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

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.

Short summary of zome-type superbuckyball part IV: Tori and miscellaneous

Lastly, we will address some other examples of this method of constructing graphitic structures from C60s. We will start with planar tori. All of the zometool models of these tori reported here are composed of entirely blue struts.

Threefold torus⊗C60 with g=1: C216 (C3v)

Fourfold torus⊗C60 with g=0: C224 (C2h)

As noted parenthetically, the rotational symmetry of this structure is actually twofold only.

Fivefold torus⊗C60 with g=0: C280 (C5v)

Sixfold torus⊗C60 with g=0: C336 (D3v)

Hyper cube⊗C60 with g=(0,3,1): C1536 (Th)

Perhaps this hypercube should be also classified as polyhedron in the previous post, since it IS a regular polyhedron in 4D. As can be seen, there are three different kinds of struts (two blues with different lengths and one yellow) in this structure. It is thus more difficult to find a reasonable set of g parameters that suits all of their intricate geometric relations. The CNTs in the outer layer are bent to accommodate this incommensurability. I have to point out that, in graph theory, 4D hypercube cannot be represented by a planar graph. This fact leads to considerable difficulty, if not impossible, in constructing corresponding graphitic structures with out previous approach of using the inner part of TCNTs. However, with the zome-type construction scheme this is nothing different than other graphically simpler structures.

Dodecahedron with V-shape edges⊗C80 with g=1: C4960 (Ih)

Note that one has to at least use C80 (or larger Ih-symmetric fullerenes) instead of C60 for the nodes, since the yellow struts joining at the same node are nearest neighbors to each other. Total fifty C80s were used, twenty for the (inner) dodecahedron and thirty for the (outer) edges. If I make it to the Bridges this year in Enschede, I'll bring a beaded molecule of this model with me.

Last but not least, how can I not play with trefoil knot?

Trefoil knot⊗C60 with g=(0,1,3): C912 (D3)

Unfortunately, so far I have not thought of any general scheme to construct arbitrary torus knots, as trefoil knot is only the simplest nontrivial case of them. In principle as long as the structure (or the space curve as for knots) can be constructed with zometool, there is also graphitic analogs of it and presumably beaded molecules as well. This concludes this series of posts. I'm currently working on a beaded molecule of C60⊗C60 with g=1 posted previously. I'm about half way there and I might talk about some specific beading strategies of this kind of structures later on.

Short summary of zome-type superbuckyball part III: Polyhedra

In this post I will present some other polyhedra built with the same principle. As you might know that the C20 and the C60 discussed in the previous post are exactly regular dodecahedron and the truncated icosahedron (an Archimedean polyhedron).

Cube⊗C60 with g=1: C624

I'd like to note that the symmetry of this structure belongs to the Th point group, although it looks as if it's got a higher symmetry of octahedral group. This is so because of the fact that locally there is only C2 rotational symmetries along each of the joining tubes. And there is no C4 rotational symmetry, not only in this structure but also in all other structures constructed with the golden ratio field where zometool is based on.

Icosahedron⊗C60 with g=2: C1560

I've posted a closely related high-genus structure quite some time ago using a different algorithm. There I treated the construction of high-genus fullerenes by replacing the faces of the underlying polyhedra by some carefully truncated inner part of a toroidal CNT. As suggested by Bih-Yaw that the current scheme of constructing superfullerenes is one another aspect of high-genus fullerene. Previously we are "puncturing holes" along the radial direction and connecting an inner fullerene with an outer one. Here we break and connect fullerenes in the lateral directions. Although topologically they are identical, as you can see the actual shapes of the resulting super-structures are quite different.

For your convenience I repost the structure here for comparison:

The construction of (regular) tetrahedron and octahedron requires the use of green struts. For now I have not come up with the corresponding strategy for green strut yet. We will move on to other polyhedra in the rest of this post.

Small Rhombicosidodecahedron⊗C60 with g=1: C5040

This Archimedean solid is of special interest since the ball of zometool is exactly it. The squares, the equilateral triangles, and the regular pentagons correspond directly to C2, C3, and C5 rotational axes, respectively. The existence of this superfullerene guarantees the possibility of building hierarchy of Sierpinski superbuckyballs. In other words, this superbuckyball can serve as nodes of a "supersuperbuckyball", with the connecting strut automatically defined. Although we are likely to stop at the current (second) level because of physical limitations, either using beads, zometool, or even just computer simulations.

Rhombic triacontahedron⊗C60 with g=1: C3120

You need red struts only for this structure.

Five compound cubes⊗C60 with g=(0,1): C6000

You need blue struts with two different lengths for this structure, which is the reason why the g factor is a two component vector here. Note that at each level the length of the strut (measured from the center of the ball at one end to the center of the other) is inflated by a factor of golden ratio. Thus, comparing to other superfullerenes introduced previously, there is additional strain energy related to the commensurability of the lengths of CNTs. It is always an approximation to use a CNT of certain length to replace the struts of a zometool model. It is also interesting to note that, comparing to the zometool model, this particular superbuckyball makes clear reference to the encompassing dodecahedron. In this perspective it is not surprising that the structure has Ih symmetry.

Dual of C80⊗C60 with g=2: C5880

This structure is obtained by inflating each of the equilateral triangles of a regular icosahedron to four equilateral triangles. An equivalent way of saying this is "inflation with Goldberg vector (2,0)".

In addition to the above mentioned, Dr. George Hart has summarized some of the polyhedra construtable with zometool here. In principle they can all be realized, at least on computers or with beads and threads, by this methodology. And there is going to be one last post in this series to cover those that are not classifiable into categories discussed so far.

Short summary of zome-type superbuckyball part II: Superfullerenes

We will move on to our main and original goal of devising this technique: constructing superfullerenes. Readers familiar with zometool would know that C60 can be built with solely blue struts (two fold rotation axes). So here it is.

C60⊗C60 with g=0: C3240

I've borrowed the notation of Kronecker product (⊗) since these two mathematical operations are in some sense similar: each entry (atom) of the matrix (fullerene) before the ⊗ sign is "expanded" into the second matrix times the original entry (the spatial location of that atom). The meaning of g will become clear once you see an example of g=1 as shown below.

C60⊗C60 with g=1: C4680

It is obvious that g indicates the length of the struts (straight CNTs). In the first case the length is essentially zero, so pairs of heptagons "merge" into octagons at the interface. For clarity the rotatable models of four connected superatoms of the above two superbuckyballs are presented below as well. Since all of these superatoms are identical and can be related through mirror symmetries, readers of interest can start with them to build your own models.

For convenience I also show the rotatable models for the superbuckyball proposed and its beaded model constructed by Bih-Yaw in the previous posts.

C60⊗C60 with g=0: C2700

C60⊗C60 with g=1: C4500

Upon close inspection, notice that there is still local threefold symmetry at each of the node in this case. While on the contrary, there are only mirror symmetries in the zome-type superbuckyball. This asymmetry leads to the fact that there is almost no strain when constructing the beaded model of these structures, or even the actual microscopic realization. This is not an issue concerning the 1D structures in the previous post, since they are all simply connected and there is no such thing as commensurability among multiple struts that join at the same node. However, the above two superstructures seem to be pretty stable and beadable, which surprise me a lot in this regard. According to Bih-Yaw, the tension of the five-member rings and the stress in the six-member rings magically balance each other. This is not so for the dodecahedron case, where tension is built everywhere in the model without being compensated by stress.

Having demonstrated the above mentioned, there is nothing so different in constructing other types of superfullerenes. Below I listed a few that I have done coding with.

C20⊗C60 with g=1: C1560

Although I have not tried to build this one with beads yet, I believe that it is quite doable in the sense of stability as mentioned above.

C80⊗C60 with g=1: C6720

C180⊗C60 with g=1: C15480

I'd like to point out in the last two cases you need red struts as well as blue ones. It can be shown that all (n,n) or (n,0) type icosahedral fullerenes are constructable from zometool (with blue and red struts). For now I just manually find out what are the atoms needing to be deleted/connected, well, in an efficient way. I hope one day I can come up with a general automatic routine that does all these for me, which should be taking account of different orbits in a symmetry group. C180 (a (3,0) Ih fullerene) is the largest one I've ever played with.

I plan to talk about other types of regular polyhedra in the next post.

Short summary of zome-type superbuckyball part I: 1D Linear and Helical C60 Polymers

Recently I've been playing with all kinds of these superbuckyballs, based on the methodology of replacing balls and struts of zometool by Ih-symmetric fullerenes and straight CNTs. Taking C60 for example, blue struts correspond to removing two atoms next to some particular C2 rotation axes. And the connecting CNTs are of chiral vector (4,0). On the other hand, yellow and red struts correspond to C3 and C5 rotation axes, respectively. In addition, due to the property of the golden field, the algebraic field of zometool, superstructures using yellow or red struts only have the possibility of being polymers of C60. This means that the number of atoms is a multiple of 60, i.e. no atoms needed to be deleted or added when constructing the superbuckyball.

Here I will briefly summarize some of the cases I've done coding with. Hopefully I'd soon come up with a short paper ready to submit to the Bridges 2013 on this topic.

First let us start with the trivial C60 dimers. As mentioned, structures with red or yellow struts cases have the atom-preserving property. The C120 isomers corresponding to joining two C60s along their fivefold and threefold axes are shown below. I should mention that they were also discussed in Diudea and Nagy's book . In particular for the C3-joined case there are two possibilities of local atomic connectivity.

C3-fused C120, case 1 (with octagons and pentagons at the interface)

C3-fused C120, case 2 (with heptagons at the interface)

C5-fused C120

For the case of blue struts (twofold rotation axes)

C2-fused C116

C2-fused C132

Some of them were already made previously by us, see here for example. But we did not realize back then this particular connection with zometool. To my knowledge, there has not been any experimental characterization of such dimers. Synthetic chemists do make C60 dimers but those are of partial sp3 characteristics, i.e. some interfacial atoms have four neighbors instead of three. Please refer to Diudea and Nagy's book for further details if you are interested.

One can come up with the one dimensional C60 chains without too much effort by enforcing periodic boundary condition. So the structure repeats itself indefinitely along the direction of polymerization. See for example below.

Also, it is one step away from constructing the 2D analog of this kind of structure.

A little bit more sophisticated extension of the above scheme is to consider helical screw symmetry. A (discrete) helical curve is defined by the angle between adjacent unit cells and the dihedral angle between next-nearest neighbors. I recommend readers of interest to play with the awesome virtual zome program vZome developed by Scott Vorthmann. You have to write Scott an email for the license of the full version of vZome. Anyway, here are some examples of helical C60 polymers.

C3-fused fourfold C60 helix

C5-fused fivefold C60 helix

Notice that if you are looking along the axes of the helices, the C60s that are four/five unit cells away lie exactly on top of each other. Curiously, this result is actually symmetry-determined, since I've tested with the relaxation scheme that does not require such symmetry. In other words, even if I optimize the geometries with full degrees of freedom of a general helix, the screw angles will still be 2*pi/4 or 2*pi/5 in the above cases.

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

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.

A workshop for a local meeting on Cognition and Digital Education

I am going to give a special exhibition and a workshop for a local meeting on Cognition and Digital Education which will be held in the main campus of National Academics of Educational Research (國家教育研究院) the Sanxia county (三峽) on Aug. 18 (tomorrow). The meeting is supported by the Math and Information Section of the National Science Council of Taiwan.

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: