Complex Projective 4-Space

Consider the Riemann sphere, obtained by adjoining a ‘point at infinity’ to the complex plane. The conformal maps from the Riemann sphere to itself form a group, the Möbius group, which is abstractly isomorphic to (and also the proper orthochronous Lorentz group , although I’ll discuss that later). Douglas Arnold and Jonathan Rogness made an excellent illustrative video:

[youtube=http://www.youtube.com/watch?v=JX3VmDgiFnY]

One way to construct the Möbius group is to embed it in a larger group generated by reflections in arbitrary lines and inversions in arbitrary circles, and to just take the index-2 subgroup of transformations that preserve orientation instead of reversing it. This is exactly analogous to the distinction between the orthogonal group and the special orthogonal group.

However, this larger group is interesting in its own right. But what should it be called? It can be thought of as ‘the group generated by Möbius transformations together with the field automorphism of complex conjugation’, and one would be very tempted to call this the projective semilinear group . So far, so good?

Actually, no.

Complex field automorphisms

In addition to complex conjugation, there are loads of silly field automorphisms of the complex numbers, obtainable by invoking the axiom of choice. Indeed, it’s a rather routine exercise:

1. Well-order the complex numbers, bijecting them to the initial ordinal of the correct cardinality. (This is the first line of most proofs involving the well-ordering principle.)
2. Cross out all of the algebraic numbers.
3. Choose the first complex number that isn’t in your well-ordering, and dump it into a set called ‘the generators’.
4. Cross out all of the numbers in the well-ordering that are not algebraically independent of ‘the generators’.
5. Repeat steps 3 and 4 uncountably many times, until we’ve exhausted the ‘list’ of complex numbers. Don’t worry; even though this sounds illicit, it is permissible by transfinite induction.

Now, permutations of the generators give silly field automorphisms of the complex numbers. Hence, the projective semilinear group is not just the Möbius group with reflections, but instead a ridiculously large group that we don’t care about.

These pathological automorphisms actually lead to some pathological geometric transformations of the complex projective plane, as described in a fascinating paper. These transformations map lines to lines, circles to circles, conics to conics and so on, yet are spectacularly discontinuous.

Hyperbolic geometry and the Lorentz group

Consider spacetime with one temporal and two spatial dimensions. It looks something like this:

The Lorentz group consists, unsurprisingly, of the Lorentz transformations, which are the linear transformations preserving the Minkowski dot product. Equivalently, they are the linear transformations fixing that hyperboloid of two sheets. If we discard one of the sheets, we obtain the orthochronous (time-preserving) subgroup.

From the perspective of the centre of the cone, the hyperboloid looks like an open disc. The orthochronous Lorentz transformations precisely correspond to distance-preserving transformations of the hyperbolic plane. These are themselves determined uniquely by a conformal (or anticonformal) transformation of the ‘circle at infinity’.

Adding an extra dimension, the orthochronous Lorentz group is isomorphic to the group of distance-preserving transformations of hyperbolic 3-space, which is again isomorphic to the group of (anti-)conformal transformations of the ‘sphere at infinity’, namely our index-2 supergroup of the Möbius group.

Moreover, this nicely generalises: the group generated by geometric inversions on the n-sphere is abstractly isomorphic to the orthochronous Lorentz group . And when n = 24, we get a very beautiful discrete subgroup, namely the automorphism group of the II(25,1) lattice intimately related to the Leech lattice.

I apologise for the sporadicity of recent cp4space posts, but I do have several legitimate excuses (including MOG marking, which occupied a significant portion of yesterday). Naturally, it would be improper for me to prematurely release the results, so you’ll have to wait until they appear on Joseph Myers’ website.

Anyway, Cantor first proved that there are multiple distinct infinite cardinals (infinite sets that cannot be bijected), such as the naturals and reals, which are countable and uncountable, respectively. It transpires that we can get ‘sets’ which are too large to be actual sets, and these are called proper classes. For instance, we cannot have a set of all sets, but we can have a proper class of all sets, namely the von Neumann universe. All proper classes are subclasses of V, some of which we shall explore:

You might be forgiven for thinking that we can form a set Ω of all ordinals, but unfortunately this is not the case. If we could, then Ω would itself be a limit ordinal and possess a successor ordinal, Ω + 1, which cannot be an element of Ω due to the Axiom of Foundation. Hence, Ω is not a set. However, it consists entirely of sets, so is a legitimate proper class.

The class of all ordinals is a subclass of the surreal numbers, a totally ordered ‘field’ (obeying all field axioms except for failing to be a set) invented by John Conway. It is the largest possible ordered field, and contains reals, ordinals and infinitesimals, amongst other things. People have also considered the surcomplex numbers, S[i], which are the algebraic closure of the surreals. I mentioned them earlier in connection with combinatorial game theory.

A less exciting proper class is the free complete lattice. This is an object with the following properties:

• Complete: For any set of points, there exists a supremum and an infimum.
• Free: No additional relations are present beyond those implied by the definition.

With just one generator, x, we get a finite singleton lattice. With two generators, x and y, we also get a finite set, namely {x, y, sup(x, y), inf(x, y)}. But with three generators, x, y and z, this suddenly explodes into a proper class. Specifically, we can construct a class of points with the same cardinality as the ordinals, by taking:

• Zero ordinal: ;
• Successor ordinal: ;
• Limit ordinal: supremum of for all β < α.

If we only define suprema and infima for finitely many arguments, then the free lattice is merely a countable set, rather than a proper class.

The Tammes problem asks for an arrangement of n points on the surface of a sphere, which maximises the minimum distance between any two points. This can be regarded as the classic problem of circle packing, but on the sphere (endowed with the usual metric) instead of a bounded region of the plane.

For n = 1, the problem is completely trivial. It should be easy to convince yourself that for 2, 3 or 4 points, they are placed at the vertices of a regular (n − 1)-simplex with unit circumradius; the last case corresponds to a regular tetrahedron. For n = 6, the best arrangement is the octahedral one. Solutions to Tammes’ problem roughly correspond to molecular configurations, although the molecular energy-minimisation problem is not quite the same as the minimax problem.

For n = 8, 12 and 24, the optimal solutions are the square antiprism (not the cube!), icosahedron and snub cube, respectively. The metric properties of the uniform snub cube involve the tribonacci constant (limiting ratio of successive terms of any non-trivial sequence such that each term is the sum of the three preceding terms), in a similar way to how the dodecahedron and icosahedron involve the golden ratio.

Anyway, you may notice that I’ve postponed discussion of the case n = 5. It transpires that we can’t arrange five points with a minimum angle greater than 90°, and with that minimum angle, it is always possible to add a sixth point (to give the octahedral vertex arrangement). It transpires that in chemistry, the favourable structure is a trigonal bipyramid (the north and south poles, together with three equidistant equatorial points).

Now, suppose we had a molecule with five distinguishable functional groups connected to a single atom in this arrangement. There are obviously 120 permutations of the functional groups, but these can be partitioned into twenty sets of six under the equivalence relation ‘rotations are equivalent’. Hence, there are precisely twenty isomers.

The Berry mechanism (shown above) gives a method to convert one isomer into another. The obvious thing to do at this point is to draw a graph, with a vertex for each isomer and an edge for each application of the Berry mechanism. We can immediately deduce that it is 3-regular and has 20 vertices. Also, note that rotations correspond to even permutations of the functional groups, wheread applications of the Berry mechanism correspond to odd permutations; hence, the graph is bipartite. Finally, if we partition the isomers into ten pairs of enantiomers, we can label each pair by the two axial (non-equatorial) functional groups, and note that the Berry mechanism must always change both axial groups.

Consequently, by the definition of the Petersen graph as a Kneser graph, we can deduce that this 20-vertex configuration must be the bipartite double cover of the Petersen graph, namely the Desargues graph. Organic chemists refer to this as the Desargues-Levi graph.

In a non-orientable universe such as a three-dimensional analogue of the Klein bottle, there are no such things as chiral compounds (enantiomers can be interchanged by rigid motions) and this graph would reduce to the Petersen graph. Indeed, the chemistry of a non-orientable universe would be very interesting. One can begin to speculate what would happen:

In the year 3000, an intrepid explorer embarks upon a voyage in a cutting-edge spaceship, endowed with the ability to travel at superluminal speeds. Upon leaving our Solar system, she overtakes the Voyager probes in an instant, passes through the Oort cloud in a few seconds, and continues in the general direction of the Andromeda galaxy. Increasing her speed yet further, the entire universe flashes past her, and she finds herself approaching a pleasantly wet planet.

She disembarks on an island, and finds a coniferous forest populated with pine trees. After convincing herself that it is safe to remove her oxygen mask, she is greeted by the smell of lemons. In spite of this, there are no lemon trees in sight. Proceeding slightly further, she notices a sign marked with strange symbols — evidence of intelligent life?

It transpires that travelling billions of lightyears is a rather exhausting experience, thus leaving our protagonist completely famished. Hence, it was serendipitous that she encountered a restaurant, and even more serendipitous that everyone there happened to speak English, despite having distinct writing. Delving into a three-course meal seemed to be a good idea at the time, and is rarely inadvisable. However, somewhere between the main course and the dessert, she passed away.

Basically, reflected food is at least incompatible with our digestive systems, and possibly even poisonous (Lewis Carroll suggested this in Alice Through The Looking Glass). As for the citric-scented pine trees, it is a consequence of the molecule limonene, the enantiomers of which have different aromas (D-limonene gives citrus fruits their distinctive smell, whereas L-limonene is found in pine needles).

Fortunately, our universe must be orientable due to a phenomenon called CP-violation, which essentially means that physics has chiral properties. Hence, the situation I envisaged above is theoretically impossible.

Chirality occurs in inorganic chemistry as well as organic chemistry, with several crystals lacking reflectional symmetry. Indeed, crystallographers count some three-dimensional crystallographic symmetry groups twice, as the left-handed and right-handed groups are not conjugate in the larger group of rigid motions. This behaviour does not have an analogue amongst any of the 17 two-dimensional wallpaper groups.

In fact, I forgot to mention that the Leech lattice is chiral, and can be characterised as the unique chiral even unimodular lattice of lowest dimension. Returning full circle, the arrangement of shortest vectors in the Leech lattice gives a unique optimal solution to Tammes’ problem in 24 dimensions.

Any fluid dynamicist would be able to tell you that fluids obey partial differential equations such as the Navier-Stokes equations. This macroscopic behaviour should, and indeed can, be derived from applying Newton’s laws of motion to the individual molecules in the fluid.

Remarkably, the rules of fluid dynamics apply macroscopically to systems much simpler than Newtonian billiard-ball mechanics. Perhaps the ultimate in simplicity is the lattice gas automaton, where space, time and motion are discretised. An example of a lattice gas is the HPP model, which can be simulated with the software Golly.

Each molecule can move in one of four directions, and collisions occur between antiparallel molecules. Each interaction can be verified to conserve momentum and energy, thus accurately representing Newtonian collisions. Unfortunately, one drawback to this model is that it is anisotropic, and vortices are square instead of circular.

This is unsurprising. A model called FHP is almost identical, but with a hexagonal grid instead of a square lattice. You may think that this would lead to hexagonal vortices, as one would expect. Remarkably, there is a sufficient amount of symmetry that it actually exhibits large-scale isotropy, and the vortices are actually circular! Consequently, FHP is a much more realistic model of fluid dynamics than HPP.

Of course, in reality fluids are three-dimensional, rather than two-dimensional. Unfortunately, the three-dimensional lattices (Z^3, D3 and D3*) are insufficiently symmetrical, and the automaton suffers from the same ‘square vortex’ effect as the HPP model. However, the D4 lattice in four dimensions is symmetrical enough, leading to an isotropic lattice gas automaton. I imagine that the same applies to the E8 Gosset lattice and the Leech lattice.

Another lattice gas is the pair-interaction lattice gas automaton, which operates on a square grid. Tim Hutton has made an open-source implementation called the Lattice Gas Explorer. By simulating flow past a circular obstacle on a massive grid (several megapixels), he was able to observe the natural phenomenon of a von Karman vortex street spontaneously emerging.

Before the advent of computers and numerical approximations, fluid dynamicists had a beautifully mathematical method of simulating fluid flow over an aeroplane wing. Firstly, they take a cross-section of the wing, which looks something like this:

Now, unfortunately that’s not circular. The fluid dynamicists reasoned that life would be much simpler if it were circular, in which case finding an analytic solution would be trivial. So, guess what they did? They made it circular, using the Riemann mapping theorem (which states that you can find a conformal mapping from any open bounded subset of the complex plane to the interior of a unit disc). There’s actually an algorithmic version of the Riemann mapping theorem called the Schwarz-Christoffel mapping, providing an explicit conformal map from your favourite polygon to the upper half-plane, which can then be transformed into the unit disc by applying a suitable Möbius transformation. I like to think of the Schwarz-Christoffel mapping as saying the following:

You can conformally map a polygonal peg into a round hole.

You may wonder why the fluid dynamicists had to take a cross-section, instead of conformally mapping the entire three-dimensional aeroplane into a sphere. The answer is that the Riemann mapping theorem only holds in two dimensions, and fails most spectacularly in higher dimensions (where Liouville’s theorem states that the only conformal maps are Möbius transformations).

Liouville’s theorem is really annoying, and (amongst other things) stifles any attempt to find an aesthetically pleasing three-dimensional analogue of the Mandelbrot set. The closest attempt is Daniel White’s Mandelbulb, but the presence of ‘whipped cream’ left its creators feeling somewhat unfulfilled. Ondřej Karlík has, nevertheless, made a stunning rendering of this beast:

The first instance of the concept of a symbolic mathematical equation is usually attributed to Robert Recorde, a Welsh mathematician, in a 1557 book with the rather verbose title ‘The Whetstone of Witte, whiche is the seconde parte of Arithmeteke: containing the extraction of rootes; the cossike practise, with the rule of equation; and the workes of Surde Nombers’.

That is to say, Recorde invented the equals sign, which is still in use today. However, over the centuries it has undergone significant length contraction. As Piers Bursill-Hall remarked (you may enjoy reading his History of Maths lecture notes or, even better, attending the aforementioned lectures), the symbol for equality was a pair of ridiculously elongated parallel lines, like so:

Even though he invented one of the most useful mathematical notations in existence, Recorde had very bizarre reasons for doing so. In The Whetsone of Witte, he explained his reasoning for choosing the symbol to represent equality, namely that ‘no two things can be more equal than a pair of parallel lines’ (!).

Actually, the hyperbolic Fermat equation shown above is not entirely in Recorde’s notation, since the Cartesian notation for powers had not been adopted. Recorde had a brilliantly clumsy method of expressing powers of quantities. We still had the Hindu-Arabic numerals at this point thanks to Fibonacci’s text, Liber Abaci (which Bursill-Hall accidentally spoonerised, resulting in him saying ‘[…] the great mathematician, Liberace’). Hence, Recorde would have expressed the quantities in this equation as:

30042907 square

So far, so good. And you can probably guess the next one:

96222 cube

Again, nothing out of the ordinary. But once we get to higher powers, the notation appears slightly comical:

43 zenzizenzizenzic

What? Zenzizenzizenzic???

I have to admit that this is definitely my favourite obsolete mathematical notation. It is based on a Germanised spelling of the Italian word censo (meaning ‘squared’), and exists in the Oxford English Dictionary, noted for possessing more copies of the letter ‘z’ than any other word. Some books feature 16th powers, described as zenzizenzizenzizenzic (starting to get ridiculous now), but this term unfortunately did not make it into the OED.

The fourth power, of course, was merely denoted zenzizenzic. The fifth power was called the first sursolid, and subsequent prime exponents were described similarly. There’s almost a sub-exponential-time algorithm for converting the Nth power into Recorde’s notation:

• Factorise the number N into a product of primes (with possible multiplicity). This can be done in sub-exponential time using the General Number Field Sieve.
• Sort the prime factors into ascending order. This can be accomplished in O(log N log log N) time with von Neumann’s merge sort.
• If there is only one instance of ‘2’ or ‘3’, replace it with ‘square’ or ‘cube’, respectively. If there is more than one instance, replace each ‘2’ with ‘zenzi’, each ‘3’ with ‘cubi’, and concatenate them with a final ‘c’ appended.
• Replace any remaining primes p with ‘(π(p) − 2)th sursolid’, where π is the prime-counting function. This can be accomplished in time O(p^(½ + ε)) with the Lagarias-Odlyzyko algorithm. Unfortunately, unless N is a smooth number, this final step is exponential in the input length, destroying what would otherwise be a sub-exponential-time algorithm.

Contrast this with linear time for expressing a power in Descartes’ notation, and it is clear that moving away from Recorde’s notation was probably a good idea.

Since the conception of the European Girls’ Mathematical Olympiad (henceforth abbreviated to EGMO), the British Mathematical Olympiad Committee has run an annual national olympiad specifically for aspiring female mathematicians. The third edition of the Mathematical Olympiad for Girls (MOG) takes place tomorrow, and over 1400 entrants have subscribed.

In its current instantiation, each candidate sits precisely five questions (and these five questions are uniform throughout the country). Unfortunately, this could be prone to collaboration and other slightly mischievous behaviour. At the opposite end of the spectrum, it would be logistically impossible to create different questions for each candidate, and to reliably compare them. In an attempt at compromise, suppose the BMO Committee decreed the following:

There shall be 24 questions in total, and each candidate is given 8 of these. Moreover, no two candidates can have 5 or more questions in common, since that would be conducive to collaboration.

This already imposes an upper limit on the number of girls who can compete. It’s not too difficult to establish an upper bound of 759 (unfortunately too low for this year’s MOG), and it transpires that this upper bound is actually attainable. Producing these 759 special octads (particular 8-subsets of {1, 2, 3, …, 24}) is a non-trivial task.

The special octads are precisely the codewords of Hamming weight 8 in the binary Golay code, which I briefly mentioned in my article about the Leech lattice. There are other ways to generate these octads, without having to endure the overkill of generating the 4096 words of the Golay code and throwing away 3337 of them. One of these is the Miracle Octad Generator of Curtis, which is a useful mathematical tool for exploring the Golay code and Mathieu groups.

Put simply, the Miracle Octad Generator (or MOG) is a 4 × 6 array of things, which may as well be the bottlecaps of these 24 empty bottles of Sandringham Apple Juice:

Here is an example of an octad, namely the eight bottlecaps that have been coloured red:

For a set of eight points to be a special octad, it needs to exhibit the following properties:

• The numbers of points in each of the six columns must have the same parity;
• The number of points in the top row must also have this parity;
• The column scores must form a hexacodeword.

The first and second conditions can be seen to hold, as the number of points in each column and the top row are all even. The third condition requires further explanation. We assign a score of 0, 1, ω and ω² (in the finite field of four elements) to each point in the first, second, third and fourth rows, respectively, and sum them in columns:

A hexacodeword is a word of six symbols over of the form:

(a, b, c, a + b + c, aω² + bω + c, aω + bω² + c)

In this case, 0101ωω² can be verified to be of this form, so that set of eight points satisfies the third condition and does indeed constitute a special octad.

The Mathieu group M24

The Mathieu group M24 is the group of permutations of the 24 bottlecaps that map special octads to special octads. A special octad is determined uniquely by a set of 5 points it contains (hence, there are octads in total), so the Mathieu group is transitive on sets of 5 bottlecaps.

From M24 to PSL(3,4)…

The MOG is ideally suited to analysing the subgroups of M24, which generally have simple descriptions in terms of the array. These are described in Chapter 11 of Sphere Packings, Lattices and Groups. Of particular interest is the following series of nested subgroups:

• (The stabiliser of the empty set is the entire group, M24);
• The stabiliser of 1 point is the Mathieu group M23;
• The stabiliser of 2 points is the Mathieu group M22;
• The stabiliser of 3 points is M21, directly isomorphic to PSL(3,4).

Indeed, the action of the projective special linear group PSL(3,4) on 21 points is the natural one, with 21 = 4² + 4 + 1 being the number of points on the projective plane over the field of four elements. Hence, this projective plane can be beautifully embedded in the MOG. The three fixed points are coloured blue, labelled I, II and III, and called Romans.

The sixteen gold bottlecaps are already in a 4 × 4 grid, which is the natural structure for an affine plane over a field with four elements, where the top-left corner is (0,0) and the top-right corner is (ω²,0) (extrapolate from there). The five green bottlecaps form the line at infinity, with the symbol referring to the value of the gradient y/x of the affine lines that pass through that point.

This can be extended to the projective general linear group PGL(3,4) if we include permutations corresponding to 3 × 3 matrices with determinants ω and ω² as well as those with determinant 1. However, those matrices cyclically permute the three Romans.

In fact, there is another outer automorphism of the projective plane, obtained by the field automorphism of complex conjugation (interchanging ω and its conjugate ω²). These operations double the size of the automorphism group, and correspond to odd permutations of the Romans. This overall group is called PΓL(3,4), or the projective semilinear group, which is a maximal subgroup of M24.

…and back again

Instead of reducing M24 to obtain PSL(3,4), we can actually use PSL(3,4) to construct the 759 octads in a way that avoids all of that tedious mucking about with the hexacode. Specifically, PSL(3,4) has 360 simple subgroups of order 168 (isomorphic to PSL(2,7)) and 168 simple subgroups of order 360 (isomorphic to the alternating group A6). There are three conjugacy classes of each.

The order-168 subgroups induce seven-point orbits (Fano subplanes), and the order-360 subgroups induce six-point orbits (hyperconics). Now, the 759 octads are:

• The 21 lines of the projective plane, each augmented with all three Romans;
• The 168 hyperconics, each augmented with two Romans depending on the conjugacy class;
• The 360 Fano subplanes, each augmented with one Roman depending on the conjugacy class;
• The 210 symmetric differences of two lines.

The octad group

The stabiliser of a special octad as a whole (without loss of generality, the standard octad comprising the three Romans together with the line at infinity) acts on the 24 points in two orbits: the octad and its complement, the 16 points of what was previously treated as an affine plane over F4. Instead, we’ll now treat it as an affine 4-space over F2, and the octad group is just the group of affine transformations. The subgroup fixing a point of the affine plane (without loss of generality, the ‘origin’ 0000) is just the general linear group GL(4,2), which is abstractly isomorphic to the alternating group A8. Indeed, it acts faithfully by even permutations of the elements of the octad, so M24 ‘explains’ the exceptional isomorphism between A8 and GL(4,2).

The dodecad group

Words in the binary Golay code consist of either 0, 8, 12, 16 or 24 points. Those with 8 points are the special octads on which we’ve concentrated so far. Equally interesting, however, are the umbral dodecads of 12 points, which can be constructed by taking the symmetric difference of any two overlapping special octads.

The stabiliser of a dodecad as a whole is the Mathieu group M12. Recall that the octad group together with fixing the origin splits the 24 points into subsets of sizes 1, 8 and 15, and ‘explains’ the exceptional isomorphism between A8 acting on 8 points and GL(4,2) acting on 15 points. Similarly, the dodecad group explains an exceptional isomorphism between M12 and M12 — that is to say, an exceptional outer automorphism. In the same way, M12 splits into two copies of A6 (or it might be S6), explaining the exceptional outer automorphism thereof. There’s actually a miniMOG (known as the kitten, for obvious reasons) to analyse M12, analogous to how the MOG can be used to analyse M24.