Two pictures from Gabala, Azerbaijan

I took a few pictures of bead models I brought to Azerbaijan.
The first one is a model of EMACs (extended metal atom chain) with 9 metal ions. This model was made by Qian-Rui Huang many years ago.

The second one is a model of zeolite which I gave to Prof. Yin-Xia Wang of Beijing Univeristy as a gift. She is an exert on zeolites.

Spinel structure

Sierpinski tetrahedron

Bead model of Kaleidocycle (萬花環)

Kaleidocycle or a ring of rotating tetrahedra was invented by originally by R. M. Stalker 1933. The simplest kaleidocycle is a ring of an even number of tetrahedra. The interesting thing about the Kaleidocycle is that you can twist it inwards or outwards continually. The geometry of kaleidocycle has been studied by many people from different fields in the last 80 years:

1. Stalker, R. M. 1933 Advertising medium or toy. US Patent 1,997,022, filed 27 April 1933 and issued 9 April 1935.
2. Ball, W. W. Rouse 1939 Mathematical recreations and essays, 11th edn. London: Macmillan. Revised and extended by Coxeter, H. S. M.
3. Cundy, H. M.; Rollett, A. R. 1981 Mathematical models, 3rd edn. Diss: Tarquin Publications.
4. Fowler, P. W.; Guest, S. Proc. R. Soc. A 461(2058), 1829-1846, 2005.
5. 全仁重, Motivation Behind the Construction of Maximal Twistable Tetrahedral Torus.
6. HORFIBE Kazunori, Kaleidocycle animation.

Typically, people use paper or other solid materials to make this kind of toy. A few months ago, I discovered that you can easily make this toy by tubular beads through the standard figure eight stitch (right angle weave).

This particular model consists of 8 regular tetrahedra. You can easily extend rings that contain 10, 12, ... tetrahedra.

The procedure I used to make this 8-tetrahedra Kaleidocycle is by the standard figure-eight stitch (right angle weave) in which one just keep making triangles. Of course, some care should be paid on the sequence of these triangle.

ε Keggin structure

δ Keggin structure

γ Keggin structure

β Keggin structure

α Keggin structure

From pentaexcavated Icosidodecahedron to Mackay polyhedra

Perovskite, 4 stella octangula

Quasicrystal (準晶)

More bead/rubber band models

I found that it is much easier to use commercial beads to make the so called styrofoam/rubber band models of L. C. King. According to Prof. H. A. Bent, every student of chemistry should make a set of this kind of models for learning the concept of chemical bond. Here, I will show how to make valence sphere models using 25mm wooden beads and rubber band with 8mm small plastic beads as endcaps.

1. To make the valence sphere model of a methane which has the tetrahedral shape, we can connect two wooden beads (two-bead unit) first as shown in the following picture:

Then cross two two-bead units with each other to get a tetrahedral methane:

2. We can easily generalize the procedure to molecules with six valencies. First make another two-bead units with rubber bands, then cross it around the tetrahedral four-bead unit we just made. Then one should get an octahedral arrangement of six beads as shown in the following picture:

3. One can also try to make a five-bead unit which represents the valence sphere model of a molecule with dsp³ hybridization.

4. Valence sphere models for molecules with more than one center such as the ethane can easily be made too. Here is my procedure for making the valence sphere model of an ethane molecule: first I connect a three-bead unit and two two-bead units as shown in the following picture.

Then, connect these three units at suitable positions, one should get the valence sphere model for the ethane.

Four face-sharing pentagonal dodecahedra

E. A. Lord, A. Mackay, and S. Ranganathan described in their book, "New geometries for new materials", a simple cluster consisting of four face-sharing pentagonal dodecahedra arranged in a tetrahedral configuration (pp.48). Here is a bead model of this cluster.

In their book, there are more clathrate structures that one might be able to construct with beads.

Clathrate cluster

I bought some more green rice-shape beads last week and managed to finish this interesting clathrate cluster of 60 dodecahedra in the last weekend. One can still see deformation of many dodecahedra in this clathrate cluster though.

VSEPR and higher-valent systems

Based on our experience, beads are best for making trivalent systems, which should be locally flat according to the VSEPR (Valence Shell Electron Pair Repulsion) commonly used by chemists to predict 3-D structures of molecules. Molecules with unconstrained tetravalent center such as methane should be locally tetrahedral. But it is difficult to weave this kind of structures. We have only tried a few of this kind of systems.

Another situation we may like to do is constraining a tetravalent or even a pentavalent (hexavalent) bonds in a plane such as boron-boron bonds in the boron fullerenes as shown in the previous post. According to our experience, it is quite hard to keep or constrain all these bonds in a plane due to the tendency to reduce the repulsion among these bonds as required by the VSEPR.

Boron buckyball: B80

I noticed a few research articles and a news report about the Boron buckyball in Science News today. This reminds me the article (The Wonderful World of Beaded Molecules, in chinese) I wrote two years ago about the potential molecules that may have the same structures as the dual structures Chuang constructed. In that article, I suggested the possible candidates should be boron clusters. Well I was not exactly right, but I was not that far away either. In the reports I read today, the smallest boron cage seems to be B₈₀, not the structure I posted before.




The structure of B₈₀ is very similar to that of C₆₀. The main difference is that all hexagons in the C₆₀ are replaced by six equilateral triangles.


(Phys. Rev. B 80, 033410, 2009)

Here is the bead model I made for this molecule. It is quite difficult to make penta- or hexa-valent local structures with rod-like beads, though. This model is not stable.
Maybe, I should try to make another one with rice-shaped beads, But I am not sure that will help.

Sierpinski Tetrahedron (Level 3)

Space-Filling Polyhedra Based on a Truncated Octahedron

I made this structure during the spring break. Chuang has made a similar structure with several truncated octahedron before. But his structure has three unit cells separated connected. So the first unit cell is not connected to the third unit cell directly. My original goal is to make a structure with six unit cells connected to each other, something like a 2x2x2 cluster. However, the structure seems to be sensitive the small deviation at each local connection. Although truncated octahedron can fill the space. the structure I made seems to have pretty large strain and distorion.

Octahedron with six nodes on each face