Tetrahedral C28 and related structures

There are only three tetrahedral fullerenes with number of carbon atoms less than that of buckyball. They are C₂₈, C₄₀, and C₄₄, respectively. The spiral code for the smallest tetrahedral fullerene, C₂₈, is [1 2 3 5 7 9 10 11 12 13 14 15]. Following this code, we can easily make its bead model using the standard figure-eight stitch. We can see that, in this molecule, there are 12 pentagons, 3 in a group located at a vertex, and 4 hexagons located on the four faces of the tetrahedron. If we replace these pentagons by heptagons, we get a tetrapod-like structure, in which tri-pentagon vertices become tri-heptagon necks as shown in the following figure.

Using these tetrapods as building blocks, we can get the following diamond-like structure. In fact, this is exactly the structure Mr. Horibe put in the postcard. OK, if we start from other tetrahedral fullerenes such as C₄₀ and C₄₄, we can find out a lot more diamond-like structures.

Two artworks for the Mathematical Exhibition of Joint Mathematical Meeting

Chern and I submitted two artworks to the Joint Mathematical Beading, which were accepted today: 1. Super Buckyball of Genus 31
2. Beaded Hilbert Curve, step two

In addition to the beadworks we submitted, we also noticed five Platonic bead models made by Ron Asherov. His bead models have multiple beads in an edge, which are similar to our works a few years ago. I labeled this type of bead models with Edge with Multiple Beads, where you can find all the posts. He doesn't seem to know our works along this direction, though.
He also mentioned that the Nylon string passes adjacent edges exactly once with carefully chosen path of string, which is simply the consequence of Hamiltonian path on the dual graph of the corresponding Platonic solids. We can view the whole beading process as a path through the face of polyhedron. Thus if there exists a Hamiltonian path through each face once (Hamiltonian path for the dual polyhedron), then the Nylon string will go through each beads exactly twice and only twice. Of course, you can also say the Nylon string will go through the adjacent edges exactly once. They are the same thing.

Here is a model made by Chern Chuang almost five years ago:

I was quite surprised by its rigidity when Chern showed me this model. At that time, people questioned me about the meaning of beads. I told some of my colleagues that spherical beads represent chemical bonds instead of atoms. Atoms are not shown in the bead model explicitly, instead, they are located at somewhere three beads meet. Most chemists feel uncomfortable with this connection. So Chern and I tried try to explore with the shape of beads and multiple beads and hope that they can better represent the shape of chemical bonds. So that is why we have these models in which multiple beads represent an edge.

But now, I have the valence sphere model of chemical bond as the theoretical foundation of bead models. Spherical beads are in fact the simplest possible approximation of electron pairs, in accord with the principle of Occam's razor. So to build a model of a molecule with only beads and strings is equivalent to performing a molecular analogue computation with beads. The result of computation is the approximate electron density of the corresponding molecule without referring to the Schrödinger equation or atomic orbitals (this is the comment I got from Prof. H. Bent). I have written an article on the connection between bead models and valence sphere model in Chinese for September issue of Science Monthly (科學月刊). I think I should write something about the molecular analogue computation with beads.

Of course, what I am saying above is to view bead models as molecular models. Results of mathematical beading do not need to have any connection to the molecular world. For instance, the beaded Hilbert curve accepted by the JMM mathematical art exhibition is a good example.

Some more amazing beadworks of Horibe with comments on colors in bead models

Most of Horibe’s beadworks are made by single color. Unlike beadworks made by my group, I usually decorated nonhexagons with different colors or use colors to represent single and double bonds. Of course, this will spoil the color of hexagons and nonhexagons. It is not possible to use colors independently for hexagons and nonhexagons. There must be some polygons, which will be decorated with more than two colors when one chooses to use more than one color. According to him, that is the reason he usually used single color to do beading to avoid this situation. Only in a few examples he used two colors to illustrate some particular features.



But, to chemists, using colored beads seems to be an obvious choice. Chemists have been using many kinds of molecular models to represent microscopic structures of molecules. It is quite common to use different colors to denote different kind of atoms. Now, in bead models, we move the emphasis from atoms to bonds. It is still apparent that one could also employ colors to mean different things. Faces in a molecular network do not have important chemical meanings as those of vertices (atoms) and edges (chemical bond). But more importantly, we would view the color pattern as a new dimension of beaded molecules, which can not only give geometric or chemical meaning, but also make them more visually pleasing.

Construction procedure of C60 (Japanese translation)

Prof. Sonoda made a nice Japanese translation of the detailed construction described in the supporting information of the article, Molecular Modeling of Fullerenes with Beads, J. Chem. Edu 2012 , 89 (3), 414–416.", by Prof. Cuccia and me.



Also, here is the original English version of the first page, the next two pages are the same.

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.

Making Beaded Buckyball Model in Chinese

If you can read Chinese, you might want to look at the following detailed tutorials by a teacher, 薛朋雨, at the Taichung First High School (國立台中第一高級中學):

化學教室活動：製作巴克球的串珠模型（Making Beaded Buckyball Model）〔I〕
化學教室活動：製作巴克球的串珠模型（Making Beaded Buckyball Model）〔II〕
化學教室活動：製作巴克球的串珠模型（Making Beaded Buckyball Model）〔III〕

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.

Figure eight stitch

I have recently found that the right angle weave we used to construct all of beadworks is commonly called the figure eight stitch （八字編, or pronounced as Baji-Bian）here in Taiwan because the weaving path looks like a Hindu-Arabic numeral eight. I guess it has the same name in China. In Japan, the same weaving technique is called the Hachinoji-Ami or Kousa-ami stitches, which has literally the same meaning as Baji-Bian or figure eight stitch.



Apparently, it is better to call this technique as the figure eight stitch than as the right angle weave because angles generated in the figure-eight stitch may not be right angle at all.

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.

Fullerenes belonging to icosahedral group

It is straightforward to make bead models for higher fullerenes with icosahedral symmetry. The simplest way is to use the Goldberg vector (see the following figure) to specify the relative position between two pentagons. Goldberg vector is very similar to the chiral vector used for defining carbon nanotubes.

Suppose we have the first pentagon located at orgin, (0,0), then we can ask where the next pentagon can we put? The answer is that any coordinate specified by (i,j) as shown in the following figure gives a unique fullerene with icosahedral symmetry. For instance, if the next pentagon is located at (i,j)=(1,1), we have a C60. It is not hard to show that the number of carbon atom for the fullerene specified by the Goldberg vector, (i,j), is N=20(i² + ij + j²).


(Figure 8 in "Jin, B.-Y.*; Chuang, C.; Tsoo, C.-C. “The Wonderful World of Beaded Molecules. 串珠分子模型的美妙世界” CHEMISTRY (The Chinese Chemical Society, Taipei) 2008, 66, 73-92, in chinese.")

A bead model of C60, Goldberg vector (1,1).



Icosahedral fullerene specified by the Goldberg vector (2,1) has 140 carbon atoms. This is the smallest chiral fullerene with icosahedral symmetry.




The following beaded fullerene is specified by (4,0) has 320 carbon atoms.

A nice mnemonic for making beaded C60s

Prof. JT. Chen forwarded me a message from Sharon, an audience of my talk early this month. Sharon has a simple mnemonic by her son for making the beaded C60. In C60, every pentagon is surrounded by 5 hexagons, and every hexagon is surrounded alternatively by 3 pentagons and 3 hexagons. (五邊形的周圍是六邊形,六邊形的周圍是一個五邊形接一個六邊形.) One can easily create a beaded C60 by following this simple rule.

Two beaded models made by Sharon:




Indeed, one does not need spiral code to make C60. But to make an arbitrary cage-like fullerene (genus=0), spiral code is the only information we need. The shape of resulting beaded structure is always similar to the shape of the corresponding microscopic fullerene. It is quite amazing that one can create the faithful structure for an arbitrary fullerene with beads so easily. A simple explanation is that hard sphere repulsion among beads effectively mimic the valence-shell electron-pair repulsion of trivalent carbon atoms in fullerene molecules.

Additionally, if one want to make a beaded C60 with two different colors, a single color for pentagons and two different colors alternatively for hexagons. Then one doesn't need to use the mnemonic as given above. One can just pay attention to the colors only. Starting with a pentagon with a single color, then hexagons with two colors alternatively, eventually, one should get a beaded C60 correctly.

A few beaded C60s (10mm faceted beads) I made in last week:


See also a discussion in the previous post.

3D printed buckyball

I found this model of buckyball created by the 3D printing technique through googling a few weeks ago. It is cool. One can simply print a 3D molecular structure with a 3D printer. I wonder how much it would cost to have a printer like this and how much it would cost to print a model.


(wiki)

It is worthy of mentioning that this model is not a beaded structure. In fact, I believe it is not possible to create a beaded structure with this kind of connectivity by using the standard beads with a single hole. The beads we normally use should represent the edges, instead of vertices of a graph.

Alternative weaving path of EMACs

Qian-Rui told me that it is easier to use the following weaving path for creating beaded EMACs. The one-end weaving is quite odd to me. This one uses one-end weaving for the pyridyl-groups at chain ends only.

Weaving path for EMACs with 1,8-naphthyridine ligands

A simple weaving path for EMACs that contain 1,8-naphthyridine (萘啶) ligands:



I am not sure if Chern Chuang followed this path when he made the first beaded EMAC. Unlike EMACs with pyridyl-ligands, here one can use the same type of two-end weaving technique we used for making bead models of fullerenes.

Weaving path of EMACs

Qian-Rui told me the ingenious weaving path of EMACs he used. I made a schematic plot to show his weave path in the following figure. This path has a major difference from the ones we used for fullerenes before. I.e. one used only one end of fishing thread to weave pyridyl groups (hexagons, green) in the ligands. In doing so, he can weave the whole structure with only one long minimal fishing thread.

Procedure of creating a high-genus fullerene

Here are a few pictures of another high-genus fullerene I made last month. This structure contains 12 necks, each of them is made of 5 octagons again.
To make this molecule, I have used approximately six long threads with around 4 meters each.

It is clear that we have to weave the inner part of a HG-fullerene first, in this case, 12 necks connected to each other.

Note also that this structure is, I believe, one of the two simplest high-genus fullerenes with icosahedral symmetry one can create. Another icosahedral HG-fullerene has 10 heptagons in each of 12 necks.