碳納米管7₄結串珠模型

每年的 Bridges數學藝術會議 (Bridges Conference) 都是一個慶祝數學與藝術之間創意連結的盛會。在 2021 年的會議上，眾多藝術家展示了他們受數學啟發的精彩作品。其中一件引人注目的作品是由 何厚勳 創作的 Carbon nanotube 7₄ knot bead model。

背景介紹：數學串珠藝術

利用串珠來具象化複雜的數學結構和分子模型，是數學藝術領域中一種獨特且迷人的表現形式。這種方法不僅能夠將抽象的概念轉化為可觸摸的實物，也展現了藝術家在數學和手工藝方面的精湛技藝。尤其值得一提的是，台灣大學金必耀教授 的實驗室長期以來在利用串珠技術構建各種分子模型方面取得了豐碩的成果。

2021年Bridges會議展品：碳納米管7₄結串珠模型

在 2021 年的 Bridges 會議上，當時身為台灣大學化學系博士生的 何厚勳 展示了他的作品 Carbon nanotube 7₄ knot bead model 。這件作品以碳納米管為靈感，並呈現了一個複雜的數學結。

作品詳情

• 作品名稱： Carbon nanotube 7₄ knot bead model
• 創作者： 何厚勳 (Hou-Hsun Ho)
• 創作年份： 2021
• 尺寸： 10 x 15 x 3 厘米
• 材料： 3 毫米塑料珠子，魚線

設計概念

根據展覽資訊，這個串珠模型是對 7₄ 結（也稱為 **纏繞的心** 結）的高度對稱表示。他利用串珠和魚線，將抽象的數學概念——拓撲學中的結——轉化為一個具體的、可視化的藝術品。作品使用了黑白兩種顏色的珠子，增強了結的結構和清晰度。

何厚勳自2019年起便在金必耀教授的實驗室工作，並遵循一篇名為 "Constructing Bead Models of Smoothly Varying Carbon Nanotori with Constant Radii and Related Intersecting Structures" 的策略，設計了基於彎折碳納米管的各種離散空間曲線。這件 7₄ 結串珠模型正是將這種設計理念付諸實踐的一個例子。

數學藝術的啟示

Carbon nanotube 7₄ knot bead model 不僅是一件精美的藝術品，更是一個探索數學概念的有趣方式。通過觀察和理解這個模型的結構，我們可以更好地認識拓撲學中結的複雜性以及數學在描述和模擬分子結構中的作用。這件作品也體現了Bridges會議的精神，即促進數學、藝術和教育之間的交流與融合。

如果您想了解更多，請訪問 Bridges數學藝術會議官方網站。

© 2025 珠璣科學|Carbon nanotube 7₄ knot bead model

數學與藝術的交匯：碳納米管三葉結 C318

在 2014 年的 Bridges 會議上，莊宸Chern Chuang（當時是麻省理工學院化學系的博士生)與國立台灣大學的金必耀教授，共同展示了一件引人注目的珠飾藝術品：碳納米管三葉結 C318。這件作品不僅是精美的視覺藝術，更是對分子結構和數學拓撲的巧妙結合。

背景知識：珠飾分子建模技術

莊宸和金必耀在過去的 Bridges 會議中一直推廣一種特定的珠飾技術。簡單來說，這種技術的核心思想是**將分子中的每一個化學鍵替換為一顆珠子，然後用魚線將這些珠子串聯起來，形成一個代表分子結構的連通圖。在他們研究的大多數案例中，所形成的圖是哈密頓圖，這意味著可以找到一條路徑遍歷圖中的每個頂點（即每個珠子）恰好一次。此外，所有組成的珠子通常會被魚線穿過兩次，以確保結構的穩定性。更多細節可以參考他們提交給 Bridges 2014 的題為 "Torus Knots with Polygonal Faces" 的手稿。

碳納米管三葉結 C318

創作者：莊宸和金必耀

• 尺寸： 5 x 5 x 3 厘米
• 材料： 3 毫米施華洛世奇水晶珠
• 創作年份： 2014

C318 碳納米管三葉結被作者認為是其家族中最優雅的作品。這種優雅性體現在兩個方面：

• 美學角度： 它的尺寸與半徑之比與最常見的三葉結呈現方式完美契合，視覺上非常協調。
• 力學角度： 它僅使用了最少數量的碳原子就實現了三葉結的結構，展現了結構上的簡潔性。

通過使用閃耀的施華洛世奇水晶珠，這件作品不僅準確地呈現了碳納米管三葉結的幾何形狀，更增添了藝術的美感。它是一個將抽象的化學結構轉化為具體的、可觸摸的藝術品的絕佳範例。

這件作品的創作正是基於前述的珠飾技術，每一個水晶珠都代表了碳納米管結構中的一個化學鍵，魚線則連接這些“鍵”，勾勒出三葉結的拓撲結構。

您可以通過以下鏈接查看該作品的更多信息：

• Bridges 2014 畫廊 - 碳納米管三葉結 C318

© 2025 珠璣科學 |碳納米管三葉結

Torus knot (1,2)

Kazunori showed me this beautiful torus knot (1,2) he made a few days ago. This structure can be classified as a torus knot, or more specifically a twisted torus without knot at all. The space curve that this tubular structure approximates can be described by the parametric equations for the torus knot (q=1, p=2). Therefore, it is reasonable to call it as the torus knot (1,2).

To me, this structure seems to be a perfect example to show the influence of the particular operation, Vertical Shift, described in the following papers:
1. Chuang, C.; Jin, B.-Y. Torus knots with polygonal faces, Proceedings of Bridges: Mathematical Connections in Art, Music, and Science 2014, 59-64. pdf
2. Chuang, C.; Fan, Y.-C.; Jin, B.-Y. Comments on Structural Types of Toroidal Carbon Nanotubes, J. Chin. Chem. Soc. 2013, 60, 949-954.
3. Chuang, C.; Fan, Y.-C.; Jin, B.-Y. On the structural rules of helically coiled carbon nanotubes, J. Mol. Struct. 2012 1008, 1-7.
Another related operation is the Horizontal shift, which is not used in this structure. Applying these two operations carefully (usually nontrivial), one can mimic the bending and twisting of many space curves in an approximate way.

作品完成時間（約）：2015/4
作者：堀部和経

Circular helix winding around a central torus

Horibe-San just constructed another beautiful beadwork, a circular helical carbon nanotube (or circular carbon spring) winding around a toroidal carbon nanotube.

作品完成時間（約）：2015/4
作者：堀部和経

Torus knot (2,9) by Kazunori Horibe

Kazunori email these photos of a beautiful bead model of (2,9)-Carbon nanotube torus knot (CNTTK) he just made the other day. To make the structure more clearly, I also use the Grapher to create the corresponding torus knot.

Trefoil knot for the Bridges Seoul 2014

Chern submitted an artwork of trefoil knot for the Bridges Seoul 2014, which can be found at this page: Carbon nanotube trefoil knot C318.

Carbon star and other clover-shaped carbon nanotori

I recently saw a post on the clover-shaped carbon tori by my facebook friend, Tetsuaki Hirata, who is an artist from Himi, Japan and seems to be a frequent visitor of this blog. After I showed Chern about his works, Chern told me he has thought about this kind of clover-shaped carbon nanotori quite some times ago. Indeed, Chern has published a number of papers on the tubular graphitic structures. He is probably one of few people who know a lot about this kind of graphitic structures, especially on how the nonhexagons could influence the structures of a carbon nanotube. I am not surprised that he thought about this kind of structures.

1. Chuang, C.; Fan, Y.-C.; Jin, B.-Y.* Generalized Classification of Toroidal and Helical Carbon Nanotubes J. Chem. Info. Model. 2009, 49, 361-368.
2. Chuang, C; Fan, Y.-C.; Jin, B.-Y.* Dual Space Approach to the Classification of Toroidal Carbon Nanotubes J. Chem. Info. Model. 2009, 49, 1679-1686.
3. Chuang, C; Jin, B.-Y.* Hypothetical toroidal, cylindrical, helical analogs of C60 J. Mol. Graph. Model. 2009, 28, 220-225.
4. Chuang, C.; Fan, Y.-C.; Jin, B.-Y.* On the structural rules of helically coiled carbon nanotubes, J. Mol. Struct. 2012 1008, 1-7.
5. Chuang, C.; Fan, Y.-J.; Jin, B.-Y. Comments on structural types of toroidal carbon nanotubes, arXiv:1212.4567, 2013 submitted to J. Chin. Chem. Soc.

In the first two and the 5th papers, we talked about general structural rules of carbon nanotori and only touched helices briefly. In the next two papers, we discussed very generally how the horizontal and vertical shifts (HS and VS) can be exploited to change the direction of a straight carbon nanotube in order to obtain an arbitrary helically coiled carbon nanotubes. In Chern's Ms thesis, he also showed how to take advantage of HS and VS to create trefoil knots or torus knots in general, which was later summarized in a brief review we wrote, "Systematics of Toroidal, Helically-Coiled Carbon Nanotubes, High-Genus Fullerenes, and Other Exotic Graphitic Materials."  (Procedia Engineering, 2011, 14,  2373-2385).

Clover-shaped TCNTs are just a special class of more general curved carbon nanotubes we considered. A simple strategy is to introduce 180 twists along the tube direction (i.e. 180 degree VS) at suitable positions. I got a few nice figures of clover-shaped TCNTs from Chern the other days.

Among all these clover-shaped tori, I particularly like the five-fold carbon star.

Beaded trefoil knot using two beads per edge

I did a tryout using two beads per edge in this beaded molecule. Also, since these assorted glow-in-the-dark beads are one of the only few kinds of beads I have here in Boston, I try to take a picture of what it looks like in the dark after suitably charged with a light bulb. Thanks to Chun-Teh Chen (陳俊德) who helped me with the photography.

Enantiomeric Pair of Graphitic Trefoil Knots C318

It's been a while since last time I wrote on this blog. Three weeks ago, I went to the Bridges 2012 conference held at Towson University, MD. It was such a terrific gathering of practitioners of mathematical art. I met a lot of new friends and enjoyed sharing our ideas with each other. The following is a repost of a graphitic structure I found last year when I was a research assistant of Bih-Yaw. It is a D3-symmetric trefoil knot structure and comprises 318 carbon atoms.

 

I figured out its construction rule while pondering on one of Escher's artwork.

 

Upon scrutinizing the one at the upper-left corner, I found that the trefoil knot can actually be decomposed into three 270-degree-arc and three inter-arc straight segments. This was a huge hint for me since I've been thinking about constructing torus knots from carbon nanotubes for years. An idea came up when I was doing my MS degree with Bih-Yaw, three years after he mentioned this project to me, which basically makes use the fact that one can divide a (p,q) torus knot (for p smaller than q). An example of this building principle applied to the trefoil knot is shown below:

 

Interested readers can also refer to the "Torus Knot" tag in this blog. A truly unfortunate fact is that this kind of molecules does not seem to be "beadible" to me, since it has got a great deal of tangential stress along the tangent of the central curve. Unless one lets loose the fish line during the course of the beading process, eventually when she or he tries to sew up the closed structure an unsurmountable stress will be encountered.

I didn't want to stop there, and finally I got the inspiration from Escher as mentioned above. The structure possesses nearly zero tangential stress when you try to bead it. Another nice point about it is that it has got much less number of atoms in it, compared to the other construction scheme. It typically takes me less than three hours to bead it.

At the Bridges conference, I met two kind Taiwanese ladies Helen Yu and Joy Hsiao. The pair of enantiomeric C318 trefoil knots as shown in the following photo was given to them as souvenirs.

They are made of 3mm glow-in-the-dark beads. I'm thinking about presenting this type of structures in the Bridges next year, hoping that I have the chance to go.

One more trefoil knot

Chern and Yuan-Chia made this bead model of trefoil this May. This one has a C2 rotational symmetry which is absent in the one I posted previously.



I gave this model to Prof. Wen-Yuan Qiu (Lan-Zhou University) when I was in An-Hui, He-Fei at the end of this May. He is an expert on the DNA polyhedra and gave me his book entitled "The Chemistry and Mathematics of DNA polyhedra" as a gift.

A new trefoil knot

Here is a new trefoil knot by Chern. Unlike the previous trefoil knot, this new construction has much less strain and distortion. In this sense, the corresponding fullerene structure should be more stable.

(I gave this model to Prof. Qian-Er Zhang 張乾二 in Xian last August. Feb. 4, 2013)

Another beaded model of CNTTK by Yuan-Chia Fan

Two more photos of Beaded Torus knots

Beaded Model of Torus Knot (4, 3)

Qian-Rui Huang showed me this remarkable beaded model for the (4,3) torus knot today. The model is not like the one for trefoil knot. It is quite soft and very crowded in the center since two tubes have to pass to each hole for the (4,3) torus knot. This problem is not very serious for the trefoil knot though. As I said before, one can in principle construct any kind of molecular graph. But only in certain situations, the resulting structure can simulate the real fullerene.

Trefoil Knot

It is interesting to make a comparison between the beaded trefoil knot and the famous wooden trefoil knot of M. C. Escher.

Beaded Trefoil Knot

I took many pictures for this elusive structure. Here are two of them.

More photos of Beaded Trefoil Knot

Here are several photos I took when I was working on the beaded trefoil knot last weekend. It is now obvious to me why we should take such a weaving path for making beading trefoil knot. I have mentioned several difficulties before. Originally, I thought that these difficulties cannot be overcome without using several different sizes of beads. But beading can be very flexiable since we don't need to make them very tight. If we allow the structure be loose, then anything kind of graphs can be weaved in principle. Of course, the resulting structures may not be rigid or similar to what we would like to have for a fullerene.


In the following photos, one can see the intermediate structures before I started to weave the final strip of polyacene is quite flexible. This gives some rooms for us to adjust the structure, so the unavoidable deformation can be redistributed more evenly on the whole structure.

Of course, it is very important to make the first loop connected correctly. :-)

Beaded model of Carbon Nanotube Trefoil Knot

Trefoil knot is one of the most beautiful geometric shapes that intrigue both mathematicians and artists alike. Previously Chuang and I have shown that one can in fact tile on trefoil knot with graphitic sheet. By introducing non-hexagons on the suitable position, the resulting carbon nanotube trefoil knots can be very stable.

The problem is whether we can really build a beaded model for this new family of carbon compounds. Previously, we have tried to create a beaded model for CNTTK by weaving the girth first and then the axial direction of the tubule. Based on this strategy, we realize that the intrinsic curvature introduced by the non-hexagons is too large, so it seems that the construction of trefoil knot with beads is not possible unless large deformation is applied.

Last weekend, Chuang gave me a figure of CNTTK 504 with only one strip of polyacene is shown (see previous post). This suggested that one may weave along the axial direction first. So construction of a beaded CNTTK 504 amounts to weaving 6 strips of polyacenes. Using this weaving algorithm, one may be able to adjust the intermediate structures more easily since they are more flexible. The extra deformation that has to be introduced can then be distributed more evenly over a larger area. This is quite different from the previous weaving algorithm we adopted. The extra deformation created in the previous method is concentrated on particular segment of the tubule. The final structure may look quite distorted. Thus, the new algorithm has better potential for making a CNTTK.

This is indeed the case. I finally made one beaded CNTTK 504. The resulting beaded structure as shown in the following figure looks great. The overall shape of this structure looks quite similar to the optimized structure we obtained before. With closer inspection, we can also see the distortion away from the intrinsic curvatures created with non-hexagons.

A new picture of CNTTK 504

Chuang gave me a new matlab fig file of CNTTK 504 last weekend, which shows more clearly on how a CNTTK is composed of six strips of polyacenes (with pentagons and octagons). In order to see the change in orientation this strip colored in red, one seem to have to use two different projections. I am not sure if one can do better than this. For instance, is it possible to use just one projection to give the relative arrangement of these polygons?

Beaded Trefoil Knot?

Carbon Nanotube Trefoil Knot is probably the most beautiful carbon structure one can make. Of course, it is extremely difficult to synthesize it if not impossible. But even though we know it is not a trivial task to make them, we still like to ask the question on the spatial arrangement of the sp2 carbons in order to make them as stable as possible. Chuang in his Ms thesis has shown a systematic way to construct this class of aesthetically beautiful molecules. He has also proved many of these molecules are indeed stable. Out of so many potential structures, we particularly like the C504 due to the similarity between this molecule and Escher's wooden knot (a poster was made to show this similarity).

Now the problem is to construct a beaded model for this molecule. We did try, but failed. Why? Previously, I used to believe that beads are generic material for constructing any 3D fullerene molecules consisting of sp2 carbon atoms. We indeed manage to build all kinds of beaded fullerenes including spheroidal, toroidal, porous, and periodic structures, in the last three years. In the mean time, the current blog was built to show the beaded models we constructed and to give some background information related to beading.

Well, trefoil knot seem to be the first class of structures we cannot use beads to construct. At the beginning, we thought this is only for C504, since the hole in the trefoil knot is too small for another segment of CNT in the same structure to pass through. So the difficulty should be removed once we move on the larger trefoil knot. I realize now this is not the real reason. In the following figure, I have shown a beaded structure for C504 (Chuang indeed has also tried to weave the similar trefoil knot as shown in the previous post yesterday).

If we inspect this structure carefully, we should see two reasons why it is not possible to make a beaded C504 trefoil knot with identical beads. The first reason is the one we just mentioned. The second reason is the number of nonhexagon pairs we need to create a 360 (2pi) loop is only about five which is too fast for the trefoil knot. To get the right structure for a trefoil knot, we need about five nonhexagon pairs to have a 3/4*2pi loop. The result is that once we have the first beaded loop constructed, the second and also third loops will be always in the wrong direction. Thus, it is almost impossible to close the final structure in a single connected way.

Another way to put it is that if we use the identical beads, the average number of beads per unit area is about the same everywhere on the beaded surface. Thus, the number of nonhexagon pairs is about slightly larger than 5. But from the structure for a real trefoil knot based on suitable force field calculations, the mean bond length in inner-rim region of loop is slightly larger than those bond lengths in the outer-region. This leads to a slower turn from the nonhexagon pairs. That is why we cannot construct this structure using beads with identical size. To avoid this problem, we have to use at least two types of beads with slightly different sizes. Of course a painful try-and-error process is necessary.