Hyperbolic buckyball

拍攝時間：2015/4/28; 地點：北一女
作品完成時間：2015/2/8

Bead model of Klein's all-heptagon network

I took a picture of single tetrahedral unit (12 heptagons) of D56 bead model on the figure depicting schematically an open network consisting only of heptagons, described by Klein in his 1879 paper.

Klein, F. (1878). "Ueber die Transformation siebenter Ordnung der elliptischen Functionen" [On the order-seven transformation of elliptic functions]. Mathematische Annalen 14 (3): 428–471. Translated in Levy, Silvio, ed. (1999). The Eightfold Way. Cambridge University Press.

C20 vs C56

Beadworks for the mathart exhibition (Mar. 15, 2014)

Hyperbolic "buckyball" with octagons

I played with carbon tetrapods described in the Diudea and Nagy's book last weekend and discovered quite accidentally that it is possible to make another type of hyperbolic "buckyball" with neighbored nonhexagons separated by only one CC bond. Particularly, all nonhexagons in this structure are octagons. So this one is different from another hyperbolic "buckyball" made of heptagons and hexagons, or simply heptagonal buckyball. The skeleton of this structure is the same as the heptagonal "buckyball". But each unit cell, which consists of two connected tetrapods, has 96 carbon atoms. I am not sure if people have mentioned this kind of hyperbolic "buckyball" or not.

Heptagonal "buckyball", C168 (This model was given to Prof. Sonoda as a gift when I was invited to Japan for a series of talks and workshops.):

(photo by Dancecology)

Octagonal "buckyball", C96:

Beadworks on display

I have many beadworks made by Chern, Wei-Chi and me on display in the faculty lounge here at the chem dept of NTU.

C60 vs C168

One can view C60 or a buckyball as the simplest graphitic sheet embedded in a sphere with every pentagon connected to five other pentagons by only single carbon-carbon bonds. In this sense, C168 can be viewed as the "buckyball" in the hyperbolic space with every heptagon connected to seven other heptagons by only single carbon-carbon bonds.

I made a bead model of C168 that consists only of one tetrapod (half unit cell). So only 84 carbon atoms are included in this structure. The structure I posted previously contains 5 unit cells.

A new model of C168

I made a new model of C168 last week.


Click the keyword C168 below to see more discussion on this structure.

My C168

I have made only one beaded C168, which I gave to Dirk Huylebrouck, a math professor in the department of architecture in Belgium, at the Bridges conference this summer. I took the following photo of this bead model at the Hotel in Pecs, Hungary. I am thinking about making another one, maybe this time with giant beads just like Mr. Horibe has used. Unlike Mr. Horibe, I prefer using different colors for nonhexagons. In this structure, all heptagons are in purple beads. One can easily see that these heptagons are separated by on beads (carbon carbon bonds). In this sense, we can call C168 is the buckyball in the hyperbolic space.


Another two photos that contains more bead models I brought to Bridges conference.



Many of these beadworks are given away as souviners for other attendee. The helically coiled carbon nanotube is given to Laura Shea and the high-genus fullerene is to M. Longuet-Higgins. Toroidal carbon nanotube (T120) with 120 carbon atoms or 180 beads (cat-eye stons) is given away to G. Hart. T120 is made by Chern Chuang. All other small beaded balls are gone too. Many of these small beaded balls are made by Q.-R. Huang. The only three left is the bead models for the P-type triply periodic minimal surface, Shoen's I-WP surface and the trefoil knot, respectively.

A few more pictures from Horibe

I found a few more pictures about Mr. Horibe's (堀部和経) beadworks and workshops in Japan.


He seems to be fond of making beaded structures similar to that of C168 with endcaps.

More pictures can be found here.

C168

C168 is quite uniqe because it correponds to C60 in the hyperbolic space.
We can view standard fullerenes as a tiling of graphene sheet on a sphere, which is a two-dimensional manifold with postive curvature everywhere. C60 corresponds to the smallest fullerene with all pentagons separated by only one CC bond. Similarly, C168 is the smallest fullerene in a hyperbolic space with all heptagons separated by only one CC bond.

In the bead model I created, purple beads stand for the edges of heptagons, and white beads are the CC bonds separating different heptagons.

From Dec 25, 2008

(I gave this model to Dirk Huylebrouck, a professor in the department of architecture at Sint Lucas (Brussels, Belgium) at the Bridges Pecs, Hungary 2010)

D168

Beaded D168

Here is the first beaded model of D168. D168 has diamond structure. So it is an extended structure. Here I only made part of the whole structure based on the admantane. We can actually infer from the factorization of 168=7*24=7*12*2=7*3*4*2 many important structural information.

Based on the fact that every heptagon in D168 is connected to seven other heptagons by a 6-6 bonds, and the local structure is a tetrepod, so these two numbers correspond to 7*4 in the factorization. The remaining two number are 3 and 2. Inspecting this structure, we know 3 corresponds to the three heptagons surrounding the neck and 2 is the two tetrapods in a unit cell. Each tetrapod has 84=7*12 carbon atoms.

Beaded model for D168!

Finally, I have made a beaded model for the famous D168 fullerene, which is essentially corresponding to the C60 in the hyperbolic space.

D168

The 3D structure of D168 we posted a few days ago is incorrect. Chuang has now the correct structure shown below:











I thought it is not a bad idea to have a beaded model for this structure. Now I am still at a very preliminary stage (see the picture shown below).


Eventually, I expect to have a D168 structure similar to Adamantane:

Graphitic D Surface C168

Chuang created this surface: