the bourbaki code

Roy Lisker (remember him from the Mormoiron post?) has written up his Grothendieck-quest(s), available for just 23$, and with this strange blurb-text:

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"The author organized a committee to search for him that led to his discovery, in good health and busily at work, in September, 1996. This committee has since become the Grothendieck Biography Project. All of this is recorded in a 300 page account in 3 parts."

Probably he refers to the trip made by Leila Schneps and Pierre Lochak, nicely described in Sam Leith's The Einstein of maths:

"One of the last members of the mathematical establishment to come into contact with him was Leila Schneps. Through a series of coincidences, she and her future husband, Pierre Lochak, learned from a market trader in the town he left in 1991 that ‘the crazy mathematician’ had turned up in another town in the Pyrenees. Schneps and Lochak in due course staked out the marketplace of the town, carrying an out-of-date photograph of Grothendieck, and waited for the greatest mathematician of the 20th century to show up in search of beansprouts.

‘We spent all morning there in the market. And then there he was.’ Were they not worried he’d run away? ‘We were scared. We didn’t know what would happen. But he was really, really nice. He said he didn’t want to be found, but he was friendly. We told him that one of his conjectures had been proved. He had no idea. He’d stopped being interested in maths at that stage. He thought his unpublished work would all have been long forgotten.’"

To city-cats this may seem an improbable coincidence, but if you live in the French mountains for some time, you learn to group your shoppings, and do them on market-days. The nearest market-town, where you can find a decent 'boulangerie' or supermarket, may be just 20 kms down the road, but it'll take you close to an hour to get there.

If you sit near the town-fountain on market-days, for some weeks, you will have seen most of the people living in the vast neighborhood.

So, we'd better try to find Leila's market-town.

One of the nicer talks on the life of Grothendieck was given by Winfried Scharlau (who also has two books on offer on Grothendieck's life, seems to become an emerging bisiness ...) at the IHES Grothendieck colloque.

This video is stuffed with unknown (at least to me) pictures of Grothendieck, his places at Mormoiron and Villecun and of his four children still living in France. Highly recommended!

But, the lecture has a very, very strange ending.

At 1hr 06.51 into the video he shows the slide reproduced on the left below and says: "Okay and here's a picture on which I will not further comment. That's the last thing I want to show you. I thank you very much for your patience."

Leila Schneps has a page with pictures on her website, including 3 pictures of her house, and then the one on the right above, merely described as 'Another house'.

And then there's this paragraph from Roy Lisker's (him again) Travelogue-France (March 8-April 5, 2005) part 2

"I left the IHP around 11 to return to the CNRS research center at 175 rue du Chevaleret. Pierre Lochak and I discussed the possibility of my going to the town of St. Giron outside of Toulouse to make another impromptu visit to La Maison d'Alexandre Grothendieck."

So, here we have three founding members of the Grothendieck circle linking publicly to the same picture of that one place they want to keep secret at all cost?

Dream on!

If you followed this series at all and have looked at the pictures of Grothendieck's houses in Mormoiron or Villecun it is hard to imagine him living in a bourgeois-house, dating from the end of the 19th century, in a medium-sized market-town.

Still, it is quite likely that the picture is indeed taken in Saint-Girons, on some saturday in 1996 when Leila and Pierre bumped into Grothendieck on the market in Saint-Girons.

After all, Saint-Girons is the market-town closest to the final Grothendieck-spot...

Previous in this series:

- Vendargues

- Mormoiron

- Massy

- Olmet-et-Villecun

- un petit village de l'Ariège

We would love to conclude this series by finding the location of the "final" Grothendieck-spot, before his 85th birthday, this thursday.

But, the road ahead will be treacherous, with imaginary villages along the way and some other traps planted by the nice people of the Grothendieck Fan Club

It is well-known that some members (if not all) of the GFC know the exact location of Grothendieck's hideout in the Pyrenees. Trying to pry this information from them, pledging to keep the name secret, is described as 'solving an equation in n unknowns' in the article Le trésor oublié du génie des maths (h/t +David Roberts):

"Cela fait aujourd’hui vingt-deux ans qu’il vit reclus au pied des Pyrénées, dans un village où personne ne va par hasard et dont le nom doit rester secret. Il le souhaite et ceux qui, de loin, le protègent le souhaitent également. Obtenir l’adresse contre l’assurance de ne pas le déranger prend le temps de résoudre une équation à «n» inconnues. Se poster devant chez lui permet de constater qu’il est bien vivant au milieu d’un village qui le regarde comme «le savant» sans chercher à en savoir plus. A 84 ans, il vient se chauffer au soleil devant son portail puis rentre dans sa maison où nul ne pénètre."

As we don't want to take this vow of secrecy, we will have to rely on the few hints they left in the literature. Presumably, the most trustworthy information is to be found in Pierre Cartier's paper A country of which nothing is known but the name, Grothendieck and “motives”:

"As I already said, he retired in 1988, and has lived since then in self-imposed exile. At first he lived near the Fontaine de Vaucluse, in the middle of a little vineyard that he cultivated, and near to his daughter Johanna and his grandchildren. But later he broke off every family relation. He didn’t seem to mind that the place where he lived was located so near to the infamous Camp du Vernet which played a sad role in his childhood. He lived for years without any contact with the outside world and only a few people even knew where he was. He chose to live alone, considered by his neighbors as a “retired mathematics professor who’s a bit mad”."

There is a small (but for our purposes important) addition to the first sentence in the French version:

"... il a pris sa retraite en 1988, et vit depuis un exil intérieur dans un petit village de l'Ariège."

This addition makes our quest a bit more 'doable'. The department of l'Ariège is one of the lesser populated ones in France (having less that 150.000 inhabitants), and has 'only' 332 villages.

One can divide this number roughly by 2, leaving out the larger villages and towns and those situated in the higher mountains, where living must be extremely difficult for an 85 year old.

An alternative reason for leaving out the more southern villages is Cartier's claim that 'le petit village' is close to the Camp du Vernet, which is the place from which Grothendieck's father was deported to Auschwitz.

This former concentration camp is located in Le Vernet, close to the town of Pamiers (central upper part of the map).

So, one can safely assume that the final G-spot must lie on the map below (click on it to navigate and explore).

Previous in this series:

- Vendargues

- Mormoiron

- Massy

- Olmet-et-Villecun

Before we start the quest for the final G-spot, hopefully in time for Grothendieck's 85th birthday, one more post on Alexandre's 'hippy-days'.

In the second part of Allyn Jackson's "The Life of Alexandre Grothendieck" she tells the story that AG, while touring the US to spread the gospel of the eco-mouvement "Survivre et Vivre" (the deal was that he gave 1 math-talk if he was allowed to give another one on ecology/politics), met a graduate student of Daniel Gorenstein, Justine Skalba, who quickly became a G-groupie and returned with him after the US-trip to France, where she lived with him for two years (and had one child with him, John, who later also became a mathematician).

Allyn Jackson writes:

"In early 1973 he (AG) and Skalba moved to Olmet-le-sec (probably she means: Olmet sec, so without any additions), a rural village in the south of France. This area was at the time a magnet for hippies and others in the counterculture movement who wanted to return to a simpler lifestyle close at hand (I would have added: and, it still is). Here Grothendieck again attempted (he did this once before in his Parisian period, setting up a commune in Chatenay-Malabry) to start up a commune, but personality conflicts led to its collapse. At various times three of Grothendieck's children came to live in the Paris commune and in the one in Olmet (probably this being: Johanna, Mathieu and Alexandre who even today maintain an alternative lifestyle). After the commune disolved, he moved with Skalba and his children to Villecum, a short distance away."

As Yves Ladegaillerie tells Jackson, Grothendieck lived an ascetic, unconventional life in an old house without electricity in Villecun, about thirty-five miles outside of Montpellier. Ladegaillerie remembered seeing Justine Skalba and her baby there. Many friends, acquantances and students went to visit Grothendieck there, including people from the ecology movement.

Here's the (in)famous house in Villecun (h/t Winfried Scharlau)

And, if you are a bit like me, wanting to see everything with G-earth or maps, here's the scenery (click on the image to be there).

Again, if someone at the Mairie d'Olmet-et-Villecun reads this, please consider adding to your list of 'Personnalités liées à la commune'

- Michel Chevalier

- Paul Dardé

this one:

- Alexandre Grothendieck

Merci infiniment!

Previous in this series:

- Vendargues

- Mormoiron

- Massy

One week from now, Alexandre Grothendieck will turn 85. Today, we'll have a glance at his 'wilder years', the early 70ties, when he resigned from the IHES and became one of the leading figures in the French eco-movement. This iconic picture is from those days

The text reads:

"Schurik entre les "frères ennemis" Gaston Galan et Dyama, rue Polonceau.

Derrière, Chantal et Motito (femme et fille de Gaston)."

Schurik (that is, AG) between the 'hostile brothers' Gaston Galan and Dyama in the 'rue Polonceau'. Behind, Chantal and Motito (wife and daughter of Gaston).

However, if you stroll down the Rue Polonceau via StreetView (note to self: high time to revisit Paris IRL) it is unclear where this picture might have been taken. One notable exception perhaps, at 38, Rue Polonceau.

Today, this address houses the feminist group Ruptures with the noble goal to establish a society based on a genuine equality between women and men.

"L’association se donne pour objectif de substituer à la société patriarcale une société fondée sur une égalité réelle et pas seulement formelle entre les femmes et les hommes dans le domaine économique, social, politique et culturel. Elle est basée sur la laïcité et la parité."

Besides, they want to encourage cooperation with other movements striving for a better world:

"Convergences des luttes féministes, altermondialistes, écologistes, antiracistes"

It is thus very well possible that this address was already used in the 70ties by similar social groups, such as the ecological movement "Survivre et Vivre" (Survive and Live), a movement founded in 1970 by three renowned mathematicians: Grothendieck, Claude Chevalley and Pierre Samuel. The origins and evolution of Survivre et Vivre are nicely described on this page at Science et Société.

So, whoever wrote that text beneath the photograph is probably right, though I'd love to hear more details. Still, this picture was the first thing on my mind when i found the place where Grothendieck lived in his IHES-years (and shortly afterwards).

The first issue of the Bulletin of Survivre et Vivre (btw. most issues are available from the Grothendieck circle and are fun reading material if you are, like me, in constant need to brush up your French) concludes with a list of the names, professions and addresses of the group's members (25 at the time, including AG's mother-in-law (Julienne Dufour, mother of his wife Mireille Dufour) and his son (from another mother) Serge):

So, here we are, Grothendieck lived with his wife, children (apart from Serge who was at the time based in Nice) and mother-in-law at 2, Avenue de Verrières, Massy, France

If you click on the picture, you can walk around this G-spot, located just across the Gare de Massy.

If some of you have better info on this or other Grothendieck-spots, please fill me in.

I'm bound to travel south, possibly in search for more information, end of next week...

Previous in this series:

- Vendargues

- Mormoiron

With Grothendieck's 85th brithday coming up, march 28th, we continue our rather erratic quest to locate the spots that once meant a lot to him.

Ever wondered what Grothendieck's last-known hideout looked like? Well, here's the answer:

(h/t gruppe eM)

And, here's the story.

One of the stranger stories to be found on the web is the Grothendieck quest by Roy Lisker. In 1988, after AG declined the Crafoord Prize, Roy convinced an editor of Le Nouvel Observateur to hire him to uncover the whereabouts of Alexandre Grothendieck and, if possible, to interview him.

The 'quest' is an hilarious account of Roy's attempts to prise AG's address out of the people from the Montpellier maths department, his subsequent travels and stay at Grothendieck's place.

He put the text online in 2008 and made it intentionally opaque wrt. AG's phone number and address:

"His phone, if in fact this notorious hermit bothered with such contrivances, was unlisted."

"...of his adopted village of Lessmoiron (after a 20 year silence it is permissible to reveal its name) , in the department of the Vacluse, a region of France long habituated to the herbergement of exiled or alienated Popes."

By that time the Grothendieck-Serre correspondence had been published for over 4 years, including a letter dated 2 september 1984, giving away this 'secret information':

So, not only do we have a phone number (today it would be 0033(0)4 90 61 88 30), but also that AG lived in the hamlet "Les Aumettes" in the village of Mormoiron (and not 'Less'moiron, duh), close to the famous (to any bicyclist) Mont Ventoux.

From Roy's quest we learn that it is about 3 kms from the center of Mormoiron and that

"Grothendieck's cottage was built up against a hillside, it's conical shape hugging the hill like the helmet of a medieval knight. The lower entrance was graced by a pair of sturdy French windows. Above these, at the level of the attic, two tiny rectangular windows filtered light into the bedroom."

If you want to explore the immediate neighborhood of Les Aumettes, click on the picture below (bonus points for anyone who is able to pinpoint the exact location on the map).

If someone at the Mairie de Mormoiron reads this, please consider adding to your list of 'Personnalités liées à la commune'

- Raymond Guilhem de Budos (? - 1363), neveu de Clément V, seigneur de Clermont, Lodève, Budos,Beaumes-de-Venise, Bédoin, Caromb, Entraigues, Loriol et Mormoiron, gouverneur de Bénévent, Maréchal de la Cour pontificale et Recteur du Comtat Venaissin de 1310 à 1317.

- Guillaume-Emmanuel-Joseph, baron Guilhem de Sainte-Croix, (1746-1809), membre de l'Institut, auteur d'un essai Examen critique des anciens historiens d'Alexandrie, couronné par l'Académie des Inscriptions et Belles-Lettres en 177224.

- Paul Vialis, ancien maire de Moirmoiron ou il est né en 1848, et député de Vaucluse.

- Albert Schou (da), (27 mars 1849 – 4 février 1900), photographe danois

this one:

- Alexandre Grothendieck (né en 1928), mathématicien français ayant reçu la Médaille Fields.

Merci!

Previous in this series:

- Vendargues

In a couple of days, on march 28th, Alexandre Grothendieck will turn 85.

To mark the occasion we'll run a little series, tracking down places where he used to live, hoping to entice some of these villages in the south of France to update their Wikipedia-page by adding under 'Personnalités liées à la commune' the line

- Alexandre Grothendieck (né en 1928), mathématicien français ayant reçu la Médaille Fields.

as did the village of Le Chambon-sur-Lignon, where Grothendieck was kept safe from 1942-1945, separated from his mother who was send to an internment camp (his father was deported by the French authorities in august 1942 and killed by the Nazis in Auschwitz).

After the war, Alexandre was reunited with his mother and, according to Allyn Jackson's As If Summoned from the Void: The Life of Alexandre Grothendieck, they "went to live in Maisargues, a village in the wine-growing region outside of Montpellier".

Amir Aczel adds to this in his book The artist and the mathematician, the story of Nicolas Bourbaki: "From 1945 until 1948, mother and son lived in the small hamlet of Mairargues, virtually hidden among the vineyards, a dozen kilometers from Montpellier. They had a marvelous small garden: they never had to work at gardening and yet the earth was so fertile, and the rains so abundant, that the garden produced a plentiful harvest of figs, spinach, and tomatoes. Their garden was at the verge of splendid poppies. Grothendieck remembers his time there with his mother as "la belle vie"."

But, there is no Maisargues nor Mairargues to be found in France.

There is the village of Caissargues, close to Nimes, about 50 kms from Montpellier, and, there is the village of Meyrargues, close to Pertuis, more than 170 kms from Montpellier.

So, where is the hamlet of "la belle vie"?

Jackson's and Aczel's info is based on a footnote in Grothendieck's Recoltes et semailles (in fact, Aczel's text is a mere translation of it):

"Entre 1945 et 1948, je vivais avec ma mère dans un petit hameau à une dizaine de kilomètres de Montpellier, Mairargues (par Vendargues), perdu au milieu des vignes. (Mon père avait disparu à Auschwitz, en 1942.) On vivait chichement sur ma maigre bourse d’étudiant. Pour arriver à joindre les deux bouts, je faisais les vendanges chaque année, et après les vendanges, du vin de grapillage, que j’arrivais à écouler tant bien que mal (en contravention, paraît-il, de la législation en vigueur. . . ) De plus il y avait un jardin qui, sans avoir à le travailler jamais, nous fournissait en abondance figues, épinards et même (vers la fin) des tomates, plantées par un voisin complaisant au beau milieu d’une mer de splendides pavots. C’était la belle vie."

Although Grothendieck misspells Mayrargues, he points to the village of Vendargues which is situated 12 kms east of Montpellier and has a hamlet called Mayrargues (foto above). Via Google Maps you can visit "l'hameau de la belle vie" by yourself (it even has streetview).

If someone at the Mairie de Vendargues comes across this post, please consider adding to your list of famous (former) inhabitants:

- Marcelin Albert (1851-1921), séjourne au mazet de Montmaris, leader de la révolte viticole, est le parrain de Marcellin Guille né en 1907 et oncle d'Archiguille.

- Sabri Allouani (1978-), raseteur (Septuple Vainqueur du Championnat de France de la Course Camarguaise au As 2000-2007)

- Archiguille (Augustin François Guille, peintre contemporain "Transfigurations") vivant en Suisse.

- Laurent Ballesta (1974-), Biologiste marin, plongeur, photographe, collaborateur de Nicolas Hulot)

- Le général Pierre Berthezène (1775-1847), baron d'Empire, pair de France (1775-1847)

- Jerôme Bonnisel (joueur de football professionnel)

- le baron Pierre Le Roy de Boiseaumarié, (1890-1967), fondateur des appellations d'origine contrôlées, vigneron à Châteauneuf-du-Pape.

this one:

- Alexandre Grothendieck (né en 1928), mathématicien français ayant reçu la Médaille Fields.

Thanks!

A classic among mathematical jokes is the paper in the August/September 1938 issue of the American Mathematical Monthly "A contribution to the mathematical theory of big game hunting" by one Hector Petard of Princeton who would marry, one year later, Nicolas Bourbaki's daughter Betti.

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There are two main sources of information on the story behind this paper. There are Frank Smithies' "Reminiscences of Ralph Boas" in the book Lion Hunting & Other Mathematical Pursuits and the transcript of an interview with John Tukey and Albert Tucker at Princeton University on 11 April 1984, part of the oral-history project on the Princeton mathematics community in the 1930s.

Smithies recalls being part of a lively group of people in Princeton during the academic year 1937/38 including Arthur Brown, Ralph Traber, Lyman Spitzer, Hugh Dowker, John Olmsted, Henry Walman, George Barnard, John Tukey, Mort Kanner (a physicist), Dick Jameson (a linguist) and Ralph Boas. Smithies writes:

"At some time that winter we were told about the mathematical methods for lion-hunting that have been devised in Gottingen, and several of us came up with new ones; who invented which method is now lost to memory. Ralph (Boas) and I decided to write up all the methods known to us, with a view to publication, conforming as closely as we could to the usual style of a mathematical paper. We choose H. Petard as a pseudonym ("the engineer, hoist with his own petard"; Hamlet, Act III, Scene IV), and sent the paper to the Americal Mathematical Monthly, over the signature of E. S. Pondiczery."

Pondiczery was Princeton's answer to Nicolas Bourbaki, and in the interview John Tukey recalls from (sometimes failing) memory:

"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. Then there's the wedding invitation done by the Bourbakis. It was for the marriage of Betty Bourbaki and Pondiczery. It was a formal wedding invitation with a long Latin sentence, most of which was mathematical jokes, three quarters of which you could probably decipher. Pondiczery even wrote a paper under a pseudonym, namely "The Mathematical Theory of Big Game Hunting" by H. Petard, which appeared in the Monthly. There were also a few other papers by Pondiczery."

Andrew Tucker then tells the story of the paper's acceptance:

"Moulton, the editor of the Monthly at that time, wrote to me saying that he had this paper and the envelope was postmarked Princeton and he assumed that it was done by some people in math at Princeton. He said he would very much like to publish the paper, but there was a firm policy against publishing anything anonymous. He asked if I, or somebody else that he knew and could depend on, would tell him that the authorship would be revealed if for any reason it became legally necessary. I did not know precisely who they were, but I knew that John [Tukey] was one of them. He seemed to be in the thick of such things. John agreed that I could accept Moulton's terms. I sent a letter with this assurance to Moulton and he went ahead and published it."

I've made a timeline of the 16 different methods contained in the paper, keeping their original numbering and formulation. Enjoy!

The footnote on page E. II.6 in Bourbaki's 1970 edition of "Theorie des ensembles" reads

If this is completely obvious to you, stop reading now and start getting a life. For the rest of us, it took me quite some time before i was able to parse this formula, and when i finally did, it only added to my initial confusion.

Though the Bourbakis had a very preliminary version of their set-theory already out in 1939 (Fascicule des Resultats), the version as we know it now was published, chapter-wise, in the fifties: Chapters I and II in 1954, Chapter III in 1956 and finally Chapter IV in 1957.

In the first chapter they develop their version of logic, using 'assemblages' (assemblies) which are words of signs and letters, the signs being $\tau, \square, \vee, \neg, =, \in$ and $\supset$.

Of these, we have the familiar signs $\vee$ (or), $\neg$ (not), $=$ (equal to) and $\in$ (element of) and, three more exotic ones: $\tau$ (their symbol for the Hilbert operator $\varepsilon$), $\square$ a sort of wildcard variable bound by an occurrence of $\tau$ (the 'links' in the above scan) and $\supset$ for an ordered couple.

The connectives are written in front of the symbols they connect rather than between them, avoiding brackets, so far example $(x \in y) \vee (x=x)$ becomes $\vee \epsilon x y = x x$.

If $R$ is some assembly and $x$ a letter occurring in $R$, then the intende meaning of the *Hilbert-operator* $\tau_x(R)$ is 'some $x$ for which $R$ is true if such a thing exists'. $\tau_x(R)$ is again an assembly constructed in three steps: (a) form the assembly $\tau R$, (b) link the starting $\tau$ to all occurrences of $x$ in $R$ and (c) replace those occurrences of $x$ by an occurrence of $\square$.

For MathJax reasons we will not try to draw links but rather give a linked $\tau$ and $\square$ the same subscript. So, for example, the claimed assembly for $\emptyset$ above reads

$\tau_y \neg \neg \neg \in \tau_x \neg \neg \in \square_x \square_y \square_y$

If $A$ and $B$ are assemblies and $x$ a letter occurring in $B$ then we denote by $(A | x)B$ the assembly obtained by replacing each occurrence of $x$ in $B$ by the assembly $A$. The upshot of this is that we can now write quantifiers as assemblies:

$(\exists x) R$ is the assembly $(\tau_x(R) | x)R$ and as $(\forall x) R$ is $\neg (\exists x) \neg R$ it becomes $\neg (\tau_x(\neg R) | x) \neg R$

Okay, let's try to convert Bourbaki's definition of the emptyset $\emptyset$ as 'something that contains no element', or formally $\tau_y((\forall x)(x \notin y))$, into an assembly.

- by definition of $\forall$ it becomes $\tau_y(\neg (\exists x)(\neg (x \notin y)))$

- write $\neg ( x \notin y)$ as the assembly $R= \neg \neg \in x \square_y$

- then by definition of $\exists$ we have to assemble $\tau_y \neg (\tau_x(R) | x) R$

- by construction $\tau_x(R) = \tau_x \neg \neg \in \square_x \square_y$

- using the description of $(A|x)B$ we finally indeed obtain $\tau_y \neg \neg \neg \in \tau_x \neg \neg \in \square_x \square_y \square_y$

But, can someone please explain what's wrong with $\tau_y \neg \in \tau_x \in \square_x \square_y \square_y$ which is the assembly corresponding to $\tau_y(\neg (\exists x) (x \in y))$ which could equally well have been taken as defining the empty set and has a shorter assembly (length 8 and 3 links, compared to the one given of length 12 with 3 links).

Hair-splitting as this is, it will have dramatic implications when we will try to assemble Bourbaki's definition of "1" another time.

According to Jean van Heijenoort, the sad state of logic in France after WW2 was largely caused by the untimely death of several key French logicians/mathematical philosophers.

Prepping for my course on the history of mathematics, starting next week, i'm trying out a couple of tools, such as Timeline JS. Below, a mini timeline of the deaths of these 5 unfortunate mathematicians.

In the spring of 2009 I did spend a fortnight dog-sitting in a huge house in the countryside, belonging to my parents-in-law, who both passed away the year before.

That particular day it was raining and thundering heavily. To distract myself from the sombre and spooky atmosphere in the house I began to surf the web looking for material for a new series of blogposts (yes, in those days I was still thinking in 'series' of posts...).

Bookmarks for that day tell me that the first picture grasping my attention was Salvador Dali's Sacrament of the last supper, in particular the depicted partial dodecahedron

I did compare it with Leonardo's last supper and in the process stumbled upon Leonardo's drawings of polyhedra, among which these two dodecahedra

From there it went on and on : the Mystery of the 2nd and 3rd Century Roman Dodecahedron and its posible use 'casting dodecahedra' in Tarot Divination Without Tarot Cards or as an astronomical instrument, a text on polyhedra and plagiarism in the renaissance, the history of the truncated icosahedron, a Bosnian pyramid and its stone balls, the sacred geometry of the dodecahedron, mathematics in the Vatican library, and on and on and on...

By noon, I felt I had enough material to post for a couple of weeks on "platonic solids through the ages".

In between two rain showers, I walked the dog, had a quick lunch, and started writing.

I wanted to approach the topic in chronological order, and as I had done already a quicky on Scottish solids, the first post of the series would have to extend on this picture of five stone balls from the Ashmolean museum (or so it was claimed).

So, I hunted for extra pictures of these stone balls from the Ashmolean, and when comparing the two, clearly something had to be wrong...

It took me a couple of hours to catch up with the scientific literature on these Scottish balls, their cataloguing system and the museums of Scotland and England that house them.

Around 4pm I had compiled a list of all potential dodecahedra and icosahedra Scottish balls: 'there are only 8 possible candidates for a Scottish dodecahedron (below their catalogue numbers, indicating to the knowledgeable which museum owns them and where they were found)

NMA AS 103 : Aberdeenshire

AS 109 : Aberdeenshire

AS 116 : Aberdeenshire (prob)

AUM 159/9 : Lambhill Farm, Fyvie, Aberdeenshire

Dundee : Dyce, Aberdeenshire

GAGM 55.96 : Aberdeenshire

Montrose = Cast NMA AS 26 : Freelands, Glasterlaw, Angus

Peterhead : Aberdeenshire

The case for a Scottish icosahedron looks even worse. Only two balls have exactly 20 knobs

NMA AS 110 : Aberdeenshire

GAGM 92 106.1. : Countesswells, Aberdeenshire'

About an hour later I'd written the post, clicked the 'Publish' button and The Scottish solids hoax, began to live a life of its own!

From the numerous reactions let me single out 3 follow-ups which I believe to be most important.

John McKay and Tom Leinster did some legwork, tracking down resp. photographer and one of the 20 knobs balls.

John Baez gave a talk at an AMS meeting dedicated to the history of mathematics on Who discovered the icosahedron? mentioning my post and extending it by:

"And here is where I did a little research of my own. The library at UC Riverside has a copy of Keith Critchlow's 1979 book Time Stands Still. In this book, we see the same photo of stones with ribbons that appears in Lawlor's book - the photo that Atiyah and Suttcliffe use. In Critchlow's book, these stones are called "a full set of Neolithic 'Platonic solids'". He says they were photographed by one Graham Challifour - but he gives no information as to where they came from!

And Critchlow explicitly denies that the Ashmolean has an icosahedral stone! He writes:

... the author has, during the day, handled five of these remarkable objects in the Ashmolean museum.... I was rapt in admiration as I turned over these remarkable stone objects when another was handed to me which I took to be an icosahedron.... On careful scrutiny, after establishing apparent fivefold symmetry on a number of the axes, a count-up of the projections revealed 14! So it was not an icosahedron."

And now there is even a published paper out!

Bob Lloyd wrote How old are the Platonic solids?, published in BSHM Bulletin: Journal of the British Society for the History of Mathematics. The full article is behind a paywall but Bob graciously send me a copy.

Bob believes the balls in the picture to belong to the Scottish ‘National Museum of Antiquities’ (NMA in the Marshall list), now the National Museum of Scotland (NMS) in Edinburgh.

He believes the third and fourth ball to be two pictures of the same object "recorded as having been discovered in Aberdeenshire" so it should be NMA AS 103 : Aberdeenshire in the above list. (Or, the other one may be NMA AS 26?).

He also attempts to identify the other 3 balls with objects in the NMS-collection. In short, he gives compelling evidence that the picture must have been taking in Edinburg and exists of genuine artifacts.

Perhaps even more important is that he finally puts the case of a Scottish icosahedron to rest. As mentioned above, there are just two candidates NMA AS 110 (Edinburg) and GAGM 92 106.1 (Glasgow). He writes:

"According to the Marshall list, there are only two balls known which have 20K; one of these is at the NMS. Alan Saville, Senior Curator for Earliest Prehistory at this Museum, has provided a photograph which shows that this object is complex, and certainly not a dodecahedron. It could be considered as a modified octahedron, with five large knobs in the usual positions, but with the sixth octahedral position occupied by twelve small knobs, and in addition there are also three small triangles carved at some of the interstices, the three-fold positions of the ‘octahedron’. These make up a total of twenty ‘protrusions’, though the word ‘knobs’ is hard to justify.

The other 20K object is at the Kelvingrove museum in Glasgow. Photographs taken by Tracey Hawkins, assistant curator, show that this also is very far from being a dodecahedron, though this time there are twenty clearly defined knobs of roughly the same size. The shape is somewhat irregular, but two six-sided pyramids can be picked out, and much of the structure, though not all, is deltahedral in form, with sets of three balls at the corners of equilateral triangles."

So, sadly for John McKay, there is no Scottish icosahedron out there!

One final comment. Both John Baez (in a comment) and Bob Lloyd (in a comment and in his paper) argue that I shouldn't have used the term "hoax" for something that is merely a 'matter of sloppy scholarship'.

My apologies.

Given Bob's evidence that the balls in the picture are genuine artifacts, I have deleted the 'fabrication or falsification'-phrase in the original post.

Summarizing : the Challifour photograph is not taken at the Ashmolean museum, but at the National Museum of Scotland in Edinburgh and consists of 5 of their artifacts (or 4 if ball 3 and 4 are identical) vaguely resembling cube, tetrahedron, dodecahedron (twice) and octahedron. The fifth Platonic solid, the icosahedron, remains elusive.