2013年Bridges會議展出的數學藝術品 C60⊗C60

Bridges會議是一個獨特的年度盛會，它匯集了數學、藝術、音樂、建築、教育和文化領域的人們。會議的藝術展覽部分尤其引人注目，展示了數學概念如何激發和融入藝術創作之中。在 **2013年的Bridges會議**上，一件名為 "**C60⊗C60**" 的藝術品以其巧妙的設計和深刻的數學意義吸引了參觀者的目光。

背景介紹：富勒烯與超富勒烯

在欣賞這件藝術品之前，我們需要了解一些關於 **富勒烯（Fullerene）** 和 **超富勒烯（Superfullerene）** 的背景知識：

• 富勒烯（Fullerene）： 富勒烯是一類完全由碳原子組成的籠狀分子，其中最著名的就是 **C60，又稱巴克球（Buckyball）**。C60分子具有足球狀結構，由20個六邊形和12個五邊形構成。
• 超富勒烯（Superfullerene）： 超富勒烯是一類假設的分子結構，其設計理念是將一個母體分子（A）的每個原子替換為一個巴克球（C60），並通過直的碳納米管保持母體分子的連接性，這種結構通常表示為 A⊗C60。

藝術品詳情：C60⊗C60

C60⊗C60

創作者：莊宸(Chern Chuang)

• 尺寸： 17 x 17 x 17 厘米
• 材料： 夜光3毫米珠子，魚線
• 創作年份： 2013

這件藝術品 "**C60⊗C60**" 正是 **將超富勒烯的概念應用於C60本身** 的一個直接例子。換句話說，它是一個以巴克球（C60）為母體分子，將其每個“原子”（即碳原子）替換為另一個巴克球（C60）所形成的更複雜結構。

根據描述，這個特定的例子可以被視為一個 **第二層的Sierpiński結構** 。Sierpiński結構是一種分形，通過不斷地自我複製和縮小產生複雜的圖案。將這個概念應用於富勒烯，創造出層層疊加的結構，展示了數學中迭代和自相似性的思想。

這件 "**C60⊗C60**" 模型包含 **4680個碳原子**，這相當於 **7020個珠子** 。作者使用了夜光3毫米珠子和魚線來構建這個精細的結構。

值得注意的是，莊宸與金必耀一起參與2013年於聖地亞哥舉行的JMM（Joint Mathematics Meetings）藝術展覽，金必耀展示過一個密切相關的結構，即 **C4500超富勒烯**，莊宸從這個結構所啟發，進而設計出此C60⊗C60。

通過使用串珠這種物理媒介來呈現複雜的分子結構，"C60⊗C60" 不僅展現了數學和化學的幾何之美，也提供了一種直觀的方式來理解抽象的分子結構和數學概念。

關於作者

• 莊宸 (Chern Chuang): 目前是UNLV化學系教授，當年是麻省理工學院化學系研究生，他的嗜好包括使用zome幾何原理構建複雜的串珠結構分子模型，尤其對超富勒烯等假想分子感興趣。

參考資料

• Chern Chuang | 2013 Bridges Conference | Mathematical Art Galleries: #### Statement. https://www.bridgesmathart.org/2013/exhibitions/gallery/chern-chuang/

© 2025 珠璣科學|C60⊗C60

2013年Bridges會議展出的超十二面體與V形連接⊗C80

Bridges會議是一個連接數學、藝術、音樂、建築等領域的國際會議。其數學藝術展覽部分是獨一無二的平台，展示了數學概念如何激發藝術創作。在 **2013年的Bridges會議**上，一件名為 "**Superdodecahedron with V-shaped Connections⊗C80**"（帶V形連接的超十二面體⊗C80）的藝術品，以其複雜的結構和巧妙的設計，引人入勝。

背景介紹：十二面體、Zome幾何與超富勒烯

為了更好地理解這件藝術品，我們需要簡要介紹以下概念：

• 十二面體（Dodecahedron）： 一種由十二個正五邊形組成的正多面體，具有20個頂點和30條邊。
• Zome幾何： 一種基於一組彩色連接件（節點）和支桿（struts）的建構系統，可以用於創建各種幾何結構。Zometool以其能夠表現複雜的對稱性和空間關係而聞名。
• 超富勒烯（Superfullerene）： 一類假想的分子，其構建方式是將一個母體分子（A）的每個原子替換為一個富勒烯分子（通常是C60），並通過連接這些富勒烯分子的碳納米管來保持原有的連接性，表示為 A⊗C60。

藝術品詳情：Superdodecahedron with V-shaped Connections⊗C80

Superdodecahedron with V-shaped Connections⊗C80

創作者：**莊宸(Chern Chuang)**

• 尺寸： 43 x 41 x 37 厘米
• 材料： 6毫米塑料珠子，魚線
• 創作年份： 2013

這件名為 "**Superdodecahedron with V-shaped Connections⊗C80**" 的藝術品，是基於 **zome幾何的十二面體模型** 進行改造的 。具體來說，作者 **將一個zometool十二面體模型中的每個藍色支桿替換為兩個通過一個額外小球連接的黃色支桿**。

由於在同一個球上連接的三個黃色支桿過於接近，為了使結構能夠順利搭建，作者 **使用了C80而不是C60**。C80是第二小的符合獨立五邊形規則的二十面體富勒烯。獨立五邊形規則是富勒烯結構穩定性的一個重要因素。

除了十二面體的 **20個頂點** 之外，這個結構還增加了 **30個作為連接器的頂點**。因此，這個結構總共包含 **4960個碳原子**，相當於 **7440個珠子**。

這個作品巧妙地利用了 **zome幾何的特性**，即C60和zometool的球都具有 **二十面體對稱性**，並且連接球的支桿總是沿著球的某些共同的對稱軸排列。這種設計思路有助於簡化構建複雜超分子結構的任務，並使其具有 **最小的應變能** 。作者在創作中特別關注一類稱為 **超富勒烯的假設分子**，並探索如何使用珠飾模型和zome幾何的原理來構建這些複雜的結構。

通過將抽象的數學和化學概念轉化為具體的藝術品，"Superdodecahedron with V-shaped Connections⊗C80" 不僅展示了數學的幾何之美，也體現了藝術家在探索複雜結構和對稱性方面的創造力。

關於作者

• 莊宸 (Chern Chuang): 目前是UNLV化學系教授，當年是麻省理工學院化學系研究生，他的嗜好包括使用zome幾何原理構建複雜的串珠結構分子模型，尤其對超富勒烯等假想分子感興趣。

參考資料

• Chern Chuang | 2013 Bridges Conference | Mathematical Art Galleries: #### Statement. https://www.bridgesmathart.org/2013/exhibitions/gallery/chern-chuang/

© 2025 珠璣科學|超十二面體與V形連接⊗C80

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

High-Genus Fullerene

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.

Bucky Helmet

High-geneus fullerene by Mr. Horibe

I just got an email from Mr. Horibe with a link to his page, which gives many photos on the different stages of making a dodecahedral high-genus fullerene. It should be useful for anyone who wish to make this model.

Mr. Horibe also emailed me a beautiful picture of the backyard of his house in Tajimi. We took a picture together at the same place this June, without snow though.

Two artworks for the Mathematical Exhibition of Joint Mathematical Meeting

Chern and I submitted two artworks to the Joint Mathematical Beading, which were accepted today: 1. Super Buckyball of Genus 31
2. Beaded Hilbert Curve, step two

In addition to the beadworks we submitted, we also noticed five Platonic bead models made by Ron Asherov. His bead models have multiple beads in an edge, which are similar to our works a few years ago. I labeled this type of bead models with Edge with Multiple Beads, where you can find all the posts. He doesn't seem to know our works along this direction, though.
He also mentioned that the Nylon string passes adjacent edges exactly once with carefully chosen path of string, which is simply the consequence of Hamiltonian path on the dual graph of the corresponding Platonic solids. We can view the whole beading process as a path through the face of polyhedron. Thus if there exists a Hamiltonian path through each face once (Hamiltonian path for the dual polyhedron), then the Nylon string will go through each beads exactly twice and only twice. Of course, you can also say the Nylon string will go through the adjacent edges exactly once. They are the same thing.

Here is a model made by Chern Chuang almost five years ago:

I was quite surprised by its rigidity when Chern showed me this model. At that time, people questioned me about the meaning of beads. I told some of my colleagues that spherical beads represent chemical bonds instead of atoms. Atoms are not shown in the bead model explicitly, instead, they are located at somewhere three beads meet. Most chemists feel uncomfortable with this connection. So Chern and I tried try to explore with the shape of beads and multiple beads and hope that they can better represent the shape of chemical bonds. So that is why we have these models in which multiple beads represent an edge.

But now, I have the valence sphere model of chemical bond as the theoretical foundation of bead models. Spherical beads are in fact the simplest possible approximation of electron pairs, in accord with the principle of Occam's razor. So to build a model of a molecule with only beads and strings is equivalent to performing a molecular analogue computation with beads. The result of computation is the approximate electron density of the corresponding molecule without referring to the Schrödinger equation or atomic orbitals (this is the comment I got from Prof. H. Bent). I have written an article on the connection between bead models and valence sphere model in Chinese for September issue of Science Monthly (科學月刊). I think I should write something about the molecular analogue computation with beads.

Of course, what I am saying above is to view bead models as molecular models. Results of mathematical beading do not need to have any connection to the molecular world. For instance, the beaded Hilbert curve accepted by the JMM mathematical art exhibition is a good example.

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.

Dendritic structures

Mr. Horibe made a number of dendritic fullerenes which are similar to the Kepler's stellated polyhedrons. By using the Euler theorem, it is quite straightforward to show that, in a cage-like fullerene without hole, N₅-N₇=12, where N₅ and N₇ are the number of pentagons and heptagons, respectively. In the typical fullerenes where the local Gaussian curvature is positive everywhere, we should have N₇=0. The total number of pentagons is 12. Another way to put it is to introduce the so-called topological charge, 1 for a pentagon and -1 for a heptagon. So the topological charge of a cage-like fullerene is 12.

Heptagons always generate a negative Gaussian curvature. For a cage-like fullerene, whenever we introduce an extra heptagon, we have to include a pentagon in order to satisfy the identity N₅-N₇=12.

One can replace a pentagon in a Goldberg icosahedron (icosahedral fullerene) by 6 pentagons (a hemisphere of C20) and 5 heptagons to get a spike. So the topological for this area is still 1 after replacement. Similarly, we can do the same replacement for all other 11 pentagons to get a dendritic structure.



Of course, we can use the same kind of trick to "grow" a spike (a carbon nanotube endcapped with the hemisphere of a C₂₀) along the normal direction of any pentagon on any kind of graphitic structure.

Incidentally, this structure without the C20 caps is just the inner part of the high-genus fullerenes we have done before.

Zometool models at the Quasicrystal international conference

The newest Nobel laureate in Chemistry, D. Shechtman, was invited to Taiwan to participate a quasicrystal international conference held in the National Taipei University of Technology (May 7-9), a university very close to the National Taiwan University where I am working.
Yuan-Chia, Qian-Rui, and Hsin-Yu moved the two large zometool models, a 3D quasicrystal consisting of two types of golden rhombohedra (prolate and oblate) and a six-layer carbon onion (C20, C80, C180, C320, C500, and C720), to the conference's lobby area. Most participants seem to enjoy these two types of quasicrystal models, cluster and tiling, a lot.

Prof. Shechtman took a picture of me in front of the zometool model of carbon onion.

I also gave Prof. Shechtman a bead model of high-genus fullerene, a topologically nontrivial quasicrystal :-), as a gift.

Super Buckyball

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

High-genus fullerenes

I made two more high-genus fullerenes with octagonal necks using 8mm beads and 0.6mm Nylon threads. If I remember everything correctly, Chern and I should have made four of this molecule (one in China and three in U.S.) before.

HG-Fullerenes with octagonal necks No. 5 & No. 6

Since this model contains less than one thousand beads, so I managed to finish one of them in a day.

One more super carbon tetrahedron

I made another super carbon tetrahedron (超級碳正四面體) consisting of four fused C60s with 8mm beads. Unlike the previous one, the two neighbored C60s are connected by a ring of hexagons.

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)

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）.