Bead model of Klein bottle 克萊因瓶富勒烯串珠模型

這些串珠模型由國立臺灣大學的學生 Zhen-Ting Liu、 I-Heng Dai、 Po-Ju Chen、與助教 Hou-Hsun Ho， 在金必耀教授的《分子美學》課程中所製作期末作品，並展示於 2021 年聯合數學會議（Joint Mathematics Meetings） 的 Bridges 數學藝術畫廊。 作品展示頁面： Bridges Math Art Gallery。

克萊因瓶與富勒烯結構

克萊因瓶（Klein Bottle） 是 1882 年由德國數學家 Felix Klein 提出的拓撲曲面，它是一種不可定向曲面（non-orientable surface），即沒有區分「內部」與「外部」的概念。克萊因瓶的基本特性包括：

• 歐拉示性數（Euler characteristic）為 0，與環面（Torus）相同。
• 不同於莫比烏斯帶（Möbius strip），它是無邊界的閉合曲面。
• 在三維空間中無法真正實現，需穿過自身才能封閉。

這些特性使克萊因瓶成為拓撲學與數學藝術的經典研究對象。在材料科學中，一些特殊的富勒烯結構（Fullerene Structures），如碳納米管（Carbon Nanotubes），可能具有類似克萊因瓶的拓撲特性。

本作品運用串珠技術來構造克萊因瓶與富勒烯結構，使抽象的數學概念變得可視化與可觸摸。

作品一：單一克萊因瓶串珠模型

• 尺寸： 12 x 7 x 5 公分
• 材料： 6 毫米塑膠珠
• 製作年份： 2020 年

作品介紹

這個串珠模型展現了一個單獨的克萊因瓶結構，學生們使用串珠技術將其轉化為三維可視化模型。 製作過程中，克萊因瓶被分解為三個部分，並利用交疊與連接技術來組裝，使其保持正確的拓撲結構。

這種設計讓觀者能夠以直覺的方式理解克萊因瓶的數學性質，例如：

• 如何從一個開口結構逐步拼接成封閉曲面。
• 如何在三維空間中視覺化一個真正意義上的四維拓撲結構。

這個模型不僅是數學與藝術的結合，也為材料科學與奈米技術提供了一種幾何參考。

作品二：三重克萊因瓶串珠模型

• 尺寸： 具體尺寸未提供
• 材料： 6 毫米塑膠珠
• 製作年份： 2020 年

作品介紹

此作品進一步發展了克萊因瓶的概念，展示了三個相互連結的克萊因瓶組成的結構。這些克萊因瓶的基底彼此相連，形成一個等邊三角形拓撲，與某些碳納米結構類似。

這樣的結構有以下幾個數學特性：

• 每個克萊因瓶仍然保持其不可定向的特性。
• 三個克萊因瓶相互連結，使得拓撲結構更加複雜且穩定。
• 該結構與某些高對稱性的分子幾何類似，例如某些富勒烯變體。

這個作品不僅展示了數學與藝術的結合，也提供了一種新穎的方式來理解分子幾何與拓撲結構的關聯性。

數學與科學應用

這兩個模型的研究對數學、物理與材料科學都有潛在影響：

• 數學上，它們展示了如何將四維拓撲結構轉化為三維可視化模型。
• 物理上，這些結構與量子環境中的拓撲效應有關，例如某些拓撲絕緣體的表面態。
• 材料科學上，克萊因瓶形態可能在奈米材料與碳基結構設計中有應用價值。

透過串珠模型，學生們成功將抽象的拓撲結構轉化為具象化的藝術作品，這樣的學習方式不僅增進對數學的理解，也啟發未來在科學與工程領域的應用可能。

© 2025 珠璣科學 |克萊因瓶

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.

Beaded Mobius band (recovered & fixed)

Broken beaded mobius band

I showed this beaded Mobius band to Prof. Yuan-Chung Cheng in our joint group meeting last Friday. Maybe it is because the talk was dull and he seemed to have concentrated more on this beaded model than on the talk. Or maybe he just like to figure out the maximal strength this band can endure. So he broke it.

Mobius Short

In addition to standard way of creating one sided surface like Mobius band through twisting the strip, another way to create a one sided surface is the Mobius short. Here we use a T-shape band to create the one sided surface without twisting. The following photos show the beaded Mobius short I constructed last weekend.

Beaded Mobius strip with three 180 degree twists

Once one got the idea to make a beaded Mobius strip, it is then straightforward to construct different kind of Mobius bands at next level of complexity.
Here is a simple beaded Mobius strip with three 180 degree twists, that I just made in the Starbucks. Similar to the standard Mobius strip with one 180 twist, this one is also a non-orientable surface. Of course, one need to weave a longer strip before one can glue two ends together. I believe that the shape of beaded model of this kind of Mobius strip should be similar to the optimal shape obtained by van der Heijden again.




An interesting little book on the Mobius strip by Pickover.

Other beaded Mobius Strip

With some googling, I found that the beaded Mobius strip has been done in different ways before as shown in the following site:

http://www.instructables.com/id/Geek_chic_Moebius_strip_earrings/

Here is the mobius strip earrings Davisjan designed:



My design is very different in the sense that I have put an emphasis on the equilibrium shape of a beaded Mobius strip originating from the hard-sphere repulsion among three trivalent beads. That's why the shape of my design has such a close resemblance to the optimal shape obtained in van der Heijden's theory.

The Shape of Beaded Mobius Strips

Although I know that one can form a mobius strip from a graphene nanoribbon for some time. I have never thought about making one by using beads. Part of the reason is that I am quite fond of making beaded fullerenes that contain nonhexagons. I have the belief that the structural stability of beaded fullerenes come from these nonhexagons and the hexagons play only a minor role in the stability.

We know that a carbon nanoribbon mobius strip (CNR-MS) can be form by twisting the strip 180 degrees and then glue two opposite ends of strip together. The CNR-MS consists of hexagons completely. Since there is no any nonhexagon in the CNR-MS, one might suspect that the corresponding beaded model may not have a stable equilibrium shape. But in fact this is not true. Due to the bending of hexagonal sheet made from the beads, the resulting beaded Mobius strip has a very stable equilibrium shape.

In the following figure, I show two beaded Mobius strips I made yesterday. These two Mobius strips have very stable equilibrium structures. This is different from the Mobius band made by a long strip of paper or belt. Of course, it is much easier to make a mobius band out of these materials. The resulting strips usually can adopt more conformations. One can easily distort the structure in more different conformations. So it is usually not easy to see the most stable conformation for the Mobius strip made by paper or belt.

Beaded Mobius strips are different since they have large hard-sphere repulsion among different beads when the corresponding surface is bended. Therefore the beaded mobius strip can adopt a more unique equilibrium shape in the absence of external pressure.

Compare the beaded mobius strip with the shape published by G. H. M. van der Heijden in Nature Material two years ago. One can see the strong resemblance between the two shapes as shown in the following figure. In my opinion, the similarity is not surprising since the beaded structures are inextensible and have their energy mainly from bending deformation, which is exactly the type of Mobius strips van der Heijden is considering about. The two beaded Mobius strips in the following figure do not have exactly the same ratio between length and width as the Mobius strip van der Heijden showed in his paper. I suspect that if I use similar ratio, for instance a longer length, the resulting shape of the beaded Mobius strip will even be closer to van der Heijden's.



(Note that the figure in the upper right is taken from a paper in Nature Material by van der Heijden, which is about the equilibrium shape of an inextensible Mobius strip. There might be some copyright problem.)

Construction of a beaded Mobius strip based on hexagonal network is straightforward. One first construct a long strip of zigzag honeycomb lattice as shown in the upper right figure. When the strip is long enough, then we can twist the strip by 180 degrees and weave these two opposite ends together with beads. To make a more stable structure, one can use the remaining fishing thread to go through beads located on the edge. Note that there is only one edge, not two edges, for the Mobius strip. Once we do that, the resulting Mobius strip will be very stable.

Mobius Carbon Tori

Prof. Mou asked me about the possiblity of making Mobius-like toroidal structures. The answer is yes, it is possible. The strain energy might be large though. Here is some discussions I wrote in my group blog on April 7, 2007.

=(REPOST)===============================================================
A few months agos, I have a discussion with Chuang on the toroidal carbon nanotubes consisted only hexagons. I mentioned that there are several possible ways to patch the two ends of a nanotube into a circular torus. Most people have discussed the simplest situation that does not allow the twisting of the nanotube. If the nanotube is either armchair or zigzag, we can view the resulting torus as consisting of many parallel armchaired or zigzagged strips.

But if the nanotube is either chiral or twisted zigzag (armchair), the situation is quite interesting from topological point of view. Firstly, if we loose the patching constraint (boundary condition), there could be many different tori that we can obtain. Secondly, the more striking thing is that the whole torus may only contain one strip, which is similar to the Mobius strip. This also remind me of the algorithm Fan used to generate the carbon nanotube by imagining that SWNT is made of polyacetylene. （I should ask Fan to comment on this point.）

Yesterday I read a section of Pickover's "This Mobius Strip" on the Mobius prismatic doughnut, in which Pickover has also noticed the same problem in different context.

Now there is no doubt that we should go ahead and look at the electronic structure, physical and optical properties for this kind of Mobius carbon tori with different boundary conditions.