Practicing graphical debugging using visualizations of the Hilbert curve

Summary

Jan 31, 2025 Practicing graphical debugging using too many visualizations of the Hilbert curve “..you don't understand things, you just get used to them.” — John von Neumann For a while now I've been advocating for a particular style of programming: Use tools that don't change too often. Use tools that don't keep historical accidents around indefinitely. Minimize moving parts. Avoid additional third-party libraries, and forswear native libraries entirely. Lua and LÖVE have been one nice way to get these properties. As I've used them, I've enjoyed an additional benefit: the ubiquitous presence of a canvas I can draw on as I program. This has been new to me with my erstwhile conservative and terminal-bound habits, and I've been pushing myself to lean more on graphics to understand what my programs are doing. Here I want to share one such experience. I'm using my run-anywhere Lua Carousel app, and you can paste the programs directly into it, but the workflow translates to any platform with a canvas. A few weeks ago Jack Rusher shared a baffling function to compute the Hilbert curve. Here it is, translated to Lua: function h(x, y, xi, yi, xj, yj, n) if n <= 0 then return {x+xi/2+xj/2, y+yi/2+yj/2} end return array_join( h(x, y, xj/2, yj/2, xi/2, yi/2, n-1), h(x+xi/2, y+yi/2, xi/2, yi/2, xj/2, yj/2, n-1), h(x+xi/2+xj/2, y+yi/2+yj/2, xi/2, yi/2, xj/2, yj/2, n-1), h(x+xi/2+xj, y+yi/2+yj, -xj/2, -yj/2, -xi/2, -yi/2, n-1)) end When I first looked at it, I could see a few superficial facts: It returns an array of points containing x and y coordinates. It's recursive. There's a base case for “leaf” calls, and each non-base case makes 4 recursive calls. Only the base case actually “draws” by adding points to the result. Non-leaf calls recursively partition the given square into 4 quadrants. Square size (the xi/yi/xj/yj) is being halved each time. But the details were still unclear. Why the swaps/rotations? Why the negative signs in one of the 4 quadrants? Looking for answers le...