u/DegreeRealistic2793

After getting into the show I went deep into the actual mathematics of the three-body problem. The whole premise is that no stable solution exists, true for generic initial conditions. But periodic exceptions do exist and they're extraordinary.

The figure-eight was proven by Chenciner and Montgomery in 2000. Three equal masses chasing each other around one curve, zero angular momentum, period T≈6.32. It genuinely shouldn't exist. This comes from integrating Newton's equations at near machine precision over one complete period.

Made a small shop putting these on shirts at Etsy for anyone who wants to carry a bit of the actual mathematics around.

This gets illustrated constantly but rarely rendered from the actual equations. Integrated from (0,1,0) with DOP853 at rtol=1e-10, transient discarded, trajectory plotted directly in phase space. The double-lobe structure, the fractal cross-section, the measure-zero set that bounds trajectories forever — it's all in the picture if you know where to look.

What physical systems do you think produce the most interesting phase space geometry? Looking for the next object to tackle. Put these on shirts at Etsy for anyone interested.

Been working on a project: rendering mathematical objects directly from their definitions. The Apollonian gasket iterating the Descartes Circle Theorem. The stereographic projection using the exact conformal map from S² to the plane. This one comes from integrating Newton's equations with DOP853 at rtol=1e-12 over one period T≈14.89. Linear stability is what makes it render cleanly. The orbit doesn't visibly drift within one period even with the 5-decimal initial conditions from the paper.

What objects do people here think are visually underrepresented or worth treating carefully? I've been working through things across topology, algebraic geometry, complex analysis, dynamical systems but curious what a mathematician would actually want to see rendered properly.

Put these on shirts at Etsy — trying to spread the gospel a bit.