Zeolite A (Cantitruncated cubic honeycomb)

For more description of this model, see the page from the Bridge conference 2016. (Nov. 11, 2016)

Bead models of four Archimedean solids

Truncated octahedron, truncated icosahedron (C₆₀), truncated cuboctahedron (great rhombidodecahedron), truncated icosidodecahedron (great rhombicosidodecahedron):

Da Vinci's Three Elevated Archimedean Solids - Cuboctahedron, small rhombicuboctahedron, and icosidodecahedron

Truss models of cubic and hexagonal closest packings

I made two more bead models of two-layer cuboctahedron and twisted cuboctahedron (anticuboctahedron) to illustrate the vector equilibrium of Buckminster Fuller. Through these two models, the connection to the cubic and hexagonal closest packings can also be visualized more easily.

Cuboctahedron

Sangaku problem and C20

In the book "算法助述" I got from Mr. Horibe, there is an interesting problem in pp. 51-52 as shown in the following picture. As I mentioned before, Mr. Horibe made two beautiful bead models with a large metal sphere inside. And more importantly, large and small balls have the correct ratio of radii, R/r=sqrt{5}. He told me that he gave one of these two models to Hidetoshi Fukagawa, the author of the book "Sacred Mathematica - Japanese Temple Geometry", because, through Dr. Fukagawa, he knew so much about these interesting Japanese sangaku problems.

This Sangaku problem was originally proposed by Ishikawa Nagamasa and written on a tablet hung in 1798 in Tokyo’s Gyuto Tennosha shrine. I was quite surprised by that I could understand this ancient problem without any translation. It was written in Chinese (or Kanji):

"今有以小球三十個如圖圍大球，小球者各切臨球四個與大球，小球徑三百零五寸，問大球徑幾何？

答曰：大球徑六百八十二寸。"

In English:

Thirty small balls cover one big ball where each small ball touches four other small balls and big balls.
The radius of small ball is 305 inches. Find the radius of large ball.

Answer: The radius of large ball is 682 inches.

In the page 52 of the book, there are some explanations on how to reach the answer by noting that thirty balls are located on the thirty vertices of an icosidodecahedron. It was written with some Japanese characters which I don't understand. But it is not hard to guess the rationale. One can see that if one starts from an arbitrarily chosen small ball, then moves to next ball along a fixed direction, and then the next one along the same direction. Eventually, one would trace over exactly 10 balls which are located on the large circle of a sphere with the radius equal to the sum of radii of small and large balls. Using some trigonometry, one can get the answer.

Truncated octahedron

C60 and extended C168 are unique because neighbored nonhexagons in them are separated by exactly one carbon-carbon bond. Is there any other graphitic structure with the similar property? The answer is yes. Chern and I have written an article on the carbon nanotori and nanohelices with this property a few years ago.

Chuang, C; Jin, B.-Y.* “Hypothetical Toroidal, Cylindrical, Helical Analogs of C60.” J. Mol. Graph. Model. 2009, 28, 220-225.

Of course, it is easy to see that there are another four Archimedean solids with this property if we allow nonhexagons to be squares or triangles. They are truncated octahedron (see the following photo), truncated cube, truncated tetrahedron, and truncated dodecahedron. Note that C60 is the truncated icosahedron. So all five truncated Platonic solids belong to this class.

Two more resonance structures of C60

As I mentioned before, two chemists, Vukicevic and Randic, gave a complete enumeration of all possible resonance forms in their paper, Detailed Atlas of Kekulé Structures of the Buckminsterfullerene, in "The Mathematics and Topology of Fullerenes". According to them, there are 158 irreducible Kekule structures for C₆₀. Following the Schlegel diagrams listed in the paper, one can easily make a bead model for any resonance structure.

I found these two intriguing resonance forms, No. 108 and 111, of C₆₀ quite accidentally yesterday. Although I knew the existence of No. 108 for a long time, I didn't know No. 111 before. I also suspect they might be the only two resonance structures which have patterns of parallel stripes along the latitude coordinates. Particularly, the Kekule structure 108 has a 5-fold rotational symmetry axis with two pentagons located at two poles and the Kekule structure 111 has a 3-fold rotational symmetry instead.

Three Kekule structures of C60

D. Vukicevic and M. Randic have figured out all possible distinct resonance (or Kekule) structures a few years ago. According them, buckminsterfullerene has 12500 Kekule structures grouped in 158 isomorphic classes. They also give a complete list of all these 158 non-isomorphic Kekuke structures in a recent paper entitled "Detailed Atlas of Kekulé Structures of the Buckminsterfullerene", in the book, "The Mathematics and Topology of Fullerenes".

This is very convenient if we want to make any particular resonance form of C₆₀. We can simply look at the Schlegel diagrams given in this paper, and pay attention to the single and double bond pattern as we bead. Here are three bead models for Kekule structures No. 134, 135 and 136 as shown in their paper.

A nice glass cuboctahedron

Alberto has this beautiful gass box with cuboctahedral shape in his living room.

Yet another bead model of rhombicuboctahedron

Another rhombicuboctahedron using beads with long-aspect ratio:

Bead model of an Icosidodecahedron

Bead model of an Icosidodecahedron:

More 3mm beaded C60

From beaded fullerene

Chuang made them.

Workshop at Taichung

I gave a one-day workshop on the beaded molecules for some junior high school students this summer (7/2-7/3/2009) at Taichung, Taiwan. Here is a picture about the workshop.



In the morning, after a brief introduction on the chemical bonding and a few simple molecular structures, I then started to teach student the platonic solids and basic weaving techniques by asking them to make a C20 (a dodecahedron). I simply asked students follow my instructions step by step without too much explanation. This is because, I think, it is important to have some hands on experience for constructing beaded fullerenes first. Some students can get the basic rules of weaving just after a few steps, others may take a longer time and kept asking me how to do the next step. But it took about an hour for all students to get the first project done. I also found that it is better to use larger beads around 10mm to 12 mm for students who have no experience in beading. To create a C20, one need 30 beads. So it is not too expensive even for a group of 50 students.

The next project in the afternoon was to construct a C60. Of course, before we started to do that, I explain the icosahedron, truncated icosahedron and a few background information on fullerenes to them. Unlike C20, where all atoms and bonds are equivalent, here we have two different bond types, 5-6 and 6-6 bonds, so it is natural to use two different color of beads for the construction. In fact, this is not a burden for weaving. Instead, color of beads can be used as a mnemonic aid for denoting the place of pentagons. One needs 90 beads to represent chemical bonds in a C60. Since students have some experience in the morning for making a C20, I find it convenient and more cost effective to use 6mm beads with two different colors for students to work on in this project. It took about 2 hours for most of students to construct his or her C60.


The picture shown below is the C60 I made in this workshop. At the end, I gave it to one of students in this workshop as a souvenir.

青城山上的 cuboctahedron

I found this cuboctahedron in a walking trail of the QingCheng mountain:

Here is the beaded cuboctahedron I made a long time ago. 青城山上清宮老蔣題的匾額： From 青城山

From 青城山

Yet another beaded buckyball (YABB)

From Dec 25, 2008

Chuang made his model with 20mm (2cm) beads. These are largest beads we can find in the local store at Taipei.

C60 and C80

truncated octahedron

C60 with cylindrical beads

Here is the C60 made from the cylindrical beads. The advantage of using this kind of beads is that the beaded representation is indeed the pi-bond network as we have emphasized repeatedly before. Pedagogically, this is very important for students to understand. Compared with spherical beads, this kind of beads may be better for beginning students since it is closer to chemical bonds introduced in the textbook.

The capsule-shaped beads are not perfect either. The most important disadvantage is the structural stability of physical models created from this kind of beads is not as stable as the same structure made from spherical beads. It is quite easy to distort the shape of this kind of model. The other advantage is that much longer fishing thread is needed to make a model. This makes the construction a little bit harder, particularly for larger systems.

rhombicuboctahedron