Two more toroidal carbon nanotubes

I made two more bead models for toroidal carbon nanotubes with 120 and 240 carbon atoms last weekend (Christmas holliday in many countries, but not here in Taiwan, :-)).

Collection of a few Goldberg polyhedra

C720 - Goldberg vector (6,0)

Fused C60 dimer - C120

Chern Chuang made a fused C60 dimer a few years ago. I scaned the model by the scanner as follows,

The neck in this structure is basically the same as the inner part of T120. The two C60s in this model is staggered, so it has D_5d symmetry just like that of ferrocene.

I made another one with two different colors last weekend. I managed to use a single color for both pentagons and heptagons. Comparing these two photos, it looks to me that the structure of this model looks a little bit longer than the one made by Chern. I don't have the original model made by Chern with me now. But I believe this is caused by these two different ways I took the photos.

Goldberg Polyhedron (5,0)

Here is the bead model for the Goldberg polyhedron (5,0) The number of vertices in this model is 20(5²) =500. The number of beads (6mm, faceted) used is 500*3/2=750.

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.

Poster for the annual meeting of Taiwan's chemical society

Here is a poster I made for the annual meeting of Chinese chemical society located at Taiwan one week ago. This poster summarizes most of what we have done in the last few years.


High-resolution poster is here.

Mr. Kazunori Horibe's wonderful site

I was informed today by Prof. Sonoda about a Japanese site which contains many interesting beaded fullerenes just like we did in this blog. This site (http://www.u-gakugei.ac.jp/~onodakk/math/beadsindex.htm) is created by a Japanese highschool mathematic teacher, Mr. Kazunori Horibe (堀部和経). In addition to the standard cage-like fullerenes (C60 and C80), kind of toroidal carbon nanotubes similar to our T120, Mr. Kazunori Horibe has also created a number of fullerenes consisted of fused C60.



(Carbon poti donut （波提甜甜圈）. I thought about making this kind of structure before, but didn't really try to do it.)


Incidentally, Chern has created a fused C60 dimer a few years ago, see here. But we didn't pursue further along this direction.

In this site http://horibe.jp/Gr2F.HTM, I also found this picture dated 1/1/2001:

(The giant structure in this picture is essentially a few unit cells of C168 with endcaps, I wish I can play with this kind of giant beads someday.)

We can also see here many more extended structures exactly the same or similar to we have created in the last few years.

Apparently, Mr. Kazunori Horibe has played systematically with mathematical beading for fullerene structures much earlier than we did. Please check his site to see the beautiful fullerene structures he has created. Unfortunately, this site is in Japanese, this may be why I didn't find it through google in the last few years.

There seem to be a commercial site by Mr. Kazunori Horibe too:
http://www.ballstructure.com/ballstructure/.
and detailed instruction on the construction of beaded C60:
http://www.ballstructure.com/ballstructure/KA-AmakingB.htm.

C540

I made another giant fullerene with Goldberg vector (3,3) last week.


Goldberg vector (3,3) gives the position where to put the next pentagon. With this in mind, one can create easily any fullerene specified by the Goldberg vector (i,j).

Bead model of the heptagonal tiling

A tiling is called Platonic if it uses only one type of regular polygons. There are only three types of Platonic tilings, triangular, square, and hexagonal tilings (see the following figure). It is not hard to make a bead model of these three kinds of Platonic solid.



Although it is not possible to tile a plane with regular heptagons, one can do it on a hyperbolic space as shown by follow figure using the Poincare disk.




I made a bead model for this tiling this summer. It is quite staightforward to make this model. One just weave heptagon by heptagon repeately along the spiral direction. After four or five layers, the whole structure just becomes so crowdy. It then becomes so tedious to have one more layer of beads.


More beautiful pictures on Hyperbolic tiling can be found here. It would be interesting to find out how many of these hyperbolic tilings are beadible by the spherical beads.

Some hyperbolic tiling from http://www.plunk.org/~hatch/HyperbolicTesselations/

Fullerenes belonging to icosahedral group

It is straightforward to make bead models for higher fullerenes with icosahedral symmetry. The simplest way is to use the Goldberg vector (see the following figure) to specify the relative position between two pentagons. Goldberg vector is very similar to the chiral vector used for defining carbon nanotubes.

Suppose we have the first pentagon located at orgin, (0,0), then we can ask where the next pentagon can we put? The answer is that any coordinate specified by (i,j) as shown in the following figure gives a unique fullerene with icosahedral symmetry. For instance, if the next pentagon is located at (i,j)=(1,1), we have a C60. It is not hard to show that the number of carbon atom for the fullerene specified by the Goldberg vector, (i,j), is N=20(i² + ij + j²).


(Figure 8 in "Jin, B.-Y.*; Chuang, C.; Tsoo, C.-C. “The Wonderful World of Beaded Molecules. 串珠分子模型的美妙世界” CHEMISTRY (The Chinese Chemical Society, Taipei) 2008, 66, 73-92, in chinese.")

A bead model of C60, Goldberg vector (1,1).



Icosahedral fullerene specified by the Goldberg vector (2,1) has 140 carbon atoms. This is the smallest chiral fullerene with icosahedral symmetry.




The following beaded fullerene is specified by (4,0) has 320 carbon atoms.

Wooden bead C60

This is a C60 constructed by eaglewood buhdda beads (烏木佛珠). I bought these beads in ChengDu, Sichuang last year.

A nice mnemonic for making beaded C60s

Prof. JT. Chen forwarded me a message from Sharon, an audience of my talk early this month. Sharon has a simple mnemonic by her son for making the beaded C60. In C60, every pentagon is surrounded by 5 hexagons, and every hexagon is surrounded alternatively by 3 pentagons and 3 hexagons. (五邊形的周圍是六邊形,六邊形的周圍是一個五邊形接一個六邊形.) One can easily create a beaded C60 by following this simple rule.

Two beaded models made by Sharon:




Indeed, one does not need spiral code to make C60. But to make an arbitrary cage-like fullerene (genus=0), spiral code is the only information we need. The shape of resulting beaded structure is always similar to the shape of the corresponding microscopic fullerene. It is quite amazing that one can create the faithful structure for an arbitrary fullerene with beads so easily. A simple explanation is that hard sphere repulsion among beads effectively mimic the valence-shell electron-pair repulsion of trivalent carbon atoms in fullerene molecules.

Additionally, if one want to make a beaded C60 with two different colors, a single color for pentagons and two different colors alternatively for hexagons. Then one doesn't need to use the mnemonic as given above. One can just pay attention to the colors only. Starting with a pentagon with a single color, then hexagons with two colors alternatively, eventually, one should get a beaded C60 correctly.

A few beaded C60s (10mm faceted beads) I made in last week:


See also a discussion in the previous post.

Pictures for the talk given at the case "台大探索講座"

I gave a talk for the special lecture series on the chemistry, Chemists' Adventure in Molecular Wonderland, given by the case of national Taiwan univeristy(國立台灣大學科學教育發展中心| Center For the Advancement of Education) on Nov. 6th. You can find the online lectures here (in Chinese).
Here are some photos I recieved today.

The Cubic Kaleidoscope I Made Today

Looking along the (1,1,1) direction:






Outside:


I gave it the name of "PlatoKaleido". Because this is in fact the three Platonic tilings, square planar(purple+red), equilateral triangular(orange+yellow) and honeycomb(green+blue) lattices, inter-penetrating one another orthogonally. The coloring is in accordance to the order of the spectrum of sunlight, if you notice. It is a great fun making this kind of kaleidoscopes, thanks to Prof. Takaaki from Kyushu University who kindly taught us earlier today.

My supervisor Bih-Yaw mentioned about the possibility of making this kind of kaleidoscope with other geometric shapes like triangular or pentagonal prisms. And Prof. Takaaki replied that they'd been trying everything possible already. However, I am thinking about using non-planar mirrors instead, e.g. concave or convex, making the "metric" of the wondering world therein non-Euclidean, maybe an interesting task. This is also related to some photos taken by Bih-Yaw at this year's Bridge conference. The artist made clever use of the curvature of the mirror, so the image of an seemingly unreasonable object on the mirror becomes a normal one (of course in this case the images are in fact the unreasonable structures that the artist tried to convey).

Photos from ICCE

I was just informed that photos for the workshop (at the ICCE) on the beaded molecules has been posted on picasaweb today.