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> Makes me wonder if there's some forth dimension that humans are quantized to ignore Interesting that your mind went to dimensionality - because the way to ge

> Makes me wonder if there's some forth dimension that humans are quantized to ignore Interesting that your mind went to dimensionality - because the way to get 8 instead of "7" is to cut in 3D! Do a + on the top, and then cut horizontally along one of the layers of icing! Voila! 8 cake pieces :D

SkyBelow 4y ago
I attempted to do a cut in 4D to create 16 pieces, but I was only managed to have 8. The 8 pieces before I ate them and the 0 pieces after, for 8 total. On a more serious (ish) note, can you cut a 3D pie in 4D space? If I go with a 2D circle in 3D space, by default you can't cut it down the extra dimension. But what if the circle is curved in 3D space, say by attempting to wrap it part the way around a sphere. Even if the circle was originally in the XY plane, you can now perform a Z plane cut which would allow going from 4 to 8 pieces. So ideally if you were to warp the 3D pie around a hypersphere, you would have a way of cutting it to make 16 pieces.
- adverbly 4y ago
  Based on what you're describing, I suppose you could have any number of pieces. Imagine you've got a 2D sheet of paper and you just fold it in half many times, and then cut through all the folds. You could have as many pieces as you want.
  - SkyBelow 4y ago
    Good point. Stretching it against some wavy object and cutting would remove any meaning from the problem as you can get however many pieces you wanted based on the wave pattern.
- wizzwizz4 4y ago
  This warping is equivalent to using a curved knife.
  - cscheid 4y ago
    And if you think sufficiently hard about curved knifes and cherries on high-dimensional pies, you end up studying machine learning theory (https://en.wikipedia.org/wiki/Vapnik%E2%80%93Chervonenkis_dimension https://en.wikipedia.org/wiki/Vapnik%E2%80%93Chervonenkis_di...)
    - TeMPOraL 4y ago
      If you're implying that any problem of N-dimensional geometry can be reduced to machine learning techniques, then maybe there is a way to reinterpret machine learning as cutting N-dimensional cakes with M-dimensional knives?
      - knodi123 4y ago
        Of course. There's a trivial proof that all neural networks can be reduced to a sufficiently complicated cake scenario. It's called Turing's Birthday Party.
      - cscheid 4y ago
        Not _any_ problem, but the _specific_ problem of determining how many arbitrarily placed points (cherries) can be split by a given shape of hypothesis classes/classifiers (knives) is literally the definition of VC-dimension, yes :)
deleted 4y ago
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jancsika 4y ago
Nice! Now I'm imagining a conspiracy among economists: to only ever show us top-down pie charts while funneling all the depth into their own slush fund. :)

SkyBelow 4y ago

I attempted to do a cut in 4D to create 16 pieces, but I was only managed to have 8. The 8 pieces before I ate them and the 0 pieces after, for 8 total. On a more serious (ish) note, can you cut a 3D pie in 4D space? If I go with a 2D circle in 3D space, by default you can't cut it down the extra dimension. But what if the circle is curved in 3D space, say by attempting to wrap it part the way around a sphere. Even if the circle was originally in the XY plane, you can now perform a Z plane cut which would allow going from 4 to 8 pieces. So ideally if you were to warp the 3D pie around a hypersphere, you would have a way of cutting it to make 16 pieces.

adverbly 4y ago
Based on what you're describing, I suppose you could have any number of pieces. Imagine you've got a 2D sheet of paper and you just fold it in half many times, and then cut through all the folds. You could have as many pieces as you want.
- SkyBelow 4y ago
  Good point. Stretching it against some wavy object and cutting would remove any meaning from the problem as you can get however many pieces you wanted based on the wave pattern.
wizzwizz4 4y ago
This warping is equivalent to using a curved knife.
- cscheid 4y ago
  And if you think sufficiently hard about curved knifes and cherries on high-dimensional pies, you end up studying machine learning theory (https://en.wikipedia.org/wiki/Vapnik%E2%80%93Chervonenkis_dimension https://en.wikipedia.org/wiki/Vapnik%E2%80%93Chervonenkis_di...)
  - TeMPOraL 4y ago
    If you're implying that any problem of N-dimensional geometry can be reduced to machine learning techniques, then maybe there is a way to reinterpret machine learning as cutting N-dimensional cakes with M-dimensional knives?
    - knodi123 4y ago
      Of course. There's a trivial proof that all neural networks can be reduced to a sufficiently complicated cake scenario. It's called Turing's Birthday Party.
    - cscheid 4y ago
      Not _any_ problem, but the _specific_ problem of determining how many arbitrarily placed points (cherries) can be split by a given shape of hypothesis classes/classifiers (knives) is literally the definition of VC-dimension, yes :)

deleted 4y ago

[deleted]

jancsika 4y ago

Nice! Now I'm imagining a conspiracy among economists: to only ever show us top-down pie charts while funneling all the depth into their own slush fund. :)