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.

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'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 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 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.

The solution is later in the article.

> The insight that leads to a solution comes from the Jacobian.

It's not a unform half disk. It has more weight away from the Y axis.

You can imagine it's painted with watercolors and you want to collect the same ammount of ink.

In an uniform disk you have

    xx
   xxxx
   xxxx
  xxxxxx
  xxxxxx
  xxxxxx

But in the weighted disk of the article the top and bottom are darker and the center lighter

    ..
   x..x
   x..x
   x..x
  Xx..xX
  Xx..xX
  Xx..xX

but there are no strips like in my ASCII art, the shade changes slowly.

He's also ignoring that the layers of the onion become significantly thinner the farther away from the center they are. So this analysis is way off even for a perfectly symmetrical onion.

Even though onions aren't perfectly symmetrical, they still optimally grow or radiate out from one axis/line through the middle. Stick a toothpick through a sphere as this line, and slice the sphere through perpendicular to the axis, you'll get circles from a sphere, or half-disks from a half onion if you keep slicing perpendicular.

I'm lazy and cut my onions perpendicularly through halves, and don't try a radial cut for uniformity.

For a moment, I thought that “the onion problem” related to some challenging issue of topology or group theory, before my brain finally sorted through its connections to identify Kenji Lopez-Alt as a chef and not a mathematician.

I share simular concern, but also think of an onion more as a bulging cylinder due to center weighted thickness variation in layers. Each layer extends from root to stalk.

I literally came in here just to make this comment. Like Ragusea, I prefer every bite to be slightly novel and different.

One of my favorite hacks for Ceasar Salad: Take a bag of packaged croutons, put it flat on the table, and crush it with the bottom of a pan. Repeatedly. Until you get a mix of various sized crouton chunks, gravel, and dust. Apply to salad.

I ate a Ceasar this way in some fancy restaurant and I've been making it that way ever since.

Adam was solving a different problem statement. Kenji's point was to have one simple rule that anyone could remember and follow to make the best cuts without having to worry about precision. This rule gets you close enough to the homogeneity that is expected in most recipes (for things like onions) without having to fuss over particular cuts. Having watched Ragusea for a while, I'm betting he would be perfectly on board with that solution to that problem.

I remember reading about the consistency of cuts from classically trained chefs. I think Adam Ragusea has a lot of niche, quirky practices that don't align with actual profession. He's more of a culinary advocate in the same way that Bill Nye is a science advocate. They're not professional chefs or scientists.

Well the logic presented in that video certainly cannot be argued against.

really isn't a right and wrong way to do it. Erring too far toward either extreme makes your food probably a bit boring versus poorly executed.

That being said, most of Ragusea's takes haven't aged all that well, some by his own admission.

it's definitely one of the more subtle tool to use when cooking, mixing heterogeneity and homogeneous!

He grates for a tomato sauce, which makes sense, but you wouldn't want that as your default onion cut.

This is the MPW way:

https://www.youtube.com/watch?v=UBj9H6z6Uxw

"Perfection is lots of little things done well."

It avoids the issue except for the remaining bit which you can't grate without risking your fingers.

For garlic, I prefer crushing them for many recipes. This creates much rougher outlines that blend better into the food and crisp nicely when fried.

At that point why not just grind them in a food processor?

“You can grate the onion, or not, its your choice really”

> Is the problem explained in text anywhere

the problem is that you want to cut up an onion in such a way as to minimize variation in the size and shape of the cut-up pieces

usually, so that the pieces will cook evenly

You would like to slice (half) an onion in a way that minimizes the variance in volume of the pieces. The problem is then simplified to slicing half an onion in a way that minimizes the variance in cross-sectional area of the pieces at the widest part of the onion.

The problem is "I have an onion that is spherical with even layers. How do I cut it into pieces with equal volume?"

It's more of a geometry thought experiment than a practical epicurean "problem".

> Is the problem explained in text anywhere?

Not very well. There are some snippets:

"to keep the pieces as similar as possible"

"The Jacobian r dr dθ gives a measure of how big the infinitely small pieces are relative to each other"

"The variance is a good measure of the uniformity of the pieces."

The problem is how to get roughly equal sized pieces from cutting an onion. If you cut towards the center the inner pieces are much smaller than the outer.

The weirder thing for me is that he makes the horizontal cut after the vertical cuts --- in fact, most cooks I've seen dicing onions do that --- and it seems completely backwards. It's safe and easy to make the horizontal cut on an intact onion half, but much harder after it's been cut up vertically.

angling the horizontal cut down is a good way to handle this. The horizontal cut is mostly only necessary for the lower sides of the onion anyways.

While narrating, Kenji says that he doesn't do the angled cuts at all, but it was an interesting math problem.

Invest in cut-resistant gloves. The few dollars will pay for themselves in non-lost time, plus you can use them on a mandolin.

NB: maybe stick a hotdog in one of the fingers to test it first.

Yes! I've been using his method for 15 years with no regrets. I'm amazed to see he's still making videos after all this time.