Genus-2 TCNT

Many years ago, Chern made this genus-2 TCNT consisting basically of two 6-fold D_6h donuts. I seem to never post it here before. It should be easy to design other similar two-layer 2-dimensional structures.

Poor little bear

I found in my iphoto library a few photos I took four years ago in St. Petersburg. I was one of four coaches of Taiwan team for the International Chemistry Olympiad(IChO), held in the Moscow State University. Our students got two gold medals and two silver medals after about 10 days of hard work in Moscow. We then went to St. Petersburg for another 3 days. St. Petersburg is a beautiful city. I wish I can visit it again someday.

Possibly nearby the winter palace square, I took these pictures of two ladies abusing a poor little bear.

The poor little bear was thrown into the sea (or harbor)!

Somewhere in St. Petersburg, I found accidentally a statue of the great Russian chemist, Mendeleev, and a huge periodic table on the wall of an adjacent house.

More pictures from Little Mama Bear

I went to the Little Mama Bear on the Yan-Ping North road last weekend and took a few photos on the beadworks shown in its display case. The designs of these beadworks are pretty cool. Some of them such as pineapples have a direct connection to the fullerenes. Some others created by right-angle weave have no direct microscopic relation.

Bucky doghouse

The north hemisphere of the super Buckyball might be used as a Bucky doghouse. I tested it with my niece's brown poodle dog last weekend. But apparently he didn't like to be put inside it. It is still too small for him.

Bead models in the corridor of TFGH

I went to the Taipei First Girls High School (TFGH) a few times last week to work on the super Buckyball with chemistry teachers and students. Outside the teachers' office, there are a number of beautiful bead models hung right under the corridor ceiling. Many students also wrote their wishes on paper strips attached to their models. Many of them are quite interesting and fun.

More information about the Taipei First Girls High School from wiki:
"Taipei First Girls' High School (臺北市立第一女子高級中學) is a prestigious Taiwanese high school, located in Zhongzheng District within Taipei City, with only the top 1% of scorers on the Basic Competence Test for Junior High School Students (國民中學學生基本學力測驗) receiving admission."

Super Buckyball (超級珠璣碳球)

After about ten days of hard working, we finally created this fabulous super Buckyball. I have to thank the alumni association of the Taipei First Girls High School (TFGH), especially the classes 1981, 1971, and 1961, who kindly sponsor this project and donate this super Buckyball to the TFGH as a gift from their 30-, 40-, and 50-years joint reunion.
Of course, the crucial collaborative effort of students (mainly from classes 2She (二射) and 2Yue (二樂)) and teachers of the TFGH makes this super Buckyball possible in about two weeks.

Explaining the weaving path to students:

Students working hard:

Super Buckyball (超級珠璣碳球)

The first super Buckyball, C₄₅₀₀, was created by students (class 3Gong 三恭) from the Taipei First Girls High School (TFGH) 北一女中 today. Each unit in this beautiful bead model is a punctured C60 with three holes surrounded by a neck of five heptagons. It took them exactly one week to construct it. The diameter of this small super Buckyball (made of 6750 6mm beads) is about 40 cm already. They might still need to clean all the loose ends up later next week.

In addition to this small super Buckyball, I am still working with teachers and some other students from the TFGH on a bigger super Buckyball made of 12mm beads. Hopefully, we can have the whole structure done early next week. Since the total weight of this model is going to be eight times of this small super Buckyball, so we need to be very careful about the rigidity of each units and necks connecting them. I name these kinds of bead models as "超級珠璣碳球" in Chinese which means literally "the super bead carbon ball".

(I found many pictures at TFGS's website. http://web.fg.tp.edu.tw/~chemistry/blog/?page_id=2&nggpage=8, 2012/9/1)

Super carbon tetrahedron

We can make a super carbon tetrahedron (超級碳正四面體) with four C60 building blocks that have three holes drilled on the three pentagons surrounding the same hexagon. Of course, four equal CNTs with suitable length are required to connect these four punctured C60s. Here, in addition to heptagons (blue), one also creates three octagons (purple) and one nonagon (red) on the C60 at each vertex. Of course, this structure was created by Mr. Horibe first.

Building blocks:

Corresponding Schlegel diagram and weaving path for creating a single vertex (punctured C60):

Super regular triangle

If the holes on C60 are located at two neighbored pentagons, the angle created between the necks at these two positions is very close to 60 degrees. So one can use three such units to make a triangle consisting of three C60s at position of vertices connected by three CNT (carbon nanotube) struts. This kind of construction scheme seems be proposed by Mr. Horibe first.

Since there are many different ways to puncture holes on a C60 or other Goldberg polyhedra, we can then use different lengths of CNTs to connect them to get complicated 2D or 3D structures. When resulting structures are cage-like, I will call them super fullerenes (超級芙類分子). Here I have a super carbon triangle （超級碳三角, 注意不是鐵三角） and the previous dodecahedron consisting of 20 C60s should be a super dodecahedron （超級十二面體）.

Two holes in a C60

There are three distinct ways to drill two holes in a C60, which can be specified by the smallest number of non-disrupted hexagons between two pentagons:
a. The shortest separation between two pentagons we can drill is one hexagon.

b. The next one is two pentagons separated by two hexagons.
c.The third one is two holes located at antipodal position.

We can see in the case (a) two octagons are created at the position of original hexagon. In addition to this octagon, there are eight other heptagons surrounding these two holes. Using this Shlegel diagram, one can easily figure out the resulting connectivity diagram when holes are introduced in the original C60.

"Super" dodecahedron consisting of 20 punctured C60s

This is the final bead model of a dodecahedron by connecting 20 C60s with three holes described in previous post. The C60 units in this structure is distorted quite significantly.

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.

Comments on type-II high-genus fullerenes

It is of interest to compare the bead model of a dodecahedron fused by 20 C60 in the previous post to high-genus fullerenes we discussed before. In fact, one can also view that bead model as a high-genus fullerene with genus=11, which is obtained by substracting one from the number of faces in a dodecahedron. The most important difference between these two types of high-genus fullerenes is the orientations of the TCNT necks relative the surface of the polyhedron. In the original high-genus fullerenes, which I will call the type-I high-genus fullerene from now on, the orientations of the TCNT necks are along the normal of the polyhedron. Here in the type-II high-genus fullerene, the orientations of the TCNT necks, which are created by fusing two neighbored C60, are lying on the surface of polyhedron (dodecahedron here) and along the directions of its edges.

Dodecahedron consisting of 20 fused C60s (still incomplete)

I am working on a beaded dodecahedral structure which is going to consist of 20 fused C60. The construction strategy is similar to certain bead models made by Mr. Kazunori Horibe. I called those models as "Carbon poti donut （波提甜甜圈）".
In the carbon poti donut, every C60 is connected to two neighbored C60 through two particular 5-fold axes. It is easy to see which two 5-fold axes I am talking about. Here I am trying to connect every C60 to three other C60 such that all 20 C60 locate on the vertices of a dodecahedron. But the bond angles generated by the trivalent C60 do not match with the angle of a dodecahedron. So the whole structure distorts quite significantly.

This model is very similar to a dodecahedron consisting of 20 dodecahedra (Clathrate cluster). I tried to make this model last month. But because each dodecahedron in this model is highly strained, I gave it up.

Quasicrystals with Zometool

Yuan-Chia Fan (范原嘉), Hsin-Yu Ko(柯星宇) and Mu-Chieh Chang (張慕傑) made a beautiful quasicrystal consisting of two types of rhombohedra with zometool last Friday. They put this model in the main lobby of our chemistry department for the alumni reunion last Saturday.

In addition to the tiling approach to the quasicrystal, one also constructs it using the cluster approach. In this sense, the carbon onion we made before is also a quasicrystal.

Originally, we didn't order enough red sticks to connect different shells of this structure. So only a few red sticks are used to hold neighbored Goldberg polyhedra along one five-fold axis (z direction). We now have enough red sticks for all twelve (or six) 5-fold axes emanating from the central dodecahedron (Goldberg vector (1,0)). So libration motion of each shell is quenched.

Yuan-Chia, Hsin-Yu, and Mu-Chieh also designed a wonderful support around pentagon at the bottom to enhance the stability for the this six-layer Goldberg polyhedra.

D surface constructed from four helical strips

There is another way to build a D-type triply periodic minimal surface (TPMS) with beads. Chern has told me previously that one can not only use helical strips to build G-type TPMS, one can also use exactly the same helical strips to build D-type TPMS. If one examine two helical strips carefully, one can find that there are exactly two different ways to put them together. One gives a D-type TPMS, the other one gives a G-type surface!

The following pictures are a bead model of D-surface consisting of four helical strips. Two of them are left handed, the other two are right handed. To build a D-surface, one has to put two helical strips together in an arrangement such that two neighbored strips are mirror-symmetric to each other. So the overall structure of D-surface is not chiral.

It is useful to look at other posts with the keyword helical strip, especially the one on the G-surface created by patching two helical strips with one strip shifted by half pitch.