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.

HG-Fullerene as a gift

I met Prof. M. Terrones in the ACS meeting yesterday in the ACS meeting. Fortunately, I brought a HG-fullerene (made by C. Chuang) with me.
I gave it (as shown in the following figure) to him.

From Dec 25, 2008

(High-genus fullerene with octagonal necks, No. 2)

碳環120 (TCNT 120)

今天到數學系給個講演，講題為「化學、幾何、藝術：芙類分

子的美妙世界」。

也把這個碳環120送給數學系，作為台大數學月的禮物。

This month is the mathematical month of NTU.
I gave a talk, "Chemisty, geometry, and art: the wonderful of fullerenes", at the department of mathematics of the Taiwan university today. Here is the TCNT 120 I gave math dept as a gift.

A C60 beaded molecule in math dept

One of the students, 林楷軒 (from math dept), in the course "general chemistry" I taught sent me a picture of C60 (shown below) he discover in math building 403. He said it was constructed by unknown author some time ago.
It looks beautiful to me for sure and reminds me that I have given away many beaded molecules to profs. 林長壽, 陳宜良 of math dept and prof. 李文卿 of math dept, Penn. State U. last June. This one is probably one of them and possibly belongs to prof. 林長壽. He probably put it there for student to play.

I check my iphoto library this morning and found this picture of C60. I guess they are the same beaded model.
Additionally, here are beads I used to construct the model.

and another beaded model, C80, made by the types of beads:

Genus-2 fullerene with local hexagonal symmetry

Chuang recently made this beaded molecule which is basically consisted of two rings fused together along their common C2 rotational axes. In the inner side of this molecule, Chuang used octagons to create negative curvatures. According to the Euler theorem, in the outer region with positive Gaussian curvature, one has to use the same number of squares for a simple torus. But here the two squares located the connected region has to be fused into an octagon in order to simulate the negative curvature in that region.
So here we need an extra two octagons in the outer side of this molecule.

Sculpture of C60 at Chem. Dept. of TKU