Beaded Entangled Cuboctahedron 珠飾纏繞的立方八面體

我們很高興為大家介紹一件新穎的數學藝術作品：珠飾纏繞的截角八面體 (Beaded Entangled Cuboctahedron) 。這件作品是由國立台灣大學的學生林軒霆、陳品希在金必耀教授開設的「分子美學」課程中創作的。

作品詳情

• 作品名稱：珠飾纏繞的截角八面體 (Beaded Entangled Cuboctahedron) [1]
• 創作者：林軒霆、何厚勳、陳品羲
• 創作年份：2022 年
• 尺寸：25.0 x 25.0 x 25.0 公分
• 材料：6 毫米塑膠珠、魚線

結構特色

這件藝術品的核心概念是利用八個由珠子串成的三角形管相互纏繞， 構成一個類似截角八面體的模型。在設計過程中，助教何厚勳提供的模擬模型對他們的創作非常有幫助。結構的穩定性和剛性完全依賴於這八個管子的存在。

為了實現管子自然的轉動，作者巧妙地在適當的位置加入了五邊形和七邊形，以創造正向和負向的曲率。此外，他們使用黑色和白色的珠子來增強整體結構的清晰度。

拓撲背景

為了更好地理解這個結構，讓我們簡單介紹一下截角八面體 (Cuboctahedron) 的拓撲性質。截角八面體是一種阿基米德立體，它有 14 個面，分別是 8 個等邊三角形和 6 個正方形。它有 12 個相同的頂點，每個頂點連接兩個三角形和兩個正方形。此外，它有 24 條邊，每條邊連接一個三角形和一個正方形。截角八面體可以看作是立方體或正八面體通過截去其所有頂點得到，直到每條邊的中點。它的對稱性很高，屬於正八面體群。

「纏繞」的意涵

這件作品被命名為「纏繞的立方八面體」，其「纏繞」的概念源於構成結構的八個三角形管的相互交織和連結。這種設計不僅賦予了作品獨特的視覺效果，也體現了結構的穩定性是如何通過這些管子的互相作用來實現的。策略性地加入五邊形和七邊形，使得這些「管子」能夠以一種看似自然的方式彎曲和連接，進一步強化了「纏繞」的感覺。

總而言之，「珠飾纏繞的截角八面體」不僅是一件視覺上引人入勝的藝術品，也巧妙地將幾何結構和手工藝技巧結合在一起。它展現了「分子美學」課程的理念，即將分子結構視為藝術品進行創作。

欲了解此作品，請訪問 2023 Bridges Conference | Mathematical Art Galleries。

© 2025 珠璣科學 |珠飾纏繞的立方八面體

Beadfinity 串珠無限

本作品由國立臺灣大學化學系博士生 何厚勳 創作，並入選 Bridges 2024 數學藝術展覽。

作品簡介

「Beadfinity」是一件基於數學曲線與碳納米管（Carbon Nanotubes, CNTs）結構的藝術創作，該作品透過珠子和魚線來模擬分子層級的幾何形狀，展現數學與化學之間的深層聯繫。

創作背景

本作品的靈感源自 Xuan-Yun Wu同學在 2023 年秋季「分子美學」課程中設計的 S 形曲線。該曲線與環面相連，從特定角度投影時呈現「∞」（無限）符號的幻象。何厚勳透過精確排列七邊形和五邊形，使碳納米管（CNTs）形成類似的曲線結構，並以珠子串聯 CNT 模型，構建出這件三維數學藝術作品。

作品特色

• 材質： 3 毫米磨砂珠子、魚線
• 尺寸： 10.0 × 5.0 × 5.0 公分
• 創作年份： 2023 年
• 數學概念： 曲線與拓撲結構、碳納米管幾何排列

何厚勳 是國立臺灣大學化學系博士生，在 金必耀教授 的指導下，研究石墨結構的幾何形狀，特別是 碳納米管（CNTs）與 富勒烯（Fullerenes）等碳分子的數學表現方式。

金必耀教授透過「數學串珠」的方法，以珠子和魚線建構分子模型，例如富勒烯的三維結構。何厚勳進一步將此方法應用於 CNT 結構，使複雜的空間曲線透過珠串藝術呈現，提供一種獨特且具有視覺衝擊力的科學探索方式。

數學藝術與分子結構

許多分子結構的幾何排列與數學中的拓撲形狀息息相關，例如：

• 富勒烯（Fullerene）： 由五邊形與六邊形組成的碳球體，類似足球的結構。
• 碳納米管（CNTs）： 由石墨層卷曲形成的奈米管，具有高度對稱性與力學剛性。
• 環面曲線（Toric Curves）： 具有特殊拓撲特性的曲線，可用於模擬分子內部的電子行為。

「Beadfinity」透過幾何模型呈現這些結構，讓觀者直觀地感受數學與化學的交會。

相關連結

• Bridges 2024 數學藝術展
• 國立臺灣大學化學系
• 金必耀教授 - 國立臺灣大學
• 碳納米管（Carbon Nanotubes, CNTs）
• 數學藝術（Mathematical Art）

「Beadfinity」不僅是一件數學與藝術結合的作品，也讓人看到納米技術與分子幾何結構的美學價值。透過珠串建構的方式，我們能更直覺地理解碳分子的排列方式，並探索其在科學與藝術中的應用。

© 2025 珠璣科學 |Beadfinity 串珠無限

Concentric carbon nanotori - a molecular gyroscope (Molecular Aesthetics 2020)

Klein bottle

Three students in my class, Molecular Aesthetics, constructed a bead model of Klein bottle.



My contribution to their work is to tell them that this model can be done with mathematical beading. Especially, I showed them how to systematically construct the interpenetration of a graphene surface and a carbon nanotube, which is essential part of this model. In addition to my part, the TA of this course, Hou-Xun Ho (何厚勳), who developed a systematic strategy to build smooth tori with large rotational number, plays an important role for successful construction of this model with beads.

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

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.

Beadworks from the viewpoint of a chemist

The difference that Mr. Horibe and I see these bead models is probably due to our different backgrounds. Mr. Horibe is trained as a math teacher, while I am a theoretical chemist. Even though I am acquainted with some basic mathematics for my own researches, I emphasized the chemical implication of these bead models as a method to realize nanoscaled molecules and materials. For instance, I recognized how to correctly interpret these bead models microscopically immediately after I my first buckyball with beads was done. And I also fully appreciated the power of mathematical beading as a new, and powerful method to make arbitrary fullerenes and, later on, further extended the method to arbitrary molecules.

Collaborating with Chern, now a graduate student of chemistry in MIT, we quickly became familiar with topological and geometric aspects of certain graphitic structures. Of course, Chern Chuang plays a critical role. He is very good at geometry, especially polyhedra and minimal surfaces. He also likes to make all kind of mathematical models since he was a high school student. He worked out the necessary mathematics for describing many interesting graphic structures systematically for his Master degree thesis. At the beginning, his focus was on the carbon nanotori with a goal to find out their general structural rules. We believe we have a good understanding of them now. The main findings are summarized in two papers:

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.

We also submitted a paper entitled “Comments on Structural Types of Toroidal Carbon Nanotubes” to Chemistry European Journal last year. In this paper, we summarized the carbon nanotori in thirteen canonical achiral structures, which are related to each other through three types of geometric maniputations, namely rim rotation, horizontal shift, and generalized Stone-Wells transformations. Unfortunately, the paper was rejected because of the criticism we had on the paper by F. Beuerls et al.

We quickly realized that we could systematically create quite generally a large family of graphitic structures using carbon nanotori as building blocks. This includes helically coiled carbon nanotori (carbon helices for simplicity), carbon torus knots, high-genus fullerenes, singly-, doubly-, and triply periodic minimal surfaces. Of course, one can also have all kind of variations of pseudo-periodic minimal surfaces by using heptagons instead of octagons. Part of this endeavor is summarized in "Systematics of Toroidal, Helically-Coiled Carbon Nanotubes, High-Genus Fullerenes, and Other Exotic Graphitic Materials" (Procedia Engineering, 2011, 14, 2373-2385) by Chuang, C., Fan, Y.-C. and me. These graphitic structures can be succinctly summarized as follows:

TCNTs with no latitude coordinates

In addition to the eight structural types of carbon nanotori, Chern also showed that there are five other different tori without latitude coordinates defined. Chern also gave the systematic transformation rule, which we call the generalized Stone-Wells transformation, to derive them from classes A, B, and E, respectively.

Of course, Mr. Horibe has made many different structural types of tori with beads as shown in the following photo taken in the Nagoya's Children and Family Center. One can see he enjoyed making tori with 7- up to 9-fold rotational symmetry quite a lot. However, I didn't check very carefully how many different types of tori (according to our classification scheme) he has done.



I also discovered that he has a few tori without latitude coordinates. Incidentally, I found that Mr. Horibe had made a particular torus which is exactly the same as the one (class K) that appeared in one of slides of my introductory talk given in the Nagoya's Children and Family Center. Even the color codings are the same. Quite amazing.

Two gifts from Mr. Horibe

I am usually the person who gave people beadworks as gifts. But I myself got a beaded C60 as a gift from Mr. Horibe on our visit to his home. And the next day, when he participated my workshop in Nagoya, I got another giant torus as a gift.



His model is quite different from mine because most often he used only a single elastic rubber band with exact length to make his beadworks. In this sense, the path of his elastic string exactly corresponds to the Hamiltonian path exactly and has, of course, the minimal length for a particular model. So, the elastic string passed every bead in the model twice and only twice. To avoid beads fall apart, Mr. Horibe then made a knot in the end. So you can pull or press his bead models to some extent.



Possibly due to his training as mathematics teacher, he always uses a single string even for structures that contain more several thousand beads. I was quite surprised when I just heard of it. Just imagine how one can use a single string to bead the giant green stellated dodecahedron as shown in the following photo. The problem is that it would become very difficult to bead with such a long string. The good thing is that the path of string in his case is really a Hamiltonian path no matter how many beads a model contains. I guess that he also needed to plan well before he started because it is not always trivial to find out the Hamiltonian path. Sometimes, we just got trapped as weaving process goes. I don't know how he managed to do it with only one string especially for certain structures that has more than several thousand beads.



Another practical aspect about using elastic string is that elastic string is much thicker than the Nylon string I typically used. I suspect that it may not easy to pass it through three times through a hole. But of course, it is not a good idea to pass string through some beads twice, but some other beads three times because that will make the tension generated by the rubber band uneven throughout the structure. There is no such problem for the Nylon string because the bead models made by Nylon string are usually quite strong and hard. So it is harder to deform them like the model made with elastic rubber band.

Eight structural types of TCNTs with latitude coordinates

Here are all eight structural types of TCNTs in the show case located in the third floor of chemistry building(NTU). Four of them belong to Dnh point group and the other four belong to Dnd group. Chern made the three in blue and white long time ago and I made the other five in red in order to have the whole set with 6mm faceted beads after I read the paper in Chem. Eur. J.

Graphitic structures consisting only of pentagons and heptagons

I made two fullerene cages consisting only of pentagons and heptagons today.

1. The first one contains 28 pentagons and 16 heptagons.

2. The second structure is created by coupling two previous structures.

Eight structural types of TCNTs

Chern and I constructed many bead models of toroidal carbon nanotubes (TCNTs) in the last few years. We understand pretty well about the relationship among different kinds of TCNTs. Based on the spatial arrangement of 5- and 7-gons, we discovered that we can classify these achiral TCNTs into 8 structural types with well-defined latitudes and 5 other miscellaneous cases which may not have well-defined latitudes.

The eight bead models shown in the following photo are models for these eight canonical structural types.

Chern and I submitted a comment about the structural rules described in a paper by Beuerle et al to the "Chem. Eur. J." last May. After seven months of extensive review, which is much longer than the typical one to two months, our paper was rejected a few days ago.
The referee suggested us that

"A revised version of the manuscript, not criticizing the paper by Beuerle et al but instead providing a detailed and at the same time accessible account of the classification of TCNTs, might be suitable for a more specialized journal."

1. F. Beuerle, C. Herrmann, A. C. Whalley, C. Valente, A. Gamburd, M. A. Ratner, J. F. Stoddart, Optical and Vibrational Properties of Toroidal Carbon Nanotubes. Chem. Eur. J. 2011, 17, 3868-3875. See also Hot topic on carbon from Wiley-VCH.

Genus-2 TCNT

Many years ago, Chern made this genus-2 TCNT consisting basically of two 6-fold D_6h donuts. I seem to never post it here before. It should be easy to design other similar two-layer 2-dimensional structures.

Two tiny T120 made by Laura Shea

Laura Shea showed me these two beautiful T120 made of tiny crystal beads in the Bridges conference. They are so small. You can really wear these two carbon nanotori as earrings.

Structural types of TCNTs

A few weeks ago, I was informed by my colleague about a paper by Beuerle et al.[1], in which they identified eight structural types of high-symmetry achiral toroidal carbon nanotubes (TCNTs) based on two of our papers in J. Chem. Inf. Model.[2,3]
These eight structural types of TCNTs are

A. Type I, Parallel Prism; outer-rim: zigzag, inner-rim: zigzag: T240, another example, detailed construction
B. Type I, Parallel Prism; outer-rim: armchair; inner-rim: armchair: Two more models
C. Type III, Parallel anti-Prism; outer-rim: zigzag; inner-rim: zigzag: T120, Another T120, T240, T240
D. Type III, Parallel anti-Prism; outer-rim: armchair; inner-rim: armchair: T140, model 2
E. Type II, Anti-parallel Prism; outer-rim: zigzag; inner-rim: armchair: T250,
F. Type II, Anti-parallel Prism; outer-rim: armchair; inner-rim: zigzag: T240
G. Type IV, anti-Parallel anti-Prism; outer-rim: zigzag; inner-rim: armchair T240, T360
H. Type IV, anti-Parallel anti-Prism; outer-rim: armchair; inner-rim: zigzag: T240, T240, detailed construction

Here, we used zigzag and armchair patterns along lattitude coordinates, instead of using chiral vectors, (1,0) and (1,1). We have all these eight structural types in our first paper without giving them names though. In addition to these TCNTs with well-defined latitudes, we also included some high-symmetry TCNTs without latitudes (not in these eight types) in our papers. They seem to ignore them. Here is an example of TCNT (with the detailed construction) that does not belong to these eight structural types. Chuang and I are working on a comment to clarify these three extra types of TCNTs and the correct relationship (transformations) among these 11 types of TCNTs.


1. F. Beuerle, C. Herrmann, A. C. Whalley, C. Valente, A. Gamburd, M. A. Ratner, J. F. Stoddart, Optical and Vibrational Properties of Toroidal Carbon Nanotubes. Chem. Eur. J. 2011, 17, 3868-3875. See also Hot topic on carbon from Wiley-VCH.


2. C. Chuang, Y.-C Fan, B.-Y Jin, "Generalized Classification Scheme of Toroidal and Helical Carbon Nanotubes." J. Chem. Inf. Model. 2009, 49, 361-368.
pdf

3. C. Chuang, Y.-C Fan, B.-Y Jin, "Dual Space Approach to the Classification of Toroidal Carbon Nanotubes." J. Chem. Inf. Model. 2009, 49, 1679-1686.
pdf

Two more toroidal carbon nanotubes

I made two more bead models for toroidal carbon nanotubes with 120 and 240 carbon atoms last weekend (Christmas holliday in many countries, but not here in Taiwan, :-)).

Metalwire stands for bead models

I bought two little metalwire stands for my beadworks from a souvenir store in the Taipei zoo, the largest zoo in Asia according to the Wikipedia.

(T120 made with 180 6mm beads and 0.6 mm nylon thread. I took this photo on the high speed railway back to Chu-Bei.)

Torus C120 with Crystal Beads

Last week Qian-Rui and I went to 延平北路 for the properly sized beads that he needed to construct the beaded EMACS in the previous post. I also bought some beads made of artificial quartz and here is the results:

Another set of beaded T240

Chuang and I are particularly interested in toroidal shape of graphitical structures or carbon donuts. Systematic structural rules of this type of molecules have been worked out by us. The details were presented in these two articles:

1. "Generalized Classification Scheme of Toroidal and Helical Carbon Nanotubes." J. Chem. Inf. Model. 2009, 49, 361-368.
pdf

2. "Dual Space Approach to the Classification of Toroidal Carbon Nanotubes." J. Chem. Inf. Model. 2009, 49, 1679-1686.
pdf

We have made many bead models for carbon donuts. Here are a few models for isomers of TCNT (Toroidal carbon nanotube) with 240 carbon atoms. Since beads stand for carbon-carbon bonds (edge of the graph), one need 360 beads to create one such structure. The basic construction procedure is basically the same for all TCNTs. The difference is where and when to put a heptagons and pentagons for a particular TCNT.


(Constructed by Chuang)

More toroidal carbon nanotubes (TCNTs)

Here are a few toroidal carbon nanotubes I made a long time ago. There are two types of TCNTs. Smaller TCNTs consist of 120 carbon atoms. So we need 180 beads to construct them.

Larger TCNTs consists of 240 atoms. So a total number of 360 beads are needed to make them.

From New Album 2007/11/6 PM 9:49