Möbius Mass-Transit

In 1950, Armin Deutsch wrote a story about the Boston subway called ‘A Subway Named Möbius’, for which spoilers will follow almost immediately.

Ready?

OK.

In this story, a tunnel is added to the MBTA subway, which curently looks something like this:

By Michael Kvrivishvili – Flickr: Boston Rapid Transit Map, CC BY 2.0, https://commons.wikimedia.org/w/index.php?curid=28660767

Sounds like a story which would at best only appeal to train spotters, right? Not a bit of it. The tunnel in question involves Boylston and changes the topology of the network in such a way that one of the trains vanishes along with everyone on it and only reappears several weeks later. When it does reappear, another train disappears. The story was adapted into the 1996 Argentine film ‘Moebius’, set on the Buenos Aires metro, Subte, whose current layout is:

There is a small amount of technical mathematical terminology in the original story, which is here. I’m not sure if this is more than the use of jargon to impress the reader, but it is genuine, valid, mathematical vocabulary relating to graph theory. Graphs in this sense are not the likes of line graphs, histograms or pie charts, but more like the maps shown above, in which pairs of items are related to each other in some way. In general, such a set of items with their relationships could be represented by dots (nodes) and lines (edges) on a piece of paper, or rather, some of them could be. I’m not sure all of them could, for instance the surface of a torus might not work. In the story, the addition of the Boylston shuttle has caused the graph to have “infinite connectivity”, and it’s this which I find most dubious.

Connectivity has a couple of meanings in mathematics. In graph theory, connectivity is the minimum number of elements (nodes or edges) which need to be removed to separate the network into at least two isolated networks. If this is the meaning used here, it sounds very doubtful that this could really happen because there is not an infinite number of either nodes or edges, and there couldn’t be in a mass-transit system. There are at least two other meanings, both topological. One is that it’s possible to move between any two points in the space, which is of course true in many mass-transit systems, but presumably not all because I would expect there to be ones which either have yet to be linked up or can’t practically be connected. The physical correspondent of either graph will be a different shape and the constraints on movement include the fact that the trains move along rails in tunnels of impenetrable brick, concrete or stone. There are other ways of considering this network – for example, flooding would effectively block off the lower layers. It also works as a template. Model railways could be constructed from these maps. They are also, famously, topological rather than geographic. This famous map:

. . . is geographically like this:

Copyright status unknown – will be removed on request

However, for the purposes of the story and the film, the topology is the relevant aspect, and it’s also the relevant aspect for passengers. Presumably for the people running the system, geography is more important for various reasons, such as power consumption, timetabling and location of depots.

The Glasgow Underground is something like the second or third oldest of its kind and despite protests has never deviated from its original plan:

As it stands, this map has a clockwise and anti-clockwise circle which happens to correspond closely to the geographic layout, or at least more so than the Tube does. However, in all these cases, the tube layout would be topologically the same if the layouts were folded in half or stretched in the middle and compressed at the edges. If Leicester Square and Covent Garden were six kilometres apart and Chesham and Chalfont & Latimer were only 260 metres apart, provided the other relations remained the same, the Tube map would still be accurate. They could also in theory, be upended and become a system of lifts linked by horizontal trains and still be topologically the same. There could be a high rise building doubled over on itself whose lifts and, well, trains, correspond to the layout of the London Underground. This is what topology is about. Of course, as soon as an extra tunnel or lift shaft was constructed, the topology would change, and depending on the geographical shape of the system, this would be more or less convenient. If the Glasgow Underground were folded over on itself it would be relatively easy, depending on its orientation, to put a lift between Partick and St Enoch, but as it stands a tunnel under the Clyde between those two stations would lead to a topologically identical system.

I presume that most people know what a Möbius strip, also known as an Afghan Band, is, but in case you don’t, take a strip of paper, put an odd number (such as one!) of twists in it and glue the ends together. Pretty simple stuff of course, but unlike a similar strip without twists or an even number thereof, such a band has only one edge and one side. This is Wikipedia’s illustration:

It might seem contrary to attempt to claim that this shape has only one side and one edge because when it comes down to it it’s just a strip of material with a twist in it, but in fact it is exactly that. If you trace the edge with your finger, you will have to go round twice to return to the original spot and if you colour it in on one side, you’ll have to do the same. As an actual physical object which takes its thickness and the interior of the sheet of paper it’s made of, this isn’t what it is, but even then it’s a three dimensional shape with only two faces, two fewer than a tetrahedron, which we usually assume to have the minimum possible number of four, but as far as I can tell, either one or two edges.

One of the peculiar features of Möbius strips is that if you cut them in half down the middle, you get a longer, narrower strip, but if you cut them a third of the way from the edge instead you get two linked strips. Hence there are three “lines of interest”, as it were. A line near the edge describes a single edge, which however is at both the top and the bottom of the strip. A line near the centre appears to loop round twice. Lines around a third of the way in start near the top, plunge down near the bottom and return to near the top. All of the lines make two complete circuits of the strip before meeting themselves. The central line doesn’t change level, assuming that the topology of the strip corresponds to minimised distances between the lines.

Now imagine each one of those lines is an underground rail tunnel, like the Glasgow underground or the London tube circle line. A cross-section of the strip at any point would appear to show five tunnels, which would be vertically arranged at one point and horizontally arranged 180° away from that point. However, there are in fact only three tunnels, even though every transverse section seems to pass through five. A relatively simple case of this system involves them vertically oriented on one side, gradually rotating to an horizontal arrangement on the other, but this is only one version of the “geographical” arrangement. They could all be rotated through right angles so the tunnels pass near the surface along half of their route and then all plunge deep underground along the other half, they could be inside a narrow tower, or each of them could be imperfect circles and meander around. However, whatever else happens, to conform to the topology of a Möbius strip, they must at some point “twist” around to the opposite side an odd number of times, the simplest case being once.

In order to make this simple, I want to imagine this system to consist of apparently five tunnels arranged vertically on one side and horizontally a semicircle away along their route. I also want there to be a system of lift shafts linking them all together and to the surface, perhaps ten of them, plus an extra set of five vertical shafts where they reach their horizontal orientation. Also at that point, the so-called “lift shaft” is horizontal, so it’s more a walkway or travelator, or perhaps a supplementary train tunnel since it is running horizontally. If we were talking about Glasgow here, which is the closest to this arrangement but still not very close, if the clockwise and anticlockwise routes were in tunnels at different levels, only one more tunnel would be needed to complete the arrangment and then each of the fifteen stations could have five different levels joined by stairs, elevators or lift shafts. If it were the Tube, the Circle Line is the most obvious, or it would be if it was actually even topologically a circle, which it isn’t:

Ignoring the western branch though, which I’m sure has an official name but I have no idea what it is, the “Circle” Line has two and a third dozen stations, by contrast with the mere fifteen of Glasgow’s. However, both systems score over Subte and the MBTA in actually having circular routes, topologically speaking.

The question arises of what the point of any of this is. Why would anyone bother to design an underground with three different levels of completely superimposed tracks? However, I kind of want to look past this. Thinking of the shops and other concessions associated with the pedestrian subways in London near the Tube stations, maybe this is an entire underground city, or rather village, with a couple of hundred subterranean chambers, some residential, some business and some administrative, and the like. But for whatever reason, this subterranean railroad exists. In my mind, anyway.

Although it would be possible for the tracks to maintain the trains at the same angle, i.e. upright, it would also be possible to make them monorails, both suspended and on a rail, with overlapping rails and beams, with the carriages rolling around in circular frames to keep them in the same position and avoid moving the passengers around in ways they might not like. Alternatively, they could all just be strapped in very securely, or perhaps in padded pods, with four doors, two in the walls, one in the ceiling and one in the floor. Magnetic levitation is another option. For safety reasons, only the doors facing the station platforms should open.

Although this does all seem rather pointless, there would be advantages to such a system. For instance, it would save wear and tear on tracks or other equipment if twice the length of tunnels were required to return to the original point. It could also allow for train arrivals and departures to be staggered so that five times as many trains were operating on three times as many tracks, although this would work against the wear and tear advantage. Trains on the same track would have more mean spacing between them, reducing the probability of collisions. However, there are also more significant electrical advantages. A resistor in such a shape will do so without causing magnetic interference. Nikola Tesla patented a similar electromagnetic device in 1894 for the wireless transmission of electricity. It’s also possible that this will enable high-temperature semiconductors, meaning that if the tracks are indeed linear induction motors, such a shape would be ideal for them. See also this video:

There is a connection between superconductivity and Möbius strips in any case because of the nature of quantum spin as mentioned previously. The way spin behaves for fermions, the particles of which matter as opposed to forces are made, can be envisaged using a strip of this kind, because as previously mentioned, the magnetic field of a fermion has to be turned through 720° to return to its original polarity, which is the same as an arrow pointing towards the edge of the strip. At low temperatures, fermions such as electrons can pair up, but the members of these pairs can be thousands of atoms apart, and because each electron is a fermion, the two together act as a boson, with integral spin. Also, because of the distance, many such bosons can occupy the same space. Because they are bosons rather than fermions in such a condition, they can have the same energy states, and this makes superconductivity possible, although not all superconductors work like this.

It also occurs to me, and this is probably nothing, that plasma of protium (ordinary hydrogen) could be suspended to form such a strip electromagnetically and this might “do something”, but this is all very vague. It’s probably nothing, but this plasma would consist entirely of fermions.

A suggestion made by a mathematician on a rather related subject was the minimum number of edges a shape could have. I only remember this very vaguely, but just as a Möbius strip has just one edge despite appearing to have to, this topologist stated that there was no known reason why a shape shouldn’t have zero edges. This is true in any case of certain shapes such as spheres and tori, but the idea was that there could be no sides or edges, and this could be constructed from the theoretical infinitely elastic “plasticine” which topological shapes are made of. It’s easy to imagine, say, taking a triangular shape with three strips, twisting each a different odd number of times (1, 3, and 5 for example), gluing them together and ending up with something weird. But intuition tells us it’s impossible, as do the laws of physics, because if you imagine folding something in the right way causing it to disappear in, to quote Douglas Adams, “a puff of unsmoke”, that clearly wouldn’t happen because matter has to go somewhere. Another way of thinking of it is to imagine it entering hyperspace, and therefore only seeming to vanish, but prima facie that doesn’t seem any more feasible. However, if such a shape started off with more than three dimensions, or it was a distortion of space itself rather than being made of matter as such, as might be achieved by negative mass or actual mass, maybe something else could arise. It’s possible that the Universe itself has a “twist” in it somewhere which makes it effectively into a multidimensional analogue of a Möbius strip, and if that is the case, a trip round the Universe through such a spatial anomaly would bring objects back as mirror images of themselves.

There are macroscopic consequences of fermion spin converting to boson spin, such as the aforementioned superconductivity but also superfluidity. What isn’t clear to me is whether a direct macroscopic manifestation of something like a Möbius strip could happen. Maybe it could. Quantum computers exist, for example. So do superfluids, superconductors and Bose-Einstein condensates. A magnification of non-integral spin would involve something like a gyroscope which needed to be inverted twice before it was spinning the other way, or indeed a tunnel which would cause a magnetically levitated train to turn upside down only if it went through it twice. It’s also rather imponderable what state such a train would be in if it had only gone through it once. Would it then be potentially inverted so that it would only need to go through it once to turn upside down? Would it mean that a passenger who changed trains or got off after one circuit would then carry the potential to turning upside down themselves if they travelled through the tunnel again? It’s very difficult to contemplate even as a thought experiment.

Armin Deutsch was primarily an astronomer. His story could itself be seen as a loop, as could the film ‘Moebius’, as a train is discovered to have vanished just after the original one disappears. The film, of course, could literally have been made as a strip with a twist in it if it were on one reel, and become a never-ending story. Alternatively, there is a way to make a story a Möbius strip by having the characters swap identities as the plot proceeds. I feel that the possibilities of the Möbius strip with respect to story writing have yet to be explored. That would be a real twist in the tale.

Spin Is Not What It Seems

Nor is isospin, but then that’s less well-known.

Most of what people say about quantum physics focusses on things like entanglement, acausality and uncertainty, with a kind of mystical bent, but there’s also something else which most people ignore which is equally weird, and on top of that is something else again which is as weird too, if not even weirder. These two things are spin and its oddly- but appropriately-named sibling isospin.

It’s been said that if you think you understand quantum mechanics, it means you don’t. This may or may not be true and there are different opinions about what it actually means, but I would say this is also true of spin and isospin. I’ll deal with spin first.

If you hold a spinning gyroscope, you can feel the difficulty of shifting it from the direction its pointing. If it’s one of those small toy ones, it won’t wrench you off your feet but its rotation will be shifted into your body if you’re standing. In a swivel chair, a sufficiently large and massive rotating object will rotate the chair if you try to move the object into a different angle. This principle is useful, and is for example employed with rifle bullets, spacecraft and compasses to stabilise them. Whereas magnetic compasses are useless near the poles, gyrocompasses can float around as they move and will therefore continue to point north if they’ve been set up that way in the first place. A spacecraft will tumble unpredictably in space unless it’s stabilised in some way, and one way of doing so is to make it spin as it launches, which keeps it pointing in the same direction. This particular spin is often counteracted by ejecting something spinning in the opposite direction to ensure the spacecraft instruments or devices stay facing the requisite direction later on.

These are all illustrations of angular momentum. Momentum in general is the tendency for an object to keep moving in the same way unless something stops it, that is, unless another force acts upon it. This is true of masses moving in straight lines, and of spinning masses. They will continue to spin in the same way around the same axis unless something acts on them to shift them or slow them down, and when this happens that momentum has to go somewhere as spin rather than in a straight line. This is called angular momentum.

We tend to think of atoms as consisting of nuclei surrounded by electrons in orbit around them: that is, rotating. Ferromagnetism happens when the atoms in a material are all lined up spinning in the same direction, and only applies to very few materials, notably iron but also cobalt and nickel. If you think of atoms as gyroscopes, which they are, what you’re doing when you magnetise something is shifting the axes of rotation of a load of gyroscopes, and that angular momentum shift has to go somewhere. And it does. If you suspend a piece of unmagnetised iron in space in zero gravity conditions, or more accessibly hang it from a thread, then apply a magnetic field to it, this will to some extent magnetise the block and shift the atoms, and it will start spinning. This is known as the Einstein-De Haas Effect. Yes, that Einstein.

This change in angular momentum can be measured quite easily because the mass of the iron is known and its rotation can be timed and observed. However, even if you take into account all the angular momentum involved in the shift, it doesn’t account for all of the spin. This is because the electrons themselves are tiny magnets pointing in a particular direction, and the magnetic field aligns not only the atoms but also the electrons. Now here’s the crucial question. How can an electron point in a particular direction? The answer is that it has an axis of rotation, and this accounts for the discrepancy in the rate of spin the lump of iron has. This difference in angular momentum just taking the orbitals into account and the actual difference allows the spin of the electron to be found.

And this is where things get really weird.

If you calculate the spin of an electron, either assuming the smallest probable size of the particle or the much more likely scenario of thinking of an electron as a point in space, there is an imponderable problem. If the electron has a size at all, in order to generate the amount of angular momentum it has, it would have to be moving faster than light. If, on the other hand, an electron is a point, it’s featureless except for location, so how can it be spinning at all? A point in space doesn’t have a direction or an axis of rotation in the conventional sense, so – huh?

This is not some abstract thing happening due to the vagaries of scientific theory either. A lump of iron really does start spinning if magnetised, and taking into account all of the rotational movement of the electrons in their orbitals shifting is not enough to account for the exact rate of rotation. In the end, then, there seems to be only one possibility: spin is a fundamental property of matter. From our usual perspective, it definitely looks as if there are just objects which are not spinning which we can rotate or might start or stop rotating, or speed up and slow down, and so on, as if it’s just another thing going on in the world, but that isn’t actually what’s happening. On a tiny scale, spin is an intrinsic property of matter like electrical charge or its absence. Moreover, it’s quantised. In the same way as there are jumps between values of something on a small scale rather than an infinitely smooth transition, so for example electrical charge is either neutral, equivalent to an electron, the reverse of an electron, and if a quark either -⅓ or + ⅔ or the opposite for their antiquarks, which add up to an equivalent charge to the electron when they form a nucleon, which is just as well because otherwise atomic matter couldn’t exist. This will become relevant.

Spin has been described as “classically non-describable two-valuedness”, as it’s indescribable in the sense that it can’t actually be properly understood but must exist. Subatomic particles don’t literally spin in the same sense as a wheel or planet, but behave as if they do. All subatomic particles have a spin of either a whole number or a multiple of ½ other than a whole number. The former are bosons, the latter fermions. Fermions are “stuff” and bosons forces, so for example quarks and electrons are fermions and gluons and photons are bosons. Non-integral spin particles have a peculiar property which doesn’t seem to make sense, which is that to reverse their spin they need to be turned through not one but two full circles. How is this possible? Well, imagine a Möbius strip, which is a joined ribbon with an odd number of twists in it, usually simplified to just one. Following the edge around with your finger pointing to the right will reach the point where it points to the left after 360°. In order to get back to pointing the finger to the right, a further 360°of the strip have to be traversed. It’s easier to do this with a strip of paper or ribbon than try to imagine it, for me anyway. This is a good model for how half-integer spin particles work and how it’s possible for them to have to be turned right round twice before they’re back to their initial state. Incidentally, there’s a short story called ‘A Subway Named Möbius’ where a complicated underground train system has one more tunnel added to it which causes a train to disappear, and it doesn’t come back again until the tunnel gets blocked off again. I’m not by any means saying that’s anything more than a fanciful story, but if a topological analogy of that kind can be made regarding such a fundamental feature of physical reality, albeit on a quantum level, it does make me wonder what’s possible. For instance, it’s possible to imagine that space as a whole is “twisted” in all three dimensions, such that any journey round the Universe ends up with one finding one’s home planet is mirrored, or rather seems to be because one has in fact reversed, because the topology of three-dimensional space could in theory be analogous to a Möbius strip. A Möbius strip with an even number of twists is effectively not one at all.

This property of fermions, for complicated reasons I don’t understand, means that no two fermions can occupy the same energy state. This is not the case with bosons. For instance, a laser consists of innumerable photons in the same energy state because they’re bosons, but it effectively means that light cannot form structured matter. It can do things like form caustics and be focussed to a point, and the like, but fermions can build themselves into atoms. Nuclei have to consist of nucleons in different energy states, although they are less obviously in shells than electrons, but neutron stars also have to be in this state – every single neutron in a neutron star is in its own distinct energy state. The electrons in an atom organise themselves much more clearly into different levels, in the form of the shells which enable the periodic table to exist, with heavier elements having more shells and different properties. Without that, there would be no chemistry and no materials as we know them. The fact that I don’t understand this is a source of discomfort to me which I feel very driven to remedy, but right now that’s how things are. It also makes me wonder about Bose-Einstein condensates. These are an unusual state of matter which happens when a low-density gas consisting of bosons is cooled to almost absolute zero and the atoms become overlapping waves and ultimately a single, collective wave comprising all the atoms because they’re larger than the distances between them. Although atoms are made of fermions, each atom as a whole can be a boson if the total number of nucleons and protons is even, so this means that the possibility of attaining this state depends on isotope number as well as what kind of element the gas is, in a similar way to how helium-4 becomes a superfluid at a higher temperature than helium-3. If they were fermions, this would be impossible because they wouldn’t be able to occupy the same energy state.

For us to exist, spin must also, and there also have to be integral and non-integral varieties. It’s a sine qua non of our reality. The Multiverse presumably means there are other universes where there are, for example, only fermions or bosons, or perhaps universes where there is no spin, but these are all very boring places. A universe with just bosons would have no structured matter but instead consist of rays of energy, and one with just fermions would have no structured matter either but simply electrons.

A particle is supposed to have mass, charge and spin. Of the charge values, these can be either positive, negative or neutral, and of the integral and non-integral varieties mentioned above depending on whether quarks, leptons or both make them up. This addition also occurs with spin. Neutral particles clearly do exist, for instance neutrons, whose existence can be deduced fairly easily with precise enough measurements. Chlorine has two common stable isotopes, and if one does something like react salt with something else in distilled water or tries to make a saturated solution of pure sodium chloride in it, one is soon confronted with the fact that the ratio between the weights of the same amount of salt and other substances has to involve a fraction. This is because all normal chlorine is a mixture of the two types of atoms with different numbers of neutral particles, and these are neutrons. Mass, charge and spin all have to be conserved in nuclear and other processes, so for example if a potassium-40 atom emits a positron, one of its protons must become a neutron and it becomes argon-40, and unstable particles decay into various different “fragments”, but they must all add up to having the same charge and spin. Hence a negatively charged muon may become an electron with the same charge but since an electron is so much less massive than a muon, the spin, i.e. the angular momentum, still has to go somewhere, which is into a muon neutrino and an electron neutrino. Likewise, when a neutron leaves the safe confines of an atomic nucleus it only has about a quarter of an hour to “live”, and will decay into a proton and an electron, conserving charge, and also an electron neutrino. They have extremely low mass but observation of the 1987 CE supernova 168 000 light years away revealed that they do have some because of the timing of their arrival here compared to light. Supernovæ produce bursts of neutrinos because the protons in them collapse into a neutron star, converting themselves to neutrons in the process and emitting neutrinos. There are three types of neutrinos, associated with tauons, muons and electrons.

Neutrinos are a bit mind-boggling because they have no mass or charge but only spin,but they must exist because otherwise the accounts wouldn’t balance, as it were. However, there was a problem with solar neutrinos detected in the 1960s, when it turned out the Sun was producing fewer of them than current physics said it should. Until this was resolved, it was possible, though of course extremely unlikely, that for some reason the Sun had stopped working and that the light and heat we were getting from it was simply the last blast of a defunct star, so in a way it was quite worrying, but it’s okay now.

Before I get to the next bit, I want to mention a much older form of philosophy than nuclear physics. Back in the day, there were supposed to be four elements opposed to each other: earth, air, fire and water. Each had two qualities opposed to each other, namely dry and wet and cold and hot. Their atoms were also supposed to correspond to the five Platonic polyhedra, which is why there are five elements rather than four. All of this makes mathematical sense and you can imagine flipping the eight-pointed star round, turning it through 90° and so on – it’s symmetrical. It could even have predictive power in that if one of them was missing, its qualities could be determined, and it has correspondances in alchemy, psychology, astrology and humoral medicine, the last of which is actually useful in herbalism. However, it isn’t applicable to science as it’s usually practiced today, and someone claiming to use it, as I just did, might be seen as undermining their ethos. Nonetheless, the symmetry is real.

By Mike Gonzalez (TheCoffee) – Work by Mike Gonzalez (TheCoffee), CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=284321

There’s also a symmetry in the physics of elementary particles, which allows one to anticipate where gaps may exist implying other particles yet to be discovered. Symmetries can be analysed by group theory in mathematics. One of the most obvious places where this crops up is with Rubik’s cubes, where certain turns may or may not be performed in a particular order to return the cube to its original state. With Rubik’s cubes there are also “orbits”. If you take one to pieces and put it back together arbitrarily, the chances are you will have placed it in another orbit in which there is no arrangement with all the faces the same colour. I think there are eight of these. Groups also apply to arithmetic, so it makes sense to introduce them with that familiar subject. A group is a set with some operations of a certain kind performable on it. It has an identity element, inverse elements and these are associative. For instance, addition and integers form a group because adding zero doesn’t change a number, adding a positive number can be undone by adding the same negative number and it doesn’t matter where you put the brackets: (2+1)+3 = 2+(1+3). Likewise with a Rubik’s cube, keeping it in the same position and turning the top row one twist to the right and then the right hand side one twist downward can be undone by turning the right hand side one twist upward and the top row one twist to the left, and there’s also an identity element in that if you leave the cube alone, it stays the same, which sounds a bit silly but these are just two examples of groups which can be easily understood. Group theory is relevant to crystallography and cryptography. Take this sentence, for example. ROT13 turns it into “Gnxr guvf fragrapr sbe rknzcyr”, and applying ROT13 to it again turns it back into “Take this sentence for example”.

Physics has various symmetries. For instance, there’s symmetry between matter and antimatter, and there are other symmetries such as the correspondence between leptons and quarks. Electrons, muons and tauons accord with up and down, charm and strangeness and top and bottom. The names of up/down and top/bottom are not accidental, although there were moves to name top and bottom truth and beauty instead.

Group theory can be used to classify different forms of symmetry. Spin falls into the SU(2) group. This is one of the Lie groups, which are groups which also behave like spaces. SU groups are “Special Unitary” groups, and I should point out here that I have never knowingly understood matrices and they were a significant hole in my mathematical knowledge at school, because I could never understand how to multiply them, so I’m just going to have to let this pass and say this is this thing, this is out there, and that’s it. I believe I can safely assume that anyone with at least a CSE in maths will get them and understand this better than I can because it’s just my personal blind spot. Having said that, I will kind of give it a go.

There are six flavours of quark: up, down, strangeness, charm, top and bottom. These can be arranged in a hexagon and can be swapped to some extent. A neutron is two down quarks and one up, and a proton two up and one down. The names seem to relate to these properties, because if up and down were swapped in an atomic nucleus it would swap the neutrons and protons. Mathematically this can be envisaged as being part of the SU(3) group, and this is the other area in which the word “spin” has been used: isospin. Isospin is another property of matter which has the same kind of symmetry as spin but is not spin. Then again, spin in the subatomic sense is really quite far from our intuitive understanding of rotating objects, so the fact that this is also called spin, relatively speaking, is not a big leap from the other kind of spin. It’s also why the words “top”, “bottom”, “up” and “down” are used. Just as an electron can be thought of as having an arrow pointing up which can be flipped through two turns to be an arrow pointing down, although it has no link with gravity which determines up and down in everyday parlance, so can some quarks be thought of as “up”, flippable conceptually to “down”, and “top”, flippable to “bottom”. If SU(3) is applied to hadrons (mesons or nucleons), they can be flipped to other hadrons with similar properties. Another application of group theory revealed a gap in the pattern which turned out to be the omega-minus particle consisting of three strange quarks, which was detected and confirmed that group theory could be fruitfully applied to isotopic spin.

Why is it called “isospin” or “isotopic spin”? Well, nuclei are isotopes of various kinds, so for example there’s helium-3, made of two protons and one neutron, as well as helium-4, consisting of two protons and two neutrons, and tritium, an isotope of hydrogen comprising two neutrons and one proton. If the nucleons in these were swapped, they would respectively be tritium, helium-4 and helium-3. This is a form of symmetry pertaining to isotopes, and it influences their stability because there are certain isotopes of elements which would be stable whether or not the neutrons in them were protons and vice versa, and these are particularly stable isotopes. Extending this into the transuranic realm of synthetic elements, it’s possible to predict which isotopes of heavy elements are likely to be the most stable.

It’s also a system of classification, because at one point in the mid-twentieth century a large number of hadrons were known, almost all of which seemed to have no prospect of being part of ordinary matter or having any special relevance to it, which was very puzzling. Another, more recent, puzzle is whether this is just a case of making pretty patterns, albeit useful ones, out of elementary particles or whether it reflects something profound about the nature of physical reality. Murray Gell-Mann, who thought this up, referred to it as the Eightfold Path à la Buddhism, and Fritjof Capra has written extensively on the idea of links between subatomic physics and Eastern spiritual concepts such as Daoism. Western philosophers tend to think of this as jejune and crass.

There is an issue regarding what appears to be the appropriation of quantum physics ideas by the New Age movement in such films as ‘What The Bleep Do We Know?’ and ‘The Secret’. On the other hand, there is also the question of whether this is an excessively proprietorial attitude on the part of some nuclear physicists and academics. But that’s a topic for another post.