Complex Projective 4-Space

The second round of the British Mathematical Olympiad was taken yesterday, with the questions available online from Joseph Myers’ website. Results should be released shortly after marking, which takes place next Saturday.

Recall that a Riemann surface is a complex 1-manifold, or equivalently an orientable real surface locally endowed with the structure of the complex plane, enabling holomorphic functions to be defined on the surface. The most familiar examples are the complex plane itself and its one-point compactification, the Riemann sphere. Another example is an elliptic curve, which is homeomorphic to a torus when viewed as a real 2-manifold.

Now, suppose we have a non-constant holomorphic function f on the sphere. We are interested in the group G of automorphisms φ which preserve the values of f (that is to say, f(φ(z)) = f(z) for all z on the surface); G can be considered to be the symmetry group of f. G must be a finite subgroup of SO(3) and, depending on f, could be either a cyclic group (take ), dihedral group, or one of the polyhedral groups. In the latter case, the largest symmetry group is given by the icosahedral group of order 60:

A function with this symmetry is Klein’s icosahedral function, which is involved in a method of solution of the general quintic.

What about symmetry groups of the hyperbolic plane? We desire the one with the most symmetry or, more precisely, the one which minimises the absolute value of the Euler characteristic of the associated orbifold. This can be found by a quick case-bash, since we only care about orientation-preserving symmetries (rotations and translations) so can ignore the more complicated reflections and glide-reflections.

Suppose our orbifold has gyration points (points of rotational symmetry) of orders . Then the orbifold has an Euler characteristic of:

For an explanation of the above identity, note that the orbifold is a sphere with n cone-points (where the ith cone point counts as of a vertex), which we can cyclically connect to form a polygon with n edges, separating the sphere into two hemispherical faces.

To tile the hyperbolic plane, we require the Euler characteristic to be negative, so we’ll need at least three gyration points. If we have four or more gyration points, the best we can manage is an Euler characteristic of -1/6, corresponding to (2,2,2,3). With three gyration points, however, we want to find the smallest positive value of:

If two of (a,b,c) are equal, we can double the amount of symmetry by applying the rule (a,b,b) → (2,b,2a). Hence, assume without loss of generality that a < b < c. If a is 3 or more, then the best we can manage is the rather pathetic (3,4,5), which is strictly inferior to (2,4,5). Consequently, we can assume a = 2.

Now, either b = 4 (and the best solution is (2,4,5) with an Euler characteristic of -1/20) or b = 3 (and the best solution is (2,3,7) with an Euler characteristic of -1/42). Clearly the latter is preferable, so the (2,3,7) triangle group has the most symmetry amongst all hyperbolic tessellations.

Now, quotients of this tiling will yield compact Riemann surfaces (multi-holed tori) with maximal symmetry (order 84(g − 1), where g is the number of holes, or genus), known as Hurwitz surfaces. The simplest of these is Klein’s quartic, a three-holed torus with symmetry group PSL(2,7) of order 168. Next is the Macbeath surface, which has seven holes and the order-504 symmetry group PSL(2,8).

The next possible genus is 14, giving the order-1092 symmetry group PSL(2,13). Interestingly, there are three distinct Hurwitz surfaces with this symmetry group, known as the first Hurwitz triplet. This can be explained by the fact that in the real subfield of the field obtained by adjoining a principal 7th root of unity to the field of rationals, the number 13 is (up to multiplication by units) a product of three prime factors.

Considering surfaces of a much higher genus, R. A. Wilson demonstrated that the Monster group can be described as the automorphism group of a Hurwitz surface, corresponding to a presentation (with an unknown number of relators omitted). Apart from 64 exceptions (the largest of which is A167), all alternating groups are also Hurwitz.

By solving the equation , where |M| is the order of the Monster group, one can deduce that the group acts on an orientable surface with g = 9619255057077534236743570297163223297687552000000001 holes.

Suppose we have a circular track occupied by finitely many trains of various lengths travelling at the same speed. The trains collide elastically with each other. If the sum of the lengths of the trains is a rational multiple of the track length, then it can be proved that the trains will eventually return to their original configuration. Here’s an animated GIF I prepared earlier:

It’s actually quite fun to prove that the system is periodic (there is a very short elementary proof, but you have to think outside the box). I’ll leave this as an exercise to the reader, and possibly give a solution in about a week or so, when you’ve had sufficient time to think about it. If you find a proof, include it in the ‘comments’ section at the foot of this page.

Now for something completely different and not at all related: solitons. Solitons are stable localised waves which propagate at a constant speed, and occur as the solutions to certain partial differential equations (e.g. the Sine-Gordon equation). When two solitons collide, they temporarily interact and pass through each other, eventually (in the limit as the elapsed time tends to infinity) recovering as if there were no interaction. Indeed, this could be given as a somewhat convincing answer to the question ‘what happens if an unstoppable force meets an immovable object?’.

It has been proposed that soliton solutions to the Sine-Gordon equation may be used to propagate impulses of information at sub-neuronal scales in cellular structures called microtubules. More information is available in D. D. Georgiev’s paper on the subject. More elaborate soliton interactions are possible, such as this collision between an antikink and a standing breather:

A much more artificial example of this type of reaction is Dave Greene’s Heisenburp device in Conway’s Game of Life, where a glider (moving localisation) collides with a complicated arrangement of machinery, which sends a fast neuronal impulse along a diagonal wire (much more quickly than the glider could have travelled in vacuo) to a receiver arrangement, which subsequently replaces the glider in exactly the same position and phase it would have occupied had the machinery not been present.

From top-left to bottom-left, in clockwise order: the glider detector (analogous to the dendrite in a neuron); the impulse transmitter (resp. soma); the impulse receiver (resp. axon); a larger view of the impulse transmitter; and the entire device.

As is visible above, the device resembles a myelinated neurone, both in layout and function. Unlike actual neurones, however, this arrangement is incredibly fragile (and can be broken, for example, by sending two gliders in quick succession, or a glider along a different lane). Indeed, configurations in Conway’s Game of Life, at least on small scales, are the antithesis of solitons — the smallest perturbation can cause chaotic devastation.

Solitons also occur in both fundamental particle physics and in fluids, as explained in this video of David Tong having a bath:

[youtube=http://www.youtube.com/watch?v=Ederft9dkag]

These vortex ring solitons also support more interesting interactions. It is possible to project a faster vortex ring through the centre of a slower one, causing them to ‘leapfrog’ over each other. I seem to recall this being the subject of a talk at the Trinity Mathematical Society, but cannot remember the title of the talk. Nevertheless, a quick search found a YouTube video showing a simulation of vortex rings exhibiting this type of behaviour:

[youtube=http://www.youtube.com/watch?v=SPBMEXX5xBI]

As I’ve mentioned before, a snark is a 3-regular bridgeless graph which cannot have its edges 3-coloured such that each vertex is incident with one edge of each colour. One particular example is curiously named the Descartes snark, and is based on the Petersen graph.

The Descartes snark has 210 vertices, and can be described by beginning with the Petersen graph and applying operations that are guaranteed to convert one snark into another. For example, a single vertex can be replaced with a triangle of vertices, with the original graph being 3-edge-colourable if and only if the new graph is 3-edge-colourable; the contrapositive is that this transforms a snark into a different snark.

Applying this simultaneously to every vertex of the Petersen graph results in the skeleton of a truncated hemi-dodecahedron. This is the projective polyhedron obtained by identifying antipodal features of the truncated dodecahedron:

Some authors insist that a snark be triangle-free, so this is a slightly degenerate example.

The second transformation is more complicated. Concentrate on the large object, and suppose we have a 3-edge-colouring. After a short case bash (hint: consider the possible 3-edge-colourings of the central ‘star’, and extrapolate from there), one can see that the two edges emanating from the left must be the same colours (in some order) as the two edges emanating from the right, and these two edges must have distinct colours.

Hence, this ‘gadget’ is equivalent to a single edge between two vertices. In fact, the Petersen graph itself can be obtained by starting from a degenerate snark (an edge between two vertices with self-loops), applying this transformation, and then contracting the two triangles which result. But, anyway, if we gratuitously apply this operation to all of the 15 original edges of the Petersen graph present in our truncated hemi-dodecahedron, then the triangles are expanded out into nonagons and the snark is no longer degenerate:

Image courtesy of Gabriel Drummond-Cole

This is the Descartes snark. But why is it called the Descartes snark? René Descartes massively predated graph theory, so clearly he did not discover it. Alternatively, it may have been named in his honour, but it transpires that the truth is much more interesting. Indeed, it was not named after René at all, but rather after a certain Blanche Descartes.

Blanche discovered the snark in 1948. One would imagine that she made the same observation that the gadget behaves identically to a single edge, and thus realised that there are, in fact, infinitely many snarks obtainable by iterating the process of truncating vertices and replacing the original edges with meta-edges.

Anyway, when I say ‘she’, there is a slight caveat: Blanche is not an individual, but rather a collective comprising Bill Tutte, Leonard Brooks, Arthur Stone and Cedric Smith (the four legendary Trinity mathmos responsible for squaring the square). They concatenated their first initials to form BLAC, which is not a name, but easily augmentable to ‘Blanche’ by the insertion of additional letters. As for the surname, this was a clever pun on the phrase carte blanche (which transliterates as ‘blank paper’, and refers to an open agreement).

As mentioned in this article, Blanche Descartes (who is undoubtedly the oldest female Trinity mathematician) also published several poems in addition to her theorems, including one to celebrate the wedding of the daughter of the (equally fictional) Nicholas Bourbaki. Despite this, the four mathematicians were meticulously careful in not revealing the non-existence of Blanche Descartes…

My favourite author, after whom I was named, is undoubtedly Douglas Adams. I particularly enjoyed Dirk Gently’s Holistic Detective Agency (sadly, not quite as famous as his bestseller, The Hitchhiker’s Guide To The Galaxy), featuring a rather disorganised and slightly unscrupulous Cantabrigian detective whose methods rely on the ‘fundamental interconnectedness of all things’.

It has been observed that I share several characteristics with Dirk, including an ability to predict exam questions (co-discovering the result of FST2 2012 Q2 several months before the paper whilst experimenting with determinants, thus trivialising the question for anyone with whom I had discussed the result), although fortunately (unlike in Dirk’s case) it did not result in expulsion from Cambridge. Also, we both famously share a penchant for fine wine and dining.

One of the themes in Dirk Gently’s Holistic Detective Agency involves a sofa* wedged in an awkward position in a tortuous staircase, from which it cannot be moved (and, since rigid motions have inverses, it was noted that it could never have even reached that position ab initio!). Dirk Gently’s assistant**, Richard MacDuff, was experimenting with a computer simulation capable of rotating the sofa in three-dimensional space to liberate it, eventually concluding its impossibility.

* Also known as a settee, couch, or Chesterfield. An example of the aforementioned ‘fundamental interconnectedness of all things’ is that I was born in Chesterfield, a town in Derbyshire whose most distinctive feature is a church afflicted by erectile dysfunction. Additionally, a Chesterfield sofa featured in one of Adams’ other novels, the identity of which I shall leave as an exercise for the reader.

** This position involves performing various tasks that Dirk is unwilling or prohibitively lazy to execute himself, and suffering frequent minor misfortune as a result of Dirk’s blithering ineptitude. I imagine that Maria Holdcroft can empathise with him, being attacked by a particularly temperamental pen that exploded as she was writing a cp4space advertisement, its volatility originating from when I accidentally left it in my trousers whilst they were subjected to the aquatic violence of a washing machine. Similarly, she had to drive 100 miles with a post-crapulent friend of hers after a rather convoluted and initially promising plan of mine failed spectacularly.

According to Wikipedia, the origin of the immovable sofa problem was an event that occurred whilst Douglas Adams was at St. John’s College, Cambridge (our rivals***). Specifically, when a particular staircase was renovated, furniture was temporarily placed in the upstairs rooms to allow the redecoration to take place. The renovation, however, made it impossible to remove the sofas from the rooms, so they remained there for decades!

*** As a result of this, the bridge over the stream between Trinity and St. John’s is permanently locked. This caused me to miss lunch on one occasion, but that’s another story…

Intriguingly, this physical problem was independent of Leo Moser’s moving sofa problem, which is a two-dimensional unsolved mathematical problem. It is of a similar vein to the problem that inspired my article about punting in clearings of arbitrarily small Lebesgue measure, but involves maximising the object rather than minimising the space. Specifically, Leo asks:

What is the maximum area of a sofa (connected subset of the plane) that can be manoeuvred around a 90° corner in a corridor of unit width?

Claudio Rocchini produced a beautiful animation of a lower bound established by John Hammersley (an Emmanuel College alumnus best known for developing the theory of Monte Carlo methods) in a paper entitled ‘On the enfeeblement of mathematical skills by ‘Modern Mathematics’ and by similar soft intellectual trash in schools and universities’:

Amazingly, that is actually a reasonable shape for a sofa! The Hammersley sofa has an area of , which he conjectured to be optimal. According to private communication between John Conway and the about-to-be-mentioned Joseph Gerver, this was very slightly optimised to give a sofa with a larger area (the Shephard piano), which was in turn optimised by Richard Guy et al to achieve an area of 2.2156. Gerver then exhibited a sofa of area 2.2195 with piecewise-smooth boundary, which he conjectured to be [very close to] optimal.

Gerver, however, demonstrated something of far greater importance than a slightly improved lower bound, namely the existence of an optimal sofa (the alternative would be an unattainable supremum on the possible area). Specifically, he considered the topological space X of all possible sofas, and proved that there exists a compact subset J of X containing sofas arbitrarily close to the supremum. Then, by compactness, the image of J under the function  mapping a sofa to its area must also be compact and therefore attain its upper bound.

This technique can be applied to many other optimisation problems in geometry to give optimal solutions. For example, given a polygon P in the plane, there is a largest circle that can be inscribed in P. (Proof: the subset of giving admissible locations and radii of circles is bounded by finitely many weak algebraic inequalities, so is compact. Proceed as in the previous argument.)

Anyway, the moving sofa problem is merely a special case of the piano mover problem.

Quite a while ago in the Eagle pub in Cambridge, James Watson and Francis Crick announced the double-helix structure of DNA. This discovery was based on earlier X-ray diffraction experiments by Rosalind Franklin. In eukaryotic organisms such as ourselves, DNA has a linear structure (admittedly tightly coiled into a chromosome, but there is no difference from a topological perspective). By comparison, bacteria have a circular loop of DNA:

In the diagram, an orientable surface (Seifert surface) is drawn, making the DNA into a ribbon. It is not immediately obvious that the surface must be orientable, without considering DNA at a more atomic level. Specifically, why can’t DNA form the boundary of a Möbius strip?

Each of the deoxyribose-phosphate backbones is directed, possessing a 3′ and 5′ end (pronounced ‘3-prime’ and ‘5-prime’, respectively, which is nice because both 3 and 5 are primes). And, as you can see, the opposite sides of the double-helix are directed in opposite directions, so the number of twists must be a full integer, rather than the half integer necessary for a Möbius band.

Now, if we forget the base pairs connecting the two deoxyribose-phosphate backbones, we obtain two linked (topological) circles. The unique topological invariant specifying how two closed curves (able to pass through themselves but not each other) are linked is the linking number. To prove this fact, we homeomorphise space so that one of the unknots is a fixed circle, and then observe that the complement of this circle has fundamental group .

Given a three-dimensional parametrisation of the unknots, the linking number can be computed by Gauss’s linking integral:

In particular, this determines the signed area of the image of the Gauss map, which is a multiple cover of the sphere.

Alternatively, we can draw a diagram of the directed curves and count signed crossings where the red curve passes over the blue curve:

Consequently, the linking number of this DNA strand is +10.

There is an enzyme called topoisomerase which changes the linking number of DNA without any net effect to the local topology of the DNA. (That is to say, the labelled graph formed by associating a vertex to each atom and edge to each bond is unchanged.) Indeed, there are several different types of topoisomerase, which increment or decrement the linking number by different amounts.

Now suppose we take the entire DNA loop and bend it into a lemniscate (this is a homotopy, thus invisible to topologists). The linking number must remain constant, but we have added what is known as a writhe. This results in a corresponding change to the number of twists in the DNA double helix, by the formula (the linking number is the signed sum of the writhes and twists).

Now, DNA minimises its energy when there is precisely one twist per 10.4 base pairs. Hence, if the linking number is different from this, the DNA will automatically introduce writhes to compensate in a process known as supercoiling. When the writhe number is significant, DNA takes on a superhelical structure. Two common structures that occur in this way are the toroidal and plectonemic structures:

You can observe supercoiling with a plain piece of string. If you keep twisting it, then eventually the strain will be too much and it will supercoil into a larger helix. Eventually, after further twisting, that helix will super-supercoil into an even larger helix, and — if the string is sufficiently long, such as a metre-long string with one end held fixed and the other attached to an electric motor — you can obtain multiple nested levels of supercoiling in this manner. Indeed, this phenomenon occurs in chromosomes, with extra structural scaffolding (known as histones) to bind it in this tight efficient structure: a human cell nucleus is only 5 microns wide, but contains nearly a gigabyte of genetic material.

It’s now New Year’s Eve, which is why the cp4space banner has changed again.

As is usual for this time of year, there is a joint Anglo-Hungarian IMO training camp in Tata, Hungary. I’ve mentioned it before at the end of 2012, but perhaps it is worth re-iterating that this is where Complex Projective 4-Space was originally popularised. I shall start by discussing a problem that appeared on the selection test at the end of this camp two years ago, which recently resurfaced on an online forum.

Given any arrangement of white and black tokens along the circumference of a circle, we’re allowed the following operations- take out a white token and change the colour of both its neighbours, or put in a white token and change the colour of both its neighbours. Is it possible to go from a configuration with just two tokens, both white, to a configuration with two tokens, both black?

You may want to attempt this problem for yourself before reading further. One of the members of the forum posted graphs of what appear to be three equivalence classes of ‘necklaces’ (we shall later prove whether or not this is actually the case), where edges represent elementary operations. Here is one of the equivalence classes:

I had seen an ‘official’ solution to this problem, but as is usual for official solutions, it was not particularly insightful. By contrast, Gabriel Gendler devised a much more interesting interpretation, which, whilst essentially equivalent to the official solution, offers far more insight and generalises beautifully.

The idea is to ‘break’ the necklace at an arbitrary prespecified point, so that the beads form a linear string. For example, we can break the following necklace at the point specified by the arrow to give a linear string:

This string will be represented as WWBWBB. Cutting the same necklace at a different point would give a different string, such as WBWBBW, but we will consider this caveat later. For now, we will focus on the simpler problem where the string is linear, rather than cyclic. This is the famous word problem, which is undecidable in general but easy in this particular case. We can summarise the elementary operations as relations:

• BWB = WW
• BWW = WB
• WWB = BW
• WWW = BB

If we allow the elements to have inverses, then this becomes a group presentation. Gabriel noticed that in particular, this is a presentation for the permutation group S3, where B = (1 2) is a transposition and W = (1 2 3) is a 3-cycle. We can convert between words only if* they represent the same group element in S3.

* Again, if we allow inverses, things become much neater and the converse also holds. Otherwise, it is necessary to consider small cases separately.

However, we now need to remember that in the original problem, the string is cyclic. Hence, where x and y are arbitrary strings, xy and yx represent the same necklace cut at different places. Consequently, as well as the local operations encoded in the group presentation, we also need to consider the global operation of ‘remove a string from the left and append it to the right’.

Gabriel remarked that if xy = e, then yx = e (where e is the identity) and therefore it is impossible to convert an identity necklace into a non-identity necklace or vice-versa, solving the original problem. However, it doesn’t solve the problem of whether we can convert between BBB and WWWW, since these are both non-identity necklaces. However, thinking for a moment results in the following observation.

• yx = y xy y’

Here y’ is the inverse of y. So, we can think of moving the string y from the left to the right as conjugating by y. Indeed, this operation of rotating the beads on the necklace is precisely the same operation as conjugation: no more; no less. This leads to this rather neat conclusion:

We can convert between two necklaces if and only if they are in the same conjugacy class.

Christmas generalisation

Gabriel attempted to construct similar problems based on other groups. I mentioned how the dihedral groups have relatively small group presentations, and this is the smallest non-trivial example. Eventually, I suggested the following problem:

We have three types of bead (red, gold and forest green, to make this problem nice and festive!), and are allowed the following operations:

• Insertion or deletion of five consecutive red beads;
• Insertion or deletion of three consecutive gold beads;
• Insertion or deletion of two consecutive green beads;
• Replacing a red-gold (in clockwise order!) pair with a green bead, or vice-versa.

Then, a nice problem is to determine necessary and sufficient conditions for it to be possible to convert a necklace of m red beads into a necklace of n red beads. Also, you may want to work out which group underlies this problem (the answer is nice and simple!), and how many conjugacy classes it possesses; post your solutions in the ‘comments’ section at the bottom of this post.

Problems, books and quotations

Anyway, Gabriel’s solution is what Paul Erdős would call a proof from the book, meaning that it is clearly the right way to think about the problem. I particularly like this problem from the Advanced Mentoring Scheme, which has a horribly ugly case-bash official solution as well as a beautiful proof from the book. See if you can find the latter (again, post your solutions as comments on this article):

A class of students takes eight exams of which each student passes exactly four. For each pair of students, there are precisely two exams passed by the first and not the second, and precisely two passed by the second and not the first. What is the maximum possible size of the class?

Completely coincidentally, whilst checking my e-mails to find the exact verbatim statement of the AMS problem, I have just received one from Gabriel entitled ‘positive consequence of cp4space’ explaining how he used the Cayley-Bacharach theorem to solve question G6 on the 2006 IMO shortlist. Again, this is shorter and more elegant than the official solutions, although I wouldn’t go as far as saying that this was a proof from the book (a term generally associated with using more advanced theory than is necessary to understand the problem). Indeed, Erdős believed that God has a book containing the most elegant proof of each theorem.

More generally, Erdős famously used rather colourful terminology to refer to various concepts. They are mentioned in this delightful comic by Mike. Nevertheless, I suppose that my favourite amusing quotation by a great mathematician is this one of Hilbert’s:

A mathematician should have both a wife and a concubine. That way your wife thinks you are with the concubine, the concubine thinks you are with the wife, and finally you have time to do some maths.

(Depending on the gender and sexuality of the mathematician, you may need to replace ‘wife’ with ‘husband’. I think that concubine is technically gender-neutral.)

More mathematical quotations can be found on MathOverflow.