The pictures are self-explanatory: the first is in Pittenweem, in Fife, Scotland, the second on the campus of the University of Adelaide, South Australia.

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*Pappus’ Theorem* states that, if alternate vertices of a hexagon are collinear, then the three intersection points of pairs of opposite edges are also collinear. In other words, if we have six points *P*_{1}, …, *P*_{6} such that *P*_{1}, *P*_{3}, *P*_{5} are collinear, as are *P*_{2}, *P*_{4}, *P*_{6}, and *Q*_{1} is the intersection of *P*_{2}*P*_{3} and *P*_{5}*P*_{6}, with *Q*_{2} and *Q*_{3} similarly, then *Q*_{1}, *Q*_{2}, *Q*_{3} are collinear. This is a classical “configuration theorem” involving nine points and nine lines, asserting that if 8 of the 9 triples are collinear then so is the 9th.

Of course, this is a theorem of projective geometry. It is valid in the Euclidean plane, but remains valid if some of the points are at infinity (recognised in the affine plane by corresponding lines being parallel). There are quite a number of cases to consider. As in various similar situations such as the classification of conics, things are much simpler in the projective plane!

At a similar time, Diophantus discovered and studied the “two squares” identity

(*a*^{2}+*b*^{2})(*x*^{2}+*y*^{2}) = (*ax*−*by*)^{2}+(*ay*+*bx*)^{2}.

The analogous four squares inequality was discovered by Euler, and the eight squares inequality by Degen, Graves and Cayley (in that order, though the last of the three somehow got his name attached to it).

The authors’ thesis is that these two pieces of work were not seen to be related until the nineteenth century. In the spirit of the work of Cauchy and Weierstrass on calculus, mathematicians turned their attention to geometry, to the task of making Euclid’s axioms more rigorous. When studying just the incidence axioms, it was discovered that (together with the usual axioms for a projective plane) Pappus’ Theorem is equivalent to the property that the plane can be coordinatised by a commutative field. On the algebraic side, the two, four and eight squares identities express the facts that there are algebras of dimenions 2, 4 and 8 over the real numbers which have multiplicative norms (and are therefore division algebras), the complex numbers, quaternions and octonions; but the last two fail to be commutative. So Diophantus’ two squares identity is the related to the only finite extension of the real numbers which coordinatises a projective plane satisfying Pappus’ theorem.

At this point, let me note that one really has to consider various “degenerate” cases of Pappus’ theorem, when some pairs of points are identified. Though tedious, this is not a serious difficulty.

After the eight squares identity was discovered, mathematicians naturally wondered whether the sequence continues. The answer is no, as was proved by Hurwitz in 1898. (His result was slightly more general; he shows it not just for sums of squares but for arbitrary non-singular quadratic forms.)

Rice and Brown also discuss finite geometries, and in particular the Steiner triple systems on 7 and 9 points; these are the projective plane over the field of two elements, and the affine plane (which can be extended to a projective plane in a standard way) over the field of three elements.

Since these fields are commutative, the Fano plane (for example) should satisfy Pappus’ theorem. Because the Fano plane only has seven points, all instances of Pappus’ theorem in it are degenerate!

It is also the case that the octonions are conveniently described by the Fano plane with arrows on some of its lines. The seven basic units (apart from the identity) are matched with the seven points of the plane, and the triples whose product is plus or minus 1 are the lines of the plane; the arrows can be put on in such a way that each line has a cyclic orientation so that the product of two of its points is the third.

There is some interesting pre-history of Steiner triple systems. As is well known, Kirkman proved the existence theorem for Steiner triple systems in 1847 in answer to a question in the *Lady’s and Gentleman’s Diary*. Unaware of this, Jakob Steiner posed the same question in *Crelle’s Journal* in 1853, and it was answered by Reiss in 1859.

The problem that had attracted Kirkman’s attention was posed by Wesley Woolhouse, the editor of the *Lady’s and Gentleman’s Diary*, in 1846. Woolhouse and Steiner may both have got the problem from Julius Plücker. He had discovered the Steiner triple system of order 9 as the structure of the nine inflection points of a plane cubic curve, in 1835; in the same paper he had posed the existence question for “Steiner triple systems”. (He mistakenly thought that they could exist only for orders congruent to 3 (mod 6); he corrected his mistake and added 1 (mod 6) in 1839.)

Should the concept be named after the person who first proposed it (so they would be *Plücker triple systems*), or the person who first constructed it (so they would be *Kirkman triple systems*)? This problem arises in many other parts of mathematics; Lyons discovered evidence for a sporadic simple group which was constructed by Sims, for example. The matter is further complicated in the present case by the fact that the term *Kirkman triple system* is used with a different meaning, based on Kirkman’s schoolgirls problem.

But there is one further connection which Rice and Brown don’t seem to mention.

The Pappus configuration has nine points and nine lines. These lines cover 27 of the 36 pairs of distinct points. The nine pairs not covered correspond to a partition of the nine points into three sets of three points each containing no collinear pairs. If we add these three sets as new “lines”, we obtain the affine plane of order 3. The uniqueness of the Pappus configuration is thus related to the fact that the parallel classes of lines in the affine plane are equivalent under symmetries of the plane.

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The system of rights of way is one of the great glories of the English countryside. Any Ordnance Survey map will show a network of walks and rides through the country; you are entitled to walk or ride along them. They come in several flavours: footpaths, bridleways, RUPPs (roads used as public paths) and BOATs (byways open to all traffic): I am not quite sure of the difference between the last two, and in any case there has been some recent change in the status.

(As I have said before, in Scotland it is different; we have a right to roam anywhere, which sounds great but in practice means that it is much harder to plan a walk since there will be no guarantee that you can get through the fences or hedges.)

Many of these rights of way date back to the early days of the rambling movement which did so much to get people out into the country. In a lot of cases, the documented use of a path as a public highway in the past was used to have it registered as a right of way.

This is the system which is about to change. On 1 January 2026, the definitive maps will be closed to the addition of rights of way on the basis of historic evidence.

I am not sure what this means for rights of way as a whole. They can be extinguished, for reasons good and bad: maybe because a nuclear power station is being built over the path, or because a rich foreigner has bought the property and doesn’t want the public walking across his land. Will it be possible for new rights of way to be created? I don’t know; but it seems that one important mechanism for this will be lost.

This is not just a triumph of evil landowners. Indeed the committee that drew up the legislation had walkers represented, and attempted to find a compromise between vested interests. What it does mean, however, is that the time to establish rights of way on the basis of historic use is running out, and a lot has to be done in ten years.

If this matters to you, see this webpage maintained by the Open Spaces Society, and support them in their action.

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A few years ago, I mentioned the new journal *Combinatorics, Physics and their Interactions*, aka *Annales de l’Institut Henri Poincaré D*.

Now they have announced a new journal starting up next year, *Journal of Combinatorial Algebra*, with Mark Sapir as editor-in-chief. The “Aims and Scope” say,

Its domain is the rich and deep interplay between combinatorics and algebra. Its scope includes combinatorial aspects of group, semigroup and ring theory, representation theory, commutative algebra, algebraic geometry and dynamical systems.

I am not completely certain how this scope differs from that of the Springer *Journal of Algebraic Combinatorics*, but no doubt this will become clearer in time; a journal is defined by its contents more than by its aims and scope.

It is particularly pleasing that both these journals are specifically about combinatorics and something else. This is combinatorics doing the job it grew up to do, in my opinion.

And also, of course, the first issue of the *EMS Surveys in Mathematical Sciences* in 2014 was devoted to a survey article by Terence Tao on “Algebraic combinatorial geometry”.

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We haven’t had time to see much of the exhibition; a trip to the antipodes looms, and it will mostly be over when we get back. But we made time for a few things.

On the railings on the Scores, the work of four Scottish photographers had been blown up and printed on waterproof material to stay out for the whole festival. These photographers were consciously documenting Scotland: one travelled along the English border photographing the scene on the Scottish side, another photographed women farming in the most extreme conditions in the highlands and islands.

In the Old Union Coffee Shop in North Street, there was an exhibition of work by Franki Raffles, a feminist and photographer who left a large body of work depicting women in various situations, mostly work. The most memorable image for me was a shepherdess on a bleak mountainside in Georgia.

We managed to catch an exhibition by the St Andrews Photographic Society at the Town Hall, although the hours were variable: a mixed bag, as one might expect. But we really wanted to see the work of the St Andrews Photographers, a group which may have some connection with the St Andrews Photographic Society (I am not sure exactly), but includes our colleague Richard Cormack. There was claimed to be an exhibition in Holy Trinity Church, but we found the church firmly locked, perhaps because of Lammas Fair (even though, by then, the fair was over).

However, in Pittenweem (on the south coast of Fife), the art festival has been on for a week, and this was the closing weekend. This festival draws artists from Fife and well beyond, who rent houses, front rooms and garages in the town to display their work, and is always worth a visit. So we went yesterday, and found the St Andrews Photographers running an exhibition in a garage in the High Street. So we did after all get to see their work (and buy a couple of prints).

The whole festival was far too large to take all of it in, with over a hundred artists on show around the town and outside. We went to maybe twenty exhibits, and saw some remarkable work. Some of the best, to my eye, were reflections of the Forth Rail Bridge by Karen Trotter, and abstract landscapes by Lynn McGregor.

But the most notable thing about the day occurred elsewhere. The weather was glorious, one of the few really beautiful days we have had in this rather dreary summer. The blue of the sea and the red of the flowers were so intense that the art did slightly pall by comparison. After we were “arted out”, we decided to walk a short stretch of the coastal path, westward to St Monans and Elie.

As we were walking along the beach at the East Links of Elie, we heard an extraordinary, out-of-the-world, music. I wondered if it might have been seals singing. On a rocky point stretching into the sea I saw some black shapes sticking up. With my camera on maximum zoom, the viewfinder revealed them as cormorants, which were certainly not responsible for the sound. We tried to imagine that it was caused by the wind moaning through the ruined tower on the point. But I took a picture anyway.

When we got home, I looked at it on the computer screen, and saw that the rocky point was indeed covered with seals, who were so well camouflaged that I simply hadn’t seen them while we were there. I am certain that it was indeed singing seals that we heard. A memorable experience!

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A couple of talks by St Andrews students involved *transducers*, finite machines which read input, change state, and write output. Casey Donoven explained that the second world war Enigma machine is an example; it is *synchronous*, that is, it writes one output symbol for every input symbol it reads. A transducer can read an infinite input string, and so defines a map on the set of such strings, which is contninuous (in the product topology induced from the discrete topology on the alphabet).

The set of binary strings corresponds naturally to the “middle third” Cantor set in the unit interval, with 0 and 1 corresponding to taking the left or right subinterval at each step. There is also a map from the set of binary strings to the unit interval, given by the “binary decimal” representation, which is not quite bijective since countably many reals have two binary representations. A transducer which preserves these pairs of strings induces a continuous map on the unit interval, as Casey explained.

Shayo Olukoya considered the group defined by synchronous transducers which are also *bi-synchronizing*. This means that there is a number *k* such that, after reading *k* symbols, the machine is in a state which depends only on the symbols read, and not on the initial state, and the same holds for a transducer realising the inverse function. He explained how to solve algorithmically the problem of whether the induced map has finite order.

Experts will see Richard Thompson’s groups lurking in the background here. Nayab Khalid described a more general situation, a “substitution and glueing” procedure on directed graphs, which produces Thompson-like groups acting on interesting fractals like the *basilica Julia set*, and proving things about these groups.

Returning to Casey’s almost-bijection between Cantor space 2^{N} and the unit interval, there is also a famous almost-bijection between *Baire space* **N ^{N}** (the set of infinite sequences of positive integers) and the unit interval, given by the continued fraction. The

Jason Dungca also used the Gauss map, with the probability of an integer *n* proportional to 1/*n*^{2}; he was interested in phase changes in the multifractal spectrum of Gibbs measures. I won’t attempt further explanation.

Wojciech Ożański started his talk about singularities of the Navier–Stokes equation in fluid mechanics by reminding us of the definitions of Hausdorff and box dimension. He was interested in bounding these dimensions of the sets of singularities and blow-up times for a solution.

Douglas Howroyd defined these two dimensions and also Assouad dimension, the one he was chiefly interested in. He showed us how to compute the dimension of fractals defined by certain sets of affine contractions. These map the unit square onto various rectangular regions obtained from a dissection of the square, and go by the name of carpets.

Demi Allen used Hausdorff dimension to refine a result of Khintchine on diophantine approximation. Given a function ψ, let A(ψ) be the set of real numbers *x* in the unit interval which satisfy |*x*−*p*/*q*| < ψ(*q*)/*q* for infinitely many rationals *p*/*q*. Khintchine showed that the Lebesgue measure of A(ψ) is equal to 1 if ∑ψ(*q*) diverges, 0 if it converges (and ψ is monotonic). So, for example, if ψ(*q*) = *q ^{−t}*, the Lebesgue measure is 0 if

Khavlah Mustafa considered the Möbius (linear fractional) groups over certain rings such as the complex, dual, and double numbers. (These are obtained from the real numbers by adjoining square roots of −1, 0 and 1 respectively.) She described fixed points and orbits of 1-parameter subgroups, and gave a dynamical classification of the fixed points.

For classical groups over finite fields, Daniel Rogers described how to find maximal subgroups in the catch-all Aschbacher class *C*_{9}, which are themselves classical groups over fields of the same characteristic. (For ease of exposition he restricted his attention to the special linear groups.) He was particularly interested in dimension 16, the smallest for which the classification of maximal subgroups is not yet done, and showed us some interesting examples. He used results of Steinberg which made it all seem much easier than in fact it is.

Scott Harper talked about 2-generated groups (a class including the finite simple groups, as we know from the classification of these groups). His basic question goes back to Netto, who conjectured in 1882 that two random permutations generate the symmetric or alternating group with high probability (tending to 1 as *n*→∞). The *spread* of such a group is the maximum *k* for which, given any *k* non-identity elements *x*_{1},…*x _{k}*, there is an element

I was delighted by Alex Rogers’ talk, which took me back to things from my distant past, forty years ago. She was calculating the PI-degree (the smallest degree of a polynomial identity) for quantum matrix algebras. (For ordinary matrix algebras of *n*×*n* matrices, the answer is 2*n*, given by the Amitsur–Levitski Theorem.) My first real job was at Bedford College (a part of the University of London which no longer exists); the head of department was Paul Cohn, and his big idea was to develop “non-commutative algebraic geometry”, not so far from modern quantum groups, and PI-degree played a role in this. Then, at the end, Alex revealed that the formula for PI-degree in some cases depends on the invariant factors of a specific matrix, and these are powers of 2. This suggested to me the chains of binary codes that my second DPhil student, Eric Lander, invented and studied.

Christian Bean demonstrated the program Struct which he and his colleagues are developing for studying (and finding patterns in) sets of permutations, and if possible enumerating these sets. As he said, the program is not yet completely user-friendly.

Waring’s problem concerns expressing positive integers as sums of *k*th powers: how many powers are required, and how many solutions are there? Kirsti Biggs extended this to Waring’s problem with shifts, where we are given small irrational numbers θ_{i} and a number η, and we want to make the difference between the given *n* and a sum of *s* terms (*x _{i}*−θ

Adelina Mânzățeanu found rational curves on a smooth cubic hypersurface over a finite field, passing through two given points. She made much use of one of my favourite theorems, the Chevalley–Warning Theorem. Her methods involved analysis in **F**_{q}(*t*) in some sense parallel to analogous arguments over **Q**, and she gave us a dictionary for comparing the two situations. (What is the analogue of a torus, of the Fourier transform, in the completion of **F**_{q}(*t*)?)

Thanatkrit Kaewtem talked about γ-Banach spaces (these are like Banach spaces but the unit ball need not be convex; a typical example is *l*^{γ} for γ < 1). He was considering inner and outer entropy measures on a bounded operator between two such spaces, involving covering the image of a unit ball by translates of a small ball, an idea going back to Kolmogorov. Various inequalities connect these numbers to the operator norm and (in the case of an operator from a space to itself) its eigenvalues. He showed us why some of these inequalities are tight.

The first time I went to the YRM (in Warwick in 2011), there were no Queen Mary students there. Things were a little better this time. The sole QM student, Wan Nur Fairuz Alwani Wan Rozali, was considering a discrete dynamical system analogoes to the pendulum, given by iteration of a function on **Z**^{2} (I didn’t catch the exact formula for the function), proving a conjecture about its first return time to the positive X-axis.

With great self-restraint (there were some great talks, authoritative and exciting), I won’t say much about the plenary and keynote talks, except for one remarkable coincidence. Both Clément Mouhout, talking about partial differential equations, and Philip Welch, talking about axiomatic set theory, saw fit to tell us the story of the famous “Scottish Book” in which mathematicians of the Lwów school such as Banach, Kuratowski, and Ulam recorded problems. It has no direct connection with the location of the conference, but was named after the Scottish Café in in Lwów which the mathematicians used to meet. There could hardly be a better demonstration of the unity of mathematics than this!

And here are a couple of quotes from keynote speakers, brazenly taken out of context:

- “You are allowed to do that”
- “There are other dimensions also”

For me it was an excellent conference, with a higher than usual proportion of talks which completed a circuit in my brain by connecting to something quite different. I have plenty of new things to think about as a result. So thanks to Oliver, Daniel, Tom, Cristina, Zoë, and Sascha for putting on such a good show.

Mathematics is in good hands!

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Last week, Neill and family came to visit us, as part of their Scottish holiday.

We had a very pleasant time, with breakfast in the White Chimneys in Pitscottie, a walk through Dura Den and over Blebocraigs, an afternoon at the St Andrews Highland Games, and a taste test comparing the two Scottish/Italian ice cream shops in St Andrews, Jannettas and Nardini (I am pleased to say the local concern got the thumbs up).

At the Highland Games, three things impressed us: tossing the caber (I had never seen this done before, I am lost in admiration), the tug of war (some epic battles in which nothing seemed to happen for ten minutes until a small crack in one team’s defence led to a quick finish), and, maybe best of all, our local MSP and leader of the Scottish Liberal Democrats, Willie Rennie, giving a good account of himself in the open 1600 metres race. Incidentally, I wondered why the distances run were 90 metres and 1600 metres, until I realised that these are just 100 yards and 1 mile.

We talked about Pokémon GO. Neill’s view is that, at last, Life is catching up with Art. (He did capture a few of the little critters in St Andrews, including winning his first fight.)

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The only report I saw in the media was on the BBC, here. This suggested that the report said that universities should put more effort into impact. So when my colleague James Mitchell kindly circulated a copy, I approached it in a somewhat prejudiced frame of mind.

I turned first to the appendix, which gave a potted history of the REF and its predecessors, the RSE and RAE, about which I know a little bit. I found some depressing distortions. Here are some, with my glosses.

- “By the time of RAE 2001, the exercise had become the principal means of assurance of the quality of research.” (p.41) Some context missing here? I do not know anyone other than the funding councils and newspaper league tables who used it in this way.
- “It was originally intended that the appropriate weighting for impact in the REF should be 25%, but this was discounted in the first REF exercise to 20%, as an acknowledgment of the views of stakeholders that this was a developmental assessment process.” (p.43) Actually, academics roundly criticised and rejected impact as part of the assessment, and as a sop to this very strong feeling the weighting of impact was slightly reduced.
- “These changes meant that REF2014 took thorough account of the ‘environment’ for both research and impact.” How? “Environment: data on research doctoral degrees awarded, the amounts and sources of external research income and research income-in-kind.” (p.44) In other words, no concern for the actual environment, but the entry of metrics by stealth. More on this later.
- “Specific changes were introduced [for 2014] that were intended to reduce the burden of REF. However, these were not entirely successful. The costs involved in undertaking the REF, both for institutions and for HEFCE/HE funding bodies, were estimated at £246m for UK HE sector, considerably more than estimates for the 2008 framework which cost around £66 million.” How are these costs measured? In 1996 HEFCE announced proudly that the cost was only £3 million; but they took account neither of the costs to the universities of preparing the submissions nor the time spent by panel members reading the papers. (And a cheap shot: £246m would fund a very great amount of research, half a dozen projects in each university.)

But the rest of the report had some sense in it. I will restrict my comments here to the conclusions. Of course you should not assume that reading what I say is any substitute for reading the real thing. The document is here, and I quote parts of it verbatim.

The review is intended to deal with

- problems of cost, demotivation, and stress associated with the selectivity of staff submitted;
- strengthening the focus on the contributions of Units of Assessment and universities as a whole, thus fostering greater cohesiveness and collaboration and allowing greater emphasis on a body of work from a unit or institution rather than narrowly on individuals;
- widening and deepening the notion of impact to include influence on public engagement, culture and teaching as well as policy and applications more generally;
- reducing the overall cost of the work involved in assessment, costs that fall in large measure on universities and research institutions;
- supporting excellence wherever it is found;
- tackling the under-representation of interdisciplinary research in the REF;
- providing for a wider and more productive use of the data and insights from the assessment exercise for both the institutions and the UK as a whole.

Some laudable aims here, some perhaps less so; some supported better than others by the content of the document.

I won’t discuss all the recommendations in detail. Specific proposals involve including all research active staff and submitting outputs “at the level of UoA”. Not entirely clear what this means. How is the problem that joint papers with authors at different institutions can be submitted by both, but not if the authors are at the same institution? A quick scan through the document gave no answer to this question.

It is also recommend that outputs should not be “portable”, in an effort to stem the “transfer market” for top performers leading up to the REF census date. They admit that this will discourage researcher mobility, but offer no concrete suggestions here.

It is recommended that impacts should be done at institutional level. It is very difficult to see the rationale for this. Impact often derives from a collaboration between individual researchers at different institutions. (Another recommendation, allowing impact case studies to depend on a body of work and a broad range of outputs, is much more sensible.)

It is also suggested that environment statements should be moved to institutional level. As I already noted, these are mostly based on metrics which do not necessarily describe the research environment of departments, so this just moves from one kind of nonsense to another. In fact it is worse than that. These figures can be changed significantly at the stroke of an administrator’s pen. See Ken Brown’s report on how PhD awards to almost all mathematics departments have been savagely reduced because of a small change in EPSRC policy. Is it sensible for REF inputs to be tied to this?

Now some more general comments.

The review recognises that the REF in its present form tends to push research in safe directions which will guarantee the production of four good papers in the prescribed period; people eschew the risky but potentially more important topics, especially at the start of their careers when they should be most adventurous. But I found no recommendations for dealing with this problem.

The review found that the REF discourages interdisciplinary research, not because it is judged unfairly by the panels, but because researchers perceive that it might be. This is a trifle arrogant: researchers are not stupid and their perceptions are based on experience. In this context, an issue that concerns me (and many of the people who will read this) is the production of software; this is interdisciplinary in the sense that software production is computer science but it is an essential component of research in almost every academic discipline. My own perception is that the REF and its predecessors have never dealt fairly with outputs in the form of software.

On impact, the report says “Studies have demonstrated how the new impact element of the REF has contributed to an evolving culture of wider engagement, thereby enhancing delivery of the benefits arising from research, as captured through the impact case studies.” I am profoundly unconvinced. The words “headless” and “chickens” spring to mind when thinking about the reaction of colleagues to the production of impact case studies. This is mainly due to the extremely narrow definition of impact which is adopted, and the extremely strict rules applying to claiming it. Someone who introduces a new design for clinical trials which will make them yield more information from a given investment with less risk to participants cannot claim an impact case, because the pharmaceutical companies (who would have to certify that they have changed their practice to use the new design) are extremely reluctant to admit that anything was less than perfect with their old procedures. But at least, the review acknowledges the problem of narrow definition of impact, without actually proposing a better one.

And here is a scary statement: “Many, but not all, universities state that they use the REF intensively to manage research performance.” Allowing managers to direct our research is surely a recipe for disaster; can’t they see that?

Overall, my view is that a research assessment involving trusted academics judging papers in their fields can be a very good thing. It should give us the opportunity to showcase our best research *and* its impact. But the absurdly restrictive rule for impact case studies make it instead seem punitive.

The overall message of the Stern review is that REF is here for the long haul, and within it, impact is in for the long haul; but there is some recognition of flaws in the present system, and some chance of making changes if we all push hard.

What happens next? Stern proposes that the vague principles enunciated in his review should be turned into concrete and specific proposals, which can be put out for consultation, by the end of the year. Can they really do it so fast, without making a terrible botch of the whole thing? Wait and see …

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Thanks to John O’Connor for the picture.

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These questions can be answered, but the answers seem to depend on five somewhat arbitrary conventions.

- First, we have to distinguish between left and right. Now certainly most people can do this; but this is the only point where we make contact with “reality”. Also, the variety of words for “left” and “right” in European languages suggests that the ability (and necessity) to distinguish does not go back to the roots of language. Admittedly, “right” seems a bit more uniform than “left”, and so is maybe older, perhaps because of its association with a different meaning of “right”, cf. the French
*droit*. As has often been remarked, the words for “left” in French and Latin, “gauche” and “sinister”, do suggest some prejudice against left-handed people; maybe one day we will become so tolerant that these layers of meaning will disappear. (In my youth it was not uncommon for children to be “converted” to right-handedness, not always successfully.) - Next, we have to remember the association “left” and “motor”, and between “right” and “dynamo” (the two problems mentioned in the opening paragraph). I am sure we learned a mnemonic for this at school, though I can’t clearly remember what it was. Perhaps it was an association with the idea that the right hand is more dynamic, another instance of the prejudice mentioned above.
- Next, we have to remember the bijection between the thumb, index, and middle fingers and force (or motion), field, and current. The mnemonic was thuMb, First, and seCond finger. (This sounds like six choices, but there are really only two, since an even permutation doesn’t change the convention).
- Next, we have to remember the direction of a magnetic field line. I remember diagrams showing the field lines leaving the north pole of a magnet and entering the south pole, which of course were brought to life by experiments with iron filings. But added confusion comes from the fact that the north magnetic pole of the earth is actually a south pole and
*vice versa*. Easily explained: the north pole of a compass magnet is the one that points north, since opposites attract. - Finally, we have to remember which way current flows. It flows in the opposite direction to the way the electrons flow. This convention, of course, was established before the discovery of electrons, and involved an arbitrary choice of which terminal of a battery is positive and which is negative.

Given all these conventions, to solve the first problem, hold the left hand so that the first finger points in the direction of the magnetic field and the second in the direction of the current; the thumb will indicate the direction of the force. For the second problem, use the right hand, with thumb in direction of motion and first finger in the direction of field, the second finger will indicate the direction of current flow.

So you have to remember 5 bits of information, or (at least) get an even number of them wrong.

There are various connections here. The right-hand rule is related to the right hand rule for the vector product (cross product) of two three-dimensional vectors: if the first and second fingers of the right hand are in the direction of the two vectors, the thumb will be in the direction of the vector product. This rule is usually formulated as a screw rule: if we turn a right-hand screw in the direction from the first vector to the second, the screw will move forward in the direction of the product. This also seems to connect with reality. It is more natural to turn a screwdriver in one direction than the other: this is presumably because we use different muscles for the two actions. The more natural direction tightens a right-handed screw if done with the right hand. (Some people transfer the screwdriver to their left hand to undo a right-hand screw.)

Also, the left-right distinction connects with the direction of the magnetic field. In the northern hemisphere, if you stand facing the midday sun, the sun will rise on your left and set on your right, and the earth’s magnetic field will come from in front of you. (These things reverse in the southern hemisphere, and the tropics require special care; also, in the region within the polar circles, it may not be clear where the midday sun is.) Of course, if the earth’s magnetic field were to reverse, the force acting on a current-carrying wire in a magnetic field would not change!

To a mathematician, of course, the cross product (or exterior product) of two vectors from the real 3-dimensional vector space *V* lives in a different 3-dimensional space, the *exterior square* *V*∧*V*. I believe that physicists do recognise the distinction, by calling the vectors of *V* *axial vectors* and those of the exterior square *polar vectors*. (I think that is right, but this is another of those things which you presumably just have to remember.) They distinguish them by the fact that they behave differently under transformations of the underlying space. So the convention here is actually a choice of identification between *V* and its exterior square.

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