Complex Projective 4-Space

I am proud to announce that Complex Projective 4-Space is celebrating its first anniversary today. This is an excellent opportunity to quantitatively summarise what has happened on cp4space in the last year:

• Complex Projective 4-Space has been viewed 104125 times since its conception, with 3304 of those being on the 18th August (i.e. last Sunday). The United Kingdom topped the list with 43454 views, with America in close second (27746 views). The top 25 countries are displayed below. Hungary is surprisingly low in 25th place, when one considers that it was a discussion in Hungary several years ago that led to the creation of cp4space.

• There have been 41 ciphers and 33 distinct solvers, with a total of 236 ordered pairs of (cipher, person who has solved that cipher). Stuart Gascoigne is currently ahead, with Joseph Myers and Sam Cappleman-Lynes previously controlling that position.
• This is the 200th article. Two of the articles were guest posts by James Cranch and Gabriel Gendler; the remaining 198 were written by me.
• Recently, cp4space acquired a Facebook page as recommended by James Aaronson and Vishal Patil.

To celebrate the anniversary, I have been informed that Maria Holdcroft is making cp4space gingerbread men. I’ll upload photographs thereof as soon as possible.

What do the Sydney Opera House, the Human Papillomavirus and the following model of artificial life have in common?

The history of artificial life has been a long one, and few would argue that it originated with the Hungarian-American mathematician John von Neumann (whom you may recognise for his research in game theory and axiomatic set theory, development of the atomic bomb and the design of the modern computer). He first proposed a kinematic model, equipped with wheels, motors, joints et cetera, but then moved away from this in favour of his more abstract model of cellular automata.

To win a pint of beer in a pub in 1968, the computer scientist Edgar Codd created a spectacularly simpler self-reproducing cellular automaton, with just eight states (if you’re unfamiliar with cellular automata, think of these as analogous to different types of elementary particles) instead of the twenty-nine states of von Neumann’s model. The size of a self-reproducing machine was still very large, with the first explicit example being Tim Hutton’s implementation of Codd’s self-replicating computer.

In the eighties, Christopher Langton had the insight that self-replication could be made much simpler. Langton’s loops, pictured at the top of this article, were loosely based on Codd’s cellular automaton, but about six orders of magnitude smaller. A loop of genetic material circulates anticlockwise, instructing a pseudopodium to extend forward and turn left. The sequence of instructions is repeated four times, completing a copy of the loop which then breaks from its parent, acquiring a copy of the genome. This repeats indefinitely, filling space with a colony of self-reproducing loops:

Update (late 2018): I wrote a sequel to this post announcing the construction of an analogous loop-like replicator in a 2-state automaton, Conway’s Life. Unlike Langton’s Loops, these loops are programmable and can send signals to each other to determine how the colony grows.

Human Papillomavirus

One of the many beautiful aspects of Langton’s loop is that the same genome is expressed four times, exploiting the symmetry of the square loop to reduce the length of its representation to a sufficient extent that it fits within the loop. This is precisely reflected in nature by a variety of viruses including Herpes simplex and the Human Papillomavirus, both common sexually-transmitted infections.

The virus has chiral icosahedral symmetry, which means that its protein coat (or capsid) is composed of sixty congruent copies of the same fundamental region. The papillomavirus actually goes one step further, in that each fundamental region is constructed from an irregular arrangement of six copies of a single protein, called L1. In other words, the protein coat of this virus self-assembles from 360 identical parts. More details about the specific geometry are on ViralZone.

Of course, the virus itself only needs to have instructions for constructing a single copy of the L1 protein, a ‘subroutine’ that is executed 360 times in the construction of the capsid. This means that its genome can be really small compared with the size of the capsid — indeed, small enough to fit within it. In addition to the gene for the L1 protein, the viral genome encodes proteins for DNA replication and for attacking our P53 protein (used for repairing DNA and preventing cancer) to leave our DNA vulnerable to modification.

Modern and classical architecture

One of the most famous examples of modern architecture, the Sydney Opera House was designed and built over a sesquidecade by the architect J∅rn Utzon.

Image courtesy of Kazuhisa Togo.

The roof was originally meant to be composed of parabolae, but the construction costs would have been phenomenal. Utzon’s solution was to instead give all of the roof segments a constant Gaussian curvature, i.e. to make them sections of the same sphere. This meant that, like the Papillomavirus capsid, it could be made from many identical subunits produced in a single mould.

When asked ‘what did the Romans ever do for us?’, one of the many accomplishments was the aqueduct. There was much debate on the math-fun mailing list about why the Romans had semicircular arches, as opposed to the structurally superior parabolae (optimal for supporting large uniform walls) and catenaries (optimal for supporting their own weight). Some people suggested that the consideration was purely aesthetic, although my preferred theory is that it is another case of using many identical subunits:

Indeed, in aqueducts such as the Pont du Gard, the arches themselves form monomers, which are repeated many times along the length of the aqueduct. These are also arranged symmetrically, but with an infinite translational symmetry group instead of the finite rotational symmetry groups observed in Langton’s loops and the Human Papillomavirus. Viruses can also have capsids arranged according to an infinite symmetry group, as in the case of the helical tobacco mosaic virus. Both this structure and the more familiar icosahedral structure were predicted by Watson and Crick, who theorised that viruses must have many identical subunits arranged in a symmetrical fashion.

It has come to my attention that there is a maths camp taking place at Balliol College, Oxford, with the intention of subjecting young aspiring mathematicians to new ideas. Since one of the volunteers at the camp is advertising cp4space there, we decided that it would be a good idea to inspire the trainees with randomly selected examples of influential mathematicians:

Hypatia

I’ve decided to adopt a vaguely chronological approach, so we shall begin in Alexandria, a city in Egypt built by the eponymous Alexander the Great. It was the centre of education in the ancient world, and thus could be regarded as a 2000-year-old counterpart of Cambridge. Nevertheless, its library was far more aesthetically pleasing than our UL, and it is absolutely tragic that it no longer exists.

Hypatia was born in the fourth century AD, and eventually became head of the Platonist school of philosophy. She shifted the focus from empirical observation to more rigorous mathematical proof, thus helping to incorporate mathematics into physics. Indeed, one of her many accomplishments was determining the motion of celestial bodies, not unlike how Gauss predicted the trajectory of Ceres in his youth.

Hypatia also developed ideas from Diophantus’ Arithmetica, Euclid’s Elements and Apollonius’ Conics (three very influential mathematical texts from Ancient Greece), amongst others.

Ada Byron

You’ve probably heard of the famous poet Lord Byron, who had a number of interesting eccentricities including keeping a pet bear in the hallowed halls of Trinity College, Cambridge, where it would roam the vast, impressive courtyards and cause something of a stir amongst fellow Trinitarians.

His daughter, Ada Byron, went into a very different discipline, becoming the first computer programmer. She developed an algorithm for computing Bernoulli numbers on the Analytical Engine, the first programmable mechanical computer, which was designed by the Lucasian Professor of Mathematics, Charles Babbage (who also went to Trinity).

Most impressively, however, she realised that the Analytical Engine had a theoretical superiority over Babbage’s earlier Difference Engines and the calculating machines of Pascal and Leibniz, in that it could be programmed to perform functions of arbitrary complexity instead of mere routine calculations. Indeed, this insight could have resulted in Turing-complete computers being built long before Alan Turing.

Unfortunately, since it was mechanical and vastly complicated, the Analytical Engine was not built within Babbage’s lifetime. Indeed, only part of it has been reconstructed, and there is currently an effort to realise it in its entirety.

Ada Byron was also somewhat promiscuous — another attribute with which I can empathise. Indeed, together with my scandalous and outlandish (albeit successful!) expenses claims from the UKMT, some people have suggested respacing my name to form ‘Ada M.P. Goucher’…

Sophie Germain

A contemporary of Gauss and Legendre, Sophie Germain originally concealed her identity and used the pseudonym M. LeBlanc. Gauss acknowledged her genius in correspondence with Olbers; hence, she was certainly a mathematician of the highest calibre. Her work on Fermat’s Last Theorem was the first non-trivial progress since Fermat and Euler, and resulted in primes of the form (p − 1)/2 being called Sophie Germain primes.

(For some reason, the factorisation of is named after her, despite being completely trivial.)

Emmy Noether

In our universe, there are many symmetries. For instance, the laws of physics are invariant under translation (homogeneity), rotation (isotropy) and Lorentz transformations (Lorentz invariance). Using the Hamiltonian formulation of mechanics, Emmy Noether was able to show that each of these symmetries gives rise to a conservation law (e.g. conservation of linear momentum, angular momentum and energy).

She also contributed to pure mathematics; a Noetherian ring is one where all ideals are finitely generated. There is an Emmy Noether Society at Cambridge named in her honour, which have hosted talks by mathematicians such as Ruth Gregory. They also gave out free cake in the CMS, which was definitely worth attending.

Vicky Neale

One of the most prolific mathematicians of the 20th century was Paul Erdős. His collaborators and doctoral advisees include many famous mathematicians; for instance, he was the advisor of Professor Béla Bollobás FRS, who was in turn the advisor of Professors Imre Leader and Timothy Gowers, the latter of which advised Professor Ben Green. (They all went to Trinity as well — not that I’m advertising this amazing college, which is far better than Balliol.)

Tragically, Ben Green is no longer with us, but will always be remembered and his legacy shall live on in his students, including Professor Vicky Neale. Vicky Neale, Vesna Kadelberg, the absolutely brilliant IMO team leader James Cranch and I have been heavily involved in the organisation of a competition called the MOG, which some of you will be sitting in September. Good luck, and I hope this cp4space post has inspired you…

A Lindenmeyer system, or L-system, involves a recursive procedure applied to a string of symbols, where each symbol in the string is simultaneously replaced with a string, dependent on that symbol. For example, one of my favourite examples involves Easter eggs and rabbits, with the following rewrite rules:

"E" --> "R" (Easter egg hatches into a rabbit.)
"R" --> "ER" (Rabbit lays an Easter egg.)

Beginning with a single egg, the following growth pattern is observed:

"E" --> "R" --> "ER" --> "RER" --> "ERRER" --> ...

It’s trivial to show that the number of symbols in a particular iteration is given by the Fibonacci sequence. Indeed, for any L-system, we can obtain a recurrence relation for the number of characters, which can then be turned into a closed form involving powers of a particular ‘transition matrix’, which can be converted into an even more closed form by diagonalising the matrix. In this case, the closed form is Binet’s formula:

where φ and ψ are the roots of the equation x² − x − 1 = 0.

Audioactive decay

The lengths of strings produced by an L-system follow a basic linear recurrence relation, so they’re quite easy to calculate. A less obvious growth is that undertaken by Conway’s audioactive decay sequence. This begins with a single ‘1’, which is repeatedly run-length encoded:

• 1
• 11
• 21
• 1211
• 111221
• 312211
• …

For example, the third string 111221 contains 3 ‘1’s, followed by 2 ‘2’s and 1 ‘1’, so the next term is 312211. Since adjacent digits can interact, the behaviour is less predictable than that of an L-system. It transpires, however, that after 24 iterations any string will ‘decay’ into a disjoint union of non-interacting ‘elements’, of which there are 92. Treating these elements as individual ‘symbols’, the audioactive decay rule is just a complicated L-system! The dominant eigenvalue is Conway’s constant, the positive real root of a particular degree-71 polynomial:

Doron Zeilberger and his pet computer Shalosh B. Ekhad proved this in their paper on the subject. He mentions how it’s a posteriori trivial, since it takes no effort to write a computer program to verify that strings eventually decay into a meta-L-system, but not a priori trivial — similar rulesets (certain Post-Tag systems) can have no computable closed form, in which case the computer program would run forever and fail to find a proof.

Double-exponential growth and bangbangs

If a character can be replaced with the previous string, we can have double-exponential growth. For instance, the following rules:

X --> $
O --> O

(where $ indicates the entirety of the previous string) applied to the initial string XOX gives the following sequence:

XOX --> XOXOXOX --> XOXOXOXOXOXOXOXOXOXOXOXOXOXOXOX --> ...

where each term has 2^2^n ‘X’s.

Of course, this is not a particularly interesting example, since it just creates alternating strings of one-dimensional noughts and crosses. A far more exciting sequence, sent to me by Volker Grabsch, arises when one experiments with the recursive functionality of Unix/Linux, which has a ‘repeat previous instruction’ command, known as a bangbang and represented by two adjacent exclamation marks.

Shadab Ahmed discovered that the resulting behaviour is quite complicated, since single quotes and double quotes can interchange roles as being string delimiters and string contents, and they have a non-trivial effect on which bangbangs are expressed at any time. Volker Grabsch wrote an algorithm to emulate this process, calculating further terms in the sequence and creating an OEIS entry for the number of bangbangs in the nth iteration. Robin Houston then ingeniously used polynomials with exponents in the symmetric group to contain the information in the strings, yielding a simple recurrence relation and an efficient way to compute further terms.

What’s remarkable is that all of this collaborative research, from initial contemplation of the problem to final recurrence relation, took place just two days ago.

Tetrational growth

In a previous article, I mentioned the finite stages of the von Neumann universe, and wondered how many pairs of braces appear in each stage. It is trivial to see that it each term is an asymptotically exponential function of the previous term, and not too difficult (by considering how many braces an average subset contains) to derive a straightforward recurrence relation.

Interestingly, this was already on the OEIS under a different definition (based on rooted trees). I decided to add the next term to the b-file, even though they don’t generally like 20000-digit entries…

I have just been involved in the production of an action thriller about the International Mathematical Olympiad. Since certain intervals of time were inactive and unthrilling, I decided to pass the time by writing a few cp4space articles, amongst other things. This particular one was distributed across an evening and several train journeys.

Anyway, when I was at the actual IMO a few years ago, the opportunity presented itself to visit the M. C. Escher museum in The Hague. In the process, I acquired five postcards featuring the art of M. C. Escher, which I intend to put to good use in the immediate future. Until then, however, I shall merely analyse them:

Snakes, 1969

This was Escher’s final print, and has always been particularly aesthetically pleasing. However, until now I haven’t analysed it in great detail. Ignoring the snakes themselves, which are intertwined throughout the structure in a strikingly elegant pattern exhibiting three-fold cyclic symmetry, reveals an elaborate chainmail of interlocking annuli.

At first glance, the overall appearance resembles one of Escher’s more familiar Circle Limit engravings, where vibrant mythical creatures tessellate the Poincaré disc model of the hyperbolic plane. However, in addition to the limiting ‘circle at infinity’, there is also a limit point in the centre of the disc. How did Escher manage that? It’s certainly unlike any projection of any Riemann surface I’ve so far encountered.

On further inspection, one can see that something is slightly amiss. Each ring is linked with n others, were n = 6, 7 or 8, depending on the position. By taking the Voronoi partition, we obtain a greatly simplified picture of exactly what is happening in this masterpiece. Colour-coding regions according to the number of neighbours gives an even clearer diagram:

Towards the edge of the disc, the tiling is locally the truncated order-6 square tiling of the hyperbolic plane, where two octagons and one hexagon meet at each vertex. This can be regarded as a hyperbolic variant of the following Euclidean tiling seen on many kitchen floors:

On the other hand, the centre of the tiling induced by Snakes is devoid of octagons, instead being composed entirely of hexagons meeting three at each vertex, as in the hexagonal honeycomb. This is locally flat, and globally has been rolled into a cylinder (also zero curvature). Indeed, it is identical to a perspective view into a carbon nanotube, extending far into the distance.

Symmetry drawing E55, 1942

Preceding Escher’s Circle Limits were analogous tessellations of the Euclidean plane, imaginatively exploiting the admissible symmetry groups of periodic planar tilings (known as wallpaper groups). In this particular watercolour, fish of three different colours are arranged to exhibit the wallpaper group with signature 632, so called because the fundamental region (a single fish, since the symmetry group is sharply fish-transitive) of the tiling has centres of six-fold, three-fold and two-fold rotational symmetry:

This is known as Thurston’s orbifold notation, and heavily popularised in The Symmetries of Things by Conway, Burgiel and Goodman-Strauss. One of the chapters explores how colour can interact with symmetry, and Escher’s tiling is an excellent exemplar for exhibiting this classification. In particular, we consider symmetries to act on the n colours, giving a homomorphism into the symmetric group Sn. The kernel of this homomorphism is simply the normal subgroup of symmetries which preserve each colour. In this case, the kernel has orbifold notation 2222, as we can find four centres of two-fold symmetry preserving all colours.

The parallelogram in the image above is not the fundamental region; it is precisely half of a fundamental region.

Reptiles, 1943

Whereas the previous postcard was a colourful tessellation of fish, this one features a ‘symmetry drawing’ of reptiles (pun intended?) of signature 333, with a colour-preserving kernel of translations.

This still image suggests the following motion: two of the tiles are developing into three-dimensional crocodiles, which embark upon an anticlockwise trek around the lithograph, climbing over a regular dodecahedron in the process. Finally, they dissociate back into the sea of reptiles, losing their three-dimensionality and completing the cycle.

In the bottom-left corner is probably the union of a spherical cactus and an Aloe Vera, but I prefer to interpret it as a decapitated pineapple. The Geometry of the Pineapple (a possible sequel to Sphere Packings, Lattices and Fruits?) certainly merits further discussion.

A loxodrome is a curve drawn on a sphere, originating at the north pole, spiralling outwards so as to intersect the meridians at equal angles, and spiralling back into the south pole. Equivalently, it is obtained by stereographic projection of a logarithmic spiral onto a sphere. If you want to use the word ‘loxodrome’ in general conversation, I find that the easiest way to do so is to peel an orange thusly and participate in the discussion that undoubtedly ensues:

Anyway, from oranges to pineapples: the surface of a pineapple features eight clockwise loxodromes intersecting thirteen anticlockwise loxodromes, positioned at regular intervals. (The directions may be reversed, and (8,13) can be replaced by any pair of consecutive Fibonacci numbers.) As 8 is not equal to 13, a pineapple is chiral; one can have left-handed and right-handed pineapples.

Consequently, John Conway complains when artists frequently draw pineapples with bilateral symmetry, as this does not accurately reflect nature. I cannot determine from the postcard whether Escher has committed the same indiscretion, although I highly doubt it given the overwhelming accuracy of the Poincaré projection in his Circle Limit opus.

Möbius Strip II, 1963

Whilst this behaviour is most popularly exhibited by processionary caterpillars, ants also follow each other to a certain extent. In this woodcut, Escher populates the surface of a Möbius strip with a set of ants. Since the Möbius strip is non-orientable, the ants can see other ants walking on the underside of the surface. With an odd number (in this case, nine) of ants, the ants are directly between their subterranean counterparts; with an even number, equidistant ants form pairs with each ant immediately below its partner. Compare this with star polygons, which are connected if and only if the parameters are coprime.

Whilst reclining on the forbidden grass of our rivals, St. John’s College, a dialogue developed about the one-sided nature of the Möbius strip. Specifically, I was looking through these Escher postcards and explained how to construct a Möbius strip by introducing a half-twist into a strip of paper before connecting the ends. The person to whom I was talking astutely remarked that the surface of an ordinary sheet of paper is also technically one-sided [since it is a topological sphere].

Meanwhile, she mentioned how one of her favourite Escherisms that was absent from my minimal collection of postcards was Drawing Hands, where two hands equipped with pencils draw each other in a paradoxical self-referential way. In the bestseller Gödel, Escher, Bach, Douglas R. Hofstadter used this as the archetypal example of a strange loop. Other examples of strange loops in Escher’s works were Ascending and Descending, along with the beautiful 1961 lithograph featured on the fifth and final postcard, unanimously and undoubtedly our favourite Escherism:

Waterfall, 1961

The centrepiece is a perpetual motion machine of the first kind, exploiting the first law of thermodynamics by allowing a continually-descending stream of water to drive a turbine and return whence it came, completing the cycle. Unfortunately (or perhaps fortunately — a violation of the first law of thermodynamics could cause a major catastrophe), this cannot be used as a source of free energy as it is unrealisable in our three-dimensional world, , due to the presence of Penrose triangles in the design.

The fact that this can be drawn but not built is a consequence of the non-invertibility of the projective transformation used to convert a three-dimensional scene into a two-dimensional photograph. It has a non-trivial kernel, which means that many points (indeed, infinitely many) in the three-dimensional space are mapped to the same point in the photograph, causing a loss of information that can conveniently be exploited to yield impossible drawings.

You may notice that the towers are each decorated with a compound of three polyhedra: cubes in one example; irregular octahedra in the other. If Escher had used regular octahedra instead, the compounds would be perfect projective duals of each other.

Concluding remarks

• In conclusion, the combination of the architectural beauty, mathematical ubiquity and general popularity mean that I shall imminently be sending the Waterfall postcard, as opposed to any of the others. Of course, rather than tell you the intended destination and purpose, I’ll leave it as an exercise for the reader…
• I’ve tried to make this article more accessible and interesting to a general audience. Hence, if you enjoyed reading this you may want to consider sharing this on whatever social networks you inhabit, in case you know others who would also find it interesting. There have been 96884 views of cp4space since the 22nd August 2012, when it was founded, and I hope it will reach 100000 before the anniversary; consequently, any publicity will be gratefully welcomed!

Let P(x) be a polynomial of degree n. Let H(i) represent the number of `1’s in the binary expansion of the integer i. Although reasonably easy to prove, it may seem surprising that the following identity holds:

Does this theorem have a name? I discovered a variation of it years ago when solving a problem given to me by James Aaronson, and applied it again this weekend to reduce an infinite problem to a finite one. One nice corollary of this theorem is that any arithmetic progression of  numbers can be partitioned into two subsets, A and B, such that:

• A and B contain equally many elements;
• The sums of the elements of A and B are equal;
• The sums of squares of elements of A and B are equal;
• (ellipsis)
• The sums of the nth powers of elements of A and B are equal.

By iterative application of this theorem, we can partition an arithmetic progression of  numbers into subsets with this property. This is similar in principle to the idea of a multimagic square, but strictly less impressive.

Clark Kimberling’s Encyclopedia of Triangle Centres contains several thousand triangle centres, many of which are defined as intersections of three concurrent lines. For example, we have:

• The three medians of a triangle intersect at the centroid;
• The three altitudes of a triangle intersect at the orthocentre;
• The three angle bisectors of a triangle intersect at the incentre;
• The three perpendicular bisectors intersect at the circumcentre.

Indeed, it was joked that any symmetrically defined lines concur, the result being proclaimed the `symmetric geometry theorem’. Unfortunately, this is actually false, as shown by James Aaronson’s example of `the angular bisector of the median and the altitude’:

Hence, we don’t get a new triangle centre in this way.

Actual theorems

Nevertheless, there are many geometrical theorems with a large amount of symmetry, which we’ll refer to as `symmetric geometry theorems’. For instance, the configuration of Desargues’ theorem has 240 symmetries (isomorphic to S5 × C2) if you include projective duality. Similarly, Miquel’s six-circle theorem has 48 symmetries, which is evident by having the incidence structure of a cube:

Conway’s Petersen graph theorem

What was most interesting, however, was the following excerpt from an e-mail by John Conway, dated October 2003:

1.  If two [sic] each vertex of the Petersen graph one assigns a line in 3-space, in such a way that all but one of the edges correspond to pairs of lines that intersect at right angles, then so does the last edge.

This seems quite remarkable. Ten lines give 40 real variables, and there are 28 linear constraints; this corresponds to 12 degrees of freedom. Also, note that if we replace ‘intersect at right angles’ with just ‘have perpendicular directions’, then the theorem must also hold: we can reduce it to the original situation by translating all lines so that they pass through the origin.

I tried to construct an example with lots of symmetry. A solution to the problem automatically induces a solution where the lines pass through the origin, so my first thought was to try the ten diameters passing through opposite vertices of a dodecahedron. Remarkably, this does have a Petersen graph incidence structure, but only when ‘at right angles’ is replaced with ‘at some other angle that I can’t be bothered to calculate’. Unfortunately, it doesn’t satisfy the original theorem, so any actual solution must have significantly lower symmetry than the Petersen graph itself.

Anyway, I found a single reference to this on a Math Overflow question, also from John Conway:

An equivalent version is the following: suppose you have a “right-angled hexagon” in 3-space, that is, six lines that cyclically meet at right angles.  Suppose that they are otherwise in fairly generic position, e.g., opposite edges of the hexagon are skew lines.  Then take these three pairs of opposite edges and draw their common perpendiculars (this is unique for skew lines).  These three lines have a common perpendicular themselves.

The other examples on the page are purely projective theorems. Pappus’ theorem, Desargues’ theorem and Penrose’s ‘conic cube’ theorem are all self-dual, by the way. Here’s one of mine that isn’t self-dual, derivable from the quartic version of Cayley-Bacharach:

• Graph: rhombic dodecahedron;
• Vertices: six vertices are conics; the other eight are pairs of points;
• Edges: incidence;
• Statement: if five of the conics are present, then so is the remaining one.

It’s a grand generalisation of Miquel’s six circle theorem, with conics instead of circles and pairs of points instead of individual points.

For the third time on cp4space, I’m going to refer to my Wythoffian polyhedron generator. This constructs a polyhedron by the use of some two-dimensional fundamental region, bounded by mirrors meeting at predetermined angles. For example, a triangle with internal angles of π/2, π/3 and π/5 generates polyhedra with icosahedral symmetry:

Now, we can compute the Euler characteristic of a fundamental region, and use this to determine the number of copies of the fundamental region necessary to tile a surface of a particular genus. For instance, when the fundamental region is a triangle with internal angles of π/p, π/q and π/r, the Euler characteristic of the region is:

More generally, for an arbitrary polygon, we have:

• V is the sum of interior angles divided by 2π;
• E is half of the number of edges;
• F = 1.

In the icosahedral case, we have , so we need 120 copies of the fundamental region to tile this sphere (which has an Euler characteristic of 2). Instantly, we can determine from a polygonal kaleidoscope whether it generates an elliptic, flat or hyperbolic tessellation, and determine how many times it covers the region.

Let’s try another one, namely the hyperbolic triangle with interior angles of π/2, π/3 and π/7. This gives an Euler characteristic of . It can tile a three-holed torus (Euler characteristic −4) with 336 copies, arranged in a nice structure with automorphism group PGL(2,7).

What about using squares to tile a torus? This ends up being indeterminate, since the Euler characteristic of the torus and the square are both 0. The same applies to tiling a Klein bottle with hexagons, or a Euclidean plane with triangles.

More complicated kaleidoscopes

To describe groups generated by reflections in higher dimensions, one generally uses a Coxeter-Dynkin diagram. I’ve briefly discussed these before, so I’ll not reintroduce these remarkable objects. Instead, we shall concentrate on simply-laced Dynkin diagrams (all edges are unlabelled) that are simply-connected and have at most one vertex with degree greater than two. For example, the E8 diagram has branches of length 2, 3 and 5:

These are analogous to the 2, 3 and 5 in the dodecahedral symmetry group, and the analogy is related to the McKay correspondence. It’s also mentioned here, although we’ll approach it with slightly more generality. We associate a Dynkin diagram with one multivalent vertex with a polygon, whose internal angles are given by π/l, where l is the length of the branch. Hence, the D4 affine diagram corresponds to a rectangular fundamental region:

It’s also well defined, since measuring angles of π on the edges of the polygon corresponds to branches of length 1 (the central node in the diagram). The thing that really makes this connection natural is the following:

• The Dynkin diagram is finite/affine/hyperbolic precisely when the surface generated by the two-dimensional kaleidoscope is spherical/flat/hyperbolic, respectively.

So, the rectangular kaleidoscope *2222 (in Thurston’s orbifold notation) corresponds to the D4 lattice, and they both generate flat tilings. Other correspondences are below:

Spherical:

• Dihedral symmetry *22n ↔ D(n+2)
• Tetrahedral symmetry *332 ↔ E6
• Octahedral symmetry *432 ↔ E7
• Icosahedral symmetry *532 ↔ E8

Flat:

• *2222 ↔ affine D4
• *333 ↔ affine E6
• Square tiling symmetry *442 ↔ affine E7
• Hexagonal honeycomb symmetry *632 ↔ affine E8

Hyperbolic:

• *732 ↔ hyperbolic E9 lattice
• *666 ↔ hyperbolic Coxeter group containing the bimonster as a quotient

Unfortunately, I can’t see how this would generalise to the affine Dn diagram (for n >= 5), since the diagram has two vertices of degree 3. Similarly, this doesn’t work particularly well for the regular An diagram, which is just a straight line:

• A7 = 4—3—2—1—2—3—4

So, according to our correspondence, this gives this hosohedron:

Now, there is a certain ambiguity in the construction. For example, what if we had labelled A7 using an off-centre root node?

• A7 = 5—4—3—2—1—2—3

This would correspond to a digonal fundamental region with angles of π/3 and π/5, which just doesn’t work properly. Hence, we had no choice but to choose the centre node. A more awkward repercussion is that this is only possible for odd n. For example, the hosohedra for A3 and A5 are shown below, but the construction fails for A4.

Well, that is at least suggestive of what A4 should correspond to: a hosohedron with five lunes, alternately coloured. When one attempts to construct it, serious difficulties are encountered. However, we can avoid those complications by using the double cover of the spherical symmetry group, in which case we recover the original McKay correspondence.

We can also make sense of some infinite Coxeter diagrams. For example, the E∞ diagram (limit of En for all n, which gives an infinitely-generated Coxeter group) below corresponds to the *23∞ symmetry group (isomorphic to the modular group PGL(2,Z)):

I don’t think that this analogy between Dynkin diagrams and two-dimensional kaleidoscopes is anything other than a convenient mathematical coincidence, although it seems to be intimately related to the McKay correspondence.