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Scottish solids, final(?) comments

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.

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From the Noether boys to Bourbaki

Next year I’ll be teaching a master course on the “History of Mathematics” for the first time, so I’m brainstorming a bit on how to approach such a course and I would really appreciate your input.

Rather than giving a chronological historic account of some period, I’d like this course to be practice oriented and focus on questions such as

  • what are relevant questions for historians of mathematics to ask?
  • how do they go about answering these questions?
  • having answers, how do they communicate their finds to the general public?

To make this as concrete as possible I think it is best to concentrate on a specific period which is interesting both from a mathematical as well as an historic perspective. Such as the 1930’s with the decline of the Noether boys (pictures below) and the emergence of the Bourbaki group, illustrating the shift in mathematical influence from Germany to France.

(btw. the picture above is taken from a talk by Peter Roquette on Emmy Noether, available here)

There is plenty of excellent material available online, for students to explore in search for answers to their pet project-questions :

There’s a wealth of riddles left to solve about this period, ranging from the genuine over the anecdotic to the speculative. For example

  • Many of the first generation Bourbakis spend some time studying in Germany in the late 20ties early 30ties. To what extend did these experiences influence the creation and working of the Bourbaki group?
  • Now really, did Witt discover the Leech lattice or not?
  • What if fascism would not have broken up the Noether group, would this have led to a proof of the Riemann hypothesis by the Noether-Bourbakis (Witt, Teichmuller, Chevalley, Weil) in the early 40ties?

I hope students will come up with other interesting questions, do some excellent detective work and report on their results (for example in a blogpost or a YouTube clip).

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Farey symbols in SAGE 5.0

The sporadic second Janko group $J_2$ is generated by an element of order two and one of order three and hence is a quotient of the modular group $PSL_2(\mathbb{Z}) = C_2 \ast C_3$.

This Janko group has a 100-dimensional permutation representation and hence there is an index 100 subgroup $G$ of the modular group such that the fundamental domain $\mathbb{H}/G$ for the action of $G$ on the upper-half plane by Moebius transformations consists of 100 triangles in the Dedekind tessellation.

Four years ago i tried to depict this fundamental domain in the Farey symbols of sporadic groups-post using Chris Kurth’s kfarey package in Sage, but the result was rather disappointing.

Now, the kfarey-package has been greatly extended by Hartmut Monien of Bonn University and is integrated in the latest version of Sage, SAGE 5.0, released a few weeks ago.

Using the Farey symbol sage-documentation it is easy to repeat the calculations from four years ago and, this time, we do obtain this rather nice picture of the fundamental domain

But, there’s a lot more one can do with the new package. By combining the .fractions() with the .pairings() info it is now possible to get the corresponding Farey code which consists of 34 edges, starting off with



Perhaps surprisingly (?) $G$ turns out to be a genus zero modular subgroup. Naturally, i couldn’t resist drawing the fundamental domain for the 12-dimensional permutations representation of the Mathieu group $M_{12}$ and compare it with that of last time.

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Aaron Siegel on transfinite number hacking

One of the coolest (pure math) facts in Conway’s book ONAG is the explicit construction of the algebraic closure $\overline{\mathbb{F}_2}$ of the field with two elements as the set of all ordinal numbers smaller than $(\omega^{\omega})^{\omega}$ equipped with nimber addition and multiplication.

Some time ago we did run a couple of posts on this. In transfinite number hacking we recalled Cantor’s ordinal arithmetic and in Conway’s nim arithmetics we showed that Conway’s simplicity rules for addition and multiplication turns the set of all ordinal numbers into a field of characteristic zero : $\mathbb{On}_2$ (pronounced ‘Onto’).

In the post extending Lenstra’s list we gave Hendrik Lenstra’s effective construction of the mystery elements $\alpha_p$ (for prime numbers $p$) needed to do actual calculations in $\mathbb{On}_2$. We used SAGE to check the values for $p \leq 41$ and solved the conjecture left in Lenstra’s paper Nim multiplication that $(\omega^{\omega^{13}})^{43} = \omega^{\omega^7} + 1$ and determined $\alpha_p$ for $p \leq 67$.


Aaron Siegel has now dramatically extended this and calculated the $\alpha_p$ for all primes $p \leq 181$. He mails :

“thinking about the problem I figured it shouldn’t be too hard to write a dedicated program for it. So I threw together some Java code and… pushed the table up to p = 181! You can see the results below. Q(f(p)), excess, and alpha_p are all as defined by Lenstra. The “t(sec)” column is the number of seconds the calculation took, on my 3.4GHz iMac. The most difficult case, by far, was p = 167, which took about five days.

I’m including results for all p < 300, except for p = 191, 229, 263, and 283. p = 263 and 283 are omitted because they involve computations in truly enormous finite fields (exponent 102180 for p = 263, and 237820 for p = 283). I'm confident that if I let my computer grind away at them for long enough, we'd get an answer... but it would take several months of CPU time at least. p = 191 and 229 are more troubling cases. Consider p = 191: it's the first prime p such that p-1 has a factor with excess > 1. (190 = 2 x 5 x 19, and alpha_19 has excess 4.) This seems to have a significant effect on the excess of alpha_191. I’ve tried it for every excess up to m = 274, and for all powers of 2 up to m = 2^32. No luck.”

Aaron is writing a book on combinatorial game theory (to be published in the AMS GSM series, hopefully later this year) and will include details of these computations. For the impatient, here’s his list






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Quiver Grassmannians and $\mathbb{F}_1$-geometry

Reineke’s observation that any projective variety can be realized as a quiver Grassmannian is bad news: we will have to look at special representations and/or dimension vectors if we want the Grassmannian to have desirable properties. Some people still see a silver lining: it can be used to define a larger class of geometric objects over the elusive field with one element $\mathbb{F}_1$.

In a comment to the previous post Markus Reineke recalls motivating discussions with Javier Lopez Pena and Oliver Lorscheid (the guys responsable for the map of $\mathbb{F}_1$-land above) and asks about potential connections with $\mathbb{F}_1$-geometry. In this post I will ellaborate on Javier’s response.

The Kapranov-Smirnov $\mathbb{F}_1$-floklore tells us that an $n$-dimensional vectorspace over $\mathbb{F}_1$ is a pointed set $V^{\bullet}$ consisting of $n+1$ points, the distinguished point playing the role of the zero-vector. Linear maps $V^{\bullet} \rightarrow W^{\bullet}$ between $\mathbb{F}_1$-spaces are then just maps of pointed sets (sending the distinguished element of $V^{\bullet}$ to that of $W^{\bullet}$). As an example, the base-change group $GL_n(\mathbb{F}_1)$ of an $n$-dimensional $\mathbb{F}_1$-space $V^{\bullet}$ is isomorphic to the symmetric group $S_n$.

This allows us to make sense of quiver-representations over $\mathbb{F}_1$. To each vertex we associate a pointed set and to each arrow a map of pointed sets between the vertex-pointed sets. The dimension-vector $\alpha$ of quiver-representation is defined as before and two representations with the same dimension-vector are isomorphic is they lie in the same orbit under the action of the product of the symmetric groups determined by the components of $\alpha$. All this (and a bit more) has been worked out by Matt Szczesny in the paper Representations of quivers over $\mathbb{F}_1$.

Oliver Lorscheid developed his own approach to $\mathbb{F}_1$ based on the notion of blueprints (see also part 2 and a paper with Javier).

Roughly speaking a blueprint $B = A // \mathcal{R}$ is a commutative monoid $A$ together with an equivalence relation $\mathcal{R}$ on the monoid semiring $\mathbb{N}[A]$ compatible with addition and multiplication. Any commutative ring $R$ is a blueprint by taking $A$ the multiplicative monoid of $R$ and $\mathcal{R}(\sum_i a_i,\sum_j b_j)$ if and only if the elements $\sum_i a_i$ and $\sum_j b_j$ in $R$ are equal.

One can extend the usual notions of prime ideals, Zariski topology and structure sheaf from commutative rings to blueprints and hence define a notion of “blue schemes” which are then taken to be the schemes over $\mathbb{F}_1$.

What’s the connection with Reineke’s result? Well, for quiver-representations $V$ defined over $\mathbb{F}_1$ they can show that the corresponding quiver Grassmannians $Gr(V,\alpha)$ are blue projective varieties and hence are geometric objects defined over $\mathbb{F}_1$.

For us, old-fashioned representation theorists, a complex quiver-representation $V$ is defined over $\mathbb{F}_1$ if and only if there is an isomorphic representation $V’$ with the property that all its arrow-matrices have at most one $1$ in every column, and zeroes elsewhere.

Remember from last time that Reineke’s representation consisted of two parts : the Veronese-part encoding the $d$-uple embedding $\mathbb{P}^n \rightarrow \mathbb{P}^M$ and a linear part describing the subvariety $X \rightarrow \mathbb{P}^n$ as the intersection of the image of $\mathbb{P}^n$ in $\mathbb{P}^M$ with a finite number of hyper-planes in $\mathbb{P}^M$.

We have seen that the Veronese-part is always defined over $\mathbb{F}_1$, compatible with the fact that all approaches to $\mathbb{F}_1$-geometry allow for projective spaces and $d$-uple embeddings. The linear part does not have to be defined over $\mathbb{F}_1$ in general, but we can look at the varieties we get when we force the linear-part matrices to be of the correct form.

For example, by modifying the map $h$ of last time to $h=x_0+x_7+x_9$ we get that the quiver-representation



is defined over $\mathbb{F}_1$ and hence that Reineke’s associated quiver Grassmannian, which is the smooth plane elliptic curve $\mathbb{V}(x^3+y^2z+z^3)$, is a blue variety. This in sharp contrast with other approaches to $\mathbb{F}_1$-geometry which do not allow elliptic curves!

Oliver will give a talk at the 6th European Congress of Mathematics in the mini-symposium Absolute Arithmetic and $\mathbb{F}_1$-Geometry. Judging from his abstract,he will also mention quiver Grassmannians.

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bookworm arXiv

One of the nicer tools around is bookworm arXiv which ‘is a collaboration between the Harvard Cultural Observatory, arxiv.org, and the Open Science Data Cloud. It enables you to explore lexical trends in over 700,000 e-prints, spanning mathematics, physics, computer science, and statistics’ posted on the arXiv.

One possible use is to explore the popularity of certain topics. Below is the graph of the number of papers submitted monthly to the arXiv in noncommutative geometry, quantum groups, cluster algebras and symplectic reflection (algebras).

The default gives the graphs in the percentage of all papers submitted, but it is better to change this to the number of papers (I think). Sadly, at present one can only search for one- and two-word phrases.

Extremely useful is that it gives you the full list of papers (with direct links to the papers) containing the search terms when you click on that months point in the graph. For example, there are 4 sheets of papers in noncommutative geometry for october 2011

Clearly, there are plenty of other fun uses for this bookworm. For example, you can graph the number of papers in a topic in function of the nationality of the submitter. Here are the papers in noncommutative geometry, submitted by people from the US, France, the UK and Italy.

Or, you can use it for vanity reasons, giving you the list of all papers containing a reference to your work, which may not always be a good idea, blood-pressure wise…

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the matrix reloaded

The dinosaurs among you may remember that before this blog we had the ‘na&g-forum’ to accompany our master-class in noncommutative algebra & geometry.

That forum ran on an early flat-panel iMac G4 which was, for lack of a better name, baptized ‘the matrix’.

The original matrix did survive the unification of the three Antwerp universities and a move to a different campus but then died around bloomsday 2007 and was replaced by an intel iMac.

This second matrix did host a number of blogs and projects started (and usually ended rather quickly) such as ‘MoonshineMath’, a muMath-site called noncommutative.org, the ‘F-un Mathematics’ blog dedicated to the field with one element and, of course, this blog.

About a month ago matrix-II was replaced by a state-of-the-art iMac running 10.7. The transition went smooth apart from the fact that 10.7 doesn’t like ‘localhost’ but prefers ‘127.0.0.1’ in setting up wordpress blogs.

Besides neverendingbooks, matrix-III runs angs@t – angs+ which is the blog of the antwerp noncommutative geometry seminar. It will be revamped over the summer and will probably be the website for our renewed master-class, starting next year.

The ‘F-un Mathematics’ blog was dropped in the transition but still survives at Ghent University where it is managed by Koen Thas.

As far as NeverendingBooks is concerned i hope to make a fresh start with blogging and will try to get more structure in this site by changing to a responsive wordpress theme (‘These responsive, fluid, or adaptive WordPress themes, automatically adjust according to the screen size, resolution and device on which they are being viewed’).

As a result this page will look weird from time to time over the next week or so. My apologies.

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noncommutative geometry at the Lorentz center

This week i was at the conference Noncommutative Algebraic Geometry and its Applications to Physics at the Lorentz center in Leiden.



It was refreshing to go to a conference where i knew only a handful of people beforehand and where everything was organized to Oberwolfach perfection. Perhaps i’ll post someday on some of the (to me) more interesting talks.

Also interesting were some discussions about the Elsevier-boycot-fallout and proposals to go beyong that boycot and i will certainly post about that later. At the moment there is still an embargo on some information, but anticipate a statement from the editorial board of the journal of number theory soon…

I was asked to talk about “algebraic D-branes”, probably because it sounded like an appropriate topic for a conference on noncommutative algebraic geometry claiming to have connections with physics. I saw it as an excuse to promote the type of noncommutative geometry i like based on representation schemes.

If you like to see the slides of my talk you can find the handout-version here. They should be pretty self-exploratory, but if you like to read an unedited version of what i intended to tell with every slide you can find that text here.

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Manin’s three-space-2000

Almost three decades ago, Yuri Manin submitted the paper “New dimensions in geometry” to the 25th Arbeitstagung, Bonn 1984. It is published in its proceedings, Springer Lecture Notes in Mathematics 1111, 59-101 and there’s a review of the paper available online in the Bulletin of the AMS written by Daniel Burns.

In the introduction Manin makes some highly speculative but inspiring conjectures. He considers the ring

$$\mathbb{Z}[x_1,\ldots,x_m;\xi_1,\ldots,\xi_n]$$

where $\mathbb{Z}$ are the integers, the $\xi_i$ are the “odd” variables anti-commuting among themselves and commuting with the “even” variables $x_j$. To this ring, Manin wants to associate a geometric object of dimension $1+m+n$ where $1$ refers to the “arithmetic dimension”, $m$ to the ordinary geometric dimensions $(x_1,\ldots,x_m)$ and $n$ to the new “odd dimensions” represented by the coordinates $(\xi_1,\ldots,\xi_n)$. Manin writes :

“Before the advent of ringed spaces in the fifties it would have been difficult to say precisely what me mean when we speak about this geometric object. Nowadays we simply define it as an “affine superscheme”, an object of the category of topological spaces locally ringed by a sheaf of $\mathbb{Z}_2$-graded supercommutative rings.”

Here’s my own image (based on Mumford’s depiction of $\mathsf{Spec}(\mathbb{Z}[x])$) of what Manin calls the three-space-2000, whose plain $x$-axis is supplemented by the set of primes and by the “black arrow”, corresponding to the odd dimension.

Manin speculates : “The message of the picture is intended to be the following metaphysics underlying certain recent developments in geometry: all three types of geometric dimensions are on an equal footing”.

Probably, by the addition “2000” Manin meant that by the year 2000 we would as easily switch between these three types of dimensions as we were able to draw arithmetic schemes in the mid-80ties. Quod non.

Twelve years into the new millenium we are only able to decode fragments of this. We know that symmetric algebras and exterior algebras (that is the “even” versus the “odd” dimensions) are related by Koszul duality, and that the precise relationship between the arithmetic axis and the geometric axis is the holy grail of geometry over the field with one element.

For aficionados of $\mathbb{F}_1$ there’s this gem by Manin to contemplate :

“Does there exist a group, mixing the arithmetic dimension with the (even) geometric ones?”

Way back in 1984 Manin conjectured : “There is no such group naively, but a ‘category of representations of this group’ may well exist. There may exist also certain correspondence rings (or their representations) between $\mathsf{Spec}(\mathbb{Z})$ and $x$.”

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