modular

the buckyball curve

Jul 2nd

Trinities

We are after the geometric trinity corresponding to the trinity of exceptional Galois groups

$\xymatrix{& L_2(7) \ar@{-}[rd] & \\ L_2(5) \ar@{-}[ru] \ar@{-}[rr] & & L_2(11) }$ <------> $\xymatrix{& \text{Klein quartic} \ar@{-}[rd] & \\ \text{Buckyball} \ar@{-}[ru] \ar@{-}[rr] & & \text{Buckyball curve} }$

The surfaces on the right have the corresponding group on the left as their group of automorphisms. But, there is a lot more group-theoretic info hidden in the geometry. Before we sketch the L_2(11) case, let us recall the simpler situation of L_2(7) .

There are some excellent web-page on the Klein quartic and it would be too hard to try to improve on them, so we refer to John Baez’ page and Greg Egan’s page for more details.

The Klein quartic is the degree 4 projective plane curve defined by the equation x^3y+y^3z+z^3x=0 . It can be tiled with a set of 24 regular heptagons, or alternatively with a set of 56 equilateral triangles and these two tilings are dual to each other

In the triangular tiling, there are 56 triangles, 84 edges and 24 vertices. The 56 triangles come in 7 bunches of 8 each and we give the 7 bunches of triangles each a different color as in the pictures below made by Greg Egan. Observe that in the hyperbolic tiling all triangles look alike, but in the picture on the left most of them get warped as we try to embed the quartic in 3-space (which is impossible to do properly). The non-warped triangles (the red ones) come into pairs, the top and bottom triangles of a triangular prism, one prism at each of the four ‘vertices’ of a tetrahedron.

The automorphism group L_2(7) acts on these triangles as S_4 acts on the triangles in a truncated cube.

The buckyball construction from a conjugacy class of order 11 elements from L_2(11) recalled last time, has an analogon L_2(7) , leading to the truncated cube.

In L_2(7) there are two conjugacy classes of subgroups isomorphic to S_4 (the rotation-symmetry group of the cube) as well as two conjugacy classes of order 7 elements, each consisting of precisely 24 elements, say C and D. The normalizer subgroup of C has order 21, so there is a cyclic group of order 3 acting non-trivially on the conjugacy class C with 8 orbits consisting of three elements each. These are the eight triangles of the truncated cube identified above as the red triangles.

Shifting perspective, we can repeat this for each of the seven different colors. That is, we have seven truncated cubes in the Klein quartic. On each of them a copy of S_4 acts and these subgroups form one of the two conjugacy classes of S_4 in the group L_2(7) . The colors of the triangles of these seven truncated cubes are indicated by bullets in the picture above on the right. The other conjugacy class of S_4 ‘s act on ‘truncated anti-cubes’ which also come in seven bunches of which the color is indicated by a square in that picture.

If you spend enough time on it you will see that each (truncated) cube is completely disjoint from precisely 3 (truncated) anti-cubes. This reminds us of the Fano-plane (picture on the left) : it has 7 points (our seven truncated cubes), 7 lines (the truncated anti-cubes) and the incidence relation of points and lines corresponds to the disjointness of (truncated) cubes and anti-cubes! This is the geometric interpretation of the group-theoretic realization that $L_2(7) \simeq PGL_3(\mathbb{F}_2)$ is the isomorphism group of the projective plane over the finite field $\mathbb{F}_2$ on two elements, that is, the Fano plane. The colors of the picture on the left indicate the colors of cubes (points) and anti-cubes (lines) consistent with Egan’s picture above.

Further, the 24 vertices correspond to the 24 cusps of the modular group $\Gamma(7)$ . Recall that a modular interpretation of the Klein quartic is as $\mathbb{H}/\Gamma(7)$ where $\mathbb{H}$ is the upper half-plane on which the modular group $\Gamma = PSL_2(\mathbb{Z})$ acts via Moebius transformations, that is, to a 2×2 matrix corresponds the transformation

$\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ <----> $z \mapsto \frac{az+b}{cz+d}$

Okay, now let’s briefly sketch the exciting results found by Pablo Martin and David Singerman in the paper From biplanes to the Klein quartic and the buckyball, extending the above to the group L_2(11) .

There is one important modification to be made. Recall that the Cayley-graph to get the truncated cube comes from taking as generators of the group S_4 the set $\{ (3,4),(1,2,3) \}$ , that is, an order two and an order three element, defining an epimorphism from the modular group $\Gamma= C_2 \ast C_3 \rightarrow S_4$ .

We have also seen that in order to get the buckyball as a Cayley-graph for A_5 we need to take the generating set $\{ (2,3)(4,5),(1,2,3,4,5) \}$ , so a degree two and a degree five element.

Hence, if we want to have a corresponding Riemann surface we’d better not start from the action of the modular group on the upper half-plane, but rather the action via Moebius transformations of the Hecke group

$H^5 \simeq C_2 \ast C_5 = \langle z \mapsto -\frac{1}{z}, z \mapsto z+ \phi \rangle$

where $\phi = \frac{1 + \sqrt{5}}{2}$ is the golden ratio.

But then, there is an epimorphism $H^5 \rightarrow L_2(11)$ (as this group is generated by one element of degree 2 and one of degree 5) and let $\Lambda$ denote its kernel. Observe that $\Lambda$ is the analogon of the modular subgroup $\Gamma(7)$ used above to define the Klein quartic.

Hence, Martin and Singerman define the buckyball curve as the modular quotient $X=\mathbb{H}/\Lambda$ which is a Riemann surface of genus 70.

The terminlogy is motivated by the fact that, precisely as we got 7 truncated cubes in the Klein quartic, we now get 11 truncated icosahedra (that is, buckyballs) in . The 11 coming, analogous to the Klein case, from thefact that there are precisely two conjugacy classes of subgroups of L_2(11) isomorphic to A_5 , each class containing precisely eleven elements! The 60 vertices of the buckyball again correspond to the fact that there are 60 cusps in this case.

So, what is the analogon of the Fano plane in this case? Well, observe that the Fano-plane is a biplane of order two. That is, if we take as ‘points’ the points of the Fano plane and as ‘lines’ the complements of lines in the Fano plane then this defines a biplane structure. This means that any two distinct ‘points’ are contained in two distinct ‘lines’ and that two distinct ‘lines’ intersect in two distinct ‘points’. A biplane is said to be of order k is each ‘line’ consist of k-2 ‘points’. As the complement of a line in the Fano plane consists of 4 points, the Fano plane is therefore a biplane of order 2. The intersection pattern of cubes and anti-cubes in the Klein quartic is this biplane structure on the Fano plane.

In a similar way, Martin and Singerman show that the two conjugacy classes of subgroups isomorphic to A_5 in L_2(11) , each containing exactly 11 elements, correspond to 11 embedded buckyballs (and 11 anti-buckyballs) in the buckyball-curve and that the intersection relations among them describe the combinatorial structure of a biplane of order three if we view the 11 buckys as ‘points’ and the anti-buckys as ‘lines’.

That is, the buckyball curve is a perfect geometric counterpart of the Klein quartic for the two trinities

At the Arcadian Functor, Kea also has a post on this in which she conjectures that the Kac-Moody algebra of E11 may be related to the buckyball curve.

References :

David Singerman, “Klein’s Riemann surface of genus 3 and regular embeddings of finite projective planes” Bull. London Math. Soc. 18 (1986) 364-370.

Pablo Martin and David Singerman, “From biplanes to the Klein quartic and the Buckyball” (note that this is a preliminary version, please contact David Singerman for the latest version).

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Galois, geometry, groups, hyperbolic, Klein, modular, Riemann, symmetry

Dedekind or Klein ?

Apr 22nd

Posted by lievenlb in groups

3 comments

The black&white psychedelic picture on the left of a tessellation of the hyperbolic upper-halfplane, was called the Dedekind tessellation in this post, following the reference given by John Stillwell in his excellent paper Modular Miracles, The American Mathematical Monthly, 108 (2001) 70-76.

But is this correct terminology? Nobody else uses it apparently. So, let’s try to track down the earliest depiction of this tessellation in the literature…

Stillwell refers to Richard Dedekind‘s 1877 paper “Schreiben an Herrn Borchard uber die Theorie der elliptische Modulfunktionen”, which appeared beginning of september 1877 in Crelle’s journal (Journal fur die reine und angewandte Mathematik, Bd. 83, 265-292).

There are a few odd things about this paper. To start, it really is the transcript of a (lengthy) letter to Herrn Borchardt (at first, I misread the recipient as Herrn Borcherds which would be really weird…), written on June 12th 1877, just 2 and a half months before it appeared… Even today in the age of camera-ready-copy it would probably take longer.

There isn’t a single figure in the paper, but, it is almost impossible to follow Dedekind’s arguments without having a mental image of the tessellation. He gives a fundamental domain for the action of the modular group $\Gamma = PSL_2(\mathbb{Z})$ on the hyperbolic upper-half plane (a fact already known to Gauss) and goes on in section 3 to give a one-to-one mapping between this domain and the complex plane using what he calls the ‘valenz’ function (which is our modular function , making an appearance in moonshine, and responsible for the black&white tessellation, the two colours corresponding to pre-images of the upper or lower half-planes).

Then there is this remarkable opening sentence.

Sie haben mich aufgefordert, eine etwas ausfuhrlichere Darstellung der Untersuchungen auszuarbeiten, von welchen ich, durch das Erscheinen der Abhandlung von Fuchs veranlasst, mir neulich erlaubt habe Ihnen eine kurze Ubersicht mitzuteilen; indem ich Ihrer Einladung hiermit Folge leiste, beschranke ich mich im wesentlichen auf den Teil dieser Untersuchungen, welcher mit der eben genannten Abhandlung zusammenhangt, und ich bitte Sie auch, die Ubergehung einiger Nebenpunkte entschuldigen zu wollen, da es mir im Augenblick an Zeit fehlt, alle Einzelheiten auszufuhren.

Well, just try to get a paper (let alone a letter) accepted by Crelle’s Journal with an opening line like : “I’ll restrict to just a few of the things I know, and even then, I cannot be bothered to fill in details as I don’t have the time to do so right now!” But somehow, Dedekind got away with it.

So, who was this guy Borchardt? How could this paper be published so swiftly? And, what might explain this extreme ‘je m’en fous’-opening ?

Carl Borchardt was a Berlin mathematician whose main claim to fame seems to be that he succeeded Crelle in 1856 as main editor of the ‘Journal fur reine und…’ until 1880 (so in 1877 he was still in charge, explaining the swift publication). It seems that during this time the ‘Journal’ was often referred to as “Borchardt’s Journal” or in France as “Journal de M Borchardt”. After Borchardt’s death, the Journal für die Reine und Angewandte Mathematik again became known as Crelle’s Journal.

As to the opening sentence, I have a toy-theory of what was going on. In 1877 a bitter dispute was raging between Kronecker (an editor for the Journal and an important one as he was the one succeeding Borchardt when he died in 1880) and Cantor. Cantor had published most of his papers at Crelle and submitted his latest find : there is a one-to-one correspondence between points in the unit interval [0,1] and points of d-dimensional space! Kronecker did everything in his power to stop that paper to the extend that Cantor wanted to retract it and submit it elsewhere. Dedekind supported Cantor and convinced him not to retract the paper and used his influence to have the paper published in Crelle in 1878. Cantor greatly resented Kronecker’s opposition to his work and never submitted any further papers to Crelle’s Journal.

Clearly, Borchardt was involved in the dispute and it is plausible that he ‘invited’ Dedekind to submit a paper on his old results in the process. As a further peace offering, Dedekind included a few ‘nice’ words for Kronecker

Bei meiner Versuchen, tiefer in diese mir unentbehrliche Theorie einzudringen und mir einen einfachen Weg zu den ausgezeichnet schonen Resultaten von Kronecker zu bahnen, die leider noch immer so schwer zuganglich sind, enkannte ich sogleich…

Probably, Dedekind was referring to Kronecker’s relation between class groups of quadratic imaginary fields and the j-function, see the miracle of 163. As an added bonus, Dedekind was elected to the Berlin academy in 1880…

Anyhow, no visible sign of ‘Dedekind’s’ tessellation in the 1877 Dedekind paper, so, we have to look further. I’m fairly certain to have found the earliest depiction of the black&white tessellation (if you have better info, please drop a line). Here it is

It is figure 7 in Felix Klein‘s paper “Uber die Transformation der elliptischen Funktionen und die Auflosung der Gleichungen funften Grades” which appeared in may 1878 in the Mathematische Annalen (Bd. 14 1878/79). He even adds the j-values which make it clear why black triangles should be oriented counter-clockwise and white triangles clockwise. If Klein would still be around today, I’m certain he’d be a metapost-guru.

So, perhaps the tessellation should be called Klein’s tessellation?? Well, not quite. Here’s what Klein writes wrt. figure 7

Diese Figur nun – welche die eigentliche Grundlage fur das Nachfolgende abgibt – ist eben diejenige, von der Dedekind bei seiner Darstellung ausgeht. Er kommt zu ihr durch rein arithmetische Betrachtung.

Case closed : Klein clearly acknowledges that Dedekind did have this picture in mind when writing his 1877 paper!

But then, there are a few odd things about Klein’s paper too, and, I do have a toy-theory about this as well… (tbc)

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Borcherds, Dedekind, google, groups, hyperbolic, Klein, modular, moonshine

the secret revealed…

Mar 26th

Posted by lievenlb in groups

3 comments

Often, one can appreciate the answer to a problem only after having spend some time trying to solve it, and having failed … pathetically.

When someone with a track-record of coming up with surprising mathematical tidbits like John McKay sends me a mystery message claiming to contain “The secret of Monstrous Moonshine and the universe”, I’m happy to spend the remains of the day trying to make sense of the apparent nonsense

Let j(q) = 1/q + 744 + sum( c[k]*q^k,k>=1) be the Fourier expansion at oo of the elliptic modular function. Compute sum(c[k]^2,k=1..24) modulo 70

I expected the j-coefficients modulo 70 (or their squares, or their partial sums of squares) to reveal some hidden pattern, like containing the coefficients of Leech vectors or E(8)-roots, or whatever… and spend a day trying things out. But, all I got was noise… I left it there for a week or so, rechecked everything and… gave up

Subject:   Re: mystery message
From:  lieven.lebruyn@ua.ac.be
Date:  Fri 21 Mar 2008 12:37:47 GMT+01:00
To:    mckayj@Math.Princeton.EDU

i forced myself to recheck the calculations i did once after receiving your mail.
here are the partial sums of squares of j-coefficients modulo 70 for the first 
100 of them

[0, 46, 26, 16, 32, 62, 38, 3, 53, 13, 63, 39, 29, 59, 45, 10, 60, 40, 30,
 10, 40, 26, 6, 56, 42, 22, 68, 48, 48, 64, 64, 45, 25, 15, 31, 31, 67,
 47, 7, 21, 51, 31, 31, 61, 21, 1, 17, 12, 2, 16, 46, 60, 20, 10, 54, 49,
 63, 63, 53, 29, 29, 23, 13, 13, 27, 27, 17, 7, 67, 43, 43, 52, 42, 42,
 16, 6, 42, 42, 42, 36, 66, 32, 62, 52, 66, 66, 0, 25, 5, 5, 35, 21, 11,
 11, 57, 57, 61, 41, 41]

term 24 is 42...
i still fail to see the significance of it all.
atb :: lieven.

A couple of hours later I received his reply and simply couldn’t stop laughing…

From:  mckay@encs.concordia.ca
Subject:   Re: mystery message
Date:  Sat 22 Mar 2008 02:33:19 GMT+01:00
To:    lieven.lebruyn@ua.ac.be

I apologize for wasting your time. It is a joke
depending, it seems, on one's cultural background.

See the google entry:

Answer to Life, the Universe, and Everything

Best, John McKay

Still confused? Well, do it!

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google, modular, moonshine

Monstrous Easter Egg Race

Mar 23rd

Posted by lievenlb in groups

1 comment

Here’s a sweet Easter egg for you to crack : a mysterious message from none other than the discoverer of Monstrous Moonshine himself…

From:  mckayj@Math.Princeton.EDU
Date:  Mon 10 Mar 2008 07:51:16 GMT+01:00
To:    lieven.lebruyn@ua.ac.be

The secret of Monstrous Moonshine and the universe. 

Let  j(q) = 1/q + 744 + sum( c[k]*q^k,k>=1) be the Fourier expansion 
at oo of the elliptic modular function.

Compute sum(c[k]^2,k=1..24) modulo 70

Background: w_25 of page x of the preface of Conway/Sloane book SPLAG 

Also in Chapter 27:
The automorphism group of the 26-dimensional Lorentzian lattice
The Weyl vector w_25 of section 2.

Jm

I realize that all of you will feel frustrated by the fact that most university libraries are closed today and possibly tomorrow, hence some help with the background material.

SPLAG of course refers to the cult-book Sphere Packings, Lattices and Groups.

26-dimensional Lorentzian space $\mathbb{R}^{25,1}$ is 26-dimensional real space equipped with the norm-map

$|| \vec{v} || = \sum_{i=1}^{25} v_i^2 – v_{26}^2$

The Weyl vector $\vec{w}_{25}$ is the norm-zero vector in $\mathbb{R}^{25,1}$

$\vec{w}_{25} = (0,1,2,3,4,\hdots,22,23,24,70)$ (use the numerical fact that $1^2+2^2+3^2+\hdots+24^2=70^2$ )

The relevance of this special vector is that it gives a one-line description for one of the most mysterious objects around, the 24-dimensional Leech Lattice $L_{24}$ . In fact

$L_{24} = \vec{w}^{\perp}/\vec{w}$ with $\vec{w}^{\perp} = \{ \vec{x} \in \Pi_{25,1}~:~\vec{x}.\vec{w}=0 \}$

where $\Pi_{25,1}$ is the unique even unimodular lattice in $\mathbb{R}^{25,1}$ . These facts amply demonstrate the moonshine nature of the numbers 24 and 70. Apart from this, the previous post may also be of use.

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Conway, groups, modular, moonshine

the McKay-Thompson series

Mar 22nd

Posted by lievenlb in groups

No comments

Monstrous moonshine was born (sometime in 1978) the moment John McKay realized that the linear term in the j-function

$j(q) = \frac{1}{q} + 744 + 196884 q + 21493760 q^2 + 864229970 q^3 + \hdots$

is surprisingly close to the dimension of the smallest non-trivial irreducible representation of the monster group, which is 196883. Note that at that time, the Monster hasn’t been constructed yet, and, the only traces of its possible existence were kept as semi-secret information in a huge ledger (costing 80 pounds…) kept in the Atlas-office at Cambridge. Included were 8 huge pages describing the character table of the monster, the top left fragment, describing the lower dimensional irreducibles and their characters at small order elements, reproduced below

If you look at the dimensions of the smallest irreducible representations (the first column) : 196883, 21296876, 842609326, … you will see that the first, second and third of them are extremely close to the linear, quadratic and cubic coefficient of the j-function. In fact, more is true : one can obtain these actual j-coefficients as simple linear combination of the dimensions of the irrducibles :

$\begin{cases} 196884 &= 1 + 196883 \\ 21493760 &= 1 + 196883 + 21296876 \\ 864229970 &= 2 \times 1 + 2 \times 196883 + 21296876 + 842609326 \end{cases}$

Often, only the first relation is attributed to McKay, whereas the second and third were supposedly discovered by John Thompson after MKay showed him the first. Marcus du Sautoy tells a somewhat different sory in Finding Moonshine :

McKay has also gone on to find these extra equations, but is was Thompson who first published them. McKay admits that “I was a bit peeved really, I don’t think Thompson quite knew how much I knew.”

By the work of Richard Borcherds we now know the (partial according to some) explanation behind these numerical facts : there is a graded representation $V = \oplus_i V_i$ of the Monster-group (actually, it has a lot of extra structure such as being a vertex algebra) such that the dimension of the i-th factor V_i equals the coefficient f q^i in the j-function. The homogeneous components V_i being finite dimensional representations of the monster, they decompose into the 194 irreducibles X_j . For the first three components we have the decompositions

$\begin{cases} V_1 &= X_1 \oplus X_2 \\ V_2 &= X_1 \oplus X_2 \oplus X_3 \\ V_3 &= X_1^{\oplus 2 } \oplus X_2^{\oplus 2} \oplus X_3 \oplus X_4 \end{cases}$

Calculating the dimensions on both sides give the above equations. However, being isomorphisms of monster-representations we are not restricted to just computing the dimensions. We might as well compute the character of any monster-element on both sides (observe that the dimension is just the character of the identity element). Characters are the traces of the matrices describing the action of a monster-element on the representation and these numbers fill the different columns of the character-table above.

Hence, the same integral combinations of the character values of any monster-element give another q-series and these are called the McKay-Thompson series. John Conway discovered them to be classical modular functions known as Hauptmoduln.

In most papers and online material on this only the first few coefficients of these series are documented, which may be just too little information to make new discoveries!

Fortunately, David Madore has compiled the first 3200 coefficients of all the 172 monster-series which are available in a huge 8Mb file. And, if you really need to have more coefficients, you can always use and modify his moonshine python program.

In order to reduce bandwidth, here a list containing the first 100 coefficients of the j-function

jfunct=[196884, 21493760, 864299970, 20245856256, 333202640600, 4252023300096, 44656994071935, 401490886656000, 3176440229784420, 22567393309593600, 146211911499519294, 874313719685775360, 4872010111798142520, 25497827389410525184, 126142916465781843075, 593121772421445058560, 2662842413150775245160, 11459912788444786513920, 47438786801234168813250, 189449976248893390028800, 731811377318137519245696, 2740630712513624654929920, 9971041659937182693533820, 35307453186561427099877376, 121883284330422510433351500, 410789960190307909157638144, 1353563541518646878675077500, 4365689224858876634610401280, 13798375834642999925542288376, 42780782244213262567058227200, 130233693825770295128044873221, 389608006170995911894300098560, 1146329398900810637779611090240, 3319627709139267167263679606784, 9468166135702260431646263438600, 26614365825753796268872151875584, 73773169969725069760801792854360, 201768789947228738648580043776000, 544763881751616630123165410477688, 1452689254439362169794355429376000, 3827767751739363485065598331130120, 9970416600217443268739409968824320, 25683334706395406994774011866319670, 65452367731499268312170283695144960, 165078821568186174782496283155142200, 412189630805216773489544457234333696, 1019253515891576791938652011091437835, 2496774105950716692603315123199672320, 6060574415413720999542378222812650932, 14581598453215019997540391326153984000, 34782974253512490652111111930326416268, 82282309236048637946346570669250805760, 193075525467822574167329529658775261720, 449497224123337477155078537760754122752, 1038483010587949794068925153685932435825, 2381407585309922413499951812839633584128, 5421449889876564723000378957979772088000, 12255365475040820661535516233050165760000, 27513411092859486460692553086168714659374, 61354289505303613617069338272284858777600, 135925092428365503809701809166616289474168, 299210983800076883665074958854523331870720, 654553043491650303064385476041569995365270, 1423197635972716062310802114654243653681152, 3076095473477196763039615540128479523917200, 6610091773782871627445909215080641586954240, 14123583372861184908287080245891873213544410, 30010041497911129625894110839466234009518080, 63419842535335416307760114920603619461313664, 133312625293210235328551896736236879235481600, 278775024890624328476718493296348769305198947, 579989466306862709777897124287027028934656000, 1200647685924154079965706763561795395948173320, 2473342981183106509136265613239678864092991488, 5070711930898997080570078906280842196519646750, 10346906640850426356226316839259822574115946496, 21015945810275143250691058902482079910086459520, 42493520024686459968969327541404178941239869440, 85539981818424975894053769448098796349808643878, 171444843023856632323050507966626554304633241600, 342155525555189176731983869123583942011978493364, 679986843667214052171954098018582522609944965120, 1345823847068981684952596216882155845897900827370, 2652886321384703560252232129659440092172381585408, 5208621342520253933693153488396012720448385783600, 10186635497140956830216811207229975611480797601792, 19845946857715387241695878080425504863628738882125, 38518943830283497365369391336243138882250145792000, 74484518929289017811719989832768142076931259410120, 143507172467283453885515222342782991192353207603200, 275501042616789153749080617893836796951133929783496, 527036058053281764188089220041629201191975505756160, 1004730453440939042843898965365412981690307145827840, 1908864098321310302488604739098618405938938477379584, 3614432179304462681879676809120464684975130836205250, 6821306832689380776546629825653465084003418476904448, 12831568450930566237049157191017104861217433634289960, 24060143444937604997591586090380473418086401696839680, 44972195698011806740150818275177754986409472910549646, 83798831110707476912751950384757452703801918339072000]

This information will come in handy when we will organize our Monstrous Easter Egg Race, starting tomorrow at 6 am (GMT)…

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Borcherds, Conway, modular, monster, moonshine, representations

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the buckyball curve

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the McKay-Thompson series

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modular

the buckyball curve

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Dedekind or Klein ?

the secret revealed…

Monstrous Easter Egg Race

the McKay-Thompson series

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