Image credit

In two words, this theory is based on the assumption that Vialism=Weilism and on textual similarities between the writings of Simone Weil and the lyrics of 21 pilots and the Clancy letters.

The Keons YouTube channel explains this in great detail.

Until now, I thought that Andre Weil was crucial to the story, and that Simone’s role was merely to have a boy/girl archetypical situation.

There’s this iconic photograph of them from 1922, taken weeks before Andre entered the ENS:

The same setting, boy on the left, girl to the right was used in the Nico and the niners-video, when they are young and in Dema

and when they are a quite a bit older, and in Trench, at the end of the Outside-video.

These scenes may support my theory that Dema was the ENS (both Andre and Simone studied there) as is explained in the post Where’s Bourbaki’s Dema?, and when they were both a bit older, and at the Bourbaki meetings in Chancay and Dieulefit, that they were banditos operating in Trench, as explained in the post Bourbaki = Bishops or Banditos.

There are two excellent books to read if you want to know more about the complex relationship between Andre and Simone Weil.

The first one is The Weil Conjectures: On Math and the Pursuit of the Unknown by Karen Olsson.

From it we get the impression that, at times, Simone felt intellectually inferior to Andre, who was three years older. She often asked him to explain what he was working on. Famous is his letter to her written in 1940 when he was jailed. Here’s a nice Quanta-article on it, A Rosetta stone for mathematics. This was also the reason why she wanted to attend some Bourbaki-meetings in order to get a better understanding of what mathematics was all about and how mathematicians think.

She was then very critical about mathematics because all that thinking about illusory objects had no immediate effect in real life. Well Simone, that’s the difference between mathematics and philosophy.

The second one is Chez les Weil, Andre et Simone written by Andre’s eldest daughter Sylvie.

From it we get another impression, namely that Andre may have been burdened by the fact that, after Simone’s death, his parents life centered exclusively around the preservation of her legacy, ignorant of the fact that their remaining child was one of the best mathematicians of his generation.

Poor Andre, on their family apartment in the Rue Auguste-Comte (which Andre used until late in his life when he was in Paris) is now this commemorative plaque

Well Andre, that’s the difference between a mathematician and a philosopher.

Let’s return to the role Simone Weil may play in Dema-lore. For starters, how did she appear in it?

She makes her appearance through a picture on Tyler’s desktop at the start of the Trench-era. This picture is a combination of two photographs from Bourbaki meetings, and Simone Weil features in both of them.

The photograph on the left is from the september 1937 meeting in Chancay, that on the right is from the september 1938 meeting in Dieulefit.

These are exactly the years crucial in Simone’s conversion to catholicism.

In the spring of 1937 she experienced a religious ecstasy in the Basilica of Santa Maria degli Angeli in Assisi.

Over Easter is 1938, Simone and her mother attended Holy Week services at the Solesmes Abbey where she had a mystic experience in which “Christ himself came down and took possession of me”.

One might ask whether there’s any connection between these religious experiences and her desire to attend these upcoming Bourbaki meetings. So, what was discussed during these conferences?

Mathematically, the 1938 meeting was not very exciting. Hardly any work was done, as they were preoccupied with all news of the Nazis invading Czechoslovakia. During the conference, Simone and Alain even escaped to Switzerland because they were convinced war was imminent. After a couple of days the Munich Treaty was signed, and Alain returned to Dieulefit, whereas Simone stayed in Switzerland, before returning to Paris.

On the other hand, the Chancay meeting was revolutionary as the foundations of topology were rewritten there with the introduction of the filter concept, dreamed up on the spot by Henri Cartan (the guy in the deckchair), while the others were taking a walk.

Simone was pretty impressed by the power of TOPology. In 1942 she wrote in her ‘Cahiers’:

One field of mathematics that deals with all the diverse sorts of orders (set theory and general topology) is a treasure-house that holds an infinity of valuable expressions that show supernatural truth.

Interestingly, she mentions the two math-subjects closest to the pilots’ universe: set theory studies all objects you can make starting from the empty set $\emptyset$, and topology studies the properties of objects and figures that remain unchanged even when you
morph them.

We’ll have to say more about this in a next post when we look into the Vialism=Weilism assumption.

Another appearance of Simone Weil in the lore might be through the cropped image you can find on the dmaorg-website.

The consensus opinion is that this is a picture of the young Clancy, next to one of the Bishops (Keons? Andre? Nico?).

In fact, the ‘little boy’ is actually a girl and her identity is unresolved as far as I know. But, given the date of the photograph (1956) the girl might be (mistakingly) taken for Andre’s daughter Sylvie.

Now, almost everyone, in particular her grandparents and Andre himself, found that Sylvie was a spitting image (almost a ‘copy’) of Simone Weil.

There are further indications that Simone Weil might be a Clancy.

Morph

In Morph there are these lines

He’ll always try to stop me, that Nicolas Bourbaki
He’s got no friends close, but those who know him most know
He goes by Nico
He told me I’m a copy
When I’d hear him mock me, that’s almost stopped me

During the meetings she attended, the other Bourbakis mocked Simone that she was a copy of het brother. From Karen Olsson’s book mentioned above:

To the others it’s startling to see his same glasses, his same face attached to this body clothed in an. unstylish dress and an off-kilter brown beret, carrying on in that odd monotone as she argues, via the chateau’s telephone, with the editors who publish her political articles.

Early in her career, Simone Weil was far from an original thinker. For her end-essay on Descartes she got the lowest score possible in order to pass from the ENS. Even Andre urged her to have a work-plan to develop her own ideas, rather than copying ideas from philosophers from the past.

Jumpsuit

Whereas Andre tried everything to avoid the draft, Simone was more of a warrior. In 1935 she volunteered to fight on the Republican side in the Spanish civil war, until a kitchen accident forced her to return to France.

Later in 1943 she left New-York to return to England and enlist in the French troupes of General de Gaulle, hoping to be parachuted behind enemy lines. Given her physical state, the military command decided against it. Upset by this refusal, she felt she had no other option than to deny herself food in empathy with the starving French.

She didn’t succeed in crossing Paladin Strait, sorry the Channel.

Overcompensate

Can this be Simone Weil?

.

In this series:

• Bourbaki and TØP : East is up
• Bourbaki = Bishops or Banditos?
• Where’s Bourbaki’s Dema?
• Weil photos used in Dema-lore
• Dema2Trench, AND REpeat
• TØP PhotoShop mysteries
• 9 Bourbaki founding members, really?
• Bourbaki and Dema, two remarks
• Clancy and Nancago

By then we will have been told why Andre Weil and the Bourbaki group show up in the Trench/Dema tale.

This leaves me a couple of weeks to pursue this series of posts (see links below) in which I try to find the best match possible between the factual history of the Bourbaki group and elements from the Dema-storyline.

Two well-known Bourbaki-photographs seem important to the pilots. The first one is from the september 1938 Dieulefit/Beauvallon Bourbaki congress:

At the time, Bourbaki still had to publish their first text, they were rebelling against the powers that be in French mathematics, and were just kicked out of the Julia seminar.

In clikkies parlance: at that moment the Bourbakistas are Banditos, operating in Trench.

The second photograph, below on the left, is part of a famous picture of Andre Weil, supposedly taken in the summer of 1956.

At that time, Bourbaki was at its peak of influence over French mathematics, suffocating enthusiastic math-students with their dry doctrinal courses, and forcing other math-subjects (group theory, logic, applied math, etc.) to a virtual standstill.

In clique-speech: at that moment the Bourbakistas are Bishops, ruling Dema.

Let me recall the story of one word, associated to the Bourbaki=Bishops era which lasted roughly twenty years, from the early 50ties till Bourbaki’s ‘death’ in 1968 : Nancago.

From the 50ties, Nicolas Bourbaki signed the prefaces of ‘his’ books from the University of Nancago.

Between 1951 and 1975, Weil and Diedonne directed a series of texts, published by Hermann, under the heading “Publications de l’Institut mathematique de l’Universite de Nancago”.

Bourbaki’s death announcement mentioned that he “piously passed away on November 11, 1968 at his home in Nancago”.

Nancago was the name of a villa, owned by Dieudonne, near Nice. Etc. etc.

But then, what is Nancago?

Well, NANCAGO is a tale of two cities: NANcy and ChiCAGO.

The French city of Nancy because from the very first Bourbaki meetings, the secretarial headquarters of Bourbaki, led by Jean Delsarte, was housed in the mathematical Institute in Nancy.

Chicago because that’s where Andre Weil was based after WW2 until 1958 when he moved to Princeton.

Much more on the history of Nancago can be found in the newspaper article by Bourbaki scholar par excellance Liliane Beaulieu: Quand Nancy s’appelait Nancago (When Nancy was called Nancago).

Right, but then, if Nancago is the codeword of the Bourbaki=Bishops era, what would be the corresponding codeword for the Bourbaki=Banditos era?

As mentioned above, from 1935 till 1968 Bourbaki’s headquarters was based in Nancy, so even in 1938 Nancy should be one of the two cities mentioned. But what is the other one?

In 1938, Bourbaki’s founding members were scattered over several places, Jean Delsarte and Jean Dieudonne in Nancy, Szolem Mandelbrojt and Rene de Possel in Clermont-Ferrand, and Andre Weil and Henri Cartan in Strasbourg. Claude Chevalley was on a research stay in Princeton.

Remember the Bourbaki photograph at the Beauvallon meeting above? Well, it was taken in september 1938 when the Munich Agreement was reached.

Why is this relevent? Well, because Strasbourg was too close to the German border, right after the Munich agreement the Strasbourg Institute was ordered to withdraw to the University of Clermont-Ferrand.

Clermont-Ferrand lies a bit south of Vichy and remained in WW2 in the ‘free zone’ of France, whereas Strasbourg was immediately annexed by Germany.

For more on the importance of Clermont-Ferrand for Bourbaki during 1940-1942 see the article by Christophe Eckes and Gatien Ricotier Les congrès de Clermont-Ferrand de 1940, 1941 et 1942.

That is, all Bourbaki members where then either affiliated to Nancy or to Clermont-Ferrand.

A catchy codeword for the Bourbaki=Banditos era, similar to Nancago as the tale of two cities, might then be:

CLermont-Ferrand + nANCY = CLANCY.

[For clikkies: rest assured, I’m well aware of the consensus opinion on the origins of Clancy’s name. But in this series of posts I’m not going for the consensus or even intended meanings, but rather for a joyful interplay between historical facts about the Bourbaki group and elements from Dema-lore.]

In this series:

• Bourbaki and TØP : East is up
• Bourbaki = Bishops or Banditos?
• Where’s Bourbaki’s Dema?
• Weil photos used in Dema-lore
• Dema2Trench, AND REpeat
• TØP PhotoShop mysteries
• 9 Bourbaki founding members, really?
• Bourbaki and Dema, two remarks
• Clancy and Nancago

A bit more on that book you can read in the topos of unconsciousness. For another take on this you can visit the blog of l’homme quantique – Sur les traces de Lévi-Strauss, Lacan et Foucault, filant comme le sable au vent marin…. There is a series of posts dedicated to the reading of ‘A l’ombre de Grothendieck et de Lacan’:

• 1. Initiation au topos
• 2. Rencontre d’une évidence
• 3. Métapsychologie du topos
• 4. Psychanalyse et mathématiques
• 5. Temps et instant
• 6. Mythes, fantasmes et topos classifiant

Alain Connes isn’t the first (former) Bourbaki-member to write a book together with a Lacan-disciple.

In 1984, Henri Cartan (one of the founding fathers of Bourbaki) teamed up with the French psychoanalyst (and student of Lacan) Jean-Francois Chabaud for “Le Nœud dit du fantasme – Topologie de Jacques Lacan”.

(Chabaud on the left, Cartan on the right, Cartan’s wife Nicole in the mddle)

“Dans cet ouvrage Jean François Chabaud, psychanalyste, effectue la monstration de l’interchangeabilité des consistances de la chaîne de Whitehead (communément nommée « Noeud dit du fantasme » ou du « Non rapport sexuel » dans l’aire analytique), et peut ainsi se risquer à proposer, en s’appuyant sur les remarques essentielles de Jacques Lacan, une écriture du virage, autre nom de la passe. Henri Cartan (1904-2008), l’un des Membres-fondateur de N. Bourbaki, a contribué à ce travail avec deux réflexions : la première, considère cette monstration et l’augmente d’une présentation ; la seconde, traite tout particulièrement de l’orientation des consistances. Une suite de traces d’une séquence de la chaîne précède ce cahier qui s’achève par : « L’en-plus-de-trait », une contribution à l’écriture nodale.”

Lacan was not only fascinated by the topology of surfaces such as the crosscap (see the topos of unconsciousness), but also by the theory of knots and links.

The Borromean link figures in Lacan’s world for the Real, the Imaginary and the Symbolic. The Whitehead link (that is, two unknots linked together) is thought to be the knot (sic) of phantasy.

In 1986, there was the exposition “La Chaine de J.H.C. Whitehead” in the
Palais de la découverte in Paris (from which also the Chabaud-Cartan picture above is taken), where la Salle de Mathématiques was filled with different models of the Whitehead link.

In 1988, the exposition was held in the Deutches Museum in Munich and was called “Wandlung – Darstellung der topologischen Transformationen der Whitehead-Kette”

The set-up in Munich was mathematically more interesting as one could see the link-projection on the floor, and use it to compute the link-number. It might have been even more interesting if the difference in these projections between two subsequent models was exactly one Reidemeister move…

You can view more pictures of these and subsequent expositions on the page dedicated to the work of Jean-Francois Chabaud: La Chaîne de Whitehead ou Le Nœud dit du fantasme Livre et Expositions 1980/1997.

Part of the first picture featured also in the Hommage to Henri Cartan (1904-2008) by Michele Audin in the Notices of the AMS. She writes (about the 1986 exposition):

“At the time, Henri Cartan was 82 years old and retired, but he continued to be interested in mathematics and, as one sees, its popularization.”

The past few weeks I’ve been looking around for topics to be included in such as course, relevant to architects/artists (not necessarily to engineers/mathematicians).

One of the best texts I’ve found on this (perhaps in need of a slight update) is the 1986-paper by Jay Kappraff: A course in the mathematics of design. He suggests the following list of topics:

• graph theory
• polyhedra
• tilings of the plane
• three dimensional packings
• proportion and the golden mean
• transformations
• symmetry
• vectors

We all know that an awful lot of math and computation is needed to design a building, but today all of the hardcore use of vectors, equations and transformations is conveniently hidden from the architect’s view by digital design platforms and CAD-programs.

These computational tools offer new creative possibilities, as illustrated in the beautiful book The new mathematics of architecture by Jane Burry and Mark Burry, also available in Dutch with a cover picture of the Möbius bridge in Bristol

In this book, about 50 recent architectural projects are clustered around these topics:

• mathematical surfaces and seriality
• chaos, complexity, emergence
• packings and tilings
• optimization
• topology
• datascapes and multi-dimensionality

In the description of the projects, cool math-topics are (sadly only) touched, including

• Amman tilings
• Aperiodic tilings
• Cellular automata
• Chaos theory
• Danzer packings
• Fractals
• Homology
• Knot theory
• Higher dimensional manifolds
• Minimal surfaces
• NURBS
• Penrose tilings
• Singularity theory
• Voronoi diagrams
• Seifert surfaces
• Quasicrystals

It will take me some time to find a balance between these two approaches. Common themes clearly are

• Shapes : what is possible/impossible in 2D and 3D, and how can mathematics help us to find new exciting shapes (think minimal and Seifert surfaces, knot complements, etc.)
• Symmetry : what is possible/impossible in 2D and 3D, and what can mathematics tell us about new symmetries (think emerging symmetries from aperiodic tilings and quasicrystals)

Over the coming months I’ll be writing the course notes and may post about it here. For this reason I’ve included a new category DesignerMaths.

If you have suggestions, please let me know.

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 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.

Chapter 4(?) :Three Studies for a Self-Portrait, (Francis Bacon, 1979)

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:

Chapter 4(?) : Three studies for portrait of Henrietta Moraes (1963)

Here are some of my best guesses:

Chapter 3(?): Portrait of Henrietta Moraes (1963)

Chapter 6(?): Three Studies for Figures at the Base of a Crucifixion (1944)

Chapter 5(?): Triptych Inspired by the Oresteia of Aeschylus (1981)

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.

It depicts beautifully daily (or better, nightly) life in mathematical and artistic circles, especially in Paris between 1900 and 1906.

Bricard, Caratheodory, Dedekind, Dehn, De la Vallee-Poussin, Frege, Godel, Hadamard, Hamel, Hatzidakis, Hermite, Hilbert, Klein, Lindemann, Minkowski, Peano, Poincare, Reynaud, Russell and Whitehead all make a brief appearance, as do Appollinaire, Casagemas, Cezanne, Degas, Derain, Max Jacob, Jacobides, Lumiere, Matisse, Melies, Pallares, Picasso, Renoir, Salmon, Toulouse-Lautrec, Utrillo, Zola.

Both lists contain names I had never heard of. But the biggest surprise, to me, was to discover the name of Maurice Princet, “le mathématicien du cubisme”.

Princet (1875-1973) was a mathematician who frequented the group around Pablo Picasso at the Bateau-Lavoir in Montmartre (at least until 1907 when his wife left him for the painter Derain).

Princet introduced the group to the works of Poincare and the concept of the 4-th dimension. He gave Picasso the book “Traité élémentaire de géométrie à quatre dimensions” by Jouffret, describing hyper-cubes and other polyhedra in 4 dimensions and ways to project them dowm to the 2 dimensions of the canvas.

This book appears to have been influential in the genesis of Picasso’s Les Demoiselles d’Avignon (the painting also appears, in an unfinished state, in “Pythagorean Crimes”).

Some other painters tried to capture movement with projections from the 4-th dimension. A nice example is Nude descending a staircase by Marcel Duchamp (mostly known for his urinoir…).

Maurice Princet loved to get the artists interested in the new views on space. Duchamp told Pierre Cabanne, “We weren’t mathematicians at all, but we really did believe in Princet”.

I don’t know whether Duchamp liked Princet’s own attempts at painting. Here’s a cubistic work by Maurice Princet himself.

The stage director Anne Bitran only discovered Grothendieck’s life by reading the front pages of French newspapers, the day after Grothendieck passed away, in November 2014.

« Rêves et Motifs » is a piece inspired by Récoltes et Semailles.

Anne Bitran: ” In Récoltes et semailles we meet a scientist who has his feet on the ground and shares our curiosity about the world around us, with a strong political engagement. This is what I wanted to share with this piece.”

Some of Grothendieck’s dessins d’enfant make their appearance. Is that one Monsieur Mathieu in the center? And part of the Hexenkuche top left? (no, see Vimeo below)

And, does this looks like the sculpture ‘Grothendieck as Shepherd’ by Nina Douglas?

More information about the production can be found at the Les Remouleurs website (in French).

In case you are interested, make sure to be in Lunéville, November 29th or 30th.

This includes the groups of symmetries of Escher’s Circle Limit drawings. Here’s Circle Limit III

And ‘Angels and Devils’ aka Circle Limit IV:

If one crawls along a mirror of this pattern until one hits another mirror and then turns right along this mirror and continues like this, you get a quadrilateral path with four corners $\frac{\pi}{3}$, whose center seems to be a $4$-fold gyration point. So, it appears to have symmetry $4 \ast 3$.

(image credit: MathCryst)

However, looking more closely, every fourth figure (either devil or angel) is facing away rather than towards us, so there’s no gyration point, and the group drops to $\ast 3333$.

Harold S. M. Coxeter met Escher in Amsterdam at the ICM 1954.

The interaction between the two led to Escher’s construction of the Circle Limits, see How did Escher do it?

Here’s an old lecture by Coxeter on the symmetry of the Circle Limits:

A central street in Helsinki has been paved with Penrose tiles.

(image credit: Sattuman soittoa)

From a Finnish paper:

“The street could also be an object to mathematical awe. The stone under one’s feet is embroidered with some profound geometry, namely, Penrose tiling.

In 1974, a British mathematician Roger Penrose realised a plane could be fully covered with a few simple rules such that the pattern constantly changes. These kind of discontinuous patterns are interesting to mathematicians since the patterns can be used to solve other geometrical problems. Together, the tiles can randomly form patterns reminding a star or the Sun but they do not regularly recur in the tiling.

Similar features are found in the old Arabic ornaments. The tiling of the Central Street prom was selected by Yrjö Rossi.

If your kid stays put to stare at the tiling, they might have what they need in order to become a mathematician.”

(via Reddit/m)