A few bead models of diamondoid molecules

Mathematical beading can be used to construct any diamondoid molecule, also known as nanodiamonds or condensed adamantanes. Here, I show three such molecular systems:

Adamantane (C₁₀H₁₆);
Diamantane (C₁₄H₂₀) also diadamantane, two face-fused cages;
One of 9 isomers of Pentamantane with chemical formula C₂₆H₃₂.

C20 vs C56

Octahedral diamond net

Octahedra connected by octahedral tunnels (38)

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.

A postcard from Mr. Horibe

A nice postcard I got from Mr. Horibe:

Hyperbolic "buckyball" with octagons

I played with carbon tetrapods described in the Diudea and Nagy's book last weekend and discovered quite accidentally that it is possible to make another type of hyperbolic "buckyball" with neighbored nonhexagons separated by only one CC bond. Particularly, all nonhexagons in this structure are octagons. So this one is different from another hyperbolic "buckyball" made of heptagons and hexagons, or simply heptagonal buckyball. The skeleton of this structure is the same as the heptagonal "buckyball". But each unit cell, which consists of two connected tetrapods, has 96 carbon atoms. I am not sure if people have mentioned this kind of hyperbolic "buckyball" or not.

Heptagonal "buckyball", C168 (This model was given to Prof. Sonoda as a gift when I was invited to Japan for a series of talks and workshops.):

(photo by Dancecology)

Octagonal "buckyball", C96:

Two hyperbolic graphitic surfaces

I made a bead model of hyperbolic graphitic surface by connecting 10 tetrahedral C₈₄ units (the model on the right, the one on the left is the C168). The skeleton of this model is similar to that of an adamantane of the smallest unit of diamond.

Building blocks for pseudo D-type Schwarzite

I just noticed that I already have a picture of punctured C₈₄ in the logo of this blog the other day. I checked my photo library and found another picture of this model which was taken almost five years ago. I possibly made this bead model after reading the paper on Nature with title "Energetics of negatively curved graphitic carbon" by Lenosky et al. (1992 vol. 355, 333-335).

However, we now know this is not a true triply periodic minimal surface. The correct D surface should partition the space into two congruent parts. It is not possible to get such a structure if one use heptagons.

Of course, we now know how to make a correct D-type Schwarzite which exactly partition the space into two identical regions. Wei-Chi made this beautiful bead model of D-type Schwarzite a few years ago.

C60 vs C168

One can view C60 or a buckyball as the simplest graphitic sheet embedded in a sphere with every pentagon connected to five other pentagons by only single carbon-carbon bonds. In this sense, C168 can be viewed as the "buckyball" in the hyperbolic space with every heptagon connected to seven other heptagons by only single carbon-carbon bonds.

I made a bead model of C168 that consists only of one tetrapod (half unit cell). So only 84 carbon atoms are included in this structure. The structure I posted previously contains 5 unit cells.

A new model of C168

I made a new model of C168 last week.


Click the keyword C168 below to see more discussion on this structure.

My C168

I have made only one beaded C168, which I gave to Dirk Huylebrouck, a math professor in the department of architecture in Belgium, at the Bridges conference this summer. I took the following photo of this bead model at the Hotel in Pecs, Hungary. I am thinking about making another one, maybe this time with giant beads just like Mr. Horibe has used. Unlike Mr. Horibe, I prefer using different colors for nonhexagons. In this structure, all heptagons are in purple beads. One can easily see that these heptagons are separated by on beads (carbon carbon bonds). In this sense, we can call C168 is the buckyball in the hyperbolic space.


Another two photos that contains more bead models I brought to Bridges conference.



Many of these beadworks are given away as souviners for other attendee. The helically coiled carbon nanotube is given to Laura Shea and the high-genus fullerene is to M. Longuet-Higgins. Toroidal carbon nanotube (T120) with 120 carbon atoms or 180 beads (cat-eye stons) is given away to G. Hart. T120 is made by Chern Chuang. All other small beaded balls are gone too. Many of these small beaded balls are made by Q.-R. Huang. The only three left is the bead models for the P-type triply periodic minimal surface, Shoen's I-WP surface and the trefoil knot, respectively.

D168

Beaded D168

Here is the first beaded model of D168. D168 has diamond structure. So it is an extended structure. Here I only made part of the whole structure based on the admantane. We can actually infer from the factorization of 168=7*24=7*12*2=7*3*4*2 many important structural information.

Based on the fact that every heptagon in D168 is connected to seven other heptagons by a 6-6 bonds, and the local structure is a tetrepod, so these two numbers correspond to 7*4 in the factorization. The remaining two number are 3 and 2. Inspecting this structure, we know 3 corresponds to the three heptagons surrounding the neck and 2 is the two tetrapods in a unit cell. Each tetrapod has 84=7*12 carbon atoms.

Beaded model for D168!

Finally, I have made a beaded model for the famous D168 fullerene, which is essentially corresponding to the C60 in the hyperbolic space.

D168

The 3D structure of D168 we posted a few days ago is incorrect. Chuang has now the correct structure shown below:











I thought it is not a bad idea to have a beaded model for this structure. Now I am still at a very preliminary stage (see the picture shown below).


Eventually, I expect to have a D168 structure similar to Adamantane:

D-Type Schwartzite

Well, here is the D-Type Schwartzite with all heptagons in blue. It is easy to see that there is a three-fold rotational symmetry axis along each pod. Totally, there are 3x4=12 heptagons in a unit cell.

This structure is similar to the molecular tetrapod I posted before. However, there are some restrictions due to the patching condition between the tube and endcaps for the same reason as the situation in the endcapped nanotubes. For instance, the tetrapod I made has larger girths.