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

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".

Sierpinski tetrahedron

Third level Sierpinski superbuckyball: C20⊗C20⊗C20

As was asked by Bih-Yaw, I think in principle one can always go on the process of using fullerenes to construct superfullerene, and then treat the result as the new module to build a supersuperfullerene etc.. The idea is the same. The problem is always physical limitations: the structure goes too heavy to support itself or you run out of memory trying to build that on a computer. Here is the simplest nontrivial case that I can do on my laptop, a third level superfullerene C20⊗C20⊗C20.

C20⊗C20⊗C20 with g=(1,1): C15920 (Ih)

It is clearer to see when there are only two adjacent nodes:

I would say this is not unbeadable. However, as mentioned, one has to make sure the structure is strong enough to hold itself up. In my experience the best shot is to go with 3mm plastic beads and 0.4mm fish lines, which I'm currently using for the construction of a C60⊗C60 superfullerene. Other possibilities are icosahedron⊗icosahedron⊗C60 or cube⊗cube⊗C60, but I think they are not as representative and illuminating as this one.

Clathrate cluster

I bought some more green rice-shape beads last week and managed to finish this interesting clathrate cluster of 60 dodecahedra in the last weekend. One can still see deformation of many dodecahedra in this clathrate cluster though.

Buckyball made of 60 dodecadedra

I made this structure with beads last weekend. Still unfinished. The finished structure should have 60 dodecahedra arranged like a buckball. One has two ways to interpret this structure:
1. If every dodecahedron represents a carbon atom, we have a standard C₆₀.
2. If we still use beads to represent CC bonds, then we have a giant molecule, C₇₅₀. In this molecule, 450 carbon atoms are sp³ hybridized or tetra-valent and 300 atoms are sp² hybridized or trivalent. But I suspect these sp² hybridized carbon atoms are not energetically favorable, so it is better to have hydrogen atoms connected to these sp2-carbons. Then we get C₇₅₀H₃₀₀!



The bead model of this structure (see here) might be first constructed by Emilie. She asked me to comment about this structure in my blog long time ago (I couldn't find the exact location though).

I decided to make one from beads after I saw the same structure made by a toy designer, Dick Esterle, who actually invented this kind of toys, in the Bridges conference last week.

Sierpinski Tetrahedron (Level 3)

The Bead Model of Sierpinski Tetrahedron

Previously, I have shown the Sierpinski buckyball made by Chuang (莊宸) and his classmates (Class 2007, Chemistry, NTU) about three years ago. It is also straightforward to make other Sierpinski Platonic solids. Here is the simplest one: the beaded Sierpinski tetrahedron.

Two more pictures for the Sierpinski buckyball

This Sierpinski buckyball is currently at the chemistry departmental office of NTU.

Comments on Sierpinski buckyball

Recently Mark left a comment on the Sierpinski's buckyball we proposed:

mark said...

That is really good and attract the Small sierpiski beaded fullerene.
butFullerenes are a family of carbon allotropes consisting of molecules composed entirely of carbon atoms arranged in the form of hollow spheres, ellipsoids, or tubes.
9/15/2009 1:41 PM

Here is my response:

You are certainly right about fullerenes. There are indeed many possible sp2 based fullerene structures. We have recently published a few papers on the systematics of toroidal CNT, helical CNT and High-Genus fullerenes.

1. Chuang, C.; Fan, Y.-C.; Jin, B.-Y. "Generalized Classification Scheme of Toroidal and Helical Carbon Nanotubes." J. Chem. Inf. 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. Inf. Model. 2009, 49, 1679-1686. DOI: 10.1021/ci900124z
3. Chuang, C.; Jin, B.-Y. "Systematics of High-Genus Fullerenes." J. Chem. Inf. Model. 2009, 49, 1664-1668. DOI: 10.1021/ci9001124, ACS News & Research, June 2009.
4. Chuang, C; Jin, B.-Y. “Hypothetical Toroidal, Cylindrical, Helical Analogs of C60.” Accepted for publication in J. Mol. Graph. Model. 2009. http://dx.doi.org/10.1016/j.jmgm.2009.07.004
5. Jin, B.-Y.; Chuang, C.; Tsoo, C.-C. "The Wonderful World of Beaded Molecules." CHEMISTRY (The Chinese Chemical Society, Taipei) 2008,66, 73-92. (in chinese).

I personally doubt the possible existence of Sierpinski fullerenes. But, mathematically speaking, it is still an interesting generalization. I have never seen any work on this possibility. As far as I know, the most popular ones are Sierpinsky tetrahedron and cube.

To see more discussion on the Sierpinski's buckyball and a simple estimation of its fractal dimension, please check several of my posts in June 2007.

http://thebeadedmolecules.blogspot.com/2007_06_01_archive.html

I still remembered that when I gave a talk on "Chemistry, Geometry and Art: The wonderful world of fullerenes" in the math department of National Taiwan University in the early July of this summer vacation, someone in the audience (an expert in the fractal geometry) was quite intrigued by the Sierpinski's buckyball I shown in one of the slides. He told me that no one has ever investigated the Siepinsky regular and semiregular polyhedra yet. I think it might be interesting if we can do something about it.

Small Sierpiski beaded fullerene

(created by Q.-R. Huang)

Spierpinski icosahedron and other fracal objects

I found the following site, in which they have created amazing artworks of Sierpinski icosahedron with modular paper folding.
However this site is in German. I am not sure how they did it. It seems to me they have a summer camp for high school or primary school kids to create this model collaboratively.
http://www.mathematik.uni-muenchen.de/~geotage/

http://www.mathematik.uni-muenchen.de/~geotage/rweber/

Here is the picture of the model they created:
http://www.mathematik.uni-muenchen.de/~geotage/rweber/PIC00025.JPG


Alternatively, Wikipedia has a long list of fractal dimensions for many different objects:
List of fractals by Hausdorff dimension


Many natural objects exhibit fractal structure too:
Check this amazing Fractal food in supermarket out.

Regular Sierpinski Polyhedra

I found an interesting article on the Sierpinski Polyhedra at "http://faculty.gvsu.edu/schlicks/phdra.pdf" entitled
"Regular Sierpinski Polyhedra1" by Aimee Kunnen and Steven Schlicker.

They made a detailed study on the Spierpinski polyhedra in general, and also report he fractal dimensions for five regular polyhedra:

1. Sierpinski tetrahedron: log(4)/log(2) = 2
2. Sierpinski hexahedron: log(8)/log(2) = 3
3. regular octahedron: log(6)/log(2) ≈ 2 585
4. Sierpinski dodecahedron: log(2)/log(d/d1) ≈ 2 32958
5. Sierpinski icosahedron: log(12)/log(d/d1) ≈ 2.581926
The meaning of d and d1 is basically the scaling factor I mentioned in the previous post, but for details, please check the original paper. The calculation of this ratio is a geometric problem, which can be solved in principle. Instead of doing complicated calculation, I just performed an estimate by straightforward inspection on the picture I obtained from the scanner.

Now the problem is what the value of log(d/d1) for the Spierpinski buckyball (Spierpinski truncated icosahedron) Chuang and his classmates made is. Or more generally, find out the contruction rules and the corresponding fractal structures, and dimension for Sierpinski Achimedean solids, etc.

Sierpinski buckyball

After a short discussion with Chuang, I now think the Sierpinski buckyball is a better name for this system than the Escher buckyball.

Interestingly, it is not hard to find out the Hausdorff (fractal) dimension of this system:

D=log 90/log scaling factor = log(90)/log(8) ~ log(90)/log(7) = 2.16 ~ 2.31

(More careful measurement indicates the scaling factor is 16 cm/2.5cm =6.5, thus D = 2.4)

where I made a rough estimation for the magnification in order to get the scaling factor from the small buckyball to the large buckyball. Detailed calculation is more involved for the 3-D geometry of truncated icosahedron is needed.


Sierpinski's pyramid from Wikipedia (Fractal dimension = log 4 / log 2 =2):

Escher's buckyball

Chern Chuang and his classmates created this amazing beaded buckyball made from 90 small beaded buckyballs. They are going to present this beautiful artwork as a gift to our chemistry department in the graduation ceremony for NTUCHEM class 2007 this coming weekend. I wish I can have a picture for this event.

As to the name of this design, we call this ball as a buckybuckyball, or level-2 buckyball. Since we can again use this buckyball as a new type of beads to create the next level of buckybuckybuckyball (level-3 buckyball), continuing in this direction recursively, we can have a very complicated fractal structure of Sierpinski type. Also, this kind of artwork containing hidden recursive structures has been first used Escher in his amazing artworks, so it is not a bad idea to call this kind of buckyball as Escher's buckyball, or Escher's ball.