Bead models of dodecahedron and icosahedron

Mr. Horibe started to play with polyhedral models very early. According to him, he has constructed Platonic models with a long cylindrical tube capped with two beads when he was still an undergraduate student in the 70s.

He demonstrated in front of us that dodecahedron and icosahedron are dual to each other using this kind of models. With long cylindrical tubes, the whole structure becomes very flexible, so you can compress it and put one inside the other. Students can realize, through this amazing way, why dodecahedron and icosahedron are dual to each other. I can only say that Mr. Horibe must be a very effective math teacher. I wish I could learn math from him when I was a high school student.



In some sense, this kind of bead models is very similar to the so-called tensegrity structures. BTW, Mr. Horibe usually uses a single elastic string for making his bead models, which makes his models even more flexible and bendable. I will comment more on this aspect later.

Cube

From Dec 25, 2008

Dodecahedron

truncated octahedron

Tetrahedron

I try to experiment with the idea that using three beads to stand for an edge of polyhedron. In this case, I made a tetrahedron. Previously, I have posted some of this kind of models. The original idea came from Chuang again.

Caged Ping-Pong Ball

Sp3 carbons (one more example)

Here is one more beaded structure defined by the same net similar to that in the previous post except that we use compounded bonds (three beads for a single bond).

rhombicuboctahedron

great rhombicuboctahedron

cuboctahedron

octahedron

truncated octahedron

Cube

truncated tetrahedron

Tetrahedron

C60 with each bond built from two spherical beads and one tube

In the following figure we show the beaded model for C60 based on the composite bead technique. The final shape is not quite satisfying as we can see it is a little bit distorted. This is probably due to that the sizes and shapes of the constituent beads and tubes are not exactly the same. I believe a better structure can be constructed if we can control the sizes and shapes in a better way.

Dodecahedral C20 with correct bond shape and force field

Previously, I have shown that beads with spherical shape can effectively mimic sp2 force field of fullerenes. However, these beads represent bonds instead of atoms. This may lead to confusion for students. We can avoid this problem by using beads with large aspect ratio, but the resulting structure usually has poor mechanical stability. Here Chuang has created a nice beaded fullerene of C20 (Ih) which can clearly exhibit the bond network of fullerene and at the same time possess great stability.