# Gallery

#### Gallery

Some mathematical art that I've created or discovered during my research and musings.

An SDE Stabilizing on Hyperbolas (May 2019)

I recently found myself studying the stochastic differential equation (SDE) $$dX_t = \cos(\pi t)dt + \sin(\pi t X_t)dB_t$$ with $$X_0 = 0$$. While no explicit solutions exist, I predicted the following interesting qualitative behavior: The noise term should have only a weak effect on the system whenever $$\sin(\pi t X_t)\approx 0$$, i.e. whenever $$X_t\approx k/t$$ for $$k\in\mathbb{N}$$. In other words, I expected that any sample path which goes near one of these hyperbolas should stay near the curve until a sufficiently large random force knocks it away. I came across the image above while checking this behavior computationally by simulating 1000 sample paths of solutions to the SDE using the Milstein scheme.

Simultaneous 2-Colorings of the Plane (August 2018)

In an introductory graph theory class we proved that any closed curve in the plane whose tangents have at most finitely many self-intersections gives rise to a 2-colorable partition of the plane. The image above is my chalkboard realization of 2 such curves (red and purple) and their respective colorings (blue and yellow) along with the regions colored by both (green).

Extremal Inscribed 5-stars in the 11-gon (September 2017)

In Vietoris-Rips Complexes of Regular Polygons, we proved that $$n \ge 4\ell + 2$$ is a necessary and sufficient condition for every point on the boundary of the regular polygon $$P_n$$ to be contained in a unique $$(2\ell+1)$$-pointed star whose vertices also lie on the boundary of $$P_n$$. Furthermore, we proved that the stars of maximal side length are those containing a vertex of $$P_n$$ and that the stars of minimal side length are those containing a midpoint of an edge of $$P_n$$. The following image realizes $$P_{11}$$ (black) along with its minimal (blue) and maximal (green) inscribed 5-pointed stars.