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To understand any mathematical concept, I first need to be able to draw it. So naturally, I like creating pictures of the objects I encounter during my research: often times the result is not what I expected, and the whole process becomes very instructive to me.
This page is a selection of my favourite encounters while doing experimental mathematics.
The unit ball of the stable norm of L(2,2).
This one requires some explanation.
The surface L(2,2) is a square-tiled surface, made of three identical squares placed in an L-shape, and whose opposite sides are glued by translation (see figure below).
The resulting surface is a translation surface of genus 2 with one singularity of angle 6π. Here the curves e1, e2, f1 and f2 are the canonical choice of a basis for the first homology group of L(2,2).
Understanding the stable norm of L(2,2) is equivalent to describing the geometry of its unit ball, which is a convex body of R^4 symmetric with respect to the origin. I like the latter approach as it allows pictures to be drawn!
Using a program made with Sagemath, I produced pictures of (3-dimensional sections of) the unit ball of the stable norm of L(2,2), that can be found in my thesis. However, the structure of this ball is difficult to grasp by eyesight, as there are a lot of things going on.
The program works by finding empirically segments and faces of the unit ball, drawing them as it finds them. The idea to produce more readable pictures is then to project radially the picture the program got onto a simplex, coloring adjacent faces using different colors to keep track of the pattern the faces creates. By doing so, we obtain a two-dimensional picture that encodes the geometry of the unit ball of the stable norm of L(2,2).
For example, on the right hand side picture, the part of the unit ball that the program has computed is shown in blue and grey. The faces (here, segments and triangles) are projected radially onto the 2-simplex in red and gold (the adjacent faces have not been colored yet, hence the two tangent golden triangles).
The initial picture is what I obtained by using this process to flatten the three dimensional section of the unit ball of L(2,2) by the space Span(e1,e2,f1) (with positive coordinates) onto the 2-simplex. We can obtain similar pictures when looking at different three dimensional sections.
I could not finish the computations in a reasonable time on this one.
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