*They’re meddling with things they don’t understand!*

That, for me, is one of the main reasons I found science interesting.

One of my first meddles… I’m not sure I ought to relate this but why not? I was about 9 and was really into having fires on this bit of waste ground near my house. Anyway someone dumped an old shed there and I discovered – much to my pleasure – that tarred roofing felt wrapped around asbestos when chucked on a fire exploded. Me and the other kids would throw it on the fire in the pit and then leap over it hoping it wouldn’t go off mid flight. It did once when Steven “Whopper” Watson made his bid for glory and caught him directly in the gentleman’s area. How we laughed as he rolled in agony.

He is now a happily married father of three so it couldn’t have been that bad.

]]>I’m sure you’re well aware of the issue, but this reminds me of what the lab technician first asked to test the Mpemba effect said. He ‘knew’ from his physical intuition that the result must be wrong. So he told his boss he’d keep working on it until he got it right.

On the one hand, this sort of thing is a good reminder that the real world is usually a lot messier than theory. There’s friction, contamination, miscalibration, interference, vibration, background noise, natural radiation, leakage, dust, fluff, and random creepy crawlies in the works. Experimental works requires a lot of experience to build up an intuition about all the things that can go wrong, to allow one to build an experiment that you can be sure works. You can only gain that confidence in the equipment by first measuring things where you know what the answer is, so you can be sure the experiment is measuring what you think it is.

But on the other hand, there is a natural tendency to relax when the equipment gives you the value you expect, and you forget the demonstrated unreliability of all the previous experiments. If you start knowing the answer, and when the experiment confirms it you stop, and when the experiment disagrees you fiddle with the equipment until the answer changes, ‘fixing’ it, then you’ll almost always just confirm what you already believed. The experiment tells you nothing. The only time you ever learn from such a set-up is if you find it impossible, after a great deal of effort, to get the right answer. You might then, grudgingly, consider the possibility that your expectation was wrong.

The experiments giving the wrong answer teach the lesson of our fallibility most directly, but the conclusion we have to draw is that we need to show the same suspicion about all our experiments, even the apparently successful ones. The techniques you learn to apply fixing a failed experiment, you also have to apply to every other experiment. Because even the curve you’re expecting is no more likely to be the curve you’re looking for, if your experimental technique is fallible. That physics common sense is absolutely necessary to *set up and test* experimental equipment capable of making the measurements you need, but it is highly dangerous to rely on when conducting the experiment itself.

I was never very good at the experimental side of physics, but had a lot of respect for those who were. I was always a theoretician. I didn’t get enough practice to develop that level of practical physical intuition. But the same principle is at work in the theory. If you manipulate mathematics blindly, with no idea of what to expect, no way to detect results that are obviously wrong or simply don’t make sense, then you can’t detect any mistakes you make, and we’re all fallible. Having a geometric intuition about what the maths is doing gives you a way to check and verify the reliability of your methods and tools. Only when you have confirmed its validity in areas you know can you go exploring the unknown.

]]>Fraser, your lack of sense of direction – your flaw is interesting. I can see how being poor at any given thing (you should hear me “sing”) gives an insight into what it’s like to be poor at things you are good at. God help me I have done private tutoring in maths and physics and there was this one guy who just couldn’t get logs at all. This was a serious problem because he was a health and safety officer and he needed to learn basic acoustics and dB is a log scale. That was an hilarity.

So, I get your point but I think there is a wider issue with human fallibility here. I suspect at a certain level things like insight and creativity require it. I suspect an eidetic memory would be quite the hindrance in grasping the essence of things. I think Borges explains this very well in his short story, Funes, the Memorious.

Personally I have some experience here. My first degree was Physics so, whilst mainly theory, 1/6 of it was experimental and that means curve fitting. And curve fitting is not necessarily all about accuracy. analysability is a big part of the game. Also some “common physical sense” such as knowing that, no matter how accurate your curve is according to the usual metrics, if you *know* it has to tend to, say, zero in a certain limit for sound physical reasons and that curve doesn’t then it ain’t the curve you want no matter how accurate it is taken over the whole range of results.

Whoever would have thought we’d end up here starting from a story about a snake eating a towel?

To get back on topic: if you think swallowing a towel is dumb, what about swallowing an alligator?

]]>Of course, the other student was called Michael Atiyah and when he told the story I knew that Michael Atiyah was, well, like * Michael Atiyah!!!* (and, for the matter of that, Roger Penrose was

Whoever would have thought we’d end up here starting from a story about a snake eating a towel? 🙂

Undoubtedly very off-topic but while people are interested…. That said, as you say geometry is geometric, visualisable, and so a difficult subject to convey in this textual format – hence my staying out of it except anecdotally.

]]>I think I know what you’re referring to, but not so. Tensors are generally expressed using a sum of dyadic projection operators. f(x) = v(v.x) means taking the component of x in the v direction, times v. Project along each eigenvector, scaled by the eigenvalue, and you’ve got your symmetric tensors.

The symmetric/antisymmetric of f depend on ∇f = ∇.f + ∇˄f, which are essentially div and curl of f. If f is antisymmetric then ∇.f = 0 and it turns out f is just f(x) = x.(∇˄f)/2, it’s x dotted with a fixed bivector, and so the antisymmetric tensor can be represented by a particular fixed bivector. It’s a big simplification for a special case. This is what people mean when they say the geometric algebra ‘includes’ antisymmetric tensors as elements. If f is symmetric, then instead ∇˄f = 0 and ∇.f is the trace of the tensor. You can’t reconstruct the tensor from that alone, so you have to stick with the dyadic expression. (See pages 23-25 here.)

*“On what to teach first: Before a student is going to have any idea what you’re talking about at a higher level of abstraction, he’s got to understand what hes doing at the level of abstraction that all this stuff is built on.”*

I agree, but the point of doing it geometrically is precisely to avoid the problems of starting with the abstraction. People already have a built-in intuition about space – lines and planes and arrows and points. They’re already familiar with rotations and reflections. You start kids with counting actual pebbles and bricks; you introduce them to the Peano axioms only once they understand *what they’re an abstraction of*.

Maths education *starts* with always doing the concrete picture before abstracting it, but eventually you reach a point where the concrete is abandoned and you launch into abstraction built on abstraction, students manipulating things blindly with no idea of what’s happening inside. Mathematicians do it far more readily than physicists, but physicists do it too. It’s the one criticism I have with regard to the the common mathematicians’ view on Geometric Algebra. They are quite correct that it’s saying nothing that wasn’t understood by mathematicians decades ago in the work on Clifford Algebras and spinors and differential geometry. It doesn’t alllow us to do anything particularly new, or that other methods can’t do as well. They’re right, but that’s not the point. The point is *the intuitive picture* – it makes the deep, abstract stuff mathematicians have invented accessible to physicists who need *simple pictures* to understand. The unique selling point of Geometric Algebra is that it is *geometric*.

*“I’ve often thought that there’s something deeply weird geometrically about half-integer-spin in quantum mechanics. That a spin should correspond to a bivector makes sense (I certainly hope it does, or angular momentum isn’t conserved in a Lorenz boost!), but there are other bits of weirdness associated with half-integer spin that are reminiscent of Riemannian branch cuts: Angular periodicity not doing what it should, etc,”*

Do you mean the whole ‘rotation-by-720-degrees’ thing? There’s a neat explanation for that.

A spinor is often described as an object that flips sign if you rotate it by 360 degrees and only returns to where it started after rotating 720 degrees, in a way that’s hard to fathom as a sensible, concrete object in space as we intuit it. Spinors in general are elements of the even subalgebra of a geometric algebra (made up entirely of products with an even number of vectors), but in 3D we can think of it as an ordered pair of vectors, representing two reflection planes. Now, if you do two reflections one after the other, you get a rotation, and it’s this rotation that we think of as what ‘spin’ is all about. But doing so loses information, because the reflection planes are *oriented* – they have a ‘front’ and a ‘back’ face – but either way round the reflection is the same.

So if you start with the two planes aligned, the reflections cancel out and you just get the identity. Start to rotate one of the reflection planes, and the pair together represent a rotation through twice the angle between them. When you’ve rotated one of the planes 180 degrees, the planes are now back to back, and the reflections again cancel. Although the reflection plane has only rotated 180 degrees, the rotation represented by *the pair* of reflections has turned through twice the angle, or 360 degrees. What the pair of reflections *does* is not the same as what they *are*. They both *do* the same, but one is two vectors pointing in *the same* direction, and the other is two vectors pointing in *opposite* directions.

Even Michael Atiyah has said “No-one fully understands spinors. Their algebra is formally understood but their general significance is mysterious.” Now, I’d certainly not claim to “fully understand” spinors – I have only a fairly casual and amateur interest in this stuff – but I think they are perhaps not as mysterious as they’re often made out to be.

*“I’m half suspicious of adopting a mathematical framework that is too “beautiful”.”*

Agreed. Evangelists for a particular approach usually grossly oversell its capabilities, and Geometric Algebra is no exception! It does have some annoying flaws and limitations. But it is *remarkably* nice, and I find it helpful to get an intuitive picture of what’s going on with the abstract stuff.

Whoever would have thought we’d end up here starting from a story about a snake eating a towel? 🙂

]]>A hole was drilled through the wall and a new cable for telephone and computer connection was installed. All is working correctly.

The system (the economic system) sometimes works and no snake brains involved.

I was even inspired to clean the house (starting the day before the Gentleman arrived – I wanted the place to be as clean as possible for a guest) – of course the house looks much the same as before I made the effort, but do have have the satisfaction of knowing I spent a lot of time and energy on the project – and only collapsed once.

]]>You can add in additional projective dimensions to handle certain kinds of singularities. You can also tweak the metric signature to do different things.

]]>On what to teach first: Before a student is going to have any idea what you’re talking about at a higher level of abstraction, he’s got to understand what hes doing at the level of abstraction that all this stuff is built on. Differential oriented areas, volumes, etc are all well and good, but he’s got to know what a vector is first. Tensor algebra inherits all its geometric properties from the properties of a vector. In addition, there are non-geometric vector quantities encountered in engineering. (Unsure of the terminology here: Abstract vectors vs. geometric vectors.) An abstract vector being a bag of numbers with no natural metric (as a consequence, differences make sense, but there is no natural rotation. When you manipulate a deformation of a medium in calculus of variations, you’re messing with an abstract vector.

I’ll read the Hestenes paper: I’ve often thought that there’s something deeply weird geometrically about half-integer-spin in quantum mechanics. That a spin should correspond to a bivector makes sense (I certainly hope it does, or angular momentum isn’t conserved in a Lorenz boost!), but there are other bits of weirdness associated with half-integer spin that are reminiscent of Riemannian branch cuts: Angular periodicity not doing what it should, etc, if you wanted to try (as I occasionally do) to put scalar particles and “spinor particles” on the same basis and explain spin as a sort of internal motion.

I’m half suspicious of adopting a mathematical framework that is too “beautiful”. You lose flexibility when you add in slick beauty in some cases: For example, you can’t naturally describe any dissipative process in Least Action or Hamiltonian mechanics. You start having to play games with the definition of momentum to shoehorn the Lorenz force into Hamiltonian mechanics. A big bag-o-numbers with no additional structural assumptions seems to be your most general-purpose mathematical tool, even if it’s ugly.

(rambling…)

]]>You’re thinking of the D’Alembert operator. It’s applied to the vector potential rather than the field. But it’s related, yes.

Whether it’s “beyond vectors” depends what you mean. Conventional physics education teaches vectors first, and a highly non-intuitive deep mathematical theory that is equivalent to Geometric Algebra is taught much later – postgrad if ever. However, its adherents argue that it’s actually a lot simpler and more intuitive than vectors, and they’ve been proposing that it be taught early on *in place of* vectors. Stuff that is mysterious and messy in vector theory cleans up beautifully in Geometric Algebra.

For example, consider the difference between a polar vector, like velocity, or the electric field, and an axial vector, like angular velocity, or the magnetic field. If you reflect a physical experiment in a mirror, a polar vector is simply reflected. An axial vector is both reflected *and reversed*. (If you reflect a spinning disc, the reflection is spinning the other way, so its angular velocity is pointing in the opposite direction.) Vector algebra tends to ignore the distinction – they’re all just vectors. But the two sorts of object follow completely different transformation laws under reflection!

Even stranger, we know that in special relativity, the electric and magnetic fields are combined into a single object: the electromagnetic field. Change your reference frame, and the electric and magnetic parts mix – they’re different aspects of one object. But one is a polar vector, and the other axial! How the hell does *that* work?

Geometric Algebra says that space has more than just directed lengths (vectors). It also has directed areas (called bivectors) which are like little patches of plane facing in a particular direction, and directed volumes (trivectors), and so on. So in 4D space we have one scalar (1), four basis vectors (x,y,z,t), six basis bivectors (xy,xz,yz,xt,yt,zt), four basis trivectors (xyz,xyt,xzt,yzt), and one quadvector (xyzt). They represent all the various combinations of lengths, areas, volumes, etc. you can have in space. They all have a direction, too, so xy = -yx because a plane segment spanned by x and y (in that order) faces the opposite direction to one spanned by y and x. The electromagnetic field is a bivector, with six dimensions. If we multiply it by a fixed vector t pointing along our time axis representing the reference frame we’re using, so this is just a constant multiple of the field, (xy,xz,yz,xt,yt,zt) turns into (xyt,xzt,yzt,x,y,z) because any unit vector multiplied by itself cancels. (You could think of it as like an area spanned by two vectors pointing in the same direction collapses to nothing. That’s not quite right, but I’ll spare you the digression.) This (xyt,xzt,yzt,x,y,z) object is the sum of a trivector part (xyt,xzt,yzt) and a vector part (x,y,z). Picking a reference frame naturally splits the electromagnetic field into two geometrically different parts. One is a directed length, the other is a directed volume *perpendicular* to a length. The trivector part is the magnetic field, and the vector part is the electric field. Pick a different time vector, get a different split.

What initially seems like a mysterious and unexplainable phenomenon (vectors and ‘pseudovectors’), quite hard to visualise and harder to keep straight notationally, is turned into a simple geometrical insight – that lengths and volumes are geometrically different sorts of things, and the split is created by looking at those components of the field parallel to our time axis xt,yt,zt, and those perpendicular to it xy,yz,xz. Vector algebra uses the coincidence of both vectors and trivectors having the same number of dimensions to conflate the two, causing utter confusion.

Lots of other bits of esoteric maths become simple, too. Complex numbers are just the combinations of (1,xy), scalar and bivector. Quaternions are combinations of (1,yz,zx,xy). They’re just geometry. If you can accept that an algebra of space ought to have elements representing not just length but also areas and volumes, everything else follows naturally.

So while it’s commonly taught as being “a bit beyond vectors”, it really shouldn’t be! 🙂

*“tomorrow is my wife’s birthday so is it OK if we take a slight hiatus?”*

Sure! Hope you both have a nice time. 🙂

I expect I’ll still be around when you get back.

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