Coxeter on Escher’s Circle Limits

Conway’s orbifold notation gives a uniform notation for all discrete groups of isometries of the sphere, the Euclidian plane as well as the hyperbolic plane.

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:

Penrose tiles in Helsinki

(image credit: Steve’s travels & stuff)

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)

the monstrous moonshine picture – 1

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.