I'm always amazed when I see how irregular some solutions look.

It would seem like this can be done by solving various optimization formulas via plane geometry? The radicals arise from solving maximization polynomials assuming it's a constructable solution.

You are right there is a field of optimization dedicated to solving this problem to guaranteed optimality.

However, Multi dimensional bin packing problems are the some of the hardest non linear optimization problems.

Depends on the shape. Irregular shape packing and optimisation is still being advanced. Newer approaches are using ai. Lots of GitHub repositories with different approaches.

Early in my generative art, I spent a lot of time packing various small shapes together, it was a fun process to figure out how to maximize the coverage without leaving large gaps. For example, a piece with almost every shape I was using: https://digcon.art/details.html?key=river-in-the-woods

What are some examples of efficient packing results that led to or inspired practical solutions?

One I can think of is circles-in-circles where you are bundling wires or optical fibre cores into cables. (And probably for tensile cables used in bridges and so on)

Bin packing everywhere! Truck loading?

Minimizing maximum to minimum ratio for structural stability.

Communications theory is largely about how to pack circles/spheres efficiently in multiple dimensions. So arguably pretty much all of the modern society is based on efficient packing.

Related:

Erich's Packing Center - https://news.ycombinator.com/item?id=29529130 - Dec 2021 (16 comments)

God he discovered so many of these? I wonder what it must have been like, to realize, “hey, no one’s found this one before either!!”

Or he wrote a constraint satisfaction resolver maybe

Great site, thanks! Improvement idea: please use SVGs instead (or in addition to) GIFs. It should be relatively easy, much better quality, probably smaller image file sizes.

related but with a more artistic slant: https://tylerxhobbs.com/essays/2016/a-randomized-approach-to...

Please explain, what am I looking at here?

I've been pondering what an algorithm would look like that does this. Given a bunch of boxes of various sizes and weights, and a room of a given dimension, what's the most efficient arrangement to make sure they all fit?

It’s a fun/NP-hard problem, this take on it ended up being useful: https://github.com/stephaniemhutson/BoxPackingAPI/blob/maste...

https://medium.com/the-chain/solving-the-box-selection-algor...