Boy surface: the immersion that has to cross — Classical Chaos
I kept waiting for a sealed outside. Boy surface never gives you one. The drawing has to pass through itself in space. That is the point: this picture is allowed to cross; a clean embedding would forbid that. Werner Boy found this immersion of the real projective plane (RP²) in ordinary 3-space in 1901, with self-intersections and a triple point — three sheets meeting in one place. The wire has to cross.
Classical Chaos samples the surface as a polar isoline wire (rings and rays), then drops those samples as a 3D point cloud. Color follows the polar pair (r, θ). Watching it fill in, the crossings stay readable because you see lines meet, not a solid shell that hides them.
Wire that grows in
The embed lights more of the wire over time until the whole cage is there. I expected a filled mesh the first time. The isoline cage is what made the triple point feel obvious instead of buried.
Two morphs, one blend
On the full Boy stage two morphs sit next to each other. Defaults land at k = 1 and bryant = 1. Knob k walks from a Roman-like cousin at 0 toward Boy at 1. Knob bryant (call it b) walks from petal shapes at 0 toward a three-balls look at 1.
Each side gets centered and scaled, then mixed:
const p =
(1 - b) * aperyPoint(k, r, theta) + b * bryantPoint(r, theta)Change k or bryant there and the cloud resamples; the grow-in restarts from the seed. I thought I was only bending what was already drawn. The wire rebuilds.
Example presets on that stage jump the same story in two clicks: Roman-ish (0, 0) to full Boy (1, 1). I often park halfway on bryant when the blend is what I am staring at.
Polar disk in, two charts out
The domain is a disk in polar (r, θ). That disk maps into Apéry’s (u, v) and into Bryant–Kusner’s complex w on the unit disk.
Apéry uses
with (u, v) in [0, π]. Bryant–Kusner works with complex w on that same unit disk. I do not need the full closed form memorized to feel the morph: k tilts toward Boy; bryant fattens petals into three balls.
