Comments on Sierpinski buckyball

Recently Mark left a comment on the Sierpinski's buckyball we proposed:

mark said...

That is really good and attract the Small sierpiski beaded fullerene.
butFullerenes are a family of carbon allotropes consisting of molecules composed entirely of carbon atoms arranged in the form of hollow spheres, ellipsoids, or tubes.
9/15/2009 1:41 PM

Here is my response:

You are certainly right about fullerenes. There are indeed many possible sp2 based fullerene structures. We have recently published a few papers on the systematics of toroidal CNT, helical CNT and High-Genus fullerenes.

1. Chuang, C.; Fan, Y.-C.; Jin, B.-Y. "Generalized Classification Scheme of Toroidal and Helical Carbon Nanotubes." J. Chem. Inf. Model. 2009, 49, 361-368.
2. Chuang, C.; Fan, Y.-C.; Jin, B.-Y. "Dual Space Approach to the Classification of Toroidal Carbon Nanotubes." J. Chem. Inf. Model. 2009, 49, 1679-1686. DOI: 10.1021/ci900124z
3. Chuang, C.; Jin, B.-Y. "Systematics of High-Genus Fullerenes." J. Chem. Inf. Model. 2009, 49, 1664-1668. DOI: 10.1021/ci9001124, ACS News & Research, June 2009.
4. Chuang, C; Jin, B.-Y. “Hypothetical Toroidal, Cylindrical, Helical Analogs of C60.” Accepted for publication in J. Mol. Graph. Model. 2009. http://dx.doi.org/10.1016/j.jmgm.2009.07.004
5. Jin, B.-Y.; Chuang, C.; Tsoo, C.-C. "The Wonderful World of Beaded Molecules." CHEMISTRY (The Chinese Chemical Society, Taipei) 2008,66, 73-92. (in chinese).

I personally doubt the possible existence of Sierpinski fullerenes. But, mathematically speaking, it is still an interesting generalization. I have never seen any work on this possibility. As far as I know, the most popular ones are Sierpinsky tetrahedron and cube.

To see more discussion on the Sierpinski's buckyball and a simple estimation of its fractal dimension, please check several of my posts in June 2007.

http://thebeadedmolecules.blogspot.com/2007_06_01_archive.html

I still remembered that when I gave a talk on "Chemistry, Geometry and Art: The wonderful world of fullerenes" in the math department of National Taiwan University in the early July of this summer vacation, someone in the audience (an expert in the fractal geometry) was quite intrigued by the Sierpinski's buckyball I shown in one of the slides. He told me that no one has ever investigated the Siepinsky regular and semiregular polyhedra yet. I think it might be interesting if we can do something about it.