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The sausage conjecture
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Say that you have a collection of tennis balls (they could also be ping-pong balls or even marbles, given that the exact size doesn't really matter). The problem is: how should you arrange them so that the surface of their collective envelop is minimum.

In three dimensions, if you have 56 balls of fewer, then the solution is to arrange the balls perfectly aligned; their collective envelop then looks like a sausage. On the other hand, if you have more than 56 balls, then the solution is to put them tightly packed in a bag (like potatoes).

And what about in 4 dimensions ? Not many people can visualize in four dimensions but the question still makes sense, because we know what it means for four-dimensional objects to touch each other without penetrating each other, and we can compute envelops and their volumes (*). The fourth dimensional space behave likes the third dimensional space: there is a number below which the (four dimensional) balls should be aligned (in the four dimensional space) and above which they should be packed in a (four dimensional) bag. We know that this number is between fifty thousand and one hundred thousand but we do not know what is it exactly.

For dimensions more than 4, i.e., dimension 5, 6, 7 etc, the conjecture is that the sausage is always the optimal way to place the balls, regardless of the number of balls. We have proven that this is true for any space of dimension more than or equal to 42, but we still don't know if the sausage arrangement is always the optimal arrangement for spaces of dimension 5 to 41.


(*) We are still talking about "surfaces" here, but in a four dimensional space a surface is a three dimensional space, so that's why I said "volume" (because to me that's what they are), but beings living and perceiving in a real four dimensional space would probably call them "surfaces".

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