math applets:
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- Parallel transport and torsion
This is a visualization of parallel transport of vectors along a line/circle in R^3 with respect to a metric-compatible but not torsion-free connection. This was created for my Math 215C Riemannian Geometry course. The code was written by Gemini 3.1 Pro.
- The convex body isoperimetric conjecture
The applet computes the isoperimetric profile of a polygon in the plane and compares it against the disk's isoperimetric profile. More precisely, let P be a compact convex subset of R^2 with area π. Let IP(A) be the length required to enclose an area A in P (the boundary of P is free). This is called the (relative) isoperimetric profile of P. The convex body isoperimetric conjecture (Wicharamala) asks: is IP(A) ≤ ID(A) for D the unit disk? See here and here. Some partial progress can be found here and here. The conjecture has been solved for A = π/2. The code was written by ChatGPT 5.4 Pro.
- Capillary deformations of Bernoulli one-phase solutions on the plane
This is a visualization of one-parameter families of capillary minimal surfaces in the upper half-space in R^3 that interpolate between Bernoulli one-phase solutions in the plane and free boundary minimal surfaces. The correspondence between one-phase and free-boundary minimal surfaces was discovered in this paper of Traizet. The Lopez–Ros deformation applied to the free-boundary minimal surfaces deforms them through capillary surfaces that limit to the original Bernoulli solution as the angle goes to zero (this observation was pointed out to me by Jacob Bernstein). See here and here for some related papers. The code was written by ChatGPT 5.4 Pro.
- The Mahler volume of a polygon
This computes the Mahler volume of a convex polygon in the plane. The applet demonstrates Mahler's sliding method to prove the polygonal case of ℳ(K) ≥ ℳ(Δ2,0) in the plane (and thus the full conjecture in the plane by approximation). This was created for the occasion of Yanir Rubinstein's Bergman lectures on this topic. The code written by Gemini 2.5 Pro.