舍恩 G 曲面富勒烯串珠模型

賞析：

本次我們將聚焦於一件曾在 2012 年 Joint Mathematics Meetings 上展出的引人入勝的數學藝術作品：舍恩 G 曲面富勒烯串珠模型 (Beaded Fullerene of Schoen's G Surface)。這件藝術品是由 Chern Chuang、Bih-Yaw Jin 和 Wei-Chi Wei 共同創作。

作品詳情

• 作品名稱：舍恩 G 曲面富勒烯珠飾模型 (Beaded Fullerene of Schoen's G Surface)
• 創作者：Chern Chuang, Bih-Yaw Jin, Wei-Chi Wei
• 展出年份：2012 Joint Mathematics Meetings
• 創作年份：2011 年
• 尺寸：18.5cm x 18.5 cm x 20cm
• 材料：刻面塑膠珠和魚線 (Faceted plastic beads and fish thread)

結構特色與概念

這件精美的珠飾模型展現了 Schoen's G 曲面 的富勒烯結構。根據作者們的陳述，他們作為化學家，對由石墨碳構成的 富勒烯 分子及其相應的珠飾模型之間的關聯性抱有濃厚的興趣，因為分子的幾何形狀極大地影響其功能。

他們認為，富勒烯分子 非常適合用於製作珠飾模型，並且最終的模型不僅忠實地再現了分子結構，還具有藝術上的吸引力。本次展出的兩個珠飾分子模型之一便是舍恩的 G 曲面，另一件是施瓦茲的 D 曲面，兩者皆為經典的 三週期極小曲面 (Triply Periodic Minimal Surfaces, TPMS)。

作者們通過在規則的六邊形蜂窩結構中巧妙地插入 八邊形，並在所有三個維度上引入週期性邊界條件，從而獲得了這些 TPMS 的富勒烯對應物。在這些珠飾模型中，八邊形由彩色珠子表示，而六邊形則為白色。

特別地，舍恩的 G 曲面 被認為是最難以捉摸的嵌入式 TPMS 之一，然而它卻廣泛存在於生物和材料科學中。如同 P 曲面和 D 曲面可以分別分解為相互連接的懸鏈線和螺旋面單元一樣，G 曲面可以被視為是連接的螺旋面，它們處於懸鏈線-螺旋面等距變形的中間狀態 。這個珠飾的螺線包含 16 個這樣的單元，每個單元的長度為兩個平移單元。模型中，作者使用了三種不同的顏色來表示八元環，因為這些環可以根據其面法線進行分類。在通常的右手笛卡爾坐標系中，面法線沿 x 軸的八元環用紫色珠子表示，而沿 y 軸和 z 軸的則分別用藍色和綠色表示。

背景介紹

富勒烯是由碳原子組成的閉合籠狀或管狀分子。最著名的富勒烯是 C60，又稱巴克球，具有足球狀的結構。富勒烯因其獨特的幾何和電子性質而在化學、材料科學和納米技術等領域引起了廣泛的關注。

三週期極小曲面 (TPMS) 是在三個方向上都具有週期性的極小曲面。極小曲面是指其平均曲率處處為零的曲面，在局部上具有類似鞍形的形狀。TPMS 在數學、材料科學（如液晶和嵌段共聚物的微觀結構）以及生物學中都有廣泛的應用。

舍恩 G 曲面 是一種複雜且重要的 TPMS，因其在自然界和材料科學中的普遍存在而備受關注。其獨特的幾何特性使其成為研究界面現象和週期性結構的理想模型。

網站連結

您可以在 Bridges Organization 的網站上找到更多關於 Chern Chuang、Bih-Yaw Jin 和 Wei-Chi Wei 的藝術作品：

• The Bridges Organization
• Chern Chuang, Bih-Yaw Jin, Wei-Chi Wei | 2012 Joint Mathematics Meetings | Mathematical Art Galleries

© 2025 珠璣科學|舍恩 G 曲面富勒烯串珠模型

Beaded (1,1) Gyroidal Surface

本作品由國立臺灣大學的 陳儀斌（Yi-Bin Chen）（農業經濟學系本科生）製作， 助教何厚勳在設計上做了重要貢獻， 並入選 Bridges 2023 數學藝術展覽。

作品簡介

「Beaded (1,1) Gyroidal Surface」是一件基於 螺旋面（Gyroid Surface） 幾何結構的藝術作品，透過數學串珠方法來呈現流線型的曲面結構，展現拓撲學與藝術設計的結合。

創作背景

在國立臺灣大學 金必耀教授 開設的「分子美學」課程中，學生們學習如何將分子結構轉化為可視化的藝術模型。金教授曾展示一個以珠子構建的 (2,0) 螺旋面 模型，啟發了學生們的創作靈感。

Yi-Bin Chen基於這個概念，選擇 (1,1) Goldberg 向量 作為設計基礎，並透過不同顏色的珠子來構建螺旋曲面結構。他特別使用 藍色、紅色與黃色 的八角形珠子，將它們排列在六角形條帶的適當位置，調整模型的曲率，使其呈現自然的螺旋形狀。

作品特色

• 名稱： Beaded (1,1) Gyroidal Surface
• 尺寸： 30.0 × 30.0 × 30.0 公分
• 材質： 6 毫米塑膠珠子、魚線
• 創作年份： 2022 年
• 數學概念： 螺旋面結構、Goldberg 向量、拓撲學

藝術家簡介

何厚勳 是國立臺灣大學化學系博士生，專注於碳分子的數學結構與藝術表現。他的研究涵蓋 碳納米管（CNTs）、富勒烯（Fullerenes） 及其幾何排列方式。

陳儀斌 是國立臺灣大學農業經濟學系的本科生，對於結合傳統工藝與數學藝術有濃厚的興趣。他透過數學串珠的方法，探索分子幾何的視覺化表達。

數學藝術與拓撲學

螺旋面（Gyroid Surface）是一種在 材料科學、結晶學與數學 中具有重要應用的幾何結構。其特點是沒有鏡面對稱性，卻能形成連續但不相交的三維曲面，這種結構在自組裝材料與光子晶體的研究中扮演重要角色。

「Beaded (1,1) Gyroidal Surface」通過簡單的串珠技術，成功模擬了這種複雜的數學結構，展現數學與藝術的完美融合。

相關連結

• Bridges 2023 數學藝術展 - 陳儀斌與何厚勳
• 國立臺灣大學化學系
• 國立臺灣大學農業經濟學系
• 金必耀教授 - 國立臺灣大學
• Gyroid 螺旋面
• 數學藝術（Mathematical Art）

「Beaded (1,1) Gyroidal Surface」透過珠子與魚線的簡單元素，展現出拓撲學的精妙之處。這不僅是一件數學與藝術結合的作品，也為分子結構的視覺化提供了新的表現方式，讓觀者感受到數學世界的美麗與秩序。

© 2025 珠璣科學 |Beaded (1,1) Gyroidal Surface

Schwarz D ("Diamond") minimal surface

Two students, 張皓 and 李于婕, in the class, Molecular Aesthetics 2020, recreated a bead model of Schwarz D ("Diamond") triply periodic minimal surface with 6mm black beads.

P-type Triply-Periodic Minimal Surface. (Molecular Aesthetics 2020)

Bead model for a P-Type Triply Periodic Minimal Surface constructed by a group students in my class, Molecular Aesthetics 2020.

Two articles about the construction of gyroid- and diamond-type triply periodic minimal surfaces

I wrote two articles in Chinese for the Journal, Chemistry Education in Taiwan (臺灣化學教育) last year. The pdf files have just come out:
1. 左家靜, 莊宸, 金必耀, 大家一起做多孔螺旋與鑽石型三度週期最小曲面的串珠模型（上）─立體幾何介紹，2014 臺灣化學教育, 328-335.
2. 莊宸, 左家靜, 金必耀, 大家一起做多孔螺旋與鑽石型三度週期最小曲面的串珠模型（下）─實作，2014 臺灣化學教育, 336-344.
The title can be translated as "Application of mathematical beading to carbon nanomaterials - A hands-on, collaborative approach to gyroid- and diamond-type triply periodic minimal surfaces with beads, I and II", literally. I described a simple modular approach which was developed mainly by Chern Chuang for making gyroid- and diamond-type Triply Periodic Minimal Surfaces.

Bead model of a Trinoid

Tsai-Rong Liu (劉采容), an undergraduate at chemistry department of the National Taiwan University, built this interesting graphitic structure which approximates the trinoid, a minimal surface with three catenoid openings. Similar graphitic surfaces with k catenoid openings, i.e. k-noids, can be built similarly.

作品完成時間（約）：2015/4/15
作者：劉采容

Bead model of the Chen-Gackstatter surface of genus 1

I made a bead model which approximates the minimal surface, Chen-Gackstatter surface of genus 1, for the spring break.

Other Chen–Gackstatter surfaces can be made with mathematical beading, in principle!
2015/4/3

D- and G-types TPMSs

28 groups of students from TFGH joined the competition designed by Ms. Chou and other teachers in the chemistry group of TFGH. They were asked to make any of these two complicated 3D models based on the slides I prepared for the G- and D-surfaces. It is still nontrivial for a beginner, who has no knowledge on the periodic minimal surfaces and graphitic structures. But most of them succeeded in creaking one of these two models. Unfortunately, when they asked local sellers about the suitable thickness of Nylon strings for 12mm beads. They were told that 0.6mm NyLong strings are best. That is why most of models they made are so soft and unable to stand on themselves. To solve the problem, students came up with the idea to hang these models on four legs of an upside-down desk they use for lectures.

However, one group discovered the cause to be the thickness of the Nylon string. Then students of that group changed the Nylon strings to 0.8mm. The two TPMS models they made are shown in the following photo. They look really nice and beautiful.

The one on the left is the G-surface. The one on the right side is the D-surface consisting of 16 helical strips. Using the decomposition technique Chern Chuang designed, we can use the same helical strips to create these two types of TPMSs.

The procedure for constructing G- and D- surfaces

Here are a few slides that show the detailed instruction for making G- and D- surfaces, which I prepared for students and teachers of TFG (Taipei) school. As I said it could be a difficult task because the gyroidal structure and D-type TPMS are complicated structures. The first bead model of a 2x2x2 G-surface took Chern and I almost five years to finally make it. Of course, I have many unfinished bead models of this structure or similar structures with different Goldberg vectors, some made by Chern and some by me, which have mistakes here or there.

In order to how to make this model successfully, we'd better to know the three-dimensional structures of G- and D-type surfaces a little bit. Additionally, it is crucial to know how two structures can be decomposed into several basic unit strips and how to connect these helical strips.

I am also working on an article in Chinese entitled "大家一起動手做多孔螺旋與鑽石型三度週期最小曲面的串珠模型 (A Hands-on, Collaborative Approach to Gyroid- and Diamond-type Triply Periodic Minimal Surfaces with Beads)", which describes in details the procedure to make G- and D-surfaces and also give some background information on TPMS. I might be able to finish the paper in a few days. Hopefully, I will find time to do it in English someday. But, even without detailed explanations, these slides together with other posts in this blog should already contain enough information for people who want to do it.

The first nine slides should give students a better picture of a gyroid:

In slide 10, we can see how a coronene unit corresponds to 1/8 unit cell. Important structural features of a beaded gyroid is summarized in slide 11. Then in slides 12-15, I describe how to make the basic construction unit, a long strip, which should be easy for student to make.

The remaining five slides, 16-20, use schematic diagrams to show how two slides can be combined to generate either D-surface or G-surface.

To create a 2x2x2 gyroidal surface, we need 16 strips, which can be easily done if many people work in parallel. To connect them is nontrivial, you need to follow slides 16-20 carefully. In total, there are about 5000 beads in the model.

P, G, and D surfaces

I am planning to have a project with students and teachers of TFG (Taipei) school later this month to construct Gyroidal and D surfaces together. It could be a difficult task because the gyroidal structure is probably the most complicated bead structure Chern and I have ever made. A simple tutorial on the three-dimensional structure of a gyroidal surface and how it can be decomposed into several basic and easily weaved units seems to be useful. So I am now preparing some slides to make the project work out smoothly. Here is one of the slides about the famous P-, D- and G-types Triply Periodic Minimal Surfaces (TPMS) which I generated with matlab:

Additionally, Chern, Wei-Chi, Chia-Chin and I also have a paper jointly for the Bridges meeting last summer. Chern made the presentation. I didn't attend it, though. This paper describes the bead models of these three structures quite generally.

Chuang, C.; Jin, B.-Y.; Wei, W.-C.; Tsoo, C.-C. "Beaded Representation of Canonical P, D, and G Triply Periodic Minimal Surfaces", Proceedings of Bridges: Mathematical Connections in Art, Music, and Science, 2012, 503-506.

P-surface with (1,0) vector

Chern made this (1,0) P-type triply periodic minimal surface with 4mm beads many years ago. I might have posted it long time ago, but forgot to make a suitable keywords for that post, so I couldn't find anywhere in this blog.
Chern used a different color coding for this bead model, i.e. orange for hexagons and black and organe alternatively for octagons. Since there is no bead which is not in octagons, one would get a single color bead model by using the color coding we adopted typically.

(1,1) and (2,0) P-type Triply Periodic Minimal Surfaces

It would be nice to compare two P-type TPMSs, one with the Goldberg vector (1,1) and the other with (2,0), together. Here the Goldberg vector, (m,n), denotes the separation between two neighbored octagons. In this sense, (1,1)-P-TPMS is similar to C60, while (2,0)-P-TPMS to C80 in the spherical space.

Building blocks for pseudo D-type Schwarzite

I just noticed that I already have a picture of punctured C₈₄ in the logo of this blog the other day. I checked my photo library and found another picture of this model which was taken almost five years ago. I possibly made this bead model after reading the paper on Nature with title "Energetics of negatively curved graphitic carbon" by Lenosky et al. (1992 vol. 355, 333-335).

However, we now know this is not a true triply periodic minimal surface. The correct D surface should partition the space into two congruent parts. It is not possible to get such a structure if one use heptagons.

Of course, we now know how to make a correct D-type Schwarzite which exactly partition the space into two identical regions. Wei-Chi made this beautiful bead model of D-type Schwarzite a few years ago.

D surface constructed from four helical strips

There is another way to build a D-type triply periodic minimal surface (TPMS) with beads. Chern has told me previously that one can not only use helical strips to build G-type TPMS, one can also use exactly the same helical strips to build D-type TPMS. If one examine two helical strips carefully, one can find that there are exactly two different ways to put them together. One gives a D-type TPMS, the other one gives a G-type surface!

The following pictures are a bead model of D-surface consisting of four helical strips. Two of them are left handed, the other two are right handed. To build a D-surface, one has to put two helical strips together in an arrangement such that two neighbored strips are mirror-symmetric to each other. So the overall structure of D-surface is not chiral.

It is useful to look at other posts with the keyword helical strip, especially the one on the G-surface created by patching two helical strips with one strip shifted by half pitch.

P, D, and G triply periodic minimal surfaces

Finally, I have all three bead models for P, D, and G minimal surfaces corresponding to the simple cubic, diamond-like, and gyroid-like structures, respectively.

Another 2x2x2 P-type triply periodic minimal surface

Here is the smallest P-type graphitic TPMS that can be constructed. Chuang made this model a long time ago.

Compare this structure with the other models of P-type surface I have posted before, one can understand why this is the smallest possible P-type TPMS.

2x2x2 bead model of Schoen's I-WP surface

Chern Chuang (莊宸) created this amazing graphitic structures consisting of a Schoen's I-WP surface decorated with graphene sheet. The I-WP surface, found by Schoen in 1970, is a TPMS of cubic symmetry Im3m. In addition to I-WP surface, there are other TPMSs of Im3m symmetry, such as Schwartz P and Neovius surfaces.

Chuang figured out the systematic structural rule for this kind of periodic minimal surface at the beginning of this year. Based on his post, I constructed a 1x1x1 bead model for this structure using three different colors of beads. Apparently, he seems to find time to bead even in the army. Now we have this beautiful 2x2x2 bead model created with 5mm semi-transparent plastic beads. Based on the previous post by Chuang, there are 3456 carbon atoms in the structure. So one need about 5000 beads to create it. It is quite a work.

I am kind of surprised by the strong mechanical strength of this structure. Unlike bead models for other minimal surfaces including Schwartz P or D surfaces, this model seems to be quite rigid, and a little bit like that of a brick. So, it may not be a bad idea to have this kind of 2x2x2 I-WP surface as a high-strength but low-density brick for purpose of construction. The only problem is that it is quite difficult to create even just one of this kind of bricks. We may solve the problem if we can invent a robot to weave them automatically.








The 2x2x2 I-WP surface and my friend's iPad:

Neovius Surface

Neovius was a student of H.A. Schwarz who contructed this surface with 'Neovius' handles towards the edges of a box.

http://www.indiana.edu/~minimal/archive/Triply/genus9/Neovius/web/index.html

Beaded Model for Scherk's Singly Periodic Minimal Surface

It is just amazing! I met Chuang unexpectedly while I was waiting for the bus home yesterday evening. Chuang told me he has figured out how to construct graphitic sheet with the structure of the Scherk's singly periodic minimal surface (SPMS) and has posted it on my group blog as shown in the following figure. It is indeed a beautiful structure. In some sense, we can view this structure as an intersection of two graphene sheets. Instead of a straight line for the intersection, here one need a chain of octagon loops to mimic this intersection.
Apparently, Chuang has worked out this when he has a shooting training in the army. I hope that he can still shoot the target in the training course.



This SPMS structure is not complicated. It is quite straightforward to make a beaded model for this kind of graphitic structure. It took me several hours to make it today.

Another P-surface

Chuang seems to have constructed many more beaded structures. I found this P-type minimal surface on his table. He is now in the military service for one year. So we probably won't have new beaded structures in this coming year .

From beaded fullerene

From beaded fullerene

From beaded fullerene