Nice catalog for the 2012 JMM Math art exhibition

I got a very nice catalog for the 2012 Exhibition of Mathematical art of JMM from Chern today. Here are two pictures, one for the cover page and the next one for our beadworks on TPMS.

Twistane

I just learned a kind of molecule called twistane the other day. It looks so interesting and I decided to make a bead model for it.

Dendritic structures

Mr. Horibe made a number of dendritic fullerenes which are similar to the Kepler's stellated polyhedrons. By using the Euler theorem, it is quite straightforward to show that, in a cage-like fullerene without hole, N₅-N₇=12, where N₅ and N₇ are the number of pentagons and heptagons, respectively. In the typical fullerenes where the local Gaussian curvature is positive everywhere, we should have N₇=0. The total number of pentagons is 12. Another way to put it is to introduce the so-called topological charge, 1 for a pentagon and -1 for a heptagon. So the topological charge of a cage-like fullerene is 12.

Heptagons always generate a negative Gaussian curvature. For a cage-like fullerene, whenever we introduce an extra heptagon, we have to include a pentagon in order to satisfy the identity N₅-N₇=12.

One can replace a pentagon in a Goldberg icosahedron (icosahedral fullerene) by 6 pentagons (a hemisphere of C20) and 5 heptagons to get a spike. So the topological for this area is still 1 after replacement. Similarly, we can do the same replacement for all other 11 pentagons to get a dendritic structure.



Of course, we can use the same kind of trick to "grow" a spike (a carbon nanotube endcapped with the hemisphere of a C₂₀) along the normal direction of any pentagon on any kind of graphitic structure.

Incidentally, this structure without the C20 caps is just the inner part of the high-genus fullerenes we have done before.

Workshop in the Hakozaki campus of Kyushu University

I gave a workshop in the chemistry department located at the Hakozaki campus of Kyushu University on Jun. 5, 2012. The workshop was hosted by Prof. Teruo Shinmyozu. Since participants are undergraduate, graduate students and some faculty members of Kyushu univ., so I gave some more details about the chemistry of molecular beading. Particularly, I talked about that the connection between beads and valence electron pairs. Bead models of molecules are possibly the only physical models that take advantage of the analogy between microscopic interactions and macroscopic hard sphere interactions. All the other physical models we commonly used have nothing to do with the microscopic interactions. With beads and suitable beading methods, one can make faithful molecular models of molecules.


Prof. Sonoda volunteered as TA in all of my workshops in Japan. He has now also become an expert of molecular beading. Here he tried to help Prof. Shinmyozu fix the beading problem.

Bead model of small stellated dodecahedron

Another class of bead models made by Mr. Horibe is the small stellated dodecahedron, which is one of the Kepler–Poinsot polyhedra.



The basic idea of making this kind of dendritic dodecahedra is to choose a suitable size of Goldberg polyhedron and then grow a short segment of endcapped carbon nanotube along each pentagon. In this particular case, the endcapped CNTs are just hemisphere of C20s. Similar trick to make dendrite-like structures is used in many of Mr. Horibe's work.

Mr. Horibe's story

Yesterday, Mr. Horibe informed me with email that he made the first regular polyhedron with duralumin tubes around 1975. After that he was busy on teaching and nothing was done for a while.

However, around 1995, he made the first bead model of dodecahedron which was consisting of 30 balls based on the problem described in the famous math book (Sangaku book, 算法助術) from the Edo period (江戶時代) of Japan. Since then, he started to make many different kinds of mathematical beadworks alone for about 17 years until we met last week. He said he will keep on making more bead models.