Friday, June 1, 2007

Sierpinski buckyball

After a short discussion with Chuang, I now think the Sierpinski buckyball is a better name for this system than the Escher buckyball.

Interestingly, it is not hard to find out the Hausdorff (fractal) dimension of this system:

D=log 90/log scaling factor = log(90)/log(8) ~ log(90)/log(7) = 2.16 ~ 2.31

(More careful measurement indicates the scaling factor is 16 cm/2.5cm =6.5, thus D = 2.4)

where I made a rough estimation for the magnification in order to get the scaling factor from the small buckyball to the large buckyball. Detailed calculation is more involved for the 3-D geometry of truncated icosahedron is needed.

Sierpinski's pyramid from Wikipedia (Fractal dimension = log 4 / log 2 =2):

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