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.
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?
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. :)
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.
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!
> First, we model the onion as half of a disc of radius one, with its center at the origin and existing entirely in the first two quadrants in a rectangular (Cartesian) coordinate system.
Can someone explain to me why a half sphere (the shape of half an onion) can be modeled as a half-disk in this problem? Why would we expect the solutions to be the same? If you think about the outermost cross-sections at the ends of the onion (closest to the heel and tip of the knife), as you get closer and closer to the ends, you approach cutting these cross-sections more vertically. I'd expect that you'd have to make the center cross-section a bit shallower to "make up" for the fact that the outsides are being cut vertically. Idk, either way I think declaring this the true "Onion constant" is probably wrong.
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.
The question I have is not about modeling an imperfect object as a perfect abstraction, it's about modeling a 3D object as a 2d object, and assuming that the optimization still holds. I think it's pretty plainly clear that it doesn't. Think about some cross-section of the onion that's closer to you and smaller than the center cross-section. Let's say it's of radius 0.25 instead of 1. The slices you take of it will be much more vertical than the center slice. This changes things. My intuition tells me it means the optimal solution is shallower than the solution found here, since you'd want the "average" cross-section to follow this constant.
Those slides show that the solution won't work on actual onions. Call the innermost layer of the onion its "biological center", and call the center of the spheroid approximately occupied by the onion its "geometrical center".
As is beautifully illustrated on slide 50, the biological center is generally not particularly close to the geometrical center, and this introduces huge distortions in slices that cut close to the biological center.* A single layer of the onion can run parallel to the knife cut for quite some distance.
* The slides also observe that in reality, before chopping an onion, you cut off the top and bottom. This same phenomenon explains why you have to do that; a vertical cut through the top or bottom end of the onion would just give you one huge piece. (You also need to get rid of the roots on the bottom and the sprouts on the top, but even if you didn't, you'd have to cut off the top and the bottom because they curve the wrong way.)
As you point out, without a perfectly symmetrical onion of course this will not work very well. You would need to use a moving geometrical center point when slicing for best results.
Further, as noted elsewhere the outer layers are thicker in a real onion, so we need to reformulate to take this into account.
The other obvious simple improvement I can think of would be to use radial cuts in both directions. Each direction with the its own optimized floating center point of course. Reformulations would need to take this into account - although the end result would be quite close, and likely well within the margin of error for almost any human being aiming at an imagined floating center point below a cutting board :).
Haven’t had enough coffee to think about this rigorously.
My intuition says that as long as you could get to the desired 3D shape from revolving the 2D shape around an axis, essentially integrating the area into a volume, the results will be valid or equivalent.
I don’t think that’s the entire story, there are probably other ways to simplify 3D shapes. And yes, onions will have non constant variations (or ones that don’t cancel out to 0) along the sweep which is what actually invalidates the real world application.
If you model the (half) onion as a stack of these slices, it’s clear that the radius of each slice varies over the height of the onion; so the points below the onion found by this method towards which you need cut will form a curve, not a straight line. That is hard to accomplish with a straight knife that makes planar cuts.
Pretty much, and if you take smaller segments, you get a more accurate answer. Exactly the same as cutting an onion, if you cut it into quarters, then seperate layer by layer and chop each individually, you get a finer result.
I believe you are supposed to calculate R*0.55... once for the max onion radius and use the same cut on the smaller disks. That way the smaller disk is cut identically to the inner part of the larger disk.
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.
J. Kenji Lopez-Alt _was_ actually mentioned (featured?) in alt-weekly The Onion this month. The problem, though, was that it was in an un-funny piece about the beef dimension, and it is not worth footnoting here. I guess they should have researched this 2021 article and spun off of it instead. But maybe a Quanta Magazine and infowars joint venture could enter the beef dimension. An onion with too many alt-layers.
He's not a chef either he's a food writer and recipe tester. I don't mean this as disrespect at all just they are very different professions, using different skills and producing different outputs.
Before he was a food writer he worked in a number of fairly high-end restaurants in Boston (which he talks about occasionally on his Youtube channel), and then he opened his own restaurant in 2017ish. Not sure how that's "not a chef"
I feel like you have an overly restrictive definition of "chef". Owning a restaurant doesn't make you a chef, cooking food professionally does. Even if he never set foot in a restaurant kitchen, the man cooks food all day long for his job. That makes him a chef.
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.
On the other hand, fellow food youtuber Adam Ragusea swears by the importance of heterogeneity. Optimizing for uniformity might not be the best strategy!
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.
Adam's never claimed to be a chef or want to do things like a chef would, he tends to focus on how someone at home could do it, where things like preparing 50 onions as quickly as possible don't matter as much, hence the difference in style. I think both practices have their place, adam just home as he's never been trained in food and so all his cooking is for the home
No, but it would have taken a lot more words to convey the same idea, so I'm happy just giving the gist. Most of the time, you don't want to grate your onions.
That's going to end up with a slush of onion fibre + onion juices. Very much different from even small bits of cut onion. Some recipes call for blended onions though.
Is the problem explained in text anywhere? (TFA delegates to a video and afaict only discusses another video-suggested solution and a novel solution in text, I don't understand what we're solving.)
You're not wrong, but I think that the author's goal was not "how do I cut an onion evenly" but rather "how would someone do this if they had only a knife". He was solving a puzzle, not trying to suggest cooking technique.
No, but they do beat the shit out of the onion. The bigger thing though is that they're a pain; you have to clean it out when you're done (also: have it handy on your counter to begin with), as opposed to just wiping down your knife, which you're already using for other things.
I don't do much with my food processor anymore besides grating cheese; even biscuit dough I'll do with a box grater at this point, just to avoid having to clean out the food processor.
I've had a food processor for maybe 20 years.. I use it at most twice a year specifically because it's a pain in the fucking ass to clean. Though thanksgiving is when I always bring it out, one bag of cranberries, one whole orange (quartered before throwing it in the food processor, peel and all), and sugar.. run it until you get a nice chunky relish.. deeeelish.
> The bigger thing though is that they're a pain; you have to clean it out when you're done (also: have it handy on your counter to begin with), as opposed to just wiping down your knife, which you're already using for other things.
I agree that cleaning the food processor is more of a pain than cleaning the knife. On the other hand, using it is far less of a pain than using the knife (especially in cases like this where you're trying to get even, small pieces). So you're really trading off one pain for another. It's not clear to me that either option is the obvious winner or loser here.
Depends on how much you are making really. If I want an entire bowl of thin cucumber slices I thank my food processor. But yeah, not worth the trouble for just one or two onions.
Don't know about your food processor but the ones I have seen are not just a plastic cylinder but also a funnel and circular blade attachment plus a couple of other bits that all tend to get pieces of food on them.
It's not impossible to clean and worth it when making larger amounts but definitely more of a hassle than a knife, even with a dish washer.
It is.. at least my box grater which can be folded out flat making it very easy to clean the inside with a sponge and a tiny amount of elbow grease (much like the outside).
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 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.
I'm surprised Kenji still does the horizontal cut at all. With the angled vertical cuts I find the horizontal cut entirely unnecessary. (Also a few years back I gave myself a nice flap avulsion doing the horizontal cut in an onion...)
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.
I'm not sure I understand. My knives are razor sharp (I keep a Shapton 1000 and 4000 on my counter along with a strop, my daily driver is a carbon steel I have to wipe down every time I cut a vegetable). They sail through the onion, but the sliced-up onion still splays out to both sides when I make the horizontal cut, and if you watch cooks doing it, it happens there too. What harm am I doing to the structure of the onion by doing it in the "wrong order"? They're the same cuts. The difference seems to be that in my order, the onion stays more stationary.
Don't get me wrong, I'm sure there's a reason everyone is doing it this way, because it's kind of clearly more annoying than the way I'm doing it?
the vertical cuts do not significantly the internal structure of the onion as each individual cut I make does not entirely sever the connection between the thin vertical slices I'm making. This means that I can do a lot of these, and not worry about harming the overall structural integrity. Then I make a single horizontal cut which does harm the overall structural integrity. This is not intrinsic to the horizontal cut itself, but the fact that I have both horizontal and vertical cuts.
If I start with the horizontal cut, again I do not signficantly harm the structural integrity of the onion. However, each subsequent vertical cut I make is now going to individually compromise the integrity of the onion.
With a sufficiently sharp knife, the single horizontal cut at the end does not really pose a significant danger overall.
This all being said I almost never do the horizontal cut out of pure laziness, and instead prefer to just do angled vertical cuts analogous to the video. They're never perfect but fine enough for me...
I still don't get why you need the horizontal cut at all. The diagram at the bottom of the blog post shows how unnecessary it is when you do the vertical cuts at a narrow range of angles like that (which I have been doing for a while now).
The point of Kenji's method (really, all radial-ish methods, but radial is strictly worse) is that you don't have to do the horizontal slice. If you slice vertically, you do --- you can see it for yourself, if you don't the dice from the edges of the onion are almost twice as big as the diece from the center.
I have them now, but's simpler to just avoid that one dangerous and unnecessary cut that proceeds towards my body instead. They taught that in Scouting, never cut towards yourself.
And either learn to sharpen your knives yourself, or take them to a sharpening service. Dull knives require more force, and slip/catch more, so are more dangerous.
The trick I use for doing freehand sharpening is to color the bevel with a sharpie, that will show you if your angle is correct. You don't need a lot of stones, I just have one Sharpal double sided diamond stone, and then I move to a leather strop with 1 micron diamond emulsion compound.
Another very useful thing is an inexpensive jeweler's loupe so you can actually diagnose issues like not having removed the burr.
I looked into sharpening services in my city a few years back and they're like dry cleaners - every one was a mix of satisfied reviews and detailed "this person completely ruined my $600 knife" reviews. It was very off putting.
It's unlikely any sharpener is going to ruin your knife; at worst, they won't put the best possible edge on it. Your knife is probably just an inert hunk of steel. :)
I've watched a lot of shows about the tools used for building log cabins in the pioneer days. I don't even know the names of them, but the tool for taking the bark off the tree by pulling the knife to you as you sit on the log is crazy. Also, the one where you straddle the log and swing the blade towards you between your legs is right up there too. Yet, I can't think of any way of making them better without using power tools.
The drawknife is the safer of the two by far. It’s fairly hard to cut yourself when your whole body is moving the same direction. Similar to using a paring knife in your palm facing your thumb.
The adz however you just have to have good aim or pay the consequences!
Draw knife. As long as you are leaning instead of pulling its relatively safe. Same as its safe to pare by contracting your hand muscles instead of pushing a knife toward yourself.
Draw knives are even safer than paring knives: the handles are placed such that they're closer to you than the blade, it's extremely difficult to get your chest far enough forward that it could contact the blade without a very large chest.
My standard housewarming gift is cut gloves and a pack of nitrile gloves to put over them. The nitrile gloves are so you don't have to wash the cut gloves so often.
Amazing, and I was soooo glad to see the integral has a closed form. I’m very curious what this looks like in the discrete case. I’d imagine it’s somewhat straightforward to code a simulation?
He had to close down his cooking school during Covid, so he started making YT videos. I watch a number of chefs on YT, and he's easily my favorite. He's a wonderful teacher, and I've learned more from him than any other because he always reinforces concepts without being repetitive and dull. It also helps that his meals are practical for home cooks, and, overall, he's just a charming guy.
In fancy cooking there's also a notion of a "perfect dice", meant literally to be that --- you pull the layers of the whole onion apart and cut/press them into sheets, so you have a rectangle to work with, and then dice that on a grid.
>To get the most even cuts of an onion by making radial cuts, one should aim towards a point 55.73066% the radius of the onion below the center. This is close, but different from, the 61.803% claimed in the Youtube video at the top.
Wife walks into kitchen with 3447 cut onions in piles: "What are you doing?!"
This guy: "I just cannot get these onions cut to a point 55.73066% below the origin! The best I have achieved is only 2 significant digits of accuracy."
Wife, mumbling: "Maybe that's why Kenji said: 60%..."
I also thought I had my finger on the pulse of some The Onion/Lopez-Alt beef, like TMZ on the Food Network.
ontopic edit: I am interested in an optimal onion cutting technique, while I'm happy with mine, the upside-down banana teaches that there's always a few ways to approach and learn something.
Besides a dice that's as even as possible, the other requirement this solution attempts to satisfy is using the minimum number of cuts. A blender doesn't satisfy that, as it's making hundreds of cuts.
Then, when you present your solution to the client, you find out there was a third, unspoken requirement: that it should involve as little cleanup as possible, which the blender also doesn't satisfy. The user researcher was on vacation, and you didn't find out about this before beginning design. Damn!
The blender solution turns out to be overoptimized on a single requirement at the expense of the others.
They're optimizing for time as knife cuts = time. A food processor will do it faster if you're more than one onion or so, assuming you can get the size you want.
Ahh, so in addition to having trouble getting consistently-sized pieces the size of a dice or chop, the other reason knives are preferred is that a food processor damages the onion, releasing more water compared to a knife. The result doesn't caramelize as well. This is why higher-end restaurants cut onions by hand, even when operating at scale.
Anecdotally I've prepared caramelized onions both ways, chopped with a knife and using a food processor and I've never noticed a difference. Onions have to release most of their water before they can begin caramelizing anyway so if anything, wouldn't that speed up the process?
You don't have to turn it into mush with a food processor, not all veggies are caramelized. High end restaurants usually optimize for speed but not over quality. Not sure why were talking that when this technique is for home chefs.
A blender will make the bottom layer into paste before the top is touched. If you want to toss the paste into a skillet and caramelize it, that'll make a good sauce.
Food processor might be better, but still won't be even.
Source: I cook onions a lot, and am lazy. This article is great!
Always bust out the food processor when making soffritto or similar very small dice. Can do onions quickly and even with the method but carrots and others take quite some time.
I don't think this gets you the texture you're looking, or even cuts. My eyes are tearing up right now thinking about scooping this out of the blender.
Spice blenders tend to have blades closer to the bottom and are fine for dry things. Still not good for dicing though, especially as onions aren't really dry but full of liquid.
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!
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?
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 think that was just an "arbitrary" constraint of the problem, inspired by the practical constraint of using it to cut an onion?
> 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.
> First, we model the onion as half of a disc of radius one, with its center at the origin and existing entirely in the first two quadrants in a rectangular (Cartesian) coordinate system.
Can someone explain to me why a half sphere (the shape of half an onion) can be modeled as a half-disk in this problem? Why would we expect the solutions to be the same? If you think about the outermost cross-sections at the ends of the onion (closest to the heel and tip of the knife), as you get closer and closer to the ends, you approach cutting these cross-sections more vertically. I'd expect that you'd have to make the center cross-section a bit shallower to "make up" for the fact that the outsides are being cut vertically. Idk, either way I think declaring this the true "Onion constant" is probably wrong.
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
But in the weighted disk of the article the top and bottom are darker and the center lighter 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.
> Even though onions aren't perfectly symmetrical
The question I have is not about modeling an imperfect object as a perfect abstraction, it's about modeling a 3D object as a 2d object, and assuming that the optimization still holds. I think it's pretty plainly clear that it doesn't. Think about some cross-section of the onion that's closer to you and smaller than the center cross-section. Let's say it's of radius 0.25 instead of 1. The slices you take of it will be much more vertical than the center slice. This changes things. My intuition tells me it means the optimal solution is shallower than the solution found here, since you'd want the "average" cross-section to follow this constant.
The author dealt with this outside the article, and posted a link to his slides in this HN post. The relevant slides begin at [1].
At the end of the day a straight cut is limiting. The next step would be to design the perfect onion dicing knife.
[1] https://drspoulsen.github.io/Onion_Marp/index.html#44
Those slides show that the solution won't work on actual onions. Call the innermost layer of the onion its "biological center", and call the center of the spheroid approximately occupied by the onion its "geometrical center".
As is beautifully illustrated on slide 50, the biological center is generally not particularly close to the geometrical center, and this introduces huge distortions in slices that cut close to the biological center.* A single layer of the onion can run parallel to the knife cut for quite some distance.
* The slides also observe that in reality, before chopping an onion, you cut off the top and bottom. This same phenomenon explains why you have to do that; a vertical cut through the top or bottom end of the onion would just give you one huge piece. (You also need to get rid of the roots on the bottom and the sprouts on the top, but even if you didn't, you'd have to cut off the top and the bottom because they curve the wrong way.)
As you point out, without a perfectly symmetrical onion of course this will not work very well. You would need to use a moving geometrical center point when slicing for best results.
Further, as noted elsewhere the outer layers are thicker in a real onion, so we need to reformulate to take this into account.
The other obvious simple improvement I can think of would be to use radial cuts in both directions. Each direction with the its own optimized floating center point of course. Reformulations would need to take this into account - although the end result would be quite close, and likely well within the margin of error for almost any human being aiming at an imagined floating center point below a cutting board :).
Haven’t had enough coffee to think about this rigorously.
My intuition says that as long as you could get to the desired 3D shape from revolving the 2D shape around an axis, essentially integrating the area into a volume, the results will be valid or equivalent.
I don’t think that’s the entire story, there are probably other ways to simplify 3D shapes. And yes, onions will have non constant variations (or ones that don’t cancel out to 0) along the sweep which is what actually invalidates the real world application.
If you model the (half) onion as a stack of these slices, it’s clear that the radius of each slice varies over the height of the onion; so the points below the onion found by this method towards which you need cut will form a curve, not a straight line. That is hard to accomplish with a straight knife that makes planar cuts.
Isn't that where calculus and intergrals come into to play? As the radius approaches infinity type of stuff?
Pretty much, and if you take smaller segments, you get a more accurate answer. Exactly the same as cutting an onion, if you cut it into quarters, then seperate layer by layer and chop each individually, you get a finer result.
I believe you are supposed to calculate R*0.55... once for the max onion radius and use the same cut on the smaller disks. That way the smaller disk is cut identically to the inner part of the larger disk.
The same cut (in terms of angle) on smaller disks would be impossible with a real knife. You'd have to bend the knife in order to achieve it.
Only some types, of course. Shallots are not spherical, though they are radial, but the centerline is often curved.
Shallots very often have two bulbs fused together which makes them very much non-radial.
I was referring to the fact that each bulb tends to expand outward in layers from a core. You are, however, correct.
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.
J. Kenji Lopez-Alt _was_ actually mentioned (featured?) in alt-weekly The Onion this month. The problem, though, was that it was in an un-funny piece about the beef dimension, and it is not worth footnoting here. I guess they should have researched this 2021 article and spun off of it instead. But maybe a Quanta Magazine and infowars joint venture could enter the beef dimension. An onion with too many alt-layers.
I thought it was funny. Probably the first funny thing I've seen from the Onion in years, actually.
He's not a chef either he's a food writer and recipe tester. I don't mean this as disrespect at all just they are very different professions, using different skills and producing different outputs.
Before he was a food writer he worked in a number of fairly high-end restaurants in Boston (which he talks about occasionally on his Youtube channel), and then he opened his own restaurant in 2017ish. Not sure how that's "not a chef"
I simply didn't realize he had ever run a restaurant.
I feel like you have an overly restrictive definition of "chef". Owning a restaurant doesn't make you a chef, cooking food professionally does. Even if he never set foot in a restaurant kitchen, the man cooks food all day long for his job. That makes him a chef.
I have a chef's definition of a chef. You're welcome to use whatever one you prefer though.
he was also a biochem phd candidate.
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.
On the other hand, fellow food youtuber Adam Ragusea swears by the importance of heterogeneity. Optimizing for uniformity might not be the best strategy!
https://www.youtube.com/watch?v=5cWRCldqrxM
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.
You brought your own croutons and pan to a fancy restaurant?? Bold.
If it was a fancy restaurant, he probably asked the waiter for pan-smashed croutons.
At normal restaurants, you can use the two-plate method to approximate the effect of pan-smashing croutons.
Gar son, jay nay pal a crooton pan? Smoosh?
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.
Adam's never claimed to be a chef or want to do things like a chef would, he tends to focus on how someone at home could do it, where things like preparing 50 onions as quickly as possible don't matter as much, hence the difference in style. I think both practices have their place, adam just home as he's never been trained in food and so all his cooking is for the home
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!
Marco Pierre-White recommends grating onions, which ingeniously avoids the entire issue.
Grating the Gordian knot, if you will.
https://youtu.be/glIUUrh6qtQ?t=40
He grates for a tomato sauce, which makes sense, but you wouldn't want that as your default onion cut.
Is there really a default onion cut? Desired sizes vary vildly depending on what you are making.
No, but it would have taken a lot more words to convey the same idea, so I'm happy just giving the gist. Most of the time, you don't want to grate your onions.
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?
That's going to end up with a slush of onion fibre + onion juices. Very much different from even small bits of cut onion. Some recipes call for blended onions though.
Some food processors with the right setting give you chopped food not blended
A good food processor will cut the material, not blend it.
“You can grate the onion, or not, its your choice really”
Is the problem explained in text anywhere? (TFA delegates to a video and afaict only discusses another video-suggested solution and a novel solution in text, I don't understand what we're solving.)
> 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
meh, the food processor usually handles that for me pretty damn well
You're not wrong, but I think that the author's goal was not "how do I cut an onion evenly" but rather "how would someone do this if they had only a knife". He was solving a puzzle, not trying to suggest cooking technique.
yes, atomization is certainly one strategy, though often people enjoy onions that are not a slurry.
i think you are confusing a food processor and a blender. a food processor has other attachments/blades that do not result in a puree.
No, but they do beat the shit out of the onion. The bigger thing though is that they're a pain; you have to clean it out when you're done (also: have it handy on your counter to begin with), as opposed to just wiping down your knife, which you're already using for other things.
I don't do much with my food processor anymore besides grating cheese; even biscuit dough I'll do with a box grater at this point, just to avoid having to clean out the food processor.
I've had a food processor for maybe 20 years.. I use it at most twice a year specifically because it's a pain in the fucking ass to clean. Though thanksgiving is when I always bring it out, one bag of cranberries, one whole orange (quartered before throwing it in the food processor, peel and all), and sugar.. run it until you get a nice chunky relish.. deeeelish.
> The bigger thing though is that they're a pain; you have to clean it out when you're done (also: have it handy on your counter to begin with), as opposed to just wiping down your knife, which you're already using for other things.
I agree that cleaning the food processor is more of a pain than cleaning the knife. On the other hand, using it is far less of a pain than using the knife (especially in cases like this where you're trying to get even, small pieces). So you're really trading off one pain for another. It's not clear to me that either option is the obvious winner or loser here.
Depends on how much you are making really. If I want an entire bowl of thin cucumber slices I thank my food processor. But yeah, not worth the trouble for just one or two onions.
We clean our food processor in the dishwasher. :-/
If you have to clean things by hand, I'd take the awkward inside of a smooth plastic cylinder over a grater any day.
Don't know about your food processor but the ones I have seen are not just a plastic cylinder but also a funnel and circular blade attachment plus a couple of other bits that all tend to get pieces of food on them.
It's not impossible to clean and worth it when making larger amounts but definitely more of a hassle than a knife, even with a dish washer.
Yes, I agree it's more hassle than a knife. I don't agree that it's more hassle than a box grater.
It is.. at least my box grater which can be folded out flat making it very easy to clean the inside with a sponge and a tiny amount of elbow grease (much like the outside).
The box grater (in the biscuit dough case, just for the butter) fits into the top rack of my dishwasher along with the cups.
You are clearly not the target audience.
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.
I'm surprised Kenji still does the horizontal cut at all. With the angled vertical cuts I find the horizontal cut entirely unnecessary. (Also a few years back I gave myself a nice flap avulsion doing the horizontal cut in an onion...)
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.
making the single horizontal cut first makes every vertical cut after more difficult to perform without harming the structure of the onion.
technique and a sharp knife enable the horizontal cut second to be vastly superior to doing it first.
I'm not sure I understand. My knives are razor sharp (I keep a Shapton 1000 and 4000 on my counter along with a strop, my daily driver is a carbon steel I have to wipe down every time I cut a vegetable). They sail through the onion, but the sliced-up onion still splays out to both sides when I make the horizontal cut, and if you watch cooks doing it, it happens there too. What harm am I doing to the structure of the onion by doing it in the "wrong order"? They're the same cuts. The difference seems to be that in my order, the onion stays more stationary.
Don't get me wrong, I'm sure there's a reason everyone is doing it this way, because it's kind of clearly more annoying than the way I'm doing it?
(I'm just nerding out on this).
Here's the way I'm thinking about this:
the vertical cuts do not significantly the internal structure of the onion as each individual cut I make does not entirely sever the connection between the thin vertical slices I'm making. This means that I can do a lot of these, and not worry about harming the overall structural integrity. Then I make a single horizontal cut which does harm the overall structural integrity. This is not intrinsic to the horizontal cut itself, but the fact that I have both horizontal and vertical cuts.
If I start with the horizontal cut, again I do not signficantly harm the structural integrity of the onion. However, each subsequent vertical cut I make is now going to individually compromise the integrity of the onion.
With a sufficiently sharp knife, the single horizontal cut at the end does not really pose a significant danger overall.
This all being said I almost never do the horizontal cut out of pure laziness, and instead prefer to just do angled vertical cuts analogous to the video. They're never perfect but fine enough for me...
I still don't get why you need the horizontal cut at all. The diagram at the bottom of the blog post shows how unnecessary it is when you do the vertical cuts at a narrow range of angles like that (which I have been doing for a while now).
The point of Kenji's method (really, all radial-ish methods, but radial is strictly worse) is that you don't have to do the horizontal slice. If you slice vertically, you do --- you can see it for yourself, if you don't the dice from the edges of the onion are almost twice as big as the diece from the center.
I’m sure I’ve seen a clip of some tv chef saying it is unnecessary. Maybe Jacques Pépin but not sure.
Probably Chef Jean Pierre.
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.
I have them now, but's simpler to just avoid that one dangerous and unnecessary cut that proceeds towards my body instead. They taught that in Scouting, never cut towards yourself.
You need to cut in the direction of your body in some cases (for example when carving wood).
Two things to prevent injuries: a) never put any force if the material resists b) do it slowly.
And either learn to sharpen your knives yourself, or take them to a sharpening service. Dull knives require more force, and slip/catch more, so are more dangerous.
The trick I use for doing freehand sharpening is to color the bevel with a sharpie, that will show you if your angle is correct. You don't need a lot of stones, I just have one Sharpal double sided diamond stone, and then I move to a leather strop with 1 micron diamond emulsion compound.
Another very useful thing is an inexpensive jeweler's loupe so you can actually diagnose issues like not having removed the burr.
I looked into sharpening services in my city a few years back and they're like dry cleaners - every one was a mix of satisfied reviews and detailed "this person completely ruined my $600 knife" reviews. It was very off putting.
It's unlikely any sharpener is going to ruin your knife; at worst, they won't put the best possible edge on it. Your knife is probably just an inert hunk of steel. :)
Grinding too much of the edge does ruin the knife. Usually not the entire blade is hard enough to hold a good edge.
It's a fair point, but how likely do you think it is that any knife sharpening service is actually going to do that?
Oft repeated, but I don’t think I’ve ever seen this actually studied in practice. And personally I suspect it’s more a clever meme by knife sellers.
> for example when carving wood
I've watched a lot of shows about the tools used for building log cabins in the pioneer days. I don't even know the names of them, but the tool for taking the bark off the tree by pulling the knife to you as you sit on the log is crazy. Also, the one where you straddle the log and swing the blade towards you between your legs is right up there too. Yet, I can't think of any way of making them better without using power tools.
Drawshave or drawknife and adz.
The drawknife is the safer of the two by far. It’s fairly hard to cut yourself when your whole body is moving the same direction. Similar to using a paring knife in your palm facing your thumb.
The adz however you just have to have good aim or pay the consequences!
Haha, jinx : )
Draw knife. As long as you are leaning instead of pulling its relatively safe. Same as its safe to pare by contracting your hand muscles instead of pushing a knife toward yourself.
Draw knives are even safer than paring knives: the handles are placed such that they're closer to you than the blade, it's extremely difficult to get your chest far enough forward that it could contact the blade without a very large chest.
My standard housewarming gift is cut gloves and a pack of nitrile gloves to put over them. The nitrile gloves are so you don't have to wash the cut gloves so often.
Do people actually use them? Seems too much of a hassle unless you are in a professianl kitchen and cutting all day.
IME kitchen knife injuries are just not common or severe enough to warrant something like that.
Another thing to get out, another thing to clean, another thing to put away.
All because we want to chew less. (I suppose nice texture too)
Just always keep them on and never wash them. Bonus: immunity to papercuts forever.
Amazing, and I was soooo glad to see the integral has a closed form. I’m very curious what this looks like in the discrete case. I’d imagine it’s somewhat straightforward to code a simulation?
I prefer the Chef Jean-Pierre method [1] because he keeps it simple. He also shows why the horizontal cut really doesn't make much sense.
[1] https://www.youtube.com/watch?v=QjZ1LFqNWRM&list=PLnujfCpADf...
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.
He had to close down his cooking school during Covid, so he started making YT videos. I watch a number of chefs on YT, and he's easily my favorite. He's a wonderful teacher, and I've learned more from him than any other because he always reinforces concepts without being repetitive and dull. It also helps that his meals are practical for home cooks, and, overall, he's just a charming guy.
In fancy cooking there's also a notion of a "perfect dice", meant literally to be that --- you pull the layers of the whole onion apart and cut/press them into sheets, so you have a rectangle to work with, and then dice that on a grid.
https://www.youtube.com/watch?v=UBj9H6z6Uxw
One thing you see in that video is how you're not going to get anything like that out of a food processor.
Does it matter? I can't remember cooking anything where it would have (though: I do want uniform dice for tacos).
>To get the most even cuts of an onion by making radial cuts, one should aim towards a point 55.73066% the radius of the onion below the center. This is close, but different from, the 61.803% claimed in the Youtube video at the top.
Wife walks into kitchen with 3447 cut onions in piles: "What are you doing?!" This guy: "I just cannot get these onions cut to a point 55.73066% below the origin! The best I have achieved is only 2 significant digits of accuracy." Wife, mumbling: "Maybe that's why Kenji said: 60%..."
Mathematicians are wired differently than engineers. To us engineers, e is approximately pi is approximately 3.
think you got it the wrong way bud
You musn't be an engineer
Not "The Onion". The original capitalization is "the Onion Problem", i.e. the problem of dicing onions into even pieces.
I was confused, especially considering this [1] is still very recent
[1] https://theonion.com/kenji-lopez-alt-returns-from-beef-dimen...
I also thought I had my finger on the pulse of some The Onion/Lopez-Alt beef, like TMZ on the Food Network.
ontopic edit: I am interested in an optimal onion cutting technique, while I'm happy with mine, the upside-down banana teaches that there's always a few ways to approach and learn something.
This is exactly what I thought of when I saw the headline!
Off topic, but the incorrect capitalization of the article misleads one to believe this is related to “The Onion” rather than “an onion.”
I saw the headline and assumed it was going to be about the Beef Dimension: https://theonion.com/kenji-lopez-alt-returns-from-beef-dimen...
[flagged]
Please don't LLM spam on here.
You want even cuts you throw it into a blender
Besides a dice that's as even as possible, the other requirement this solution attempts to satisfy is using the minimum number of cuts. A blender doesn't satisfy that, as it's making hundreds of cuts.
Then, when you present your solution to the client, you find out there was a third, unspoken requirement: that it should involve as little cleanup as possible, which the blender also doesn't satisfy. The user researcher was on vacation, and you didn't find out about this before beginning design. Damn!
The blender solution turns out to be overoptimized on a single requirement at the expense of the others.
They're optimizing for time as knife cuts = time. A food processor will do it faster if you're more than one onion or so, assuming you can get the size you want.
Ahh, so in addition to having trouble getting consistently-sized pieces the size of a dice or chop, the other reason knives are preferred is that a food processor damages the onion, releasing more water compared to a knife. The result doesn't caramelize as well. This is why higher-end restaurants cut onions by hand, even when operating at scale.
> The result doesn't caramelize as well.
Anecdotally I've prepared caramelized onions both ways, chopped with a knife and using a food processor and I've never noticed a difference. Onions have to release most of their water before they can begin caramelizing anyway so if anything, wouldn't that speed up the process?
You don't have to turn it into mush with a food processor, not all veggies are caramelized. High end restaurants usually optimize for speed but not over quality. Not sure why were talking that when this technique is for home chefs.
That’s the engineering solution.
You could also hire two interns to do it layer by layer, call it the consultant‘s solution.
If you want an extremely fine and even brunoise that's exactly what you do.
Then it seems you need consultants to get a guide Michelin star
That sounds like a cost plus defense contract if there ever was one
The consultant solution would be to buy precut onions, so cutting perfect slices becomes someone else's problem.
A blender will make the bottom layer into paste before the top is touched. If you want to toss the paste into a skillet and caramelize it, that'll make a good sauce.
Food processor might be better, but still won't be even.
Source: I cook onions a lot, and am lazy. This article is great!
Always bust out the food processor when making soffritto or similar very small dice. Can do onions quickly and even with the method but carrots and others take quite some time.
That’s really not true, unless you are really mincing it or making a paste .
I don't think this gets you the texture you're looking, or even cuts. My eyes are tearing up right now thinking about scooping this out of the blender.
two words: Slap Chop.
Blender's not really well suited for dicing dry foods, they typically need some sort of liquid to bring the solids down to the blades.
Food processor might be more what you're thinking about but it's more so for dice or mince. You won't ever really get an even chop out of a processor.
Spice blenders tend to have blades closer to the bottom and are fine for dry things. Still not good for dicing though, especially as onions aren't really dry but full of liquid.
Ah yes, I wonder why the chefs haven’t thought of this.