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 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)

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!

There seems to be a slight chance that the next US-administration may (finally) be joining the rest of the civilized world and sign the Kyoto-treaty. Here’s an appeal to Flock and other webbrowsers : please add blackle.com to our Search Engine Preferences!

The idea is simple : you Google as you’d do anyway but … you save a lot of energy. Via PD2 (for Pseudonymous Daughter 2).

When more than 200.000 websites are using google analytics, a one-man action proposed last time to de-activate GA on neverendingbooks is pretty useless. Perhaps a better alternative is :

When you use google analytics on your site, announce this and add a link to CustomizeGoogle BLOCK GOOGLE ANALYTICS COOKIES. Your visitors can then decide whether or not to block google analytics cookies only.

Or link to this post, as I’ll give a detailed step-by-step instruction so that even web-newbies can protect themselves agains google analytics stalking cookies.

1. Surf using a Firefox-clone Because CustomizeGoogle “is a Firefox extension that enhances Google search results by adding extra information (like links to Yahoo, Ask.com, MSN etc) and removing unwanted information (like ads and spam)” it only works on Firefox-like browsers. Screenshots below use Flock, the mac “social web browser”.

2. Go to CustomizeGoogle BLOCK GOOGLE ANALYTICS COOKIES and click on the Install Now! link (upper right)

3. A warning message will appear saying “Flock prevented this site from asking you to install software on your computer”. As we do want to install, click on the Edit Options button (on the left).

4. A pop-up window appears and click on the allow-button.

5. Click again on the Install Now! link from CustomizeGoogle. A pop-up will appear asking you to confirm installation of the extension. Click on the install-buttom (right below).

6. Installation complete! But you have to restart before you can use it. Click on the restart button and Flock will do it.

7. Select under Flock/Tools CustomizeGoogle Options.

8. A pane pops up with plenty of configurable options.

9. Select “Privacy” and mark the ‘Don’t send any cookies to GoogleAnalytics’ option.

10. Done! You can now surf to any of the 200.000+ google-analytics-powered sites without being stalked!

Using this feature makes you more anonymous. But your visit on a single webpage can still be logged. This way, both Google and the owner of the website knows that someone visited a webpage, but it’s difficult to track all pages you’re visiting. And it’s really really hard for Google to track that you visited both Website A and Website B.

Ever since my accidental ‘discovery’ of the word CEILIDH written on a numb3rs blackboard, I keep an eye on their blackboards whenever I watch a new episode, and try to detect terms I might know. Here’s todays screen-shot

I had to choose one frame from a minute long shot (the ‘link’ on the left hand side is more recognizable in other frames). Anyway, here’s what I thought to recognize : a link, a quaternion-algebra over a number field,

to be precise, and a in it. So did someone construct link-invariants from maximal orders in quaternion algebras? I surely didn’t know, but when in doubt there is always… google. I searched for ‘link invariant quaternion algebra maximal order’ and the third hit on page 2 gave me a pdf-file of a paper which seemed to have the relevant terms in it.

The paper is Automorphic forms and rational homology 3–spheres by Frank Calegari and Nathan Dunfield. Bingo! Their first figure is the ‘link’ drawn on the blackboard

which actually turns out to be a graph… This couldn’t be a coincidence, so as in the ceilidh-story, there had to be a connection with one of the authors. ‘Calegari+numb3rs’ didnt look promising but ‘Dunfield+numb3rs’ returned the hit Crime and Computation from CalTech News.

Krumholtz’s star turn as a math genius belies his dismal record as an algebra student. He explained that in preparation for his role he hung around Caltech last fall, “wandering the hallways and campus for two to three weeks,” to soak up the academic ambiance. To plumb character motivation he talked to a real-life youthful math guy, Caltech’s 30-year-old professor Dunfield.
In a phone interview shortly after the show’s television debut Dunfield recalled spending about an hour with Krumholtz, who plays 29-year-old Charlie. “He wanted to know what it’s like to do mathematics and work in academia, what types of things his character would likely be concerned about, like tenure or other issues.”
The professor, who was so un-starstruck that he hadn’t even made a point of watching the premiere, added, “He wanted to know, why would somebody choose to become a mathematics professor. Would they have to love math?” What was his response? The professor said he does not recall.