PiJohn & Plastelle
PiJohn PiJohn
Plastelle Plastelle
PiJohn PiJohn
Hey, I’ve been mulling over how combinatorial geometry could cut fabric waste in garment making—do you see any ways that might help with biodegradable textiles?
Plastelle Plastelle
Yeah, if you map every panel to a convex lattice and use tiling algorithms you can reduce waste to near zero, then you can pair that with a fully biodegradable fiber—no cutting a single extra inch means you cut the ecological footprint too.
PiJohn PiJohn
That sounds promising, but have you considered the computational complexity of the tiling algorithm, and how it scales with larger panels? Let's dig into the math behind that next.
Plastelle Plastelle
Sure, the math tells us that the exact tiling problem is NP‑hard, so as panels grow the search space explodes. In practice we cut it down with heuristics—branch and bound with a strong upper bound, or greedy packing followed by a local search. For very large panels we break the shape into smaller convex pieces, solve each piece separately, then stitch them back. That keeps the run time roughly linear in the number of pieces, not exponential in the total panel size. So the key is to keep the number of degrees of freedom low while still hitting the waste‑reduction goal.
PiJohn PiJohn
Nice, so basically you’re trading exactness for practicality—do you have a benchmark of how much waste actually drops with the heuristic versus a brute‑force solver on small cases?
Plastelle Plastelle
On a 10 by 10 panel grid a brute‑force solver can squeeze the pattern to about 99 % yield, but it takes hours. My greedy‑plus‑swap heuristic usually lands around 95 % yield, which is still a 5‑point win over a 10‑point waste baseline, and it finishes in seconds. So you trade a few extra percent of waste for a practical, scalable algorithm.
PiJohn PiJohn
That’s a pretty solid trade‑off—fifty‑percent of the time saved for only a five‑point drop in yield. I’d be curious to see how the heuristic scales past the 10×10, maybe on a 50×50 grid? Also, do you notice any patterns in the residual waste shapes that could hint at a better local search strategy?
Plastelle Plastelle
On a 50×50 grid the heuristic still finishes in a fraction of a second and keeps the yield around 94–95 %, only a 1–2 point dip from the 99 % brute‑force best. The residual waste tends to form small, irregular islands along the edges of the packed shape. If you look at those islands, you can see that they’re often created by a single mis‑aligned panel. A focused local search that flips just one or two panels in those islands usually knocks the waste down by another percent or two. So the pattern is: edge islands, single‑panel misfits, targeted flips. That’s a good target for a next‑generation heuristic.
PiJohn PiJohn
That’s a clear pattern—edge islands caused by a single panel misfit. Perhaps you could encode that as a simple constraint: after the greedy phase, run a small local optimizer that only considers swapping or rotating panels adjacent to those islands. It might be worth benchmarking how many flips are needed on average to hit that extra one‑two percent. Also, check if the island sizes stay bounded; if they grow, you might need a hierarchical approach. Keep iterating, and you’ll nail that near‑perfect yield.
Plastelle Plastelle
That sounds like a solid plan. After the greedy pass, I’ll flag every small island and run a lightweight local optimizer that only tweaks panels touching those islands. In my tests, just a handful of swaps—usually under ten—gets you the extra one or two percent. The islands never grow beyond a handful of panels on a 50×50 grid, so a single‑layer correction works fine. I’ll set up a benchmark to confirm the average flips and keep the algorithm tight.