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Everything’s wrappable to a sphere

Published January 11, 2018 by lieven

One of the better opening quotes of a paper:

“Even quite ungainly objects, like chairs and tables, will become almost spherical if you wrap them in enough newspaper.”

The paper in question is The orbifold notation for surface groups by John Conway.

Here’s Conway talking leisurely about Thurston’s idea to capture the acting group via the topology of the orbifold space and his own notation for such orbifolds.



Here’s another version of the paper, with illustrations: The orbifold notation for two-dimensional groups, by Conway and Daniel H. Huson.

A very accessible account are these lecture notes:

A field guide to the orbifolds, notes from class on “Geometry and the Imagination” in Minneapolis, with John Conway, Peter Doyle, Jane Gilman and Bill Thurston, on June 17–28, 1991.

And, here are notes by Thurston on The Geometry and Topology of Three-Manifolds, including stuff about orbifolds.

I came across these papers struggling my way through On the discrete groups of moonshine by Conway, McKay and Sebbar.

On the genus $0$ property of moonshine groups they have this to say:

“As for groups of the form $(n|h)+e,f,\dots$, the genus can be determined from the fundamental regions using the Riemann-Hurwitz formula. Since most of the groups are not subgroups of the modular group, the calculations of the genus, which cannot be produced here because of their length, are carried out by finding the elliptic fixed points and the cone points in the orbifolds attached to the fundamental regions. The Euler characteristic of the orbifold determines the genus of the group. See [paper] for more details on orbifold techniques.”

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Chomp and the moonshine thread

Published January 10, 2018 by lieven

Chomp is a 2-player game, usually played with chocolate bars.

The players take turns in choosing one chocolate block and “eat it”, together with all other blocks that are below it and to its right. There is a catch: the top left block contains poison, so the first player forced to eat it dies, that is, looses the game.

If you start with a rectangular bar, the first player has a winning strategy, though it may take you too long to actually find the correct first move. See this post for the strategy-stealing argument.

If you label the blocks of the rectangular bar by $(a,b)$ with $0 \leq a \leq k$ and $0 \leq b \leq l$, with the poisonous one being $(0,0)$, then this can be viewed as choosing a divisor $d$ of $N=p^k q^l$ and removing all multiples of $d$ from the set of divisors of $N$. The first person forced to name $1$ looses.

This allows for higher dimensional versions of Chomp.

If you start with the set of all divisors of a given natural number $N$, then the strategy-stealing argument shows that the first player has a winning move.

A general position of the game corresponds to a finite set of integers, closed under taking divisors. At each move the player has to choose an element of this set and remove it as well as all its multiples.

The thread of $(N|1)$, relevant in understanding a moonshine group of the form $(n|m)+e,f,\dots$ with $N=n \times h$, consists of all divisors of $N$.

But then, the union of all threads for all 171 moonshine groups is a position in higher dimensional Chomp.

Who wins starting from this moonshine thread?

Perhaps not terribly important, but it forces one to imagine the subgraph of the monstrous moonshine picture on the $97$ number-lattices way better than by its Hasse diagram.

Click on the image for a larger version.

By the way, notice the (slight) resemblance with the ‘monstrous moonshine painting’ by Atria

Here’s how the Hasse diagram of the moonshine thread was produced. These are ‘notes to self’, because I tend to forget such things quickly.

1. Work though the list of 171 moonshine groups in Monstrous Moonshine, pages 327-329. Add to a list all divisors of $N$ for a group of type $N+e,f,\dots$ or $n|h+e,f,\dots$ with $N=n \times h$. This should give you these $97$ integers:

1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,
31,32,33,34,35,36,38,39,40,41,42,44,45,46,47,48,50,51,52,54,55,56,57,59,60,62,
63,64,66,68,69,70,71,72,78,80,84,87,88,90,92,93,94,95,96,104,105,110,112,117,
119,120,126,136,144,160,168,171,176,180,208,224,252,279,288,360,416

2. Let $L$ be this list and use Sage:

P=Poset((L,attrcall("divides")),linear_extension=True)
H=P.hasse_diagram()
H.graphviz_string()

3. Copy the output to a file, say chomp.dot, and remove all new-line breaks from it.

4. Install Graphviz on Mac OS X.

5. In Terminal, type
dot -Tpng chomp.dot -o chomp.png

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the monstrous moonshine picture – 1

Published January 9, 2018 by lieven

We’re slowly closing in on the elusive moonshine picture, which is the subgraph of Conway’s Big Picture needed to describe all 171 moonshine groups.

About nine years ago I had a first go at it, drawing a tiny fraction of it, just enough to understand the 9 moonshine groups appearing in Duncan’s realization of McKay’s E(8)-observation.

Over the last weeks I’ve made enough doodles to feel confident that the full picture is within reach and is less unwieldy than I once feared it might be.

The moonshine picture only involves about 212 lattices and there are about 97 snakes crawling into it, the dimension of the largest cell being 3.

I write ‘about’ on purpose as I may have forgotten a few, or counted some twice as is likely to happen in all projects involving a few hundreds of things. I’ll come back to it later.

For now, I can only show you the monstrous moonshine painting, which is a work by the Chilean artist Magdalena Atria.

Here’s a close up:

It is a large scale painting made with plasticine, directly attached to the wall of the Alejandra Von Hartz gallery where it was exhibited in 2010.

What does it have to do with monstrous moonshine?

From the press release:

“In mathematics ‘monstrous moonshine’ is a term devised by John H. Conway and Simon P. Norton in 1979, used to describe the (then totally unexpected) connection between the monster group M and modular functions.

The term ‘monstrous moonshine’ was picked to convey the feelings from the bizarre relations between seemingly unrelated structures. The same spirit of connecting apparently unrelated situations, at times revealing deeper links and at times constructing them, permeates through Atria’s work in this exhibition.”

I was impressed by the first sentence until I read the Wikipedia article on monstrous moonshine which starts off with:

“In mathematics, monstrous moonshine, or moonshine theory, is the unexpected connection between the monster group M and modular functions, in particular, the j function. The term was coined by John Conway and Simon P. Norton in 1979.”

It appears that curators of art-exhibitions, and the intended public of their writings, are familiar with modular forms and functions, but fail to grasp the $j$-function.

When they speak about ‘modular forms’, I fear they’re thinking of something entirely different.

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