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Given n+1 points in n-dimensional Euclidean space, how many polytopes (generalizations of polygons of n to as few as 2 dimensions) may be defined by the representation of each point as a

*possible*vertex?
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- Thread starter Loren Booda
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- #1

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- #2

damgo

So 2^(n+1) - 1 - (n+1) = 2^(n+1) - n - 2

- #3

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...n......k.....................0.......1..........n+1

sum...C......=2^(n+1)-C.......- C........- C.......= 2^(n+1)-n-2;

.k=2....n+1..................n+1....n+1.....n+1

(using Newton's bynom...(bad english));

Exactly as damgo said...

Take the . as space (' ')

sum...C......=2^(n+1)-C.......- C........- C.......= 2^(n+1)-n-2;

.k=2....n+1..................n+1....n+1.....n+1

(using Newton's bynom...(bad english));

Exactly as damgo said...

Take the . as space (' ')

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- #4

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Can anyone show the

- #5

damgo

- #6

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I believe that

- #7

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- #8

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- #9

damgo

If you want to see *really* quickly increasing things, try Ackerman's function, or the Busy Beaver function.

- #10

Hurkyl

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Hurkyl

- #11

damgo

Roughly, it's the biggest number a computer with n states and infinite storage can create. It's known for n<5... BB(6) is crazy huge, at least 10^100. It's provably uncomputable, and IIRC you can prove that it grows faster (in a rough sense) than any computable function.

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