Schwarz P 曲面的串珠模型

繼我們之前探討過的 Schwarz's D 與 Schoen G 曲面的串珠富勒烯模型之後，現在讓我們將注意力轉向一個由 莊宸與金必耀 協力製作的早期串珠模型。這個 2x2x2 Schwarz P 曲面的串珠模型 大約在2007 年左右完成，比他們之後於 2011 年完成的 Schoen G 曲面以及 2008 年完成的 Schwarz D 曲面模型還要早。您現在可以在下方看到這個模型的照片：

理解 Schwarz P 曲面

與 Schoen G 曲面和 Schwarz D 曲面類似，Schwarz P 曲面 是 三週期極小曲面 (Triply Periodic Minimal Surface, TPMS) 的一個經典例子。TPMS 是在三個方向上都具有週期性，並且在所有點上的平均曲率都為零的曲面，這意味著它們局部看起來像馬鞍形。這些曲面在數學、材料科學乃至生物學中都引起了極大的興趣。

以 Hermann Schwarz 命名的 P 曲面，其特點是具有立方對稱性。它將空間劃分為兩個全等的迷宮狀區域。其基本單元可以想像成一個立方體，相鄰面中心的點之間由彎曲的表面連接。

受到使用串珠技術構建數學結構的啟發，莊宸與金必耀 在大約 2008 年合作完成了這個 2x2x2 Schwarz P 曲面的串珠模型。由於 P 曲面的結構相對 單純，這使得它成為比結構更複雜的 D 曲面和 G 曲面更早完成的作品。這個模型使用了 與後續作品相同材質與大小的珠子，採用了與其他串珠數學雕塑類似的 角編織 方法。由 八個晶胞 組成的 2x2x2 結構，可以讓人們更具體地理解 P 曲面的週期性。

模型的意義

創建這樣的串珠模型具有多重意義。首先，它提供了一種 動手實踐的方式來可視化 一個複雜的數學概念。極小曲面的抽象性可能難以理解，而物理模型則可以讓人們更直觀地掌握其幾何和拓撲結構。其次，它突出了 數學與藝術的交叉，展示了數學原理如何激發出美觀而複雜的結構，就像 Bridges 展覽中探索的富勒烯模型一樣。這個早期的 P 曲面模型也為後續更複雜的 TPMS 串珠模型的創作奠定了基礎.

關於展覽歷史的說明

需要特別指出的是，這個 2x2x2 Schwarz P 曲面的串珠模型並未在 Bridges 會議或 Joint Mathematical Meetings 的數學藝術展覽中單獨展出。這篇博客文章旨在介紹這個早期合作完成的個人作品。

© 2025 珠璣科學 |Schwarz P 曲面的串珠模型

Beadworks for the mathart exhibition (Mar. 15, 2014)

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.

Shing-Tung Yau's visit (丘成桐)

Famous mathematician, Shing-Tung Yau (Winner of Fields Medal 1982 and Wolf Prize in Mathematics 2010), and some of his friends (Prof. Mu-Tao Wang 王慕道, Math. Dept., Columbia university; Prof. I-Liang Chern, 陳宜良, Math. Dept., National Taiwan Unversity; another one I don't know) came to see the beadworks in our chemistry department yesterday morning. He seemed to enjoy these mathematical beadworks and asked me a few questions like whether these nontrivial structures have been really synthesized etc. ...

I told him I would feel honor to give him any model in the display case if he like it as a gift. He chose the bead model of P-type TPMS with (2,0).

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.

Another unfinished (1,1) P-surface

I found an unfinished (1,1) P-surface made by Chern before. I should have used it as the starting point last week.

P-surface with the Goldberg vector (1,1)

In principle, it is possible to construct a graphitic P-surface with neighbored octagons separated by any Goldberg vector, (h, k). Most of the beaded P-surfaces I posted before have the vector (2,0). The smallest graphitic P-surface has a vector of (h,k)=(1,0) and its bead model was built by Chern long time ago.

Here is the bead model of P-surface with (1,1) I made this weekend. Unlike (2,0)-surface, the bead model of (1,1)-surface is a little bit crowded though.

Single unit cell:

2x2x2:

Exhibition at the JMM meeting

There will be an exhibition about mathematics and art in the Joint Mathematics Meetings held at the Hall D (second level) of Hynes Convention Center, Boston from 1/4 to 1/7, 2012. Chern, Wei-Chi and I have three artworks for this exhibition. If any of you lives or happens to be in the New England close to the Boston area these few days, you are more than welcome to come and share your comments with us!! However, I will stay here in Taipei. Chern is now a graduate student at MIT and will show up in the exhibition room.

Photo took by Chern:

Three beadworks for the Joint Mathematical Meeting

Chern and I submitted three beadworks, P-, D- and G-TPMSs, for
the mathart exhibition of Joint mathematical Meeting which is going to be held in Boston next January.

Here I took the two photos from the JMM site:
the first one is the G-TPMS view from another angle:

Beaded Fullerene of Schoen's G Surface
18.5cm x 18.5 cm x 20cm
Faceted plastic beads and fish thread
2011

and also the D-TPMS

Beaded Fullerene of Schwarz's D Surface
23cm x 21cm x 18 cm
Faceted plastic beads and fish thread
2008 (constructed by my former student Wei-Chi Wei)

Exhibition : The Fabulous World of Beaded Molecules 串珠幾何的異想世界

In conjunction with the special Marie Curie’s exhibition for the international year of chemistry, I am going to have a joint exhibition, "The fabulous world of beaded molecules (串珠幾何的異想世界)", for my beadworks here in the chemistry department of National Taiwan University from 10/9-10/20.




I made a few posters for this event.

P-surface:


D-surface:




G-surface:

Network of perpendicular carbon nanotubes (2x2x2)

I cut off all loose ends of fishing threads. It looks pretty. This model is the largest one (about 22.5cm x 22.5cm x 22.5cm) among all beadworks I had now. Again, I used two different colors of 6mm beads to make it. Red beads are used to make 7-bead groups (heptagons) and white beads for the other 6-bead groups (hexagons). CNTs in this structure belong to the zigzag form of CNTs.

Network of perpendicular carbon nanotubes (2x2x2)

I spent time to finish up this structure made of 12 interpenetrating carbon nanotubes (CNTs) last weekend. Compared to the G-surface, this structure is not that complicated. Once one figures out the structure of node that connects three different perpendicular CNTs, the rest is hard work. One should be able to see how the arrangement of heptagons in a node is related to the inner part of a toroidal carbon nanotubes (TNCTs).

Beadworks on display

I have many beadworks made by Chern, Wei-Chi and me on display in the faculty lounge here at the chem dept of NTU.

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.

Schwarz's P, Schoen's I-WP and Neovious's surfaces

P-surface

This is a better photo for the third P-surface I made recently.

It is interesting to compare this structure with the one in previous post. This one has octaongs in the neck, while the previous pseudo P-surface has heptagons within its unit cell.

Pseudo P-surface

This structure is the same as the P-surface I posted yesterday. Chuang made the other one a long time ago with 4mm beads. I have tried to remade just one unit cell of this structure with 6mm beads (with the all of necks included) last night. Unlike the true P-surface which divides the 3-D space into two congruent regions, the two disconnected regions in Pseudo P-surface are not identical. In this structure, we used heptagons (blue beads) to simulate the region with negative Gaussian curvature. The neck region in this structure consists of hexagons (rings that contain white beads).

Another quasi-periodic minimal structure

From beaded fullerene

From beaded fullerene

Only one unit cell in this structure.

New P-Type surface

I created a new P-surface with eight unit cells (2x2x2) using white and red beads last weekend. In total, 2880 beads are used. (Note that 90 beads are needed to create a C60.) Unlike previous two P-surfaces, I use red beads for octagons this time. The weaving process for creating a large beaded structure usually takes a long time and is prone to error.

Since the whole weaving process is essentially serial, the beads have to be added one by one along a single string of thread. So, once an error was made in the past, it is difficult to correct it without untieing all the beads up to that point the error was made.

The errors I made are usually something like using 9 beads to create an octagons (3 or 4 times out 128 octagons in the structure) and using 7 beads for hexagon (only 1 times). It is easy to understand why these errors occur more frequently for octagons, since it is more difficult to distinguish octagon and nonagon.

The first P-type structure Chuang and I made together has been given away to Bob Silbey as a gift in the summer of 2008. I made the remaining two P-surfaces by myself.




Silbey's 65 Birthday Symposium. I was in the second row far left.

(photo taken from http://web.mit.edu/newsoffice/2005/silbey.html)