This time of year I’m usually in France, or at least I was before Covid. This might explain for my recent obsession with French math YouTube interviews.
Today’s first one is about Bourbaki’s golden years, the period between WW2 and 1975. Alain Connes is trying to get some anecdotes from Jean-Pierre Serre, Pierre Cartier, and Jacques Dixmier.
If you don’t have the time to sit through the whole thing, perhaps you might have a look at the discussion on whether or not to include categories in Bourbaki (starting at 51.40 into the clip).
Here are some other time-slots (typed on a qwerty keyboard, mes excuses) with some links.
Again, if you don’t have the energy to sit through it all, perhaps I can tempt you with Serre’s reaction to Connes bringing up the subject of toposes (starting at 14.36 into the clip).
The literary sensation that spring of 1939 no doubt was the publication of Finnegans Wake by James Joyce. On May 4th 1939 FW was published simultaneously by Faber and Faber in London and by Viking Press in New York, after seventeen years of composition.
In 1928-29, Joyce started publishing individual chapters from FW, then known as ‘Work in Progress’, including chapter II.2 ‘The Triangle’, of which a brief excerpt was already published in February 1928. The name comes from the only diagram in FW, the classical Euclidian construction of an equilateral triangle (FW, p. 293)
This Vesica piscis has multiple interpretations in FW, most of them sexual. The triangle $\Delta$ is the Sigla for Anna Livia Plurabelle throughout FW, but it also refers to the river Liffey through Dublin.
Here’s Anthony Burgess explaining some of the Sigla, the relevant part starts at 14.20 into the clip.
In fact, many of FW’s Sigla are derived from mathematical symbols, such as $\exists$ (Earwicker), $\perp$ and $\vdash$ (Issy). For more on this, please read The logic of the doodles in Finnegans Wake II.2.
Not only does the equilateral triangle $\Delta$ refer to the river Liffey, the entire Euclidian diagram can be seen as a map for Dublin and its surroundings, as emphasised by the words “Vieus Von DVbLIn” (views from Dublin) in FW right under the diagram.
Here’s Dublin with the Liffey running through it, and Phoenix Park, which also features prominently in FW, see for example Phoenix Park in Finnegans Wake.
There’s a lot more to say about Joyce’s uses of geometry and topography in Ulysses and Finnegans Wake, in fact Ciaran McMorran wrote an entire Glasgow Ph. D. about it, but perhaps I’ll save some of that for a future post.
But what does this have to to with the Bourbaki Code, the puzzles contained in the Bourbaki-Petard wedding announcement?
Well, I claim that Andre Weil hid the Vesica Piscis/Euclidian diagram into the ‘faire part’. The challenge is to view the wedding announcement as a partial city- map. Clearly this time, the city of Dublin should be replaced by the city of Paris. Se non e vero …
Probably, there are enough hints contained in the previous posts in this series for you to spot the triangle(s) on the map of Paris. If you do so, please leave a comment, or email me.
Meanwhile, we’ll unravel first the more obvious levels of interpretation of the wedding announcement.
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.
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’.
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.
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.’
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?
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.
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.
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.
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.
In this series I’ll mention some books I found entertaining, stimulating or comforting during these Corona times. Read them at your own risk.
In an attempt to raise the level of this series, I tried to get through the latest hype in high-brow literature: The Death of Francis Bacon by Max Porter.
It’s an extremely thin book, just 43 pages long, hardly a novella. My Kindle said I should be able to read it in less than an hour.
Boy, did that turn out differently. I’m a week into this book, and still struggling.
A few minutes into the book I realised I didn’t know the first thing about Bacon’s death, and that the book was not going to offer me that setting. Fortunately, there’s always Wikipedia:
While holidaying in Madrid in 1992, Bacon was admitted to the Handmaids of Maria, a private clinic, where he was cared for by Sister Mercedes. His chronic asthma, which had plagued him all his life, had developed into a more severe respiratory condition and he could not talk or breathe very well.
Fine, at least I now knew where “Darling mama, sister oh Dios, Mercedes” (p.7) came from, and why every chapter ended with “Intenta descansar” (try to rest).
While I’m somewhat familiar with Bacon’s paintings, I did know too little about his life to follow the clues sprinkled throughout the book. Fortunately, there’s this excellent documentary about his life: “Francis Bacon: A Brush with Violence” (2017)
Okay, now I could place many of the characters visiting Bacon, either physically sitting on the chair he offers at the start of each chapter (“Take a seat why don’t you”), or merely as memories playing around in his head. It’s a bit unclear to me.
Then, there’s the structure of the book. Each of the seven chapters has as title the dimensions of a painting:
One: Oil on canvas, 60 x 46 1/2 in.
Two: Oil on canvas, 65 1/2 x 56 in.
Three: Oil on canvas, 65 x 56 in.
Four: Oil on canvas, 14 x 12 in.
Five: Oil on canvas, 78 x 58 in.
Six: Oil on canvas, 37 x 29 in.
Seven: Oil on canvas, 77 x 52 in.
Being the person I am, I hoped that if I could track down the corresponding Bacon paintings, I might begin to understand the corresponding chapter. Fortunately, Wikipedia provides a List of paintings by Francis Bacon.
Many of Bacon’s paintings are triptychs, and the dimensions refer to those of a single panel. So, even if I found the correct triptych I still had to figure out which of the three panels corresponds to the chapter.
And often, there are several possible candidates. The 14 x 12 in. panel-format Bacon often used for studies for larger works. So, chapter 4 might as well refer to his studies for a self portrait (see above), or to the three studies for a portrait of Henrietta Moraes:
No doubt, I’m just on a wild goose chase here. Probably, Max Porter is merely using existing dimensions of Bacon paintings for blank canvases to smear his words on, as explained in this erudite ArtReview What Does It Mean To Write a Painting?.
Here’s the writer Max Porter himself, explaining his book.
Raymond Smullyan‘s logic puzzles frequently involve Knights (who always tell the truth) and Knaves (who always lie).
In his book Logical Labyrinths (really a first course in propositional logic) he introduced islands where the lying or truth-telling habits can vary from day to day—that is, an inhabitant might lie on some days and tell the truth on other days, but on any given day, he or she lies the entire day or tells the truth the entire day.
An island is said to be Boolean if is satisfies the following conditions:
$\mathbf{N}$ : For any inhabitant $A$ there is an inhabitant who tells the truth on all and only those days on which $A$ lies.
$\mathbf{M}$ : For any inhabitants $A$ and $B$ there is an inhabitant $C$ who tells the truth on all and only those days on which $A$ and $B$ both tell the truth.
$\mathbf{J}$ : For any inhabitants $A$ and $B$ there is an inhabitant $C$ who tells the truth on all and only those days on which either $A$ tells the truth or $B$ tells the truth (or both). (In other words, $C$ lies on those and only those days on which $A$ and $B$ both lie.)
On any given day there are only Knights and Knaves on the island, but these two populations may vary from one day to the other. The subsets (of all days) for which there is an inhabitant who is a Knight then and a Knave on all other days form a Boolean algebra with operations $\wedge = \cap$ ($\mathbf{M}$eet), $\vee= \cup$ ($\mathbf{J}$oin) and $\neg=$ set-complement ($\mathbf{N}$egation).
Here’s a nice puzzle from Smullyan’s book:
Solomon’s Island also turned out to be quite interesting. When Craig arrived on it, he had the following conversation with the resident sociologist:
Craig : Is this island a Boolean island?
Sociologist : No.
Craig : Can you tell me something about the lying and truth-telling habits of the residents here?
Sociologist : For any inhabitants $A$ and $B$, there is an inhabitant $C$ who tells the truth on all and only those days on which either $A$ lies or $B$ lies (or both).
Show that the sociologist didn’t go native, and that his research is lousy.
(My wording, not Smullyan’s)
Smullyan’s version: This interview puzzled inspector Craig; he felt that something was wrong. After a while he realized for sure that something was wrong, the sociologist was either lying or mistaken!
Extending Smullyan’s idea, we can say that an island is Heyting if, in addition to $\mathbf{M}$ and $\mathbf{J}$ is satisfies the following rules
$\mathbf{T}$ : at least one inhabitant tells the truth on all days.
$\mathbf{F}$ : at least one inhabitant lies on all days.
$\mathbf{I}$ : For any inhabitants $A$ and $B$ there is an inhabitant $C$ sharing Knight-days with $A$ only when $B$ tells the truth, and there are no inhabitants doing this while telling the truth on more days than $C$.
Let’s give an example of an Heyting island which is not Boolean.
On Three-island there are only three kinds of people: Knights, Knaves and Alternates, who can neither lie nor tell the truth two days in a row. All Alternates tell the truth on the same days.
Here’s a riddle:
You meet John, who is a Knight, James, an Alternate, and William, a Knave. You don’t know who is who. You can only ask one question containing at most four words, giving you a Yes or No answer, to just one of the three. The answer must tell you whether that person is James or not.
You may like to watch Smullyan on the Carson show for a hint.
Or, you might just watch it reminiscing long forgotten times, when talkshow-hosts still listened to their guests, and could think for themselves…
Two lattices $L$ and $L’$ in the same vector space are called neighbours if their intersection $L \cap L’$ is of index two in both $L$ and $L’$.
In 1957, Martin Kneser gave a method to find all unimodular lattices (of the same dimension and signature) starting from one such unimodular lattice, finding all its neighbours, and repeating this with the new lattices obtained.
In other words, Kneser’s neighbourhood graph, with vertices the unimodular lattices (of fixed dimension and signature) and edges between them whenever the lattices are neighbours, is connected.
Last time, we’ve constructed the Niemeier lattice $(A_1^{24})^+$ from the binary Golay code $\mathcal{C}_{24}$
\[
L = (A_1^{24})^+ = \mathcal{C}_{24} \underset{\mathbb{F}_2}{\times} (A_1^{24})^* = \{ \tfrac{1}{\sqrt{2}} \vec{v} ~|~\vec{v} \in \mathbb{Z}^{\oplus 24},~v=\vec{v}~mod~2 \in \mathcal{C}_{24} \} \]
With hindsight, we know that $(A_1^{24})^+$ is the unique neighbour of the Leech lattice in the Kneser neighbourhood graph of the positive definite, even unimodular $24$-dimensional lattices, aka the Niemeier lattices.
Let’s try to construct the Leech lattice $\Lambda$ from $L=(A_1^{24})^+$ by Kneser’s neighbour-finding trick.
Sublattices of $L$ of index two are in one-to-one correspondence with non-zero elements in $L/2L$. Take $l \in L – 2L$ and $m \in L$ such that the inner product $l.m$ is odd, then
\[
L_l = \{ x \in L~|~l.x~\text{is even} \} \]
is an index two sublattice because $L = L_l \sqcup (L_l+m)$. By definition $l.x$ is even for all $x \in L_l$ and therefore $\frac{l}{2} \in L_l^*$. We have this situation
\[
L_l \subsetneq L = L^* \subsetneq L_l^* \]
and $L_l^*/L_l \simeq \mathbb{F}_2 \oplus \mathbb{F}_2$, with the non-zero elements represented by $\{ \frac{l}{2}, m, \frac{l}{2}+m \}$. That is,
\[
L_l^* = L_l \sqcup (L_l+m) \sqcup (L_l+\frac{l}{2}) \sqcup (L_l+(\frac{l}{2}+m)) \]
This gives us three lattices
\[
\begin{cases}
M_1 &= L_l \sqcup (L_l+m) = L \\
M_2 &= L_l \sqcup (L_l+\frac{l}{2}) \\
M_3 &= L_l \sqcup (L_l+(\frac{l}{2}+m))
\end{cases}
\]
and all three of them are unimodular because
\[
L_l \subsetneq M_i \subseteq M_i^* \subsetneq L_l^* \]
and $L_l$ is of index $4$ in $L_l^*$.
Now, let’s assume the norm of $l$, that is, $l.l \in 4 \mathbb{Z}$. Then, either the norm of $\frac{l}{2}$ is odd (but then the norm of $\frac{l}{2}+m$ must be even), or the norm of $\frac{l}{2}$ is even, in which case the norm of $\frac{l}{2}+m$ is odd.
That is, either $M_2$ or $M_3$ is an even unimodular lattice, the other one being an odd unimodular lattice.
Let’s take for $l$ and $m$ the vectors $\lambda = \frac{1}{\sqrt{2}} (1,1,\dots,1) \in L – 2L$ and $\mu = \sqrt{2}(1,0,\dots,0) \in L$, then
\[
\lambda.\lambda = \frac{1}{2}\times 24 = 12 \quad \text{and} \quad \mu.\lambda = 1 \]
Because $\frac{\lambda}{2}.\frac{\lambda}{2} = \frac{12}{4}=3$ is odd, we have that
\[
\Lambda = L_{\lambda} \sqcup (L_{\lambda} + (\frac{\lambda}{2} + \mu)) \]
is an even unimodular lattice, which is the Leech lattice, and
\[
\Lambda_{odd} = L_{\lambda} \sqcup (L_{\lambda} + \frac{\lambda}{2}) \]
is an odd unimodular lattice, called the odd Leech lattice.
Let’s check that these are indeed the Leech lattices, meaning that they do not contain roots (vectors of norm two).
The only roots in $L = (A_1^{24})^+$ are the $48$ roots of $A_1^{24}$ and they are of the form $\pm \sqrt{2} [ 1, 0^{23} ]$, but none of them lies in $L_{\lambda}$ as their inproduct with $\lambda$ is one. So, all non-zero vectors in $L_{\lambda}$ have norm $\geq 4$.
As for the other part of $\Lambda$ and $\Lambda_{odd}$
\[
(L_{\lambda} + \frac{\lambda}{2}) \sqcup (L_{\lambda} + \mu + \frac{\lambda}{2}) = (L_{\lambda} \sqcup (L_{\lambda}+\mu))+\frac{\lambda}{2} = L + \frac{\lambda}{2} \]
From the description of $L=(A_1^{24})^+$ it follows that every coordinate of a vector in $L + \frac{\lambda}{2}$ is of the form
\[
\frac{1}{\sqrt{2}}(v+\frac{1}{2}) \quad \text{or} \quad \frac{1}{\sqrt{2}}(v+\frac{3}{2}) \]
with $v \in 2 \mathbb{Z}$, with the second case instances forming a codeword in $\mathcal{C}_{24}$. In either case, the square of each of the $24$ coordinates is $\geq \frac{1}{8}$, so the norm of such a vector must be $\geq 3$, showing that there are no roots in this region either.
If one takes for $l$ a vector of the form $\frac{1}{\sqrt{2}} v = \frac{1}{\sqrt{2}}[1^a,0^{24-a}]$ where $a=8,12$ or $16$ and $v \in \mathcal{C}_{24}$, takes $m=\mu$ as before, and repeats the construction, one gets the other Niemeier-neighbours of $(A_1^{24})^+$, that is, the lattices $(A_2^{12})^+$, $(A_3^8)^+$ and $(D_4^6)^+$.
The Leech lattice has a unique neighbour, that is, among the $23$ remaining Niemeier lattices there is a unique one, $(A_1^{24})^+$, sharing an index two sub-lattice with the Leech.
How would you try to construct $(A_1^{24})^+$, an even unimodular lattice having the same roots as $A_1^{24}$?
The root lattice $A_1$ is $\sqrt{2} \mathbb{Z}$. It has two roots $\pm \sqrt{2}$, determinant $2$, its dual lattice is $A_1^* = \tfrac{1}{\sqrt{2}} \mathbb{Z}$ and we have $A_1^*/A_1 \simeq C_2 \simeq \mathbb{F}_2$.
Thus, $A_1^{24}= \sqrt{2} \mathbb{Z}^{\oplus 24}$ has $48$ roots, determinant $2^{24}$, its dual lattice is $(A_1^{24})^* = \tfrac{1}{\sqrt{2}} \mathbb{Z}^{\oplus 24}$ and the quotient group $(A_1^{24})^*/A_1^{24}$ is $C_2^{24}$ isomorphic to the additive subgroup of $\mathbb{F}_2^{\oplus 24}$.
A larger lattice $A_1^{24} \subseteq L$ of index $k$ gives for the dual lattices an extension $L^* \subseteq (A_1^{24})^*$, also of index $k$. If $L$ were unimodular, then the index has to be $2^{12}$ because we have the situation
\[
A_1^{24} \subseteq L = L^* \subseteq (A_1^{24})^* \]
So, Kneser’s glue vectors form a $12$-dimensional subspace $\mathcal{C}$ in $\mathbb{F}_2^{\oplus 24}$, that is,
\[
L = \mathcal{C} \underset{\mathbb{F}_2}{\times} (A_1^{24})^* = \{ \tfrac{1}{\sqrt{2}} \vec{v} ~|~\vec{v} \in \mathbb{Z}^{\oplus 24},~v=\vec{v}~mod~2 \in \mathcal{C} \} \]
Because $L = L^*$, the linear code $\mathcal{C}$ must be self-dual meaning that $v.w = 0$ (in $\mathbb{F}_2$) for all $v,w \in \mathcal{C}$. Further, we want that the roots of $A_1^{24}$ and $L$ are the same, so the minimal number of non-zero coordinates in $v \in \mathcal{C}$ must be $8$.
We now know that there is a unique such code, the (extended) binary Golay code, $\mathcal{C}_{24}$, which has
one vector of weight $0$
$759$ vectors of weight $8$ (called ‘octads’)
$2576$ vectors of weight $12$ (called ‘dodecads’)
$759$ vectors of weight $16$
one vector of weight $24$
The $759$ octads form a Steiner system $S(5,8,24)$ (that is, for any $5$-subset $S$ of the $24$-coordinates there is a unique octad having its non-zero coordinates containing $S$).
Witt constructed a Steiner system $S(5,8,24)$ in his 1938 paper “Die $5$-fach transitiven Gruppen von Mathieu”, so it is not unthinkable that he checked the subspace of $\mathbb{F}_2^{\oplus 24}$ spanned by his $759$ octads to be $12$-dimensional and self-dual, thereby constructing the Niemeier-lattice $(A_1^{24})^+$ on that sunday in 1940.
John Conway classified all nine self-dual codes of length $24$ in which the weight
of every codeword is a multiple of $4$. Each one of these codes $\mathcal{C}$ gives a Niemeier lattice $\mathcal{C} \underset{\mathbb{F}_2}{\times} (A_1^{24})^*$, all but one of them having more roots than $A_1^{24}$.
