虧格31的超級巴克球 Super Buckyball of Genus 31

聯合數學會議（Joint Mathematics Meetings, JMM）是世界上最大的數學學術會議之一，除了數學研究的交流，會議的藝術展覽也提供了一個獨特的平台，展示數學與藝術之間的奇妙聯繫。在 **2013 年的聯合數學會議**上，一件名為 "**Super Buckyball of Genus 31**" 的藝術品吸引了眾多目光。

背景介紹：富勒烯與拓撲虧格

在欣賞這件藝術品之前，我們先簡單了解一下相關的背景知識：

• 富勒烯（Fullerene）： 富勒烯是一類完全由碳原子組成的中空的球狀、橢球狀、或管狀分子。最著名的富勒烯是 **C60，又稱足球烯或巴克球（Buckyball）**，其結構與足球相似，由 20 個六邊形和 12 個五邊形構成。
• 拓撲虧格（Genus）： 在拓撲學中，一個曲面的虧格是指它包含的“洞”的數量。例如，一個球面（如普通的巴克球）的虧格為 0，而一個環面（如甜甜圈）的虧格為 1。**Genus 31** 意味著這個結構在拓撲上相當複雜，擁有 31 個“洞”。

藝術品詳情：Super Buckyball of Genus 31

Super Buckyball of Genus 31

創作者：**金必耀 (Bih-Yaw Jin) 及台北第一女子高級中學的師生**

• 尺寸： 20 英寸 x 20 英寸 x 20 英寸（約 60 厘米 x 60 厘米 x 60 厘米）
• 材料： 塑料珠子
• 創作年份： 2011

這件 "**Super Buckyball of Genus 31**" 是一個使用塑料珠子製作的大型多面體模型。它並非一個普通的巴克球（Genus 0），而是一個 **虧格為 31 的超級巴克球**。

這個模型的每一個頂點本身都是一個帶有三個孔的巴克球，並且通過三個最短的碳納米管連接到三個相鄰的頂點。

另一種理解這個結構的方式是將其視為第二層的 Sierpinski 巴克球，並且這種結構可以無限擴展。Sierpinski 結構是一種分形，通過不斷地自我複製和縮小形成複雜的圖案。將 Sierpinski 的概念應用於巴克球，創造出更複雜、更高虧格的結構，體現了數學中迭代和自相似性的思想。

這件藝術品是由金必耀教授與台北第一女子高級中學的師生於2011 年 11 月共同製作完成的。這也展現了數學和科學概念在教育和公眾推廣中的藝術表達。

通過將抽象的數學概念（如拓撲虧格和分形）與具體的物理模型相結合，"Super Buckyball of Genus 31" 不僅是一件引人注目的藝術品，也是探索複雜幾何結構的一種有趣方式。

參考資料

• Bih-Yaw Jin | 2013 Joint Mathematics Meetings | Mathematical Art Galleries: #### Artworks ##### Super Buckyball of Genus 31. 2013 Joint Mathematics Meetings Bih-Yaw Jin

© 2025 珠璣科學｜Super Buckyball of Genus 31

Beautiful photo of Chern's C60@60 at the Columbia Secondary School

The superbuckyball, C60@C60, made by Chern is now on exhibition at the Columbia Secondary School, New York. This is for an event called MoSAIC—Mathematics of Science, Art, Industry, Culture—the festival, an offshoot of the annual Bridges Organization international conference dedicated to the connections between art and mathematics.

There is a nice photo of this C60@C60 in the Columbia Spectator.

Evolution of superbuckyballs

Since the last month of 2011, I started to work on the so-called Sierpinski buckyballs or superbuckyballs, which belong to a particular family of fullerenes created by treating C60s as supernodes and carbon nanotubes as superbonds. Using this idea, an unlimited number of hierarchical super-structures of sp²-hybridized (3-coordinated) carbons can be constructed. Before this task was really started, I have managed to build some simpler structures such as super-triangle, super-tetrahedron, and other related structures. With the experience, I firmly believed in the feasibility of creating bead models of much larger superbuckyballs. But it is too tedious to construct bead models for this kind of super-structures, especially the so-called C60xC60, alone. So I designed a modular approach to build these models collaboratively. I told a few chemistry teachers, especially Dr. Chou (周芳妃), at a local high school, The Taipei First-Girl School (TFGH), about this structure. They were glad to try this idea out together. The results are two beautiful superbuckyballs (or C60xC60) made by 6mm and 12 mm beads, respectively. Both of the structures were on public display for the anniversary of TFGH and a simultaneous event of the TFGH's 30-year alumni reunion. Alumni association of TFGH kindly supported the whole project. Dr. Tsoo was one of alumni that year, that was why we got supported from them.

Later on, I made another bead model of C60xC60 for the JMM held in San Diego about two years ago (Jan. 2013). I used the photo of the giant bead model students and I made in the JMM description though. I met Chern (莊宸) in the meeting. We discussed the structural rules for this family of compounds. Particularly, I commented on that the particular model I made cannot be constructed by Zometool. After returning back to Cambridge, MA, Chern solved the problem by carefully puncturing holes along certain symmetry axes in order to be consistent with the Zometool requirements.

Yuan-Jian Fan (范原嘉) then proved Chern's idea by building a virtual C60xC60 super-buckyball with the zometool construction software, vZome, which was kindly given to us by its author, Scott Vorthmann, a few years ago. With everything ready, a few enthusiastic students from the theoretical chemistry group of the National Taiwan University started to build the first zometool super buckyball after the Chinese new year.

Soon, a number of practical issues on the construction of a real zometool model of C60xC60 super-buckyball appeared. The first issue is the structural stability against gravity. The original neck structures (shortest situations) designed by Chern consisted of a number of octagons were too weak and simply cannot hold the whole structure due to its own weight. Another issue is still weight, without extra stands, the southern hemisphere of super-buckyball constructed by zometool simply cannot hold the northern hemisphere. Finally, how to put those parts on top without scaffold is also a question. All these problems were solved beautifully by Yuan-Jia Fan. Of course, local dealer, Helen Yu, of Zometool in Taiwan is also helpful. She always responded us with the necessary zometool pieces upon our requests in a very short time. So, we can have the first zometool sculpture of C60xC60 erected in the NTU campus around mid-March.

Chern then proposed to Paul Hildebrand to have a family-day activity for the 2013 Bridges which will be held in Enschede that year. Paul agreed to provide us with the necessary materials. At the family day, we got more help from a few Bridges participants and their family members from Taiwan. These included Profs. Liu (劉柏宏) and Tung-Shyan Chen (陳東賢). Without them the final C60xC60 structure couldn't be finished in such a short period.

In addition to the construction of huge zometool superbuckyball, Chern also presented a small bead model to the Bridges meeting as shown in the following pictures based on the same construction rule he designed. Another bead model in his right hand is an edge-elevated dodecahedron assembled from fifty C80s, twenty for the vertices of dodecahedron and thirty for the elevated edges. The idea for making this one is similar to C60xC60. I brought them back to Taiwan and put them on exhibition in the NTU Chemistry Museum for about a year until the last July when Chern got an email from George Hart asking him about the possibility of donating this small C60xC60 model for MoSAIC (Mathematics of Science, Art, Industry, and Culture) traveling exhibition. In the email, George commented on this model as " having the right combination of artistic expression, mathematical content, and practical transportability".

The two beaded superfullerenes showing up at the Bridges 2013

Now that the big week of our society, the Bridges 2013, is coming. Embarrassingly, I completed the beaded molecules submitted to the art exhibition this year only recently. So here they are. If you are coming to the event, you are more than welcome show up at Bih-Yaw's and my talks on next Tuesday afternoon. Also with the help from Paul of the Zometool inc., we will hold an event for the Family Day (Sunday) constructing the giant superfullerene appeared earlier on this blog (blog entry link, Family Day event link). Feel free to join us at the scene!

C60⊗C60 with g=1: C4680. 7020 3mm phosphorescent beads used. Viewed from one of the fivefold rotational axes.

My friend Chun-Teh (陳俊德), who is also a grad student at MIT, helped me shooting the photos. He managed to do a long exposure shot on this one glowing in the dark. It's pretty amazing when you see it glow. Way much brighter than I thought. This particular photo was shot nearly along one of the threefold axes.

V-substituted Dodecahedron⊗C80 with g=1: C4960. 7440 6mm faceted plastic beads were used. Viewed from one of the fivefold rotational axes. There are 20 supernodes in the inner shell (purple) and 30 in the outer shell (blue). If you look careful enough and follow the colored beads representing the non-hexagons, you would find that all of the 50 nodes have the same orientation. This is a manifestation of the zome geometry property. 

Viewed from one of the threefold rotational axes. 

At the very beginning of the constructing this model, I was worrying about whether the structure could hold itself or not. When the first ring of supernodes (five on the inner side and five on the outer side) was completed, it was so soft that I could easily bend it to a degree. And also I was concerned about if the structure can be built at all. Since it is a very different story than constructing them hypothetically on a computer: strain can be built alongside the construction and one might not be able to fit the later pieces in near completion. Gratefully none of the above issues was really an issue. I can really tell that there is minimal strain since before the last supernode comes in, the whole structure is already holding itself up so that the piece will fit just about right. Eventually the structure, though is not as strong as other giant beaded molecules I've built, is pretty OK of supporting itself without additional scaffold. 

A comparison of the scale. See you soon in Enschede!

Superfullerene with zometools

After months of planning, we finally created the zometool model of superfullerene, a giant buckyball consisting of 60 smaller buckyballs.


The main lobby, 勝凱廳, of chemistry department of National Taiwan University.

New superbuckyball for math art exhibition of JMM 2013

I made a new superbuckyball for the Mathart exhibition of Joint Mathematical Meeting JMM held in San Diego this few days. The original one made by students of TFGS is too big (~60cm wide) to bring to the US. The new one is made by 8mm beads and is about 40cm wide.

One more picture of super buckyball

Super Buckyball as a Molecular Sculpture

I have written an article about super buckyball, Super buckyball as a molecular sculpture - its structure and the construction method (分子雕塑─超級珠璣碳球的結構與製作), in Chinese recently. I guess it will appear in the next issue of CHEMISTRY (The Chinese Chemical Society, Taipei) (化學季刊), a local chemistry journal in Taiwan. I tried to describe the construction method of the super buckyball in details in this paper. Also, in the reference 1 of this paper, I commented on how this paper was inspired by the Horibe's works, particularly, the idea of fusing many C60s into structured super fullerenes, which is the way I understand many of his beautiful models. I wrote it in Chinese because I hope local high-school students in Taiwan can read the paper more easily and reconstruct the model as a school activity.

分子雕塑 ⎯ 超級珠璣碳球的結構與製作

金必耀

臺灣大學 化學系

摘要:串珠是最適合用來建構各種芙類分子模型的材料,珠子代表芙類分子中的碳碳鍵,珠子的硬殼作 用正好模擬微觀芙類分子內的化學鍵作用。本文將介紹以模組化方式,讓許多對基本串珠模型建構有一 定認識的人,親手一起協同製作大型的超級芙類分子模型,非常適合作為中學化學與立體幾何教育的活 動,所製作的巨型模型不僅是一個為微觀分子模型,更可以說是一件具有科學含意的雕塑藝術品。

Super Buckyball as a Molecular Sculpture − Its Structure and the Construction Method

Bih-Yaw Jin

Department of Chemistry, Center of Theoretical Sciences and Center for Quantum Science and Engineering, National Taiwan University, Taipei 10617, Taiwan

ABSTRACT

Mathematical beading can be exploited to construct faithful physical model of any fullerene. The hard sphere interactions among different beads effectively mimic the ligand close packing of carbon-carbon bonds in fullerenes. Here we show a simple modular approach for students to build complicated graphitic structures together. Particularly, we describe in details the structure of the so-called super buckyball, which consists of sixty fused buckyballs, and our hands-on experience in making its bead model by the students of the Taipei First Girls High School collaboratively.

Super Buckyball

Mr. Hwang took this picture of my group the next day after I had the workshop last month.

Photos of Workshop for Exchange Students from Okayama Univeristy

Mr. Hwang took many photos for my workshop last Tuesday.

As usual, I gave a 30-min short talk before we started to make beaded C₂₀ and C₆₀.

While I was talking, students were free to examine the beadworks I brought to the workshop.

Prof. Sonoda (Kyushu University) also visited Taiwan in those few days. So I invited him to this workshop. He seemed to have fun making a beaded C₂₀.

Almost all of students succeeded in making their own C₂₀ and C₆₀ in this 3-hour workshop. You can see how happy they were when they finally made their own buckyballs.

After the workshop, I took a picture with my colleagues.

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.

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)

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.