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Comment by jancsika | Hacker News Reader

> or the “lazy caterer’s sequence” (1, 2, 4, 7, 11, 16, 22, 29,...), the maximum number of pie pieces that can be achieved by n cuts.

My brain just did not want to accept "7" as the fourth number in the series.

I even drew it, and it was still damned difficult to make my hand offset the line to leave that shit piece in the middle!

Makes me wonder if there's some forth dimension that humans are quantized to ignore. Like right now, there is an alien traffic jam on some sprawling inter-dimensional superhighway caused by a human who plopped down like a duck to check on the price of an NFT.

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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 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

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 :)

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. :)

The pattern of the 3 lines creating 7 pieces always reminded me of a poorly graffitied anarchy symbol. which, in its own fun way is ironic, as cutting pie that way is pure disorder and not forming the top vertex of the “A” along the circle does not fully conform to the standard anarchy symbol! (fwiw, I do know anarchy means no ruler/hierarchy rather than no rules/chaos, so pardon the relaxed definition in search of humor)

I felt similarly about the maximal number of regions you can obtain by joining N points on the circumference of a circle by straight lines. That sequence [1] starts:

  1, 2, 4, 8, 16, 31

[1] https://oeis.org/A000127

    2^0
    2^1 
    2^2 
    2^3
    2^4
    2^5 ... -1 ?

I think it's more that stuff you deal with in normal life brings along substantial training in favor of fairness.

Dividing a shape means "divide fairly", because that's what you've done thousands of times. Drawing some lines means "in an aesthetically pleasing way" because that's what you've seen a billion times. Etc.

Math / what's possible doesn't care one bit about that though. It takes work to train yourself to move past other training.

> I think it's more that stuff you deal with in normal life brings along substantial training in favor of fairness.

> Dividing a shape means "divide fairly", because that's what you've done thousands of times.

There are fundamental game-theoretic reasons for a fair split, yes. But there are also, arguably, more "pure" aspects at play, unrelated to agent dynamics. For example, out of the possible division ratios, there is an infinite number of them that give you two different pieces of cake, but only one ratio that gives you two indistinguishable pieces. What it means, among other things, is that[0] the 50/50 split is a singular one[1] that removes one bit of information from the "cake split into two pieces" system! Any other split requires you to track which piece is which (if only to be able to tell whether I just took the smaller or the larger one); the 50/50 one does not (the pieces are the same, it's pointless to ask which one I ate).

> Drawing some lines means "in an aesthetically pleasing way" because that's what you've seen a billion times. Etc.

The cake example above involves a special solution that removes bits from the problem. I feel the same is the case with a lot of aesthetics - we seem to have a preference for patterns that compress well. I don't think it's a peculiarity specific to human minds - it seems to me to be fundamental enough that all minds would naturally end up with a bias towards well-compressible pattern.

[0] - Take this piece of cake with a grain of salt, because I'm just a smartass, not an actual mathematician.

[1] - Ignoring the degenerate cases, i.e. the 0/100 and 100/0 splits.

As long as no two lines are parallel and no three lines meet at a single point, you will get the maximum number of pieces.

You just have to be careful not to have two lines with similar slope, or you'll require a very large pie!

... and, every line intersects with every other? (I mean, when cutting a finite pizza...)

Yes, definitely. I like to think of it as: consider you've got infinite space, see where your lines intersect and then draw your pie around everything.

Now I want to try this by placing the three cuts (chords) symmetrically so that they form a neat equilateral triangle at the center.

I wonder if it is possible to make just the right cut so all pieces are "equal" (in area)?

Is there a mathematical tool to make experiments like this graphically / visually? I define three cuts as lines that are offset by "x" from the center of the circle and as I vary the value of x from zero to r, the areas of each cut piece should be computed and plotted.

Better not look at the related sequence https://oeis.org/A000127

1, 2, 4, 8, 16, 31, 57, 99, 163, 256, 386...

> Maximal number of regions obtained by joining n points around a circle by straight lines.

I was ready to attempt drawing that, but then I saw the other possible definition:

> Also number of regions in 4-space formed by n-1 hyperplanes.

So much easier to visualize it like this!

> I even drew it, and it was still damned difficult to make my hand offset the line to leave that shit piece in the middle!

Make the offset bigger and then the middle piece can be plenty big!

It felt more natural after I renamed it the "inept" or "a*hole" caterer's sequence.

I wonder how much of this is due to a cultural sense of fairness and what proper pie cutting looks like.

A friend mentioned being at a party once where the guest of honor cut the first slice for themself and they chose to make a glancing slice that maximized their surface frosting to volume ratio.

> > or the “lazy caterer’s sequence” (1, 2, 4, 7, 11, 16, 22, 29,...), the maximum number of pie pieces that can be achieved by n cuts.

> My brain just did not want to accept "7" as the fourth number in the series.

The sequence that you get from the difference of successive numbers in the list is rather predictable: 1,2,3,4,5,6,7, ...

A picture of a pancake cut into 7 pieces in 3 cuts:

https://en.m.wikipedia.org/wiki/File:PancakeCutThrice.agr.jp...

Lazy Caterer make Grug angry. Grug smash stupid caterer in stupid face.

i think this is a simple consequence of euler characteristic, and the property that any addition cut intersects each of the others at most once. You can set up a sinple equation that gives you an upper bound and a simple configuration which achieves it.

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