one of the most vexing things for me to illustrate is a planar vector field, especially when there is a singularity. there are so many distractions that arise from the canonical square-grid sampling that is endemic to software. here is an example with the linear vector field dx/dt = x ; dy/dt = -y

now for something completely different: the *loxodromic grid*, obtained by intersecting two circle’s worth of counterspinning spirals. you see these in sunflowers, pinecones, and the like. it seems as though nobody uses the term * loxodromic grid* (why was it in my head when i wanted to call this something? why did i know to use 137.5 degrees? intuition is a frustrating thing…). it also seems as though nobody ever plots vector fields on such a grid. pity, since that meant it took me a long time to figure out how to do it. but: it is worth the trouble…

i really like the fact that the sampling gets increasingly tight near the equilibrium. i am simultaneously disturbed and pleased that the stable and unstable manifolds are “hidden” (as it were). it is fitting that you have a hard time visually getting on and staying on the stable manifold, since that’s exactly what happens in the flow. even the visual artefacts are instructive: note that near the origin, the tiny vectors are packed in such a way as to suggest not a circle (where the sampling points are) but a squashed ellipse. this is right: the flow squeezes and stretches areas.

this is not a perfect plot. but it is satisfying to me, after having tried and failed to make square and hex grids not look awful.