C84 with three and four heptagonal holes

As I said in the previous post, one can puncture holes on a C₈₄ and use the resulting structures as basic units for building larger graphitic structures. Here I show two C₈₄-derived bead models with three and four holes surrounded by heptagonal.

The Schlegel diagram and a simple beading path for three-hole structure can be worked out easily.

Similarly, here is the Schlegel diagram and a beading path for a four-hole (tetravalent) unit.

Of course, one can use these building blocks to make many interesting structures. I am planning to make a dodecahedron consisting of 20 tetravalent units. It is not hard to see the angle between two holes in a unit is about 109 degree which is very close to 108 degree for the inner angle of a pentagon. So one can expect that these units should be happily fitted in the resulting dodecahedron without too much distortion.

C84 - a tetrahedral fullerene

I posted a few bead models of C₈₄ before, but I had never given the detailed beading procedure for this molecule. C₈₄ is the smallest achiral fullerene with tetrahedral shape that satisfies the independent pentagon rule (i.e. no two pentagons are connected).

Here is another bead model of C84 consisting of 8mm beads I made yesterday.

The easiest way to make it is to follow the beading path as shown in the following Schlegel diagram. Note that this path does not correspond to the path give by a spiral code.

In principle, one can puncture holes on this molecule and use the resulting structures as building blocks to create more complicated structures or super fullerenes (fused C84 in this case). I will show how this can be done later.

Graphitic structures consisting only of pentagons and heptagons

I made two fullerene cages consisting only of pentagons and heptagons today.

1. The first one contains 28 pentagons and 16 heptagons.

2. The second structure is created by coupling two previous structures.

Gyroidal surface on top of a Macbook air

There are 8 helical strips right now. I still need to make the other half.

Workshop next month and one more gyroidal graphitic structure

I sent the first bead model of gyroidal graphitic structure (GGS) for the math art exhibition of Joint Mathematical Meeting last month. The model is still in Chern's place in Cambridge, MA.

But, I am going to have a three-hour workshop, Molecular Modeling of Fullerenes with Mathematical Beading: a Hands-on Approach, next month in my department for 40 students from Okayama University (Japan) and National Taiwan Universities.

I plan to show students how to construct molecular models of fullerenes with beads and how to correctly interpret microscopic meaning of these bead models. Particularly, students will learn simple beading techniques and create a dodecahedron and a truncated icosahedron, which are C20 and C60 respectively, in this workshop. Additionally, I also wish to have an exhibition of bead models we made in the last few years. Since the bead model of GGS is not with me now, so I decided to make one more bead model of gyroidal graphitic structure.

The bead model is now almost half-done. I wish I can finish the whole model at the end of this month. The model shown in the following photos contains six helical strips. It is quite subtle to make this structure even though I have made one GGS already. At the beginning, I thought I was familiar with the structural rules of the GGS, but I still incorrectly made a D-type TPMS, instead of G-type TPMS, the first time.

Four more valence bond structures of C70

There could be thousands of VB (valence bond) structures or resonance forms of C70. Here are four possible VB structures.

One more super carbon tetrahedron

I made another super carbon tetrahedron (超級碳正四面體) consisting of four fused C60s with 8mm beads. Unlike the previous one, the two neighbored C60s are connected by a ring of hexagons.

Cube kaleidoscope (Wonder Mirror Box, or CUMOS cubic cosmos scope)

N asked me about the cube kaleidoscope Chern posted before. If you are interested in getting instructions and materials for making your own cube kaleidoscope, you should contact Prof. Takaaki Sonoda (園田 高明) of Kyushu University (九州大學). (reports about him in Czechish??? and in Japanese).

Here I just want to show the other two cube kaleidoscopes, kaleidostereochemistry and kaleidogarden, I made when Prof. Sonoda visited Taiwan about one year ago.

It is really fun to make your own kaleidoscope. Basically, one needs to have six mirrors to make a cube kaleidoscope. Once you have the necessary materials, you need to make patterns on two or three mirrors by scratching part of mirror away by using suitable stick. For details, please contact Prof. Sonoda.

In addition to the two kaleidoscopes I made by myself, I also got another professionally made cube kaleidoscope from Prof. Sonoda as a gift.

You can also find extra information at this site (CUMOS cubic cosmos scope). Here is one I found at that site:

More images is in this gallery.