Tuesday, November 30, 2010

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/

Friday, November 26, 2010

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.

Wednesday, November 24, 2010

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.

Thursday, November 18, 2010

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.

Wednesday, November 3, 2010

The Cubic Kaleidoscope I Made Today

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


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).

Tuesday, November 2, 2010

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.