數學與藝術的交匯：基於三角多面體的可伸縮負泊松比拉脹（Auxetics）結構

Bridges 會議是一個獨特的年度盛會，匯集了數學、藝術、音樂、教育、文化等不同領域的學者和藝術家，共同探索數學與藝術之間的深刻聯繫。在 2023 年的 Bridges 會議藝術展覽上，一件名為“Extendable auxetic structure based on deltahedron made by stringing bugle beads”（基於三角多面體的弦式伸縮負泊松比結構）的作品引起了廣泛關注。

背景知識：負泊松比結構與三角多面體

在深入了解這件藝術品之前，我們需要簡要介紹兩個相關的概念：負泊松比結構（Auxetic Structure） 和 三角多面體（Deltahedron）。

• 負泊松比結構： 大多數材料在受到拉伸時，其垂直於拉伸方向的尺寸會縮小；反之，受到壓縮時則會膨脹。然而，負泊松比材料（也稱為膨脹材料）卻表現出相反的特性——當它們被拉伸時，垂直方向也會膨脹；當被壓縮時，垂直方向則會收縮。這種獨特的性質使得負泊松比結構在工程和材料科學領域具有廣闊的應用前景。
• 三角多面體： 三角多面體是指所有面都是等邊三角形的凸多面體。例如，正四面體、正八面體和正二十面體都是三角多面體。由於其結構的穩定性和幾何特性，三角多面體在數學和結構設計中扮演著重要的角色。

作品詳情：基於三角多面體的可伸縮負泊松比拉脹（Auxetics）結構

Extendable auxetic structure based on deltahedron made by stringing bugle beads

創作者：林嘉陽（Jiayang Lin），施宣光 (Shen-Guan Shih)，周昌裕(ChangYu Chou)，金必耀(Bih-Yaw Jin)

• 尺寸： 20.0 x 30.0 x 30.0 厘米
• 材料： 串珠（Bugle beads）和魚線
• 創作年份： 2023

這件作品的核心創新在於**利用串接的管狀珠子（bugle beads）來創造基於三角多面體的負泊松比結構，並且這種結構可以在水平和垂直方向上無限延伸**。

作者們的主要發現是，**通過串聯管狀珠子，可以實現基於三角多面體的負泊松比結構，並使其在水平和垂直方向上都能夠無限擴展** 。他們首先利用**正三角形的負泊松比平面構型**，然後將其映射到三維的三角多面體結構中。這個三維結構的構建是通過**結合使用四面體和八面體**來實現的。

作品的一個關鍵特性是，四面體和八面體在負泊松比結構的壓縮過程中能夠相互補充。由於這種互補性，當結構完全壓縮時，可以形成基於具有相同邊長的凸多面體的最密集結構。

該模型的物理實現使用了等長的管狀珠子，長度均為 3 厘米。 作者使用了不同的顏色來展示結構的組裝方式。

這件藝術品巧妙地將數學幾何中的三角多面體概念與材料科學中的負泊松比特性相結合，並通過精巧的珠飾工藝將其呈現出來。它不僅展示了數學在藝術創作中的潛力，也為負泊松比結構的設計提供了一種新穎的思路。

關於作者

• 林嘉陽（Jiayang Lin）: 國立台灣科技大學建築系碩士生，對數字和幾何領域充滿熱情，並希望將跨領域知識應用於建築設計。
• 施宣光(Shen-Guan Shih): 國立台灣科技大學建築系教授。
• 周昌裕（ChangYu Chou）: 國立台灣科技大學建築系碩士生 。
• 金必耀(Bih-Yaw Jin): 國立台灣大學化學系教授 ，長期致力於將數學概念融入藝術創作。

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

© 2025 珠璣科學 |負泊松比結構

探索五重對稱之美：二十面體複合桁架模型

在 2015 年的 Bridges 會議中，金必耀與左家靜不僅展示了鈣鈦礦結構的精美模型，還呈現了另一組引人入勝的藝術作品——「Truss models of icosahedral complexes」（二十面體複合桁架模型）。這些模型以其獨特的結構和對非週期性有序結構的探索，再次展現了數學與藝術的優雅結合。

這個模型的尺寸為 12 x 12 x 12 厘米，同樣是在 2014 年 使用 空心玻璃珠 (Tubular glass beads) 製作而成。

什麼是二十面體複合體？又為何與準晶體相關？

二十面體 (icosahedron) 是一種具有 20 個三角形面、30 條邊和 12 個頂點 的柏拉圖立體。它擁有令人著迷的 五重旋轉對稱性。然而，這種五重對稱性與傳統晶體結構中常見的平移對稱性是不相容的，這意味著單獨的二十面體無法像傳統晶胞那樣在三維空間中進行週期性排列來形成晶體。

這就引出了準晶體 (quasiperiodic crystal)的概念。準晶體是一種有序但不具週期性的結構，其原因正是五重旋轉對稱性與平移對稱性之間的不相容性。準晶體雖然具有長程的有序性，但其原子排列並不呈現出像傳統晶體那樣的重複單元。

金必耀與左家靜的「Truss models of icosahedral complexes」正是對這種複雜概念的可視化嘗試。 這些模型可以被視為 有限的準晶體，是通過在 中心二十面體周圍添加足夠數量的四面體、八面體和五角雙錐體 而構建的。

更令人驚嘆的是，模型中 珠子的顏色被精心挑選，用於描繪具有二十面體對稱性的 四種不同的多面體：

• 二十面體 (icosahedron)：使用帶有螺旋圖案的珠子。
• 二十-十二面體 (icosidodecahedron)：使用白色珠子。
• 頻率為 2 的 Mackay 十二面體 (frequency-2 Mackay dodecahedron)：使用藍色珠子。
• 六十面體 (hexecontahedron)：使用紅色珠子。

這些模型的意義何在？

通過將這些不同的多面體以特定的方式組合在一起，金必耀與左家靜的作品生動地展現了構成類晶體的局部結構單元以及它們之間的複雜關係。使用空心管珠和桁架結構的方式，不僅使得模型輕巧且具有一定的剛性，更清晰地展示了這些幾何體的骨架和連接方式。

這件二十面體複合桁架模型不僅是美麗的藝術品，更是理解準晶體結構這一複雜科學概念的寶貴工具。它們將抽象的數學和物理原理轉化為具體的、可供觀賞和思考的藝術形式，激發我們對自然界中非週期性有序結構的好奇心。如同他們之前的作品一樣，金必耀與左家靜再次證明了 空心管珠是構建複雜多面體結構和探索深奧數學概念的理想媒介。

透過這些模型，我們得以一窺準晶體那既有序又非重複的奇特世界，並欣賞數學與藝術在探索物質結構本質時所展現的創造力。

欲了解更多關於 金必耀與左家靜在 2015 Bridges 會議上的作品，請訪問 Bih-Yaw Jin & Chia-Chin Tsoo | 2015 Bridges Conference | Mathematical Art Galleries。

© 2025 珠璣科學 |二十面體複合桁架模型

探索材料之美：串珠 Perovskite 結構模型

在 2015 年的 Bridges 會議上，金必耀與左家靜帶來了令人著迷的數學藝術作品。其中一件名為 鈣鈦礦結構（The perovskite structure） 的串珠模型， 以其色彩分明的結構和對重要材料科學概念的視覺化呈現，引發了人們對微觀世界的好奇。

這件 鈣鈦礦結構 模型尺寸為 16 x 16 x 16 厘米，於 2014 年 使用 空心玻璃珠 (tubular glass beads) 製作而成。這個模型以 兩種不同的顏色 的空心管狀珠子構成，呈現了一個 2x2x2 的晶胞結構 。

什麼是 Perovskite 結構呢？

在材料科學中，鈣鈦礦 (perovskite) 指的是 具有特定晶體結構的一類材料 。其名稱源於最早發現的礦物——鈣鈦礦 (CaTiO₃)。更廣泛地說，具有鈣鈦礦結構的化合物通常具有通式 ABX₃，其中 A 和 B 是不同尺寸的陽離子，而 X 通常是陰離子（如氧化物或鹵化物）。

關鍵在於這些離子的排列方式： 鈣鈦礦結構可以視為由 八面體 (octahedrons) 和 立方八面體 (cuboctahedrons) 以 1:1 的比例 在三維空間中 均勻且密鋪 (space-filling tessellation) 而形成的。金必耀與左家靜的模型使用了兩種不同顏色的空心管狀珠子來建構這個結構.

金必耀與左家靜的模型正是對這種微觀排列的宏觀呈現。他們使用不同顏色的空心管珠來代表不同的結構單元，清晰地展示了八面體和立方八面體如何交織在一起，形成一個重複的、空間填充的整體。

Perovskite 材料的重要性：

值得一提的是，鈣鈦礦材料在近年來引起了科學界的廣泛關注。這是因為它們在諸如太陽能電池、發光二極體 (LEDs)、感測器和催化劑等領域展現出優異的性能。特別是在太陽能電池領域，鈣鈦礦太陽能電池因其製備成本低、效率高等優點，被認為是下一代太陽能技術的有力競爭者。

透過金必耀與左家靜的這個串珠模型，我們不僅能夠欣賞到一件精美的藝術品，更能直觀地理解鈣鈦礦這種重要材料的基礎結構。它展示了科學概念如何透過藝術的媒介變得更加具體和易於理解，也讓我們對構成我們周圍世界的微觀結構產生更深的認識。藝術與科學的結合，往往能開啟我們探索未知世界的新視角。

這件作品再次印證了空心管珠是構建由剛性結構單元（如八面體和立方八面體）組成的多面體複合體的理想材料。金必耀與左家靜持續探索利用傳統的串珠技術來呈現奈米級無機化合物和非分子晶體固體的空間排列。他們的作品不僅具有藝術價值，也為科學教育提供了一種新穎而有趣的方式。

欲了解更多關於 Bih-Yaw Jin 和 Chia-Chin Tsoo 在 2015 Bridges 會議上的作品，請訪問 Bih-Yaw Jin & Chia-Chin Tsoo | 2015 Bridges Conference | Mathematical Art Galleries。

© 2025 珠璣科學 |鈣鈦礦

Straw model of the elevated icosahedron (himmelis)

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

Two icosahedral complexes derived from an icosahedron

Starting from a bead model of icosahedron, one can make a few beautiful rigid polyhedral complexes by adding more regular octahedra and tetrahedra surrounding the central icosahedron. Here are two examples:

Icosahedron + Icosidodecahedron

Icosahedron + Icosidodecahedron + Rhombic Hexecontahedron

Trefoil knot for the Bridges Seoul 2014

Chern submitted an artwork of trefoil knot for the Bridges Seoul 2014, which can be found at this page: Carbon nanotube trefoil knot C318.

C60 and T120 with 5mm cateye stones

It's been a while since I made T120 and C60 sometime ago. Also, I have given away all other cateye T120 models made by Chern many years ago. Hopefully, I can show more models with this kind of beads in the future.

Quasicrystal (準晶)

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.

Beaded trefoil knot using two beads per edge

I did a tryout using two beads per edge in this beaded molecule. Also, since these assorted glow-in-the-dark beads are one of the only few kinds of beads I have here in Boston, I try to take a picture of what it looks like in the dark after suitably charged with a light bulb. Thanks to Chun-Teh Chen (陳俊德) who helped me with the photography.

Enantiomeric Pair of Graphitic Trefoil Knots C318

It's been a while since last time I wrote on this blog. Three weeks ago, I went to the Bridges 2012 conference held at Towson University, MD. It was such a terrific gathering of practitioners of mathematical art. I met a lot of new friends and enjoyed sharing our ideas with each other. The following is a repost of a graphitic structure I found last year when I was a research assistant of Bih-Yaw. It is a D3-symmetric trefoil knot structure and comprises 318 carbon atoms.

 

I figured out its construction rule while pondering on one of Escher's artwork.

 

Upon scrutinizing the one at the upper-left corner, I found that the trefoil knot can actually be decomposed into three 270-degree-arc and three inter-arc straight segments. This was a huge hint for me since I've been thinking about constructing torus knots from carbon nanotubes for years. An idea came up when I was doing my MS degree with Bih-Yaw, three years after he mentioned this project to me, which basically makes use the fact that one can divide a (p,q) torus knot (for p smaller than q). An example of this building principle applied to the trefoil knot is shown below:

 

Interested readers can also refer to the "Torus Knot" tag in this blog. A truly unfortunate fact is that this kind of molecules does not seem to be "beadible" to me, since it has got a great deal of tangential stress along the tangent of the central curve. Unless one lets loose the fish line during the course of the beading process, eventually when she or he tries to sew up the closed structure an unsurmountable stress will be encountered.

I didn't want to stop there, and finally I got the inspiration from Escher as mentioned above. The structure possesses nearly zero tangential stress when you try to bead it. Another nice point about it is that it has got much less number of atoms in it, compared to the other construction scheme. It typically takes me less than three hours to bead it.

At the Bridges conference, I met two kind Taiwanese ladies Helen Yu and Joy Hsiao. The pair of enantiomeric C318 trefoil knots as shown in the following photo was given to them as souvenirs.

They are made of 3mm glow-in-the-dark beads. I'm thinking about presenting this type of structures in the Bridges next year, hoping that I have the chance to go.

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.

Helically coiled carbon nanotube derived from T140

I made one more HCCNT that was derived from parent torus, T140.

The inner part should be weaved first. Here I keep the relative position of these heptagons unchanged.

The next step is to determine the HSP (Horizontal Shift Parameter) on the outer part of the torus. The pitch of the HCCNT will depend on the magnitude of HSP. For detail, check the papers mentioned in previous post.

Helically coiled carbon nanotube derived from torus 120

I made another HCCNT (Helically coiled carbon nanotube) derived from the parent molecule, carbon nanotorus with 120 carbon atoms yesterday.

The construction of this carbon helix is quite straightforward. First we should know that this structure can be decomposed into six strips. To simplify the weaving process, one should start from the inner part of HCCNT.

To make a helical tube, one still need to finish the remaining two strips. Particularly, we need to be careful about the relative position between two neighbored pentagons. The systematic way to generate a whole family of HCCNTs from a parent TCNT is based on the concept of horizontal shift parameters (HSP). By applying a suitable HSP, one can create a whole family of HCCNTs.
The details of structural rules of HCCNTs can be found in the following three papers we published:

Chuang, C.; Fan, Y.-C.; Jin, B.-Y.* Generalized Classification of Toroidal and Helical Carbon Nanotubes J. Chem. Info. Model. 2009, 49, 361-368.
Chuang, C; Jin, B.-Y.* Hypothetical toroidal, cylindrical, helical analogs of C60 J. Mol. Graph. Model. 2009, 28, 220-225.
Chuang, C.; Fan, Y.-C.; Jin, B.-Y. On the Possible Geometries of Helically Coiled Carbon Nanotubes J. Mol. Struct. 2012, 1008, 1-7.


In fact, Chern made a bead model of the same structure a few years ago. But in the Bridges conference held in Pecs, Hungary, I met Laura Shea and gave that model to her as a souvenir. Since then, both Chern and I didn't make any new model of helically coiled carbon nanotubes.

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.

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.

Bridges 2011

I gave a talk this morning on the designing sculpture inspired by high-genus fullerenes with mathematical beading (pdf file) in the Bridges conference (Coimbra, Portugal).



Here are some bead models I brought to this meeting. These include three large pieces (a dodecahedra, a tetrahedron, and a trefoil knot), four TCNTs (120, 140, 240), and around 25 C60s.