Tag: nonsense

Scottish solids, final(?) comments

Published June 5, 2012 by lieven

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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Art and the absolute point (3)

Published May 19, 2011 by lieven

Previously, we have recalled comparisons between approaches to define a geometry over the absolute point and art-historical movements, first those due to Yuri I. Manin, subsequently some extra ones due to Javier Lopez Pena and Oliver Lorscheid.

In these comparisons, the art trend appears to have been chosen more to illustrate a key feature of the approach or an appreciation of its importance, rather than giving a visual illustration of the varieties over $\mathbb{F}_1$ the approach proposes.

Some time ago, we’ve had a couple of posts trying to depict noncommutative varieties, first the illustrations used by Shahn Majid and Matilde Marcolli, and next my own mental picture of it.

In this post, we’ll try to do something similar for affine varieties over the absolute point. To simplify things drastically, I’ll divide the islands in the Lopez Pena-Lorscheid map of $\mathbb{F}_1$ land in two subsets : the former approaches (all but the $\Lambda$-schemes) and the current approach (the $\Lambda$-scheme approach due to James Borger).

The former approaches : Francis Bacon “The Pope” (1953)

The general consensus here was that in going from $\mathbb{Z}$ to $\mathbb{F}_1$ one looses the additive structure and retains only the multiplicative one. Hence, ‘commutative algebras’ over $\mathbb{F}_1$ are (commutative) monoids, and mimicking Grothendieck’s functor of points approach to algebraic geometry, a scheme over $\mathbb{F}_1$ would then correspond to a functor

$h_Z~:~\mathbf{monoids} \longrightarrow \mathbf{sets}$

Such functors are described largely by combinatorial data (see for example the recent blueprint-paper by Oliver Lorscheid), and, if the story would stop here, any Rothko painting could be used as illustration.

Most of the former approaches add something though (buzzwords include ‘Arakelov’, ‘completion at $\infty$’, ‘real place’ etc.) in order to connect the virtual geometric object over $\mathbb{F}_1$ with existing real, complex or integral schemes. For example, one can make the virtual object visible via an evaluation map $h_Z \rightarrow h_X$ which is a natural transformation, where $X$ is a complex variety with its usual functor of points $h_X$ and to connect both we associate to a monoid $M$ its complex monoid-algebra $\mathbb{C} M$. An integral scheme $Y$ can then be said to be ‘defined over $\mathbb{F}_1$’, if $h_Z$ becomes a subfunctor of its usual functor of points $h_Y$ (again, assigning to a monoid its integral monoid algebra $\mathbb{Z} M$) and $Y$ is the ‘best’ integral scheme approximation of the complex evaluation map.

To illustrate this, consider the painting Study after Velázquez’s Portrait of Pope Innocent X by Francis Bacon (right-hand painting above) which is a distorded version of the left-hand painting Portrait of Innocent X by Diego Velázquez.

Here, Velázquez’ painting plays the role of the complex variety which makes the combinatorial gadget $h_Z$ visible, and, Bacon’s painting depicts the integral scheme, build up from this combinatorial data, which approximates the evaluation map best.

All of the former approaches more or less give the same very small list of integral schemes defined over $\mathbb{F}_1$, none of them motivically interesting.

The current approach : Jackson Pollock “No. 8” (1949)

An entirely different approach was proposed by James Borger in $\Lambda$-rings and the field with one element. He proposes another definition for commutative $\mathbb{F}_1$-algebras, namely $\lambda$-rings (in the sense of Grothendieck’s Riemann-Roch) and he argues that the $\lambda$-ring structure (which amounts in the sensible cases to a family of endomorphisms of the integral ring lifting the Frobenius morphisms) can be viewed as descent data from $\mathbb{Z}$ to $\mathbb{F}_1$.

The list of integral schemes of finite type with a $\lambda$-structure coincides roughly with the list of integral schemes defined over $\mathbb{F}_1$ in the other approaches, but Borger’s theory really shines in that it proposes long sought for mystery-objects such as $\mathbf{spec}(\mathbb{Z}) \times_{\mathbf{spec}(\mathbb{F}_1)} \mathbf{spec}(\mathbb{Z})$. If one accepts Borger’s premise, then this object should be the geometric object corresponding to the Witt-ring $W(\mathbb{Z})$. Recall that the role of Witt-rings in $\mathbb{F}_1$-geometry was anticipated by Manin in Cyclotomy and analytic geometry over $\mathbb{F}_1$.

But, Witt-rings and their associated Witt-spaces are huge objects, so one needs to extend arithmetic geometry drastically to include such ‘integral schemes of infinite type’. Borger has made a couple of steps in this direction in The basic geometry of Witt vectors, II: Spaces.

To depict these new infinite dimensional geometric objects I’ve chosen for Jackson Pollock‘s painting No. 8. It is no coincidence that Pollock-paintings also appeared in the depiction of noncommutative spaces. In fact, Matilde Marcolli has made the connection between $\lambda$-rings and noncommutative geometry in Cyclotomy and endomotives by showing that the Bost-Connes endomotives are universal for $\lambda$-rings.

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Who dreamed up the primes=knots analogy?

Published May 16, 2011 by lieven

One of the more surprising analogies around is that prime numbers can be viewed as knots in the 3-sphere $S^3$. The motivation behind it is that the (etale) fundamental group of $\pmb{spec}(\mathbb{Z}/(p))$ is equal to (the completion) of the fundamental group of a circle $S^1$ and that the embedding

$\pmb{spec}(\mathbb{Z}/(p)) \subset \pmb{spec}(\mathbb{Z})$

embeds this circle as a knot in a 3-dimensional simply connected manifold which, after Perelman, has to be $S^3$. For more see the what is the knot associated to a prime?-post.

In recent months new evidence has come to light allowing us to settle the genesis of this marvelous idea.

1. The former consensus

Until now, the generally accepted view (see for example the ‘Mazur-dictionary-post’ or Morishita’s expository paper) was that the analogy between knots and primes was first pointed out by Barry Mazur in the middle of the 1960’s when preparing for his lectures at the Summer Conference on Algebraic Geometry, at Bowdoin, in 1966. The lecture notes where later published in 1973 in the Annales of the ENS as ‘Notes on etale cohomology of number fields’.

For further use in this series of posts, please note the acknowledgement at the bottom of the first page, reproduced below : “It gives me pleasure to thank J.-P. Serre for his vigorous editing and his suggestions and corrections, which led to this revised version.”

Independently, Yuri I. Manin spotted the same analogy at around the same time. However, this point of view was quickly forgotten in favor of the more classical one of viewing number fields as analogous to algebraic function fields of one variable. Subsequently, in the mid 1990’s Mikhail Kapranov and Alexander Reznikov took up the analogy between number fields and 3-manifolds again, and called the resulting study arithmetic topology.

2. The new evidence

On december 13th 2010, David Feldman posted a MathOverflow-question Mazur’s unpublished manuscript on primes and knots?. He wrote : “The story of the analogy between knots and primes, which now has a literature, started with an unpublished note by Barry Mazur. I’m not absolutely sure this is the one I mean, but in his paper, Analogies between group actions on 3-manifolds and number fields, Adam Sikora cites B. Mazur, Remarks on the Alexander polynomial, unpublished notes.“

Two months later, on february 15th David Feldman suddenly found the missing preprint in his mail-box and made it available. The preprint is now also available from Barry Mazur’s website. Mazur adds the following comment :

“In 1963 or 1964 I wrote an article Remarks on the Alexander Polynomial [PDF] about the analogy between knots in the three-dimensional sphere and prime numbers (and, correspondingly, the relationship between the Alexander polynomial and Iwasawa Theory). I distributed some copies of my article but never published it, and I misplaced my own copy. In subsequent years I have had many requests for my article and would often try to search through my files to find it, but never did. A few weeks ago Minh-Tri Do asked me for my article, and when I said I had none, he very kindly went on the web and magically found a scanned copy of it. I’m extremely grateful to Minh-Tri Do for his efforts (and many thanks, too, to David Feldman who provided the lead).”

The opening paragraph of this unpublished preprint contains a major surprise!

Mazur points to David Mumford as the originator of the ‘primes-are-knots’ idea : “Mumford has suggested a most elegant model as a geometric interpretation of the above situation : $\pmb{spec}(\mathbb{Z}/p\mathbb{Z})$ is like a one-dimensional knot in $\pmb{spec}(\mathbb{Z})$ which is like a simply connected three-manifold.”

In a later post we will show that one can even pinpoint the time and place when and where this analogy was first dreamed-up to within a few days and a couple of miles.

For the impatient among you, have a sneak preview of the cradle of birth of the primes=knots idea…

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