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Comment by drspoulsen | Hacker News Reader

Hi everyone, the author of the blog here. I'm glad to see the interest here on this piece!

I have slides that detail the problem setup and the mathematics, as well as a consideration of three-dimensional onions, here: https://drspoulsen.github.io/Onion_Marp/index.html

I have submitted a formal write-up of the details of the problem and the solution to a recreational mathematics journal.

I'm also happy to answer any questions about this!

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The thing I love about Hacker News is that someone can post an article like this, then the author of the paper shows up to answer any questions. Keep being awesome.

You should send a "continue and persist" letter. From a HN post a few days ago: https://continueandpersist.org/

This is tremendously fun, thank you!

Your solution seems to assume that all cuts need to be directed towards a single point, but doesn't it seem likely that an even more optimal solution increases h (depth of cut target) as the cuts move outward? Or did I miss a reason that's not the case?

I think that was just an "arbitrary" constraint of the problem, inspired by the practical constraint of using it to cut an onion?

Hey now, even the practical application of cutting a typical 10 layer onion was left as en excercise for the reader. Quoth:

"So, the best depth for an onion with ten layers would be somewhere between 0 and 0.5573066. I have not investigated this in depth, but this seems like a fun next step."

You are suggesting something even more advanced. :)

Another onion enthusiast has sent me python code that considers finite-layer onions, and that code will be featured in the upcoming journal article.

The two cases this solution generalizes are the vertical and radial cut method, which both aim towards a single point (you can think of the vertical method as aiming to a point infinitely far beneath the center of the onion). There may be other more optimal ways (cutting each layer individually for example), but they are not conducive to an ultimately simple strategy.

> I'm also happy to answer any questions about this!

If you're still checking, I have a semi-related question:

You're solving the problem for a circle in a plane (actually, a semicircle in a plane), and the reduction in dimensions is related to something that has bothered me.

I can easily segment a circle into a bunch of identical arcs (say, by making each arc 3 degrees long and getting 120 identical copies). Polar coordinates are great for this.

But spherical coordinates are terrible for accomplishing the same thing on a sphere, and my understanding is that the analogous effect - tiling the surface of a sphere with a single shape - can't be achieved?

What motivated me to thinking about this was the idea of a coordinate system that would allow every "square" on a map to be the same as the other squares, regardless of how much distortion there might be between the shape of the region on the spherical surface and the shape of the same region as a square on this fancy map. But it also seems relevant to the question of how well your two-dimensional analogue to the onion problem answers the original three-dimensional question. (I'm writing this comment in the middle of reading your article, so I don't know if the 3D solution is ultimately addressed.)

I'd be happy for any comments you might have related to this.

I agree that spherical coordinates are not good for the 3D onion. In the slides I linked, I use cylindrical coordinates with appropriate bounds to encompass the problem within a sphere.

I love how deeply nerd-sniped you have been by this topic. It's wonderful to be able to observe your delight in solving this. Thank you for sharing.

Would be really interesting if you could reverse engineer the model which yields 1/phi as the correct answer. Evidently for some non-uniform measure on the onion you could do it. What about for considering the onion as a half-ball? (Although if you're cooking it really is primarily the thickness that matters.)

I've thought about this. Unfortunately, everything I have tried (changing dimensions and layers, for example) has not yielded anything. This is still something one could explore, though!

I feel like mathematics and many other rigorous field-friends have tons of great questions like this that are ripe for fun research. Thanks for publishing this and contributing to that world of curiosity!

Thanks, this has been my go to technique after watching Kenji's video about it.

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