Saturday, August 18, 2012

Enantiomeric Pair of Graphitic Trefoil Knots C318

It's been a while since last time I wrote on this blog. Three weeks ago, I went to the Bridges 2012 conference held at Towson University, MD. It was such a terrific gathering of practitioners of mathematical art. I met a lot of new friends and enjoyed sharing our ideas with each other. The following is a repost of a graphitic structure I found last year when I was a research assistant of Bih-Yaw. It is a D3-symmetric trefoil knot structure and comprises 318 carbon atoms.
 


I figured out its construction rule while pondering on one of Escher's artwork.
 

Upon scrutinizing the one at the upper-left corner, I found that the trefoil knot can actually be decomposed into three 270-degree-arc and three inter-arc straight segments. This was a huge hint for me since I've been thinking about constructing torus knots from carbon nanotubes for years. An idea came up when I was doing my MS degree with Bih-Yaw, three years after he mentioned this project to me, which basically makes use the fact that one can divide a (p,q) torus knot (for p smaller than q). An example of this building principle applied to the trefoil knot is shown below:

 


Interested readers can also refer to the "Torus Knot" tag in this blog. A truly unfortunate fact is that this kind of molecules does not seem to be "beadible" to me, since it has got a great deal of tangential stress along the tangent of the central curve. Unless one lets loose the fish line during the course of the beading process, eventually when she or he tries to sew up the closed structure an unsurmountable stress will be encountered.

I didn't want to stop there, and finally I got the inspiration from Escher as mentioned above. The structure possesses nearly zero tangential stress when you try to bead it. Another nice point about it is that it has got much less number of atoms in it, compared to the other construction scheme. It typically takes me less than three hours to bead it.

At the Bridges conference, I met two kind Taiwanese ladies Helen Yu and Joy Hsiao. The pair of enantiomeric C318 trefoil knots as shown in the following photo was given to them as souvenirs.


They are made of 3mm glow-in-the-dark beads. I'm thinking about presenting this type of structures in the Bridges next year, hoping that I have the chance to go.



2 comments:

Bih-Yaw Jin said...

Very interesting. It seems that you really enjoyed the Bridges meeting.

I recently read a paper about polyhedral tori, which might be useful for classifying trefoil knots systematically in the same way as we did with carbon nanotori. I will mail it to you later.

city said...

thanks for sharing.