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.