Friday, December 29, 2006

A simple estimate of scaling factor for the minimum length of thread

I made some simple measurements in order to get an estimate for the length of thread I need. The results for three fullerenes are

Isomer 50:24 theoretical length, 60 cm; measurement: 68 cm; scaling factor s = 68/60=1.1
Isomer 50:43 theoretical length, 60 cm; measurement: 69 cm; scaling factor s = 68/60=1.2
Isomer 80:7 theoretical length, 96 cm; measurement: 106 cm; scaling factor s = 68/60=1.1

From Craft Projects

The scaling factor is about 1.1-1.2. I guess this result should hold for other fullerene too, at least for me. But this estimate could vary for other people. Those people who can make an even tighter structure are going to have a smaller s. On the contrary, measurements on the loose structure will lead to larger scaling factor. I suspect that structures with scaling factor around 1.4 are already very loose.

Since we need some extra length about 20 to 30 cm to tight up the final beaded molecule, so the total length of thread to use for building a fullerene with N carbon atoms can be chosen as

L = 2*(3/2)*N*d*s + K = s*3*N*d + K
where s = 1.1 to 1.2, K = 20 to 30 cm and d (= 4mm) is the diameter of the bead.

However, I think it would be interesting that we can perform more measurements to get a better estimate for s.

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