Thursday, December 28, 2006

Spiral code for creating fullerenes

As I have mentioned that the details of the beading procedure for creating a fullerene molecule describable by a spiral should be completely determined by the sequence of pentagons and hexagons. Once this sequence is given, then we can just carry out the weaving process by making the 5 or 6-bead group using the RAW according the recipies. Sound simple, right. But we still need to know the sequence in order to make the fullerene we intend to create. Fortunately, the complete list of all possible isomers for fullerenes in the range C20 to C50 and isolated-pentagon isomers of fullerenes in the range C60 to C100 are tabulated in Fowler's "An Atlas of Fullerenes". Instead of giving the whole sequence of 5- and 6-gons. Fowler also gives another simplified notation for the sequence of 5- and 6-gons in the spiral. Since there are exactly twelve pentagons in a fullerene, and the others are hexagons, therefore we only need to know the positions for the pentagons in the spiral.
Here I will illustrate his notation with two simple examples. The first one is an isomer of C80, Fowler's 80:7 isomer (the 7th isomer out 7 isolated-pentagon isomers of C80), with the spiral code, 1 8 10 12 14 16 28 30 32 34 36 42. According this spiral code, I have to make a pentagon first, then 6 hexagons, and then a pentagon, hexagon, pentagon, and so on, in a clockwise spiral. Finally a C80 is created. In the process of making this fullerene, I don't need to worry about the connectivity or geometry, the resulting beaded C80 is in good agreement with the actual geometry of C80 due to the repulsion between different beads.
From Craft Projects

Another one I made is isomer 50:24 with spiral code 1 2 3 4 7 12 17 22 24 25 26 27, which is an isomer of C50. The shape of this isomer looks like a bean or cocoon.

From Craft Projects

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