5 ms·

> 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 (Ca

by yunwal 2y ago

> 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.

Cerium 2y ago
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.
dole 2y ago
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.
- yunwal 2y ago
  > 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.
  - StrangeDoctor 2y ago
    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.
    - jameshart 2y ago
      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.
  - dylan604 2y ago
    Isn't that where calculus and intergrals come into to play? As the radius approaches infinity type of stuff?
    - ktallett 2y ago
      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.
  - sgc 2y ago
    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 https://drspoulsen.github.io/Onion_Marp/index.html#44
- PittleyDunkin 2y ago
  Only some types, of course. Shallots are not spherical, though they are radial, but the centerline is often curved.
  - account42 2y ago
    Shallots very often have two bulbs fused together which makes them very much non-radial.
    - PittleyDunkin 2y ago
      I was referring to the fact that each bulb tends to expand outward in layers from a core. You are, however, correct.
dhosek 2y ago
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.
- deleted 2y ago
  [deleted]
- glompers 2y ago
  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.
  - bigstrat2003 2y ago
    I thought it was funny. Probably the first funny thing I've seen from the Onion in years, actually.
- giraffe_lady 2y ago
  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.
  - grgbrn 2y ago
    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"
    - giraffe_lady 2y ago
Maken 2y ago
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.
gus_massa 2y ago
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.

by yunwal 2y ago

Cerium 2y ago

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.

dole 2y ago

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.

yunwal 2y ago
> 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.
- StrangeDoctor 2y ago
  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.
  - jameshart 2y ago
    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.
- dylan604 2y ago
  Isn't that where calculus and intergrals come into to play? As the radius approaches infinity type of stuff?
  - ktallett 2y ago
    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.
- sgc 2y ago
  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 https://drspoulsen.github.io/Onion_Marp/index.html#44
PittleyDunkin 2y ago
Only some types, of course. Shallots are not spherical, though they are radial, but the centerline is often curved.
- account42 2y ago
  Shallots very often have two bulbs fused together which makes them very much non-radial.
  - PittleyDunkin 2y ago
    I was referring to the fact that each bulb tends to expand outward in layers from a core. You are, however, correct.

dhosek 2y ago

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.

deleted 2y ago
[deleted]
glompers 2y ago
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.
- bigstrat2003 2y ago
  I thought it was funny. Probably the first funny thing I've seen from the Onion in years, actually.
giraffe_lady 2y ago
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.
- grgbrn 2y ago
  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"
  - giraffe_lady 2y ago

Maken 2y ago

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.

gus_massa 2y ago

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.