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# Category: stories

In this series I’ll mention some books I found entertaining, stimulating or comforting during these Corona times. Read them at your own risk.

This must have been the third time I’ve read The genius in by basement – The biography of a happy man by Alexander masters.

I first read it when it came out in 2011.

Then, in conjunction with Genius at play – The Curious Mind of John Horton Conway Conway’s biography by Siobhan Roberts, in july 2017, which is probably the best way to read this book.

And, then again last week, as Simon Norton‘s work pops up wherever I look, as in the previous post.

It takes some time to get used to the rather chaotic style (probably used because that’s how Masters perceives Norton), and all attempts at explaining Simon’s mathematics can better be skipped.

The book tries to find an answer as to why a child prodigy and genius like Simon Norton failed to secure a safe place in academics.

Page 328:

Simon’s second explanation of his loss of mathematical direction is heartbreaking. Now that Conway has fled to America, there is no one in the mathematical world who will work with him.

They say he is too peculiar, too shabby, too old.

His interests are fixed in mathematics that has had its day. His brilliance is frigid. His talent, perfectly suited to an extraordinary moment in algebraic history (the symmetry work at Cambridge during the early 1970s and 1980s) is out of fashion.

This may give the impression that Norton stopped doing good math after Conway left for Princeton in 1985. This is far from true.

Norton’s Wikipedia page mentions only post 1995 publications, which in itself is deplorable, as it leaves out his contributions to the ATLAS and his seminal paper with Conway on Monstrous moonshine.

Here’s Alexander Masters talking about ‘Genius in my basement’

I’ll leave you with a nice quote, comparing Monstrous Moonshine to a Sainsbury’s bag on Jupiter.

Page 334:

This much I do know: Monstrous Moonshine links the Monster to distant mathematics and the structure of space in ways that are as awe-inspiring to a man like Simon as it would be to an astronaut to step out of his space machine on Jupiter, and find a Sainsbury’s bag floating past. That’s why it’s called ‘Moonshine’, because mathematicians can even now hardly believe it.

‘I think’, said Simon, standing up from his berth and shaking crumbs and clotted blobs of oil and fish off his T-shirt onto the covers, ‘I can explain to you what Moonshine is in one sentence.’

When he really tries, Simon can be a model of clarity.

‘It is,’ he said, ‘the voice of God.’

Ps, wrt. SNORT.

Stairways feature prominently in several drawings by Maurits Cornelis (“Mauk”) Escher, for example in this lithograph print Relativity from 1953.

Relativity (M. C. Escher) – Photo Credit

From its Wikipedia page:

In the world of ‘Relativity’, there are three sources of gravity, each being orthogonal to the two others.
Each inhabitant lives in one of the gravity wells, where normal physical laws apply.
There are sixteen characters, spread between each gravity source, six in one and five each in the other two.
The apparent confusion of the lithograph print comes from the fact that the three gravity sources are depicted in the same space.
The structure has seven stairways, and each stairway can be used by people who belong to two different gravity sources.

Escher’s inspiration for “Relativity” (h/t Gerard Westendorp on Twitter) were his recollections of the staircases in his old secondary school in Arnhem, the Lorentz HBS.
The name comes from the Dutch physicist and Nobel prize winner Hendrik Antoon Lorentz who attended from 1866 to 1869, the “Hogere Burger School” in Arnhem, then at a different location (Willemsplein).

Stairways Lorentz HBS in Arnhem – Photo Credit

Between 1912 and 1918 Mauk Escher attended the Arnhem HBS, located in the Schoolstraat and build in 1904-05 by the architect Gerrit Versteeg. The school building is constructed around a monumental central stairway.

Arnhem HBS – G. Versteeg 1904-05 – Photo Credit

Plan HBS-Arnhem by G. Versteeg – Photo Credit

If you flip the picture below in the vertical direction, the two side-stairways become accessible to figures living in an opposite gravitation field.

Central staircase HBS Arnhem – Photo Credit

There’s an excellent post on the Arnhem-years of Mauk Escher by Pieter van der Kuil. Unfortunately (for most of you) in Dutch, but perhaps Google translate can do its magic here.

In the first half of 1937, Andre Weil visited Princeton and introduced some of the postdocs present (notably Ralph Boas, John Tukey, and Frank Smithies) to Poldavian lore and Bourbaki’s early work.

In 1935, Bourbaki succeeded (via father Cartan) to get his paper “Sur un théorème de Carathéodory et la mesure dans les espaces topologiques” published in the Comptes Rendus des Séances Hebdomadaires de l’Académie des Sciences.

Inspired by this, the Princeton gang decided to try to get a compilation of their mathematical ways to catch a lion in the American Mathematical Monthly, under the pseudonym H. Petard, and accompanied by a cover letter signed by another pseudonym, E. S. Pondiczery.

By the time the paper “A contribution to the mathematical theory of big game hunting” appeared, Boas and Smithies were in cambridge pursuing their postdoc work, and Boas reported back to Tukey: “Pétard’s paper is attracting attention here,” generating “subdued chuckles … in the Philosophical Library.”

On the left, Ralph Boas in ‘official’ Pondiczery outfit – Photo Credit.

The acknowledgment of the paper is in true Bourbaki-canular style.

The author desires to acknowledge his indebtedness to the Trivial Club of St. John’s College, Cambridge, England; to the M.I.T. chapter of the Society for Useless Research; to the F. o. P., of Princeton University; and to numerous individual contributors, known and unknown, conscious and unconscious.

The Trivial Club of St. John’s College probably refers to the Adams Society, the St. John’s College mathematics society. Frank Smithies graduated from St. John’s in 1933, and began research on integral equations with Hardy. After his Ph. D., and on a Carnegie Fellowship and a St John’s College studentship, Smithies then spent two years at the Institute for Advanced Study at Princeton, before returning back ‘home’.

In the previous post, I assumed that Weil’s visit to Cambridge was linked to Trinity College. This should probably have been St. John’s College, his contact there being (apart from Smithies) Max Newman, a fellow of St. John’s. There are two letters from Weil (summer 1939, and summer 1940) in the Max Newman digital library.

The Eagle Scanning Project is the online digital archive of The Eagle, the Journal of St. John’s College. Last time I wanted to find out what was going on, mathematically, in Cambridge in the spring of 1939. Now I know I just had to peruse the Easter 1939 and Michaelmas 1939 volumes of the Eagle, focussing on the reports of the Adams Society.

In the period Andre Weil was staying in Cambridge, they had a Society Dinner in the Music Room on March 9th, a talk about calculating machines (with demonstration!) on April 27th, and the Annual Business Meeting on May 11th, just two days before their punting trip to Grantchester,

The M.I.T. chapter of the Society for Useless Research is a different matter. The ‘Useless Research’ no doubt refers to Extrasensory Perception, or ESP. Pondiczery’s initials E. S. were chosen with a future pun in mind, as Tukey said in a later interview:

“Well, the hope was that at some point Ersatz Stanislaus Pondiczery at the Royal Institute of Poldavia was going to be able to sign something ESP RIP.”

What was the Princeton connection to ESP research?

Well, Joseph Banks Rhine conducted experiments at Duke University in the early 1930s on ESP using Zener cards. Amongst his test-persons was Hubert Pearce, who scored an overall 40% success rate, whereas chance would have been 20%.

Pearce and Joseph Banks Rhine (1932) – Photo Credit

In 1936, W. S. Cox tried to repeat Rhine’s experiment at Princeton University but failed. Cox concluded “There is no evidence of extrasensory perception either in the ‘average man’ or of the group investigated or in any particular individual of that group. The discrepancy between these results and those obtained by Rhine is due either to uncontrollable factors in experimental procedure or to the difference in the subjects.”

As to the ‘MIT chapter of the society for useless research’, a chapter usually refers to a fraternity at a University, but I couldn’t find a single one on the list of MIT fraternities involved in ESP, now or back in the late 1930s.

However, to my surprise I found that there is a MIT Archive of Useless Research, six boxes full of amazing books, pamphlets and other assorted ‘literature’ compiled between 1900 and 1940.

The Albert G. Ingalls pseudoscience collection (its official name) comprises collections of books and pamphlets assembled by Albert G. Ingalls while associate editor of Scientific American, and given to the MIT Libraries in 1940. Much of the material rejects contemporary theories of physical sciences, particularly theoretical and planetary physics; a smaller portion builds upon contemporary science and explores hypotheses not yet accepted.

I don’t know whether any ESP research is included in the collection, nor whether Boas and Tukey were aware of its existence in 1938, but it sure makes a good story.

The final riddle, the F. o. P., of Princeton University is an easy one. Of course, this refers to the “Friends of Pondiczery”, the circle of people in Princeton who knew of the existence of their very own Bourbaki.

One of the few certainties we have on the Bourbaki-Petard wedding invitation is that it was printed in, and distributed out of Cambridge in the spring of 1939, presumably around mid April.

So, what was going on, mathematically, in and around Trinity and St. John’s College, at that time?

Well, there was the birth of Eureka, the journal of the Archimedeans, the mathematical society of the University of Cambridge. Eureka is one of the oldest recreational mathematics publications still in existence.

Since last year the back issues of Eureka are freely available online, unfortunately missing out the very first two numbers from 1939.

Ralph Boas, one of the wedding-conspirators, was among the first to contribute to Eureka. In the second number, in may 1939, he wrote an article on “Undergraduate mathematics in America”.

And, in may 1940 (number 4 of Eureka) even the lion hunter H. Petard wrote a short ‘Letter to the editors’.

But, no doubt the hottest thing that spring in Cambridge were Ludwig Wittgenstein’s ‘Lectures on the Foundations of Mathematics’. Wittgenstein was just promoted to Professor after G.E. Moore resigned the chair in philosophy.

For several terms at Cambridge in 1939, Ludwig Wittgenstein lectured on the philosophical foundations of mathematics. A lecture class taught by Wittgenstein, however, hardly resembled a lecture. He sat on a chair in the middle of the room, with some of the class sitting in chairs, some on the floor. He never used notes. He paused frequently, sometimes for several minutes, while he puzzled out a problem. He often asked his listeners questions and reacted to their replies. Many meetings were largely conversation.

These lectures were attended by, among others, D. A. T. Gasking, J. N. Findlay, Stephen Toulmin, Alan Turing, G. H. von Wright, R. G. Bosanquet, Norman Malcolm, Rush Rhees, and Yorick Smythies.

Here’s a clip from the film Wittgenstein, directed by Derek Jarman.

Missing from the list of people attending Wittgenstein’s lectures is Andre Weil, a Bourbaki member and the principal author of the wedding invitation.

Weil was in Cambridge in the spring of 1939 on a travel grant from the French research organisation for visits to the UK and Northern Europe. At that time, Weil held a position at the University of Strasbourg, uncomfortably close to Nazi-Germany.

Weil not attending Wittgenstein’s lectures is strange for several reasons. Weil was then correcting the galley proofs of Bourbaki’s first ever booklet, their own treatment of set theory, which appeared in 1939.

But also on a personal level, Andre Weil must have been intrigued by Wittgenstein’s philosophy, as it was close to that of his own sister Simone Weil

There are many parallels between the thinkers Simone Weil and Ludwig Wittgenstein. They each lived in a tense relationship with religion, with both being estranged from their cultural Jewish ancestry, and both being tempted at various times by the teachings of Catholicism.

They both underwent a profound and transformative mystical turn early into their careers. Both operated against the backdrop of escalating global conflict in the early 20th century.

Both were concerned, amongst other things, with questions of culture, ethics, aesthetics, epistemology, science, and necessity. And, perhaps most notably, they both sought to radically embody their ideas and physically ‘live’ their philosophies.

Andre and Simone Weil in Knokke-Zoute, 1922 – Photo Credit

Another reason why Weil might have been interested to hear Wittgenstein on the foundations of mathematics was a debate held in Paris of few months previously.

On February 4th 1939, the French Society of Philosophy invited Albert Lautman and Jean Cavaillès ‘to define what constitutes the ‘life of mathematics’, between historical contingency and internal necessity, describe their respective projects, which attempt to think mathematics as an experimental science and as an ideal dialectics, and respond to interventions from some eminent mathematicians and philosophers.’

Among the mathematicians present and contributing to the discussion were Weil’s brothers in arms, Henri Cartan, Charles Ehresmann, and Claude Chabauty.

As Chabauty left soon afterwards to study with Mordell in Manchester, and visited Weil in Cambridge, Andre Weil must have known about this discussion.

The record of this February 4th meeting is available here (in French), and in English translation from here.

Jean Cavaillès took part in the French resistance, was arrested and shot by the Nazis on April 4th 1944. Albert Lautman was shot by the Nazis in Toulouse on 1 August 1944.

Jean Cavailles (2nd on the right) 1903-1944 – Photo Credit

A book review of Wittgenstein’s Lectures on the Foundations of Mathematics by G. Kreisel is available from the Bulletin of the AMS. Curiously, Kreisel compares Wittgenstein’s approach to … Bourbaki’s very own manifesto L’architecture des mathématiques.

For all these reasons it is strange that Andre Weil apparently didn’t show much interest in Wittgenstein’s lectures.

Had he more urgent things on his mind, like prepping for a wedding?

The fictitious life of Nicolas Bourbaki remains a source of fascination to some.

A few weeks ago, Michael Barany wrote an article for the JStor Daily The mathematical pranksters behind Nicolas Bourbaki.

Here’s one of the iconic early Bourbaki pictures, taken at the Dieulefit-meeting in 1938. More than a decade ago I discovered the exact location of that meeting in the post Bourbaki and the miracle of silence.

Bourbaki at Beauvallon 1938 – Photo Credit

That post was one of a series on the pre-war years of Bourbaki, and the riddles contained in the invitation card of the Betti Bourbaki-Hector Petard wedding that several mathematicians in Cambridge, Princeton and Paris received in the spring of 1939.

A year ago, The Ferret made the nice YouTube clip “Bourbaki – a Tale of Mathematics, Lions and Espionage”, which gives a quick introduction to Bourbaki and the people mentioned in the wedding invitation.

This vacation period may be a good opportunity to revisit some of my older posts on this subject, and add newer material I discovered since then.

For this reason, I’ve added a new category, tBC for ‘the Bourbaki Code’, and added the old posts to it.

In the strange logic of subways I’ve used a small part of the Parisian metro-map to illustrate some of the bi-Heyting operations on directed graphs.

Little did I know that this metro-map gives only a partial picture of the underground network. The Parisian metro has several ghost stations, that is, stations that have been closed to the public and are no longer used in commercial service. One of these is the Haxo metro station.

Haxo metro station – Photo Credit

The station is situated on a line which was constructed in the 1920s between Porte des Lilas (line 3bis) and Pré-Saint-Gervais (line 7bis), see light and dark green on the map above . A single track was built linking Place des Fêtes to Porte des Lilas, known as la voie des Fêtes, with one intermediate station, Haxo.

For traffic in the other direction, another track was constructed linking Porte des Lilas to Pré Saint-Gervais, with no intermediate station, called la voie navette. Haxo would have been a single-direction station with only one platform.

But, it was never used, and no access to street level was ever constructed. Occasional special enthusiast trains call at Haxo for photography.

Apart from the Haxo ‘station morte’ (dead station), these maps show another surprise, a ‘quai mort’ (dead platform) known as Porte des Lilas – Cinema. You can hire this platform for a mere 200.000 Euro/per day for film shooting.

For example, Le fabuleux destin d’Amelie Poulin has a scene shot there. In the film the metro station is called ‘Abbesses’ (3.06 into the clip)

There is a project to re-open the ghost station Haxo for public transport. From a mathematical perspective, this may be dangerous.

Remember the subway singularity?

In the famous story A subway named Mobius by A. J. Deutsch, the Boylston shuttle on the Boiston subway went into service on March 3rd, tying together the seven principal lines, on four different levels. A day later, train 86 went missing on the Cambridge-Dorchester line…

The Harvard algebraist R. Tupelo suggested the train might have hit a node, a singularity. By adding the Boylston shuttle, the connectivity of the subway system had become infinite…

Now that we know of the strange logic of subways, an alternative explanation of this accident might be that by adding the Boylston shuttle, the logic of the Boston subway changed dramatically.

This can also happen in Paris.

I know, I’ve linked already to the movie ‘Moebius’ by Gustavo Mosquera, based on Deutsch’s story, set in Buenos Aires.

But, if you have an hour to spend, here it is again.

Sunday, January 28th 1940, Hamburg

Ernst Witt wants to get two papers out of his system because he knows he’ll have to enter the Wehrmacht in February.

The first one, “Spiegelungsgruppen und Aufzählung halbeinfacher Liescher Ringe”, contains his own treatment of the root systems of semisimple Lie algebras and their reflexion groups, following up on previous work by Killing, Cartan, Weyl, van der Waerden and Coxeter.

(Photo: Natascha Artin, Nikolausberg 1933): From left to right: Ernst Witt; Paul Bernays; Helene Weyl; Hermann Weyl; Joachim Weyl, Emil Artin; Emmy Noether; Ernst Knauf; unknown woman; Chiuntze Tsen; Erna Bannow (later became wife of Ernst Witt)

Important for our story is that this paper contains the result stating that integral lattices generated by norm 2 elements are direct sums of root systems of the simply laced Dynkin diagrans $A_n, D_n$ and $E_6,E_7$ or $E_8$ (Witt uses a slightly different notation).

In each case, Witt knows of course the number of roots and the determinant of the Gram matrix
$\begin{array}{c|cc} & \# \text{roots} & \text{determinant} \\ \hline A_n & n^2+n & n+1 \\ D_n & 2n^2-2n & 4 \\ E_6 & 72 & 3 \\ E_7 & 126 & 2 \\ E_8 & 240 & 1 \end{array}$
The second paper “Eine Identität zwischen Modulformen zweiten Grades” proves that there are just two positive definite even unimodular lattices (those in which every squared length is even, and which have one point per unit volume, that is, have determinant one) in dimension sixteen, $E_8 \oplus E_8$ and $D_{16}^+$. Previously, Louis Mordell showed that the only unimodular even lattice in dimension $8$ is $E_8$.

The connection with modular forms is via their theta series, listing the number of lattice points of each squared length
$\theta_L(q) = \sum_{m=0}^{\infty} \#\{ \lambda \in L : (\lambda,\lambda)=m \} q^{m}$
which is a modular form of weight $n/2$ ($n$ being the dimension which must be divisible by $8$) in case $L$ is a positive definitive even unimodular lattice.

The algebra of all modular forms is generated by the Eisenstein series $E_2$ and $E_3$ of weights $4$ and $6$, so in dimension $8$ we have just one possible theta series
$\theta_L(q) = E_2^2 = 1+480 q^2+ 61920 q^4+ 1050240 q^6+ \dots$

It is interesting to read Witt’s proof of his main result (Satz 3) in which he explains how he constructed the possible even unimodular lattices in dimension $16$.

He knows that the sublattice of $L$ generated by the $480$ norm two elements must be a direct sum of root lattices. His knowledge of the number of roots in each case tells him there are only two possibilities
$E_8 \oplus E_8 \qquad \text{and} \qquad D_{16}$
The determinant of the Gram matrix of $E_8 \oplus E_8$ is one, so this one is already unimodular. The remaining possibility
$D_{16} = \{ (x_1,\dots,x_{16}) \in \mathbb{Z}^{16}~|~x_1+ \dots + x_{16} \in 2 \mathbb{Z} \}$
has determinant $4$ so he needs to add further lattice points (necessarily contained in the dual lattice $D_{16}^*$) to get it unimodular. He knows the coset representatives of $D_{16}^*/D_{16}$:
$\begin{cases} [0]=(0, \dots,0) &~\text{of norm 0} \\ [1]=(\tfrac{1}{2},\dots,\tfrac{1}{2}) &~\text{of norm 4} \\ [2]=(0,\dots,0,1) &~\text{of norm 1} \\ [3]=(\tfrac{1}{2},\dots,\tfrac{1}{2},-\tfrac{1}{2}) &~\text{of norm 4} \end{cases}$
and he verifies that the determinant of $D_{16}^+=D_{16}+([1]+D_{16})$ is indeed one (btw. adding coset $[3]$ gives an isomorphic lattice). Witt calls this procedure to arrive at the correct lattices forced (‘zwangslaufig’).

So, how do you think Witt would go about finding even unimodular lattices in dimension $24$?

To me it is clear that he would start with a direct sum of root lattices whose dimensions add up to $24$, compute the determinant of the Gram matrix and, if necessary, add coset classes to arrive at a unimodular lattice.

Today we would call this procedure ‘adding glue’, after Martin Kneser, who formalised this procedure in 1967.

On January 28th 1940, Witt writes that he found more than $10$ different classes of even unimodular lattices in dimension $24$ (without giving any details) and mentioned that the determination of the total number of such lattices will not be entirely trivial (‘scheint nicht ganz leicht zu sein’).

The complete classification of all $24$ even unimodular lattices in dimension $24$ was achieved by Hans Volker Niemeier in his 1968 Ph.D. thesis “Definite quadratische Formen der Dimension 24 und Diskriminante 1”, under the direction of Martin Kneser. Naturally, these lattices are now known as the Niemeier lattices.

Which of the Niemeier lattices were known to Witt in 1940?

There are three obvious certainties: $E_8 \oplus E_8 \oplus E_8$, $E_8 \oplus D_{16}^+$ (both already unimodular, the second by Witt’s work) and $D_{24}^+$ with a construction analogous to the one of $D_{16}^+$.

To make an educated guess about the remaining Witt-Niemeier lattices we can do two things:

1. use our knowledge of Niemeier lattices to figure out which of these Witt was most likely to stumble upon, and
2. imagine how he would adapt his modular form approach in dimension $16$ to dimension $24$.

Here’s Kneser’s neighbourhood graph of the Niemeier lattices. Its vertices are the $24$ Niemeiers and there’s an edge between $L$ and $M$ whenever the intersection $L \cap M$ is of index $2$ in both $L$ and $M$. In this case, $L$ and $M$ are called neighbours.

Although the theory of neighbours was not known to Witt, the graph may give an indication of how likely it is to dig up a new Niemeier lattice by poking into an already discovered one.
The three certainties are the three lattices at the bottom of the neighborhood graph, making it more likely for the lattices in the lower region to be among Witt’s list.

For the other approach, the space of modular forms of weight $12$ is two dimensional, with a basis formed by the series
$\begin{cases} E_6(q) = 1 + \tfrac{65520}{691}(q+2049 q^2 + 177148 q^3+4196353q^4+\dots \\ \Delta(q) = q-24 q^2+252q^3-1472q^4+ \dots \end{cases}$

If you are at all with me, Witt would start with a lattice $R$ which is a direct sum of root lattices, so he would know the number of its roots (the norm $2$ vectors in $R$), let’s call this number $r$. Now, he wants to construct an even unimodular lattice $L$ containing $R$, so the theta series of both $L$ and $R$ must start off with $1 + r q^2 + \dots$. But, then he knows
$\theta_L(q) = E_6(q) + (r-\frac{65520}{691})\Delta(q)$
and comparing coefficients of $\theta_L(q)$ with those of $\theta_R(q)$ will give him an idea what extra vectors he has to throw in.

If we’re generous to Witt (and frankly, why shouldn’t we), we may believe that he used his vast knowledge of Steiner systems (a few years earlier he wrote the definite paper on the Mathieu groups, and a paper on Steiner systems) to construct in this way the lattices $(A_1^{24})^+$ and $(A_2^{12})^+$.

The ‘glue’ for $(A_1^{24})^+$ is coming from the extended binary Golay code, which itself uses the Steiner system $S(5,8,24)$. $(A_2^{12})^+$ is constructed using the extended ternary Golay code, based on the Steiner system $S(5,6,12)$.

The one thing that would never have crossed his mind that sunday in 1940 was to explore the possibility of an even unimodular 24-dimensional lattice $\Lambda$ without any roots!

One with $r=0$, and thus with a theta series starting off as
$\theta_{\Lambda}(q) = 1 + 196560 q^4 + 16773120 q^6 + \dots$
No, he did not find the Leech lattice that day.

If he would have stumbled upon it, it would have simply blown his mind.

It would have been so much against all his experiences and intuitions that he would have dropped everything on the spot to write a paper about it, or at least, he would have mentioned in his ‘more than $10$ lattices’-claim that, surprisingly, one of them was an even unimodular lattice without any roots.

In this neverending pandemic there’s a shortage of stories putting a lasting smile on my face. Here’s one.

If you are at all interested in chess, you’ll know by now that some days ago IGMs (that is, international grandmasters for the rest of you) Magnus Carlsen and Hikaru Nakamura opened an official game with a double bongcloud, and couldn’t stop laughing.

The bongcloud attack is the chess opening in which white continues after

1. e2-e4, e7-e5

with

2. Ke1-e2 !

thereby blocking the diagonals for the bishop and queen, losing the ability to castle, and putting its king in danger.

If you are left clueless, you should download the free e-book Winning with the bongcloud immediately.

If you are (or were) a chess player, it is the perfect parody to all those books you had to suffer through in order to build up an ‘opening repertoire’.

If you are new to chess (perhaps after watching The queen’s gambit), it gives you a nice selection of easy mate-in-one problems.

Every possible defence against the bongcloud is illustrated with a ‘game’ illustrating the massive advantage the attack gives, ending with a situation in which … black(!) has a one-move mate.

One example:

In the two knight Copacabana tango defense against the bongcloud, that is the position after

the Haight-Asbury (yeah, well) game (Linares, 1987) continued with:

4. Kg3, Nxe4?
5. Kh3, d6+
6. Qg4!, Bxg4
7. Kxg4, Qf6??
8. Ne2, h5+??
9. Kh3!, Nxf2
10. Kg3!

giving this position

which ‘Winning with the bongcloud’ evaluates as:

White continues his textbook execution of a “pendulum,” swinging back and forth between g3 and h3 to counter every Black threat. With the dynamically placed King, Black’s attack teeters on the edge of petering out. Nxh1 will trap the Black Knight and extinguish the threat.

In the actual game, play took a different turn as Black continued his h-file pressure. Nonetheless, this game is an excellent example of how 2. …Nc6 is often a wasted tempo in the Bongcloud.

Here’s Nakamura philosophising over the game and the bongcloud. Try to watch at least the first 30 seconds or so to see the commentators reaction to the actual Carlsen-Nakamura game.

Now, that put a smile on your face, didn’t it?