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        riddles >> hard >> packing wires in a sleeve
        
(Message started by: JocK on Sep 4^th, 2004, 1:07pm)

Title: packing wires in a sleeve
Post by JocK on Sep 4^th, 2004, 1:07pm

An electrical wire manufacturer wants to construct a wire of unit cross-sectional area such that any number of these wires can be tightly packed in a sleeve with a sleeve circumference that is as small as possible.

The chief engineer of the company has worked out a lower bound for the sleeve circumference S as function of N, the number of wires: S > 3.5449 [sqrt]N, and starts out to calculate for various cross-sectional wire-shapes the smallest value [alpha] for which any number N = 1, 2, 3, ... of wires of that particular shape can be packed in a sleeve with circumference S [le] [alpha][sqrt]N. She finds that for square wires [alpha] can not reach values smaller than 6/[sqrt]2 [approx] 4.243.

Further study reveals that for a rectangular wire cross-section, smaller values down to [alpha] [approx] 4.060 can be reached.

Can you find cross-sectional shapes that lead to values [alpha] < 4 ? How close can you get to 2[sqrt][pi] [approx] 3.5449 ? What is the minimum value that can be reached?

J8)CK