Visual Insight

I have another blog, called Visual Insight. Over here, our focus is on applying science to help save the planet. Over there, I try to make the beauty of pure mathematics visible to the naked eye.

I’m always looking for great images, so if you know about one, please tell me about it! If not, you may still enjoy taking a look.

Here are three of my favorite images from that blog, and a bit about the people who created them.

I suspect that these images, and many more on Visual Insight, are all just different glimpses of the same big structure. I have a rough idea what that structure is. Sometimes I dream of a computer program that would let you tour the whole thing. Unfortunately, a lot of it lives in more than 3 dimensions.

Less ambitiously, I sometimes dream of teaming up with lots of mathematicians and creating a gorgeous coffee-table book about this stuff.


Schmidt arrangement of the Eisenstein integers


Schmidt Arrangement of the Eisenstein Integers - Katherine Stange

This picture drawn by Katherine Stange shows what happens when we apply fractional linear transformations

z \mapsto \frac{a z + b}{c z + d}

to the real line sitting in the complex plane, where a,b,c,d are Eisenstein integers: that is, complex numbers of the form

m + n \sqrt{-3}

where m,n are integers. The result is a complicated set of circles and lines called the ‘Schmidt arrangement’ of the Eisenstein integers. For more details go here.

Katherine Stange did her Ph.D. with Joseph H. Silverman, an expert on elliptic curves at Brown University. Now she is an assistant professor at the University of Colorado, Boulder. She works on arithmetic geometry, elliptic curves, algebraic and integer sequences, cryptography, arithmetic dynamics, Apollonian circle packings, and game theory.


{7,3,3} honeycomb

This is the {7,3,3} honeycomb as drawn by Danny Calegari. The {7,3,3} honeycomb is built of regular heptagons in 3-dimensional hyperbolic space. It’s made of infinite sheets of regular heptagons in which 3 heptagons meet at vertex. 3 such sheets meet at each edge of each heptagon, explaining the second ‘3’ in the symbol {7,3,3}.

The 3-dimensional regions bounded by these sheets are unbounded: they go off to infinity. They show up as holes here. In this image, hyperbolic space has been compressed down to an open ball using the so-called Poincaré ball model. For more details, go here.

Danny Calegari did his Ph.D. work with Andrew Casson and William Thurston on foliations of three-dimensional manifolds. Now he’s a professor at the University of Chicago, and he works on these and related topics, especially geometric group theory.


{7,3,3} honeycomb meets the plane at infinity

This picture, by Roice Nelson, is another view of the {7,3,3} honeycomb. It shows the ‘boundary’ of this honeycomb—that is, the set of points on the surface of the Poincaré ball that are limits of points in the {7,3,3} honeycomb.

Roice Nelson used stereographic projection to draw part of the surface of the Poincaré ball as a plane. The circles here are holes, not contained in the boundary of the {7,3,3} honeycomb. There are infinitely many holes, and the actual boundary, the region left over, is a fractal with area zero. The white region on the outside of the picture is yet another hole. For more details, and a different version of this picture, go here.

Roice Nelson is a software developer for a flight data analysis company. There’s a good chance the data recorded on the airplane from your last flight moved through one of his systems! He enjoys motorcycling and recreational mathematics, he has a blog with lots of articles about geometry, and he makes plastic models of interesting geometrical objects using a 3d printer.

7 Responses to Visual Insight

  1. John Baez says:

    This is a test to see if you can now use Markdown in comments, for example to create links, italicized or boldface text, or quotes:

    Markdown is not a replacement for HTML, or even close to it. Its syntax is very small, corresponding only to a very small subset of HTML tags. The idea is not to create a syntax that makes it easier to insert HTML tags. In my opinion, HTML tags are already easy to insert. The idea for Markdown is to make it easy to read, write, and edit prose. HTML is a publishing format; Markdown is a writing format. Thus, Markdown’s formatting syntax only addresses issues that can be conveyed in plain text.

    For any markup that is not covered by Markdown’s syntax, you simply use HTML itself. There’s no need to preface it or delimit it to indicate that you’re switching from Markdown to HTML; you just use the tags.

    The only restrictions are that block-level HTML elements — e.g. <div>, <table>, <pre>, <p>, etc. — must be separated from surrounding content by blank lines, and the start and end tags of the block should not be indented with tabs or spaces. Markdown is smart enough not to add extra (unwanted) <p> tags around HTML block-level tags.

  2. arch1 says:

    Thanks for sharing these, John. They are beautiful! In the 3rd picture above, using Chrome in Win7, I see circles filled with at least 3 different shades of blue, while in your AMS article they are all shaded identically; unless I’m missing something you may want to delete the qualifier “light-colored” in the description, or tweak the shading.

    • arch1 says:

      Correction – I’m using IE9 in Win7, not Chrome in Win7.

    • John Baez says:

      Well, the circles are always lighter blue than their surroundings, but they get darker as they get smaller. I suppose I can just drop “light-colored” and it will be less confusing.

      I went with Roice’s black-and-white image on Visual Insight because it lets you see more fine detail, and it looks spooky—a nice change from the colorful pictures that I usually feature:

  3. tcmJOE says:

    I learned that Danny Calegari was a professor at Caltech JUST AFTER he left, much to my chagrin. I really wanted to put a poster next to his office that said “Du musst Calegari werden”.

You can use Markdown or HTML in your comments. You can also use LaTeX, like this: $latex E = m c^2 $. The word 'latex' comes right after the first dollar sign, with a space after it.

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