently moving slower than light to reach the speed of light, any object currently moving faster than light to decelerate to the speed of light and any object currently moving at the speed of light to accelerate or decelerate. This is not the same thing as it being impossible to move faster than light. There are also a few anomalies that suggest superluminal travel, such as the fact that when a particle moves through barriers their location is “blurred” such that the time taken to travel the distance has a low but not zero probability of being ahead of where it would be if it had moved at the speed of light from its previous location, and there are jets emitted from galaxies which seem to move faster than light, although that’s an optical illusion caused by foreshortening, because the speed of light is finite and a fast jet approaching us will be visible earlier than expected due to the shorter distance travelled by the light leaving it.
Before I get going on the other bit, I want to make an observation which I’m sure can be explained in accordance with relativity but whose explanation I’m unaware of. As an object accelerates, it becomes foreshortened in the direction of movement and increases in mass. I would expect a sufficiently foreshortened and massive object to be smaller than the size required to make it a black hole, which would then warp space. If this happens, what stops objects near the speed of light from opening wormholes in space and slipping through them faster than light? I can’t have been the first person to have thought of this so I presume there’s an answer. I just don’t know what it is.
Geometry as it actually is, as opposed to Euclidean geometry, which maintains falsely that parallel lines meet at infinity rather than converging or diverging as they really do, is substantially the study of what follows from distances and angles between items. Movement is not the same thing as an increase in distance. This crucial point is what allows the Universe to expand at a rate which over great distances is greater than the speed of light. No actual matter within the Universe needs to move faster than light. I’ll try to illustrate what I mean. If two rocks are located just outside the event horizon of a black hole and it moves away from them, the distance between them will change but neither of them will have moved, because the space warp created by the black hole will lessen.
This is the principle on which the Alcubierre Warp Drive is based, and at this point it’s fair to point out that a warp drive could also be used to travel, or rather modify one’s location, slower than light. It works by changing the geometry of space around the object to be moved. Clearly objects will tend to fall towards massive bodies in their vicinity, which is because they warp space in front of them, but this doesn’t help them get places unless those places are somewhere between the object and the massive body. However, this also contracts space. If space could also be expanded behind the object, relocation is possible over a period less than that required for light to travel between the initial and final locations of the object. The only trouble is, this requires negative mass. If positive mass, such as a black hole, reduces the space around it, negative mass should increase it. I’m personally suspicious of this idea for all sorts of reasons. If this kind of warp drive is possible, it also makes gravity control, antigravity, tractor beams and practically limitless energy possible, and this just sounds too good to be true. It sounds like the kind of thing which ought to be ruled out by the laws of physics because it would solve so many problems. It means we would be able to effectively travel faster than light, have antigravity, spacecraft with their own vertical gravity fields and we’d never need to worry about generating electricity again. I realise this is not a scientific objection, but so much hangs on it, it just feels wrong. The catch is that nobody knows if negative mass is a thing. Also, relocating something faster than light is stepping outside the light cone and this influences the order in which things happen. This means that a simultaneous event can become earlier and be interfered with even though it’s known by observation what its consequences are already. This is still an issue even with the Alcubierre warp drive: it kind of turns a spacecraft into a time machine. It’s not a good thing, incidentally. It suggests there’s a reason it wouldn’t work.
I’ll turn now to the related subject of cosmic strings. A few comments need to be made here about the relationship between these and the strings of string theory. I’ll talk about string theory first, also known as “superstring theory”.
According to string theory, the fundamental component of the Universe is loops of string which vibrate in different ways. The differences in vibration manifest as particles with different properties. These loops operate in a ten-dimensional space, or possibly eleven, but six of those dimensions are a maximum of 10-33 centimetres in size. They are, like the three dimensions of the space we’re familiar with, curved back on themselves. One of the main points of string theory is to provide a grand unified theory which accounts for both the standard model (all that quantum stuff and particles) and gravity, and it does do that, but one drawback is that it seems untestable. It’s also been criticised for predicting the existence of 10500 universes, each with their own laws of physics, and fails to explain why we’re in this one. I’m no expert, but I would’ve thought that the answer is that the others are uninhabitable and that life and intelligence can’t arise in them, or that there are rather few of them. Objecting to it on those grounds is a bit like objecting to the idea of outer space because we live on a particular planet. The theory is also far from elegant, but that objection is kind of æsthetic. Other physicists claim that we are too attached to elegance and that there’s no reason why the Universe should be like that. However, more seriously no version of string theory explains the expansion of the Universe, and it doesn’t make useful predictions about the nature of physical reality.
The mathematics of string theory, however, can be applied elsewhere, including to other kinds of “string”, and the question arises of whether there is a direct connection between superstrings and cosmic strings or merely a mathematical one, and of course the main focus of this post is the latter type of string. I can see a similar incident in the early Universe causing both superstrings and strings, but I lack the scientific “knowledge” (which it isn’t because it’s empirical science, hypothetically) to know if I’m saying something sensible about it. Cosmic strings are basically topological entities, and are one of four types of object which emerged just after the Big Bang, and I’m going to present my idea here for what it’s worth. What if the same kind of process led to the formation of extremely small cosmic strings before the large ones emerged? Maybe not, I don’t know.
There are, as I said, supposed to be four types of entity known as topological defects in space. These defects can be studied to some extent because they also occur in other situations, such as in liquid crystals, which makes me wonder if there are some in front of me right now, and microvortices in helium II and other superfluids as mentioned here. Strings are thought to have appeared 10-35 seconds after the Big Bang and are “defects” in space, which appeared because the Big Bang was a phase change analogous to a liquid freezing in the sense that the primordial chaos became an ordered Universe, and just as cracks appear in ice as it freezes from the apparently homogenous water, so do topological defects appear in space. They’re similar to whorls in hair and partings as well, in the sense that there’s a polarity to each point in space which may be oriented in different directions in different regions.
I’m going to have to confess to not being confident that I’ve got the following right.
Space, according to string theory, has ten or eleven dimensions. At the point of the Big Bang, all of these dimensions were of zero size, so in other words they didn’t exist and presumably immediately after, or rather as soon as size had any meaning, they were equal in size but also non-Euclidean, curled up on themselves. At 10-35 seconds, over most of space the extra dimensions collapsed, but not the the same extent in certain regions, as far as anyone can tell at random, and the regions where space was “normal” began to expand until they almost came into contact with each other. Where they did this, they stayed very slightly separated, by a distance smaller than the size of an atomic nucleus. These took different forms. They may be almost points, corresponding somewhat to particles, lines, which are the cosmic strings, domain walls, which are two-dimensional and textures, which are regions of variation liable to collapse. Presumably textures no longer exist because they were formed 13.8 æons ago and being unstable can surely not have survived. Incidentally, the initial topological defects are now much larger and may, for example, stretch across the entire observable Universe because it’s expanded.
There are a few things I don’t understand at all here. In particular, I think I don’t understand why a point defect would be a magnetic monopole. If all this is about is magnetism, like the magnetic field of the Universe as it were, I can see that there could be points from which everywhere is north or south in the magnetic sense, and an analogy could even be made with the poles of a planet as positions whence everywhere is south or north. However, this seems to be about more than mere magnetism, which leads me to contemplate whether there are other kinds of “monopole”, which we would be familiar with involving other forces such as quantum black holes, which are tiny black holes theorised to have existed since the early Universe. But I honestly don’t get this bit.
It’s also important to note that although these things are in a sense one- or two-dimensional, this is not the same as them being literally perfectly straight or flat. Rather, they’re crooked lines, able to swirl around, and irregular bumpy surfaces. Where a cosmic string intersects with itself, it pinches off a loop because it’s able to penetrate itself, leaving the rest of the string to “heal” and continue.
A circle drawn round a cross-section of a cosmic string would not have 360°. This is, I think, because the space in the vicinity to one of these objects is far from Euclidean, in turn because the multiple dimensions of the Universe have been retained at a larger size than in most of space. Hence it’s a spatial anomaly – a small piece of hyperspace. However, just as there can be north and south magnetic monopoles, there can be cosmic strings with more than 360° circumference and others with less than 360°. Because gravity is the warping of space-time, this means two things, depending on the type of string (and I presume this also applies to domain walls). One type is extremely dense – one quote is that an inch of cosmic string, with the width of a proton, is as massive as Mount Everest, which is around 160 gigatonnes an inch or 50 gigatonnes a centimetre. They’re also under a lot of tension and vibrate, meaning that they’re going to give off gravitational waves if they exist, and these can be detected now. Thus it’s possible to look for a confirming instance in gravitational waves, which are disturbances in the curvature of space-time. Another way to find them would be to look for a line of stars with twin images, where the light has been refracted either side of the string.
As is probably clear, I don’t know what these things are in detail but the relevant aspect is that the ones whose circumference is less than that of a circle would have negative mass. Now imagine a situation where a cosmic string, or possibly a domain wall, with negative mass, is near either a black hole or a topological defect with positive mass. This is a “warp field”, or rather the space between the two is. It’s tipped ana (four-dimensional direction number 1) behind and kata (four-dimensional direction number 2) in front. Therefore items small enough to be completely covered on both sides would be able to “move” faster than light. This could be taken to mean that cosmic strings can’t exist. There’s also a problem with movement, as it seems it would be very difficult to move a cosmic string with positive mass at all, let alone near the speed of light. On the other hand, it might be moving of its own accord, and this makes me wonder whether the mass that a topological defect has is the same as that of ordinary matter, because rather than being matter it just is a warp in space, which we already know can move faster than light. I don’t know the answer to this.
Then there’s time travel, and this is not my idea. Apparently there are two ways to travel in time using cosmic string. Firstly, because they’re so dense, time slows down near them in the same way as it does near a black hole, meaning that just staying close to one would be tantamount to travelling faster downstream in time than one is anyway, although the chances of much physical matter of the familiar variety, such as the stuff our bodies are made of, seem pretty slim. That doesn’t stop a signal from travelling though. Secondly, in a mechanism I don’t understand and with an enormous amount of energy, a spaceship near two crossing strings could travel into the past, and again the spaceship can be replaced by a signal. Information from the future is just as likely to cause paradoxes as matter, so this is a problem. Also, teleportation is often thought of as using signals, so if teleportation through matter transmission is possible, physical objects would be able to travel in time, or at least be cloned. As usual with time machines, you wouldn’t be able to travel back before the formation of the machine, but it’s possible that these could form by chance, and since they date from the early Universe, that’s not really a problem.
And no, I do not know how to overcome the paradoxes this would apparently cause. I merely present it as I received it.
I don’t know if any of you blog using WordPress, but one of the things you get after a while of using tags (I only started doing that fairly recently) is a list of the ones you use most often. Probably because of the decimal bias of our cultural hegemony, it lists the ten. In my case, this is probably not a good guide to getting more readers but then I’m not particularly interested in doing that, except maybe as a kind of game in which I hope I wouldn’t become emotionally invested. It makes me want to draw a diagram, or rather a pattern:
Apparently this is called a “complete graph” and is described as a simple undirected graph in which each pair of distinct vertices is connected by a distinct edge. The above image shows a K12 , apparently. Because of the decimal bias, my ten tags can be linked up in a similar diagram with rather fewer edges. I used to have hours of “fun” getting computers to draw ever more complicated complete graphs. The distinction also ought to be made between undirected and directed complete graphs of this kind.
There is bound to be an equation which tells you how many edges are needed for a given number of vertices, and in fact there is. It’s:
wn+2=n!en
. . . where “e” is Euler’s constant. No, hang on a minute, that isn’t it apparently as it isn’t necessarily an integer and these obviously will be, so it’s:
(n(n-1))/2
Okay, so plugging in my ten tags gets me (10(10-1))/2, which is forty-five. So much for my title then! I’d worked it out at a hundred and ten but it seems it’s smaller. So then: ninety blog posts.
Here’s what I’m thinking. I have ten tags listed. A fairly crude way of generating blog post ideas would be to combine pairs of them, perhaps in both directions. They are: Philosophy, Ethics, Christianity, Judaism, Veganism, Racism, Evolution, History, Star Trek, Politics. Most of the time, if I blog on one subject on that list it’s likely to involve more than one of the others, which adds to the number of possible combinations in the graph, but it would also be interesting to see what I’ve missed, using those as major foci for a post. For instance, veganism and racism is something I’ve written about before, but not in a “pure”, more focussed sense, and there’s also racism and veganism, which could be something quite different. In pursuit of that combination, there is a lot to be said. For instance, veganism is perceived as a very White project even though, for example, I-Tal diet in its most complete form is RastafarIan and there’s also the question of the growth of supposèdly vegan products in the Third World as cash crops for export and forcing up the prices of something like quinoa, putting it out of reach of the communities which have traditionally eaten it. All very fruitful subjects. There are apparently forty-five pairs of tags in one direction and another forty-five in reverse. Judaism and Christianity is another interesting subject which it would be very easy to write something about, but writing something original and respectful might be a lot harder.
Thinking about writing in this way links mathematics and composition, but as a fairly naïve mathematician I may not be the person to do that. I often find that when I try to connect mathematical activity to something usually considered non-mathematically, I come up with a lot of mind game-type ideas but not much which is particularly applicable, or sometimes something which fits quite well into a particular mathematical activity but is also amenable to common sense. The question in my mind right now is, how useful is it to think of pairs of blog tags as a complete digraph? Is “evolution and Star Trek” a different topic to “Star Trek and evolution”?
Incidentally, the reason “Star Trek” crops up in that list is that I’ve reviewed every episode of “Star Trek TOS” and written several other more general posts on the series. It’s the kind of thing you might expect to generate a lot of views, or maybe not because so many people must be writing about it. I feel, unfortunately, that although it’s a major cultural phenomenon it’s also quite naff to write too much about it.
The above graph apparently also forms the net of an eleven-dimensional simplex, because every complete graph is a projection (the way it’s represented here, in two dimensions) of a simplex of Kn-1 dimensions. Hence this image:
is the net of a tetrahedron. And it clearly is: you can see the faces at the front and back, paired off and seemingly at right angles to each other. Each vertex connects to each other by three edges, and that gives the essence of the simplex in a way. My K10 graph would presumably have each vertex joined to the other nine, each edge forming a polygon enclosing a face, each such polygon enclosing a tetrahedral cell, each tetrahedral cell forming the solid limiting a four-dimensional simplex, and so on. Each one of these encloses a possible combination of tags, more than one this time, and we’re in the realm of factorials and the possibility of more than three and a half million possible blog posts which can be appropriately tagged in various ways from that list, and will be found in the depths, if that’s the right word (it isn’t). This, then, is the hyperspatial approach to blogging. Each tag is located at a precise location relative to the others in hyperspace and since the links between them need not be mere edges but triangles, each blog post can be considered to be written on one of the faces of this nine dimensional simplex, either tapering towards the bottom or getting longer and longer lines as it goes on. You can hold this cluster of blog posts in your nine-dimensional hand-things and turn it this way and that to read each one of the ninety posts, all of which are on the surface of the polytope. If you happen to be a nine-dimensional entity, that is. Some of these are probably already written but I don’t know what they are.
This suggests a way of viewing blog posts via a virtual tesseract, merely four-dimensional and with each face of each of the eight cubes having a post written on both sides, four dozen in all, manipulable via one’s viewing device while wearing 3-D glasses or a VR headset. But all of this is fanciful and it isn’t clear how it would help one blog.
Leaving all that aside, it’s also possible to use the same old AI as I’ve been using for a lot of other things to finish my list of tags with others. It’s quite interesting what happens when I do this, because it fills my list in with the subjects I deliberately avoid on this blog, such as gender identity and trans stuff. InferKit just now gave me this:
Harry Potter Animals Politics Military Religion Science Food Smart People Animals and Animals Writings David Icke Family Values Hot Car Deaths Holocaust Asian-American
“Animals and Animals” is a little like “Vulcan And Vulcan” even though it hasn’t seen it. I don’t really want to blog about Harry Potter, although “Hot Car Deaths” is a depressing but possible subject. “Asian-American” strikes me as something you really should be in order to write about it, except that it is interesting how America sometimes seems like the extreme Far East even beyond Asia, so that has possibilities. DeepAI gives me “Science, Education, Welfare, Vacation, Innocent and Damn Law,”, then it seems to turn into a government form of some kind with things like “Pregnancy”, “Birth Year” and the like. This is not very useful and probably reveals the kind of text it thinks I’m writing.
I’ve done all this before, of course.
This blog is naturally a meandering mess of brain dumps, and consequently these two methods vaguely reveal some topics I might want to write about but they’re unlikely to get much readership, and that’s fine. However, I would say this. I suspect that if you’re serious about blogging and already have a blog which has a direction, a focus and a significant readership, you could do worse than to use these techniques. Maybe you’ve written about every combination of tag pairs. Finding out which ones you have and haven’t and colouring in the edges on the resultant complete graph would probably reveal where the large gaps are in your coverage, although some might be nonsensical. I don’t think any of mine would be though, so I suspect yours wouldn’t be either. Just two tags is rather limited, and if you open it up to all combinations, unless you’ve automated the process in some way you just will not have written hundreds of thousands of blog posts, meaning that some of the combinations will be stimulating and novel. As far as predicting tags is concerned, I found it tended to fill in things that I was genuinely interested in but hadn’t blogged about. This would also seem useful. You could also take all the AI-completed tags and build your own complete graph from those. It seems to me that there are likely to be other applications of graph theory to blogging which I have yet to become aware of. Worth investigating maybe?
“Dimensionally transcendental” was initially a cool-sounding phrase mentioned by, I think, Susan Foreman in the first episode of ‘Doctor Who’. It meant “bigger on the inside”, and definitely sounds like technobabble. TARDIS stands, as we all know, for “Time And Relative Dimensions In Space”, but even in the Whoniverse this is probably a backronym because why would something from Gallifrey have an English initialism? I think most people who think about it would probably say that Susan came up with the abbreviation, which probably explains why it doesn’t make much sense.
The BBC, and also Terry Nation’s estate, are quite protective about their intellectual property with respect to ‘Doctor Who’, which has led to a couple of disputes over the use of the likeness of police boxes and the word “Tardis”. Therefore I’ve posted a picture of a Portaloo up there instead of a Tardis or police box. In 2013, the portable toilet hire company Tardis Environmental came into dispute with the BBC over the use of the word, which was registered as a trademark by the Corporation in 1976. The BBC claimed that the company might end up seeming to be endorsed by them, to which they responded, “we don’t roam the universe in little police boxes from the 1930s, we actually hire out portable toilets and remove waste.”. I think we can all be grateful to them for clearing that up. I suppose it does make sense that the taboo against human excrement is not a positive association for this word. There was also a dispute with the Met. In 2002, after six years, the BBC won a case against the Metropolitan Police who took them to court over their use of the police box in ‘Doctor Who’ merchandise because they claimed that since they were responsible for the original boxes, it rightly belonged to them. I think I’ve seen two or possibly three police boxes, in Glasgow, Bradgate Park and London, this last being the one I’m least confident about, and I don’t think any of them look very like the Tardis. The one in Bradgate Park I’ve seen on a regular basis, and looks like this:
This is a listed building and is apparently still in use. It doesn’t look like a Tardis to me really but it’s a nice shade of blue. It’s 9 646 metres from where I’m sitting right now. The one in Glasgow is rather further away. It was the Met against which the BBC won the case, but the Tardis props are clearly wooden, a different shade of blue and have different windows, at least compared to the one I’m familiar with, so it seems a bit unfair. To be honest I don’t understand why this dispute even happened. It was between two publicly-funded bodies, I think, and seems to be a bit of a waste of money and time. Even if it was BBC Worldwide or BBC Enterprises, the Met was still involved.
Anyway, this is not what I came here to talk about today, but the concept of dimensional transcendentality. I’ve previously mentioned the fact that extremely large spheres are appreciably larger on the inside than their Euclidean volume because space is non-Euclidean – parallel lines always meet, at a distance of many gigaparsecs. This is possible because Euclid’s Fifth Postulate is based on observation rather than axiomatic or deduction, and the observation turned out to be incorrect. A sphere whose radius is equivalent to that of the Universe’s has a volume of five thousand quintillion (long scale) cubic light years, but if it were to be considered a sphere in Euclidean space, its volume would be only four hundred and twenty quintillion cubic light years, a difference of a dozenfold. This is quite counter-intuitive and I’ve ended up checking the calculation about five times to ensure it’s correct, but it starts to indicate how very confounding to the human mind higher dimensions really are.
I want to consider three cases of curved shapes in hyperspace to illustrate what I mean. Well, actually one of them is rotary motion rather than a literal curved shape, and I’ll go into that first. Here’s a circle with a dot in the middle:
(I’m drawing all of these in a ZX Spectrum emulator because Chromebooks rule out the use of more sophisticated graphics programs as far as I know). The circle can be rotated around the dot, so in a sense that dot is the “axis” of rotation of that circle. Now consider this as a cross-section down the middle of a sphere:
This is an axis of symmetry and also of rotation. Spinning the sphere through which this is a cross-section would lead to it turning round this line, which would be the only stationary part of the sphere just as the point is the only stationary part of the circle. Geometrically speaking, these are infinitely thin and infinitely small, so it’s rather abstract, but in the real world the closer you get to the centre of a spinning circle or sphere, the less you’d move.
Now consider the hypersphere, i.e. a four-dimensional version of a sphere: that which is to a sphere as a sphere is to a circle. If that rotates, doesn’t that mean its “axis” is a circular portion of a plane bisecting it? Can we even imagine something rotating about a two-dimensional axis? Also, just as two-dimensional objects have lines or points of symmetry and three-dimensional ones lines or planes of symmetry, surely that means that four-dimensional ones can have solids of symmetry? A hypersphere could be divided into two hemihyperspheres along a central sphere touching its surface, and since it’s symmetrical in that way, just as points on or in a sphere describe circles when they spin, doesn’t that mean line segments on or in a hypersphere would describe spheres? I find this entirely unimaginable, but is that a failure of my three-dimensional imagination or a flaw in the idea of hyperspace. It’s probably the former but this brings up a surprising recent finding about the nature of the human brain, which is that small cliques of neurones form which are best modelled topologically in up to eleven dimensions. No, I don’t really understand that either.
This hints the nature of hyperspace is very counter-intuitive, which isn’t that surprising really. Another issue is that of the torus. This is a Clifford Torus:
Travelling across the surface of a torus, one would find oneself disappearing off the top or bottom of a map and appearing on the bottom or top of it, or doing the same at the right and left hand sides. This is not like a cylindrical map projection of a planet, where the poles are either at an infinite distance or one would traverse horizontally exactly half way across the map and appear 180° of longitude away vertically but do the same as on a torus horizontally. With a four-dimensional torus, one would be in an apparently three-dimensional warped space forming the analogue of its surface, which you might think of as a cube with linked opposite faces, but the faces could be linked in different ways. One of the dimensions could be like a spherical map, with the concomitant traversal near the faces, or two of them could be, so there seem to be at least two different four-dimensional toroidal analogues. I confess at this point that this may not be what the above two animations represent.
The third problem relates to what ‘Doctor Who’ calls dimensional transcendentality, and it’s this which I’ve only recently heard about, from Numberphile. To illustrate this, I’ll go back to the Spectrum:
These are supposed to be four circles fifty pixels in radius touching each other. Now the question arises of what the biggest circle fitting among those four would be. The answer is quite straightfoward because squares can be drawn around each circle whose diagonals touch at the centre of these four circles. If you think of each circle as having a radius of one, the diagonal of the containing square has a length of the square root of 22 +22, or roughly 2.8. The radius of the circles is one, so subtract that from 1.4, or half the length of that diagonal, and you have 0.6. In other words, the square root of two is involved.
If you then extend this into three dimensions and imagine eight spheres stacked together in a similar manner, there’s a bit more room. The hypotenuse of a right angled triangle from the centre of an outer sphere to the inner one’s is then the square root of the sum of the squares of the three sides, which is root three, so the radius of the inner sphere is just over 73% of the outers’. This makes sense intuitively, for the last time, because it’s easy to understand that the diagram above shows a cross-section of the equators of all the spheres and therefore the minimum space between them, so a larger sphere is possible than one with the same circumference as the central circle in two dimensions.
The radius of the hyperspheres at the centres of analogous arrangements in higher dimensions is always going to be one less than the square root of the number of dimensions involved. At four dimensions, the central hypersphere’s radius is one less than root four, also known as 2-1, which is one, so rather surprisingly perhaps, it’s possible to fit seventeen equally sized spheres into a hypercubic arrangement. At five dimensions, the central “sphere” is actually 23% larger in radius, as root 5 minus 1. This is actually nearly three times the size in terms of a five-dimensional “bulk”, if that’s the right word. At nine dimensions, even the radius is double that of the surrounding hyperspheres, which makes it five hundred and twelve times larger altogether. There’s no limit to the increase in radius at all. I find this highly counterintuitive.
Moreover, these sphere analogues don’t even occupy the whole space. What does is a peculiar pointed shape which starts off like a square with concave sides in two dimensions (whose bottom point I’ve accidentally cut off) and a kind of inwardly-curved octahedron in three. In three, it has to be greater than the area of the largest circle in six different directions. In four, it resembles a concave version of a cross polytope, which is the higher-dimensional counterpart to the octahedron. Cross polytopes always have twice the number of vertices as they have dimensions, whereas measure polytopes, also known as hypercubes, always have twice the number of faces as dimensions.
Now consider a nine-dimensional stack of hyperspheres intersecting with our three dimensional space at one of its equators, with the centres of the hyperspheres aligned at the vertices of a nine-dimensional measure polytope. This would appear to be a stack of eight spheres, so this can be simplified by cutting off the outer spheres and converting them to hemidemisemispheres, if that’s the word, stacked together. Similar slicing could occur in hyperspace. So, it’s converted to a cube, then you put a door in the middle of one of the faces of the cube and find that it opens into a space which is quite a bit larger than the volume of the cube. The trimmed cube is only an eighth of the volume of the original, but it contains a “sphere” which is four thousand and ninety six times larger. With a mere four dimensions this becomes a mere eight times the size. This is starting to sound very like dimensional transcendentality.
The term has two words in it. “Dimensional” is fairly straightforward if one sticks to a simple definition instead of the non-integral dimensions used with fractal geometry. “Transcendental” brings to mind transcendental meditation, which is probably one reason for using it along with the fact that it was also used to refer to a particular set of numbers. What, then, are transcendental numbers?
A transcendental number is defined as a number which is not the root of a non-zero polynomial of finite degree with rational coefficients. The numbers e and π are both transcendental. All such numbers are irrational, that is, they cannot be expressed as the ratio of two whole numbers, since all rational numbers can be expressed in the way transcendental numbers can’t. Π is sometimes approximated by such values as 22/7, but these are not accurate values of the constant in question. Since the value is in fact involved in calculations of these volumes and hypervolumes, there might be a way of including the word “transcendental” in the description of this property of being “larger on the inside”. The square root of two is involved in two dimensions, but that’s merely irrational and not transcendental because it can be expressed using algebra – it’s a square root. This also means that the method of calculating the volume of a central sphere within a stack of hyperspheres is not transcendental either, so a good bet for including the concept would be to use π instead.
Although I can see that π is useful in calculating the surface area of the shape between the spheres, I don’t know what this thing is called. There’s a gallery of similar shapes here but they don’t include this one. I find it hard to believe this thing neither has a name nor has been extensively studied. I can assert various things about it. Its volume is greater than the largest sphere it can contain. It’s also greater than six times the spheres which can be placed touching the equators of the spheres it can occur within. I don’t know if the central sphere overlaps with its neighbours in the points. Each of its eight curved surfaces has an area equal to ½(πr2), meaning that its total surface area is equal to a sphere whose diameter is equal to the length of its largest diameters. Similar criteria apply to its higher dimensional friends. Hence I could perhaps be allowed to say that it’s dimensionally transcendental because its volume or hypervolume, or the volume of its hypersurface in higher-dimensional space can be calculated using the transcendental number π. And it can be, as I will now show.
Up until now I’ve been describing the central spheres and hyperspheres as if they’re three dimensional, and it is possible to lodge three dimensional spheres in there if you want, although it would be rather a waste of space. However, the actual volume of a four-dimensional space is not its bulk but its surface. I’m going to consider this nameless shape as having a length of two units, which is the same as the cube it’s found inside. The surface area of a sphere is 4πr2 and the circumference of a circle is 2πr. If it just carries on like this, it makes the volume of the hypersurface of this shape in four dimensions 8πr3 (spot today’s deliberate mistake with the volume of a sphere half the size of the Universe, incidentally). This means the volume of this shape is a bit more than twenty-five cubic metres, which is equivalent to that of a cube 2.9 metres on a side. For a nine-dimensional version, this would be over eight hundred cubic metres, which is a nine-metre cube. That’s about the size of a three-story house.
The TARDIS is of course bigger than that, although as far as I can remember Nu-Who has never shown its real internal size. If the door was located at a point where it was at the end of one of the projections and located in three-dimensional space, it would be accessible to a three-dimensional being. In fact it could have up to six such doors, though if it had there’d be one in the roof and another underneath it, and there could also be two other doors opening into four-dimensional space. If, however, it had nine dimensions, it could have a total of eighteen doors, only a third of which would be accessible from normal space and the majority of which wouldn’t even open into four-dimensional hyperspace.
I think it makes more sense for the police box to be closer to a cube than just a cuboid, for the sake of neatness, so maybe the chamæleon circuit should’ve got stuck on the Bradgate Park police box after all, with two secret trap doors and two hypersecret doors for which there is no name because they’re ana and kata 3-space.
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:
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.
Yahoo Answers is, as I mentioned previously, about to die, although it’s a death by a thousand cuts. In the past I’ve used this blog to put more thoroughly thought-out answers to frequently-asked questions on the site, so I’ve probably addressed this before, but right now I have a different and perhaps less dogmatic take on this question than I usually adopt. Before I go on, I should probably insert the standard diagram people put in nowadays when talking about the Big Bang:
Strictly speaking, this diagram is inaccurate because it shows a two-dimensional projection of a three-dimensional model of a four-dimensional set of circumstances. Take the barred spiral galaxy at top right. If the X-axis is supposed to be time, we should be concluding that the left hand arm of that galaxy happens first, then the end of the right hand arm and the nucleus, and finally the middle of the right hand arm. Also, space is two-dimensional in this picture when for most practical large-scale purposes it really has three dimensions. In other words, this isn’t so much a diagram as an illustration intended to communicate the history of the Universe since the Big Bang. You can’t take it too seriously. It has an artistic, creative aspect.
One possibly inaccurate, because it isn’t really intended to be that accurate, feature of this diagram is the way it shows space. It’s a black rectangle into which the Universe is expanding. There is an outside to this Universe, and at that point you’d be forgiven for asking, if the Universe is everything, what’s the blackness outside it supposed to be? Why is that not also the Universe? The Jains, of all people, had an answer to this. They believed that the Universe as we know it was suffused with a substance which made movement possible, but was surrounded by infinite space from which this was absent. Nowadays, maybe we could do something similar with the idea of dark energy, the apparent force which causes the Universe’s expansion to accelerate. The above picture has a literal “bell end”. It flares out rather than widening steadily or perhaps slowing down from left to right. This is the influence of dark energy, as it represents accelerating expansion. I suppose it’s possible to think of the Universe as infinite space with at least one region where dark energy is active. However, this is neither how I think of it nor, as far as I know, the way scientists do.
Before I go on, I want to make a point about the nature of science at this scale. In certain circumstances, rational thought is “bigger” than science. Maths is one example of that. There’s plenty of pure mathematics which seems to have no practical application and even applied maths doesn’t need to be tested by observation if it’s proper pure maths. For instance, it’s a mathematical truth that any roughly spherical planet covered by an atmosphere must have at least two points on its surface where there’s no wind at any moment, although these points may move. However, our oceans needn’t have any points where there’s no current because there’s land on this planet. Likewise, a doughnut-shaped planet needn’t have any such locations, nor need any planet with at least two mountains high enough to stick up into the stratosphere. There’s no need to observe any planets to prove this because it’s a mathematical fact. I’m not entirely sure about this, but I suspect that cosmology may also have aspects of this: it may not be possible to approach the nature of the Universe entirely scientifically because there’s by definition only one example of the Universe and it can’t be compared to others. This is a particular view of the nature of the Universe which either includes the Multiverse as part of the Universe or in some way demonstrates that this Universe is all there is. There are a number of conceivable ways in which there could be other universes, but some of the arguments for it not only rely on logic and maths but also require that they cannot be observed even in principle. For this reason, without disrespecting the field, there’s a way in which cosmology cannot be scientific. James Muirden once said:
The Universe is a dangerous place – a sort of abstract wildernessembracing the worlds of physics, astronomy, metaphysics, biology and theology. These all subscribe to the super-world of cosmology, to which students of these various sciences can contribute. Strictly speaking there is no such person as a ‘cosmologist,’ for the simple reason that nobody can be physicist, astronomer, metaphysicist, biologist and theologian at the same time.
James Muriden, ‘The Handbook Of Astronomy’ 1964.
It isn’t clear though whether something which is outside the realm of science will always remain there, and in this view, it may be that there’s not in principle something imponderable about cosmology if the mind pondering it is sufficiently powerful, but simply that the span of disciplines is too broad for anyone to grasp. There certainly seem to be cosmologists nowadays, but maybe they’re cosmologians.
Although I don’t want to dwell on that, I do want to point out that it isn’t immediately obvious what space and time are. The nature of space in particular seems to depend on observation. It’s possible to doubt the existence of space but not the passage of time, since as far as we know we are disembodied viewpoints imagining the world but we can only do that imagining if time passes. This is in spite of the fact that spacetime is unified, so it isn’t clear how we’re immediately confronted with time but not space. Maybe there are more advanced minds in the Universe who experience both with the same immediacy. But there are, in any case, at least two different ways of thinking of space and this is what I usually based my answer on.
Space can be thought of as a thing or a relationship. That is, it could be understood as a container, as it were, in which objects are located, but also an object in itself. The Universe clearly is an object, but that doesn’t mean it’s made of space and studded with galaxies like spotted dick. There is a famous “balloon” analogy applied to space, which views the galaxies as spots on the surface which move apart from each other as the balloon inflates. This makes it sound like there’s a hyperspace into which the Universe is expanding, but this may not be the case.
In maths and physics, the concept of space is often used to make arcane ideas simpler. For instance, up, down, top and bottom quarks seem to refer to direction and location, but of course they don’t. They’re just called that to indicate that they are related to each other more closely than they are to other quarks. Likewise, we might talk about the temperature rising and falling, but that doesn’t mean there’s a spatial dimension called temperature. This can even be taken into the realm of space itself. We impose the idea of several dimensions on the idea of direction and temporal precedence, but there are reasons to suppose that this is mere convenience.
Suppose space is an actual thing. What would happen if there was a tear in it? It would surely mean that one could go into that tear, wouldn’t it? But how could that happen if there was no space there, since it’s torn? Does it mean anything to say that you can take a one metre sphere out of space? What happens when you move “into” it? How would it be different from a point? This suggests that there’s a flaw in thinking of space as the fabric of the Universe.
Consequently, space can be thought of as a combination of direction and location. Location can be described, more or less, using three numbers, although since there are higher dimensions this doesn’t work perfectly. It is, however, true, that relative to one’s current position a list of numbers is sufficient to describe where something else is. This tells you how far away something else is and in what direction. However, there is no absolute position. The Universe has no centre, or its centre is everywhere. This would also be true if space is infinite but it isn’t. However, as I’ve just said, space cannot have an outside, so how can this be?
The answer is that there is a maximum distance between two points, after which the direction between them reverses. This follows from the fact that the parallel postulate is incorrect: parallel lines do in fact meet at an enormous distance in most circumstances, and nearer than that in special circumstances to do with extremely high gravity. These are just properties of that group of qualities we refer to as space or spacetime, in a similar sense to addition working the same way either way round and subtraction not. When it’s said that space is expanding, all that means is that the maximum possible distance between two locations is increasing. That doesn’t imply that any actual object is expanding. A further clue to this being so is that although it’s impossible to travel faster than light, sufficiently distant objects do recede from each other at superluminal speeds. This would be impossible if space was an object unless the mass of such an object could only be expressed by a number on the complex number plane, but the distance between nearby locations increases at less than the speed of light, at a specific distance at the speed of light and at a greater distance greater than the speed of light. This is impossible for a single object because it would have to have real mass in small quantities, zero mass at the volume of the observable Universe and imaginary mass at greater than that volume. I have to say that’s an interesting set of properties and I’m not sure if it really is impossible.
The point is that in this view the Universe has no outside or, in terms of hyperspace, no interior. It clearly does have a three-dimensional interior, but not an interior in terms of a larger set of large dimensions. This account is slightly complicated by the fact that as well as time there are tiny further dimensions, but it usually makes more sense to measure the length of a pencil line than its area.
That’s an expanded version of my usual answer to the question “what is the Universe expanding into?” but it could be wrong. The reason it might be wrong is fascinating, and therefore probably not valid, but here it is anyway: ‘Brane Theory.
You might think at first that Brane Theory is just “Brain Theory” spelt wrong. That would be funny, but sadly it’s not so. Brane Theory is an extension of string theory and although I’m not afraid of maths, I can’t understand it fully. I’ve already mentioned the issue of extra dimensions which are, however, tiny. Brane theory uses this idea to explain why gravity is so much weaker than the other forces, if indeed it is a force. It isn’t immediately clear to observation, but there seem to be three major forces in the Universe plus gravity: electromagnetism, the strong force and the weak force. Of these, electromagnetism is obvious except that it may not be realised that light is part of electromagnetism. The strong force prevents atoms other than hydrogen from exploding as soon as they form, since their nuclei are made up of positively charged particles which repel each other. The weak force is a bit more obscure, and might be better described as the weak interaction because it doesn’t involve attraction or repulsion. It amounts to a tiny force field which occurs when radioactive decay involves atoms emitting beta particles, which are fast electrons. When a nucleus releases an electron, because it’s negatively charged and there are no negatively charged particles in the nucleus, a neutron becomes a proton, or the nucleus emits a positron and a proton becomes a neutron. In the former case it means the element moves one place up the periodic table. But nothing is pushing or pulling, which makes it confusing. The strong and weak nuclear forces are very small scale in their range, only operating within atomic nuclei, and for some reason the strong nuclear force is 128 times weaker at double the distance. Electromagnetism is more straightforward, probably because we experience it ourselves directly and obviously in the form of light, current, magnets, compasses, lightning and so on, and it diminishes like gravity, following the inverse square law. That is, for example, a light source emitting light all around it such as the Sun will do so in a sphere and because a sphere twice the size has four times the volume, it will be a quarter as bright from twice as far away. Gravity may not even be a force at all, but the distortion of spacetime by mass, and is anomalously weak. A magnet can pick up a piece of iron against gravity even if the magnet only has a mass of one gramme, yet Earth’s mass is nearly six quintillion (long scale) times the mass of the magnet. That’s ridiculously weak.
Brane theory, at least sometimes, attempts to solve the problem of gravity being as weak as it is by using extra dimensions. Instead of exerting a force in three-dimensional space, gravity may be doing so in hyperspace, which means that instead of weakening due to the geometry of a sphere, it does so due to the geometry of a higher, multidimensional cousin of a sphere, but the other forces are confined to three-dimensional space, in a thin membrane, hence the name “Brane Theory”, which is of course expanding in hyperspace. It’s also theorised that just after the Big Bang, in the part of the above diagram labelled “inflation”, this Universe collided with another one, causing this inflation.
So in other words, perhaps it isn’t a silly question to ask what the Universe is expanding into. This still doesn’t require space to be a thing, but makes the galaxies and stars into a thin, three-dimensional skin on a four-dimensional or multidimensional bubble. The answer is therefore possibly that the Universe is expanding in hyperspace, which is also not a thing but a way of describing distances and directions which need more than three numbers relative to where you are.
A few bits and pieces I want to clear up. This might all be thrown up in the air by the recent discovery of the way muons precess, because that suggests that the standard model of particle physics is wrong. And finally, I may have got this wrong myself. That is, what I just said might turn out to be nothing like what Brane Theory actually is. But note this: it’s maths and I’m not afraid of it. Lots of people are afraid of maths, and think they’re no good at it. I may well also be no good at maths, but I’m not afraid of it. This is a tangential point but very important, and probably has more bearing on everyday life that Calabi-Yau manifolds and stuff have anyway.
A Box Of Chocolates 