My apologies, I nearly forgot I started this Tumblr. You see, I’m in my last year at university and so the past few months have been all about applying to grad to school. I think I got three solid letters of recommendations, a beautifully written personal statement, and a well rounded CV; but I honestly don’t know where exactly I fit in the spectra of math students. And don’t tell me to lurk on gradcafe because that leads to undue anxiety. Anyways, I applied to 19 schools, which is too many, but hopefully one of them sees fit to admit me. I don’t know if I’m going to keep this tumblr or just create a professional math blog. Maybe both. I have some life organizing to do after that hellstorm of applications. Anyways, happy new years. 2014 is squarefree! And only good things can come in a squarefree year!

-Bradley.

The above picture was generated by Dr. David Dumas of University Illinois at Chicago in order to visualize projective measured laminations of the five punctured sphere. A lamination is fairly simple, it is layering curves onto a topological space (a surface). What the adjectives “projective” and “measured” indicate are beyond me. Click through to find the above picture animated.

Jim Fowler

Dr. Jim Fowler is a professor of topology at The Ohio State University (my home institution!), he wrote this interesting Java applet that allows you to move triangle reflection generated lattices.

People were more likely to solve a problem incorrectly when it conflicted with their political beliefs

I was reading Gil Kalai’s Combinatorics blog when I came upon this surprising post tagged “art.” These three are paintings by French artist Bernar Venet. These painting belong to a series called “Equation Paintings” (click his name to find more paintings), and from what I can tell are an instance of artistic appropriation, since these are actually diagrams from published works of mathematics. This is exactly the type of thing that this blog has set out to do, to set images generated for the elucidation of mathematics in a context generally reserved for artistic images.

The title of these works are as follows: 1. “Three-Dimensional Faces and Quotients of Polytopes,” 2. “Canonical Field Quantization,” and 3. “Zig-Zag Path Z_uv between Nodes u and v in a Planar Mesh.”

The only image I have found the origin of is the first, which comes from a paper that Kalai wrote (with Kleinschmidt and Meisinger): "Three Theorems, with Computer-Aided Proofs, on Three-Dimensional Faces and Quotients of Polytopes.” The image selected is only a portion of the computer generated output included in the proof of one of the papers main theorems. You can jump to Kalai’s post to read his explanation of the theorem, but I will briefly try myself. We all know what polygons are (triangles, rectangles, pentagons), and we all know what polyhedra are (cubes, tetrahedrons, icosahedrons). Well a polytope is just a higher dimensional version of this, its just mathematicians got tired of affixing different words to “poly-” so they just say “polytopes.” You can make a polyhedron with an arbitrarily large face (for example a thousand sided polygon). It turns out if a polytope has dimension greater than 8, then it must have a “small” three dimensional face. One does not expect this to be the case, but it is. This theorem (Kalai admits) has absolutely no purpose. But it is one of those great little self contained mysteries of mathematics.

The final image is the original text from Kalai’s paper.

These doodles are taken from Milnor’s paper "Whitehead Torsion". In it, he sets out some basic algebraic constructions for chain complexes. He then goes on to prove that these constructions are a combinatorial invariant for cellular complexes (like the penguin from my previous post). It turns out this construct is essential in higher homotopy theory, though it can basically be defined as the determinant of a change of basis matrix.

These types of pictures are very common for early topology papers, they were produced by hand. What’s so lovely about them is that they are representing arbitrary dimensions, and are somehow clear to those experts who would read this. Do not ask me what they mean, for, I have yet to make sense of the latter half of this paper.