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November 15th, 2007, 11:36 AM   #1
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Maximum and minimum area for n-gon

Say I have n sticks of lengths a1, a2, a3, ..., an.
What is the n-gon with the biggest area that I can build with all of my sticks? And the smallest?
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November 15th, 2007, 05:36 PM   #2
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Good question, I have no idea.

For n < 3, you can't make a polygon. For n = 3, the minimum and maximum will be the same -- if you can make a triangle at all, its area is determined by the lengths of the three sides (Heron's rule).

For n > 3, I expect the minima will decrease rather than increase with increasing n, especially if degenerate polygons are allowed.
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November 16th, 2007, 10:30 AM   #3
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I tried writing a mathematical explanation, but I decided words would be better here. So, here's the idea: find a set of n-3 sticks (S1) and a set of 3 elements (S2) where the element's in one exclude their existence in the other but with certain rules. But first, a notation:

∑S - the sum of all element's lengths in S

Rules:

1) you can split S1 into 2 sets (Sa and Sb) that have the following property:

∆ = |∑Sa-∑Sb|

and ∆ is minimal for all possible sets S1, Sa and Sb.

2) for set S2 where a, b and c are it's elements:

l(a)+l(b)>l(c)+∆

for some combinations of a, b and c.

Then, for such set's S1 and S2 do the following:

1) connect all elements of Sa and Sb into two straight lines, then connect both of them at one end and point the remaining two in the same direction (or, in simpler words, visually make a single line of the same length as ∑Sa or ∑Sb, depending witch one is bigger)

2) connect c form S2 with the end of greater line and the smaller one of a and b with the end of the smaller line, and the remaining one with c and the other one.

That's it for the minimum! (I know it's complicated, but I gave a picture)



And for maximum, connect all of them into a n-gon and make the angles in between each one of them large as can be (I think this should work)!
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