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.

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.

Building blocks for the type-II high-genus fullerenes

The building block of type-II high-genus fullerenes can be chosen to be an arbitrary Goldberg polyhedron. Puncturing three holes along three carefully chosen pentagons can create a basic unit with three coordination (or a trivalent unit).
I use C60 and its Schlegel diagram to illustrate how to puncture a hole on an arbitrary pentagon.

1. Schlegel diagram of C60

2. C60 with a hole punctured on a pentagon: one pentagon and five hexagons are replaced by five heptagons.

In principle, one can connect two this kind of unit with one hole to create a fused C120 with dumbbel-shape.

3. Of course, if we like, we can puncture two holes on a C60. There are three possible ways. Here I only show the situation with two pentagons separated by two hexagons. The resulting structure will contain two holes connected (or separated) by two heptagons.

There are two other different ways to puncture second hole. If the second pentagon separated from the first one by one CC bond are punctured, the resulting structure will have an octagon. The third situation is that the second pentagon is located at the antipodal position. I will talk about these situations later.

4. Punctured C60 with three holes:

It is easy to see that there are five heptagons and five more bonds are introduced around each hole. So one needs 105 beads for creating a single unit.

5. Here are two possible weaving path. I usually used the first path though. a. non-spiral path

b. spiral path

6. I am working on a project with teachers and students of the Taipei First Girls High School (北一女). We are going to make a giant buckyball consisting of sixty units of punctured C60s. Here are a few basic units I made:

105 12mm faceted beads are used for each unit.

C70 beading procedure

C70. Point group D5h.
Shape: like a Rugby ball.
60 8mm red beads and 45 8mm white beads are used.
Ring spiral: [1 7 9 11 13 15 27 29 31 33 35 37]








The red and green parts of spiral are exactly the same as the spiral of C60. The orange part of the spiral is a ring of 10 hexagons inserted in between two hemispheres of C60! Of course, instead of inserting one ring of 10 hexagons, one can do it repeatedly to get different length of endcapped carbon nanotubes! So C70 is the shortest endcapped carbon nanotube.

Note also that the blue circles in this graph are not beads! Beads represent edges or chemical bonds of fullerenes.

I made a few bead models for Prof. Gillespie, who proposed the the famous VSEPR method, last week. This C70 model is one of them.

C60 beading procedure

I took a few photos for making bead model of C60. This will be helpful for learning the so-called figure-eight stitch (or right-angle weave) and the beading rule of C60 I mentioned before.



South hemisphere:


North hemisphere:




* General instruction:
1. There are 32 polygons consisting of 12 pentagons and 20 hexagons in a C60.
2. Every pentagon is separated from neighbored pentagons by eactly one CC bond.
3. If we choose one color of beads for pentagons and the other for the rest, one would find hexagons made of two colors alternatively.
4. It is better to view C60 as a sphere consisting of six layers of polygonal strips. For the south semisphere, they are basically a pentagon for the south pole, five hexagons next, 10 polygons consisting of 5 pentagons and 5 hexagons. Reverse the beading sequence, one gets the north semisphere. (This is what people called the spiral code.)
5. One has to check how many beads of group one wants to create in the next step. Some beads are already done, so one has to thread the fishing cord through these beads first, then add the remaining beads through the other end of fishing cord, and finally, form the n-bead group by threading the fishing cord through the last bead just added along the opposite direction.
Of course, one should always check the sequence of colors of beads one is going to bead and make sure that they satisfy the color coding mentioned in 3.
For beginners, this is usually the hardest part and mistakes occur easily. Most often, wrong number of beads are added or some beads are not threading through first. But, if one can pay attention to the number of beads in the next group one is going to make, it should be trivial to figure out how many beads are already there and how many more one should add.
If two colors of beads are used, one can simply pay attention to color. Beading process for a C60 becomes trivial.


The beading procedure can be summarized by the spiral code on the following Schlegel diagram of C60. For C60, it is [1 7 9 11 13 15 18 20 22 24 26 32], which is essentially the positions of pentagons along the spiral path starting from south pole to the north pole. It is not hard to find this code is the only information we need to create a bead model of C60. Similarly, one can create any other cage-like fullerene by its spiral code only if the fullerene possesses it.

A list of fullerenes up to 100 carbon atoms is given in the appendix of the book "An Atlas of Fullerenes" by P. W. Fowler and D. E. Manolopoulos. So one can create any fullerene easily by following its spiral code. The shape of the resulting bead model is basically consistent with the corresponding fullerene.

Slides for making a beaded C60

I would like to thank Rochelle for pointing out that the procedure in the tabular form I posted before is incorrect. I have to admit that I have never used this kind of table for making C60. I think it is easy to make mistakes by just following this table literally and without thinking. If you are a little bit careful, you should be able to see the hidden rules for making the C60 just after about 10 steps.

Below is a few slides I used to teach people how to make a buckyball.


But, it is much easier to follow the simple mnemonic for making a buckyball.


If one wants to make a beaded C₆₀ with two different colors, a single color for pentagons and two different colors alternatively for hexagons, one can use these two colors as a mnemonic for deciding whether one need to make a pentagon or hexagon in the next step. Remember that in a C₆₀ every pentagon is surrounded by 5 hexagons and every hexagon is surrounded by 3 pentagons and 3 hexagons alternatively. Then one can start with a pentagon with a single color, then hexagons with two colors alternatively, eventually, one should get a beaded C₆₀ correctly without using any other information.

C80

The spiral code of C80:

C60 Isomer with D3 symmetry

Michel Deza, Ecole Normale Superieure, Paris; Mathieu Dutour, Rudjer Boskovic Institute, Zagreb

Fullerene 528 (Td)

Spiral code: 123579 10 11 12 13 14 15