Comment by SkyBelow | Hacker News Reader

Comment by SkyBelow | Hacker News Reader

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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.

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.

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.

This warping is equivalent to using a curved knife.

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_di...)

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?

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.

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 :)

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