Bead model of Klein's all-heptagon network

I took a picture of single tetrahedral unit (12 heptagons) of D56 bead model on the figure depicting schematically an open network consisting only of heptagons, described by Klein in his 1879 paper.

Klein, F. (1878). "Ueber die Transformation siebenter Ordnung der elliptischen Functionen" [On the order-seven transformation of elliptic functions]. Mathematische Annalen 14 (3): 428–471. Translated in Levy, Silvio, ed. (1999). The Eightfold Way. Cambridge University Press.

C20 vs C56

A postcard from Mr. Horibe

A nice postcard I got from Mr. Horibe:

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.

G- and D-surfaces in TFGH

Fang-Fei Chou and other teachers of chemistry section of the Taipei First Girl High school (TFGH) started a new bead project based on the slides I made for the anniversary of their school early next month. Using these slides only, they are going to make 2x2x2 G and D surfaces by themselves. Fang-Fei told me that there are about 30 teams in this project, which means they are going to have about 30 giant bead models of TPMS.

Attached is a photo that shows their current progress.

As you can see that their strips are quite long because they use 12mm beads. I made two G surfaces with 6mm and 8 mm beads, respectively. The one made of 6mm beads is about 20x20x20cm. So the G surfaces they are going to make are about 40x40x40cm. I wonder where they are going to put so many gigantic bead models.

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.

Another way to view D surface

There is another way to partition the D-surface to its constituents. It looks quite different.

It would be interesting to compare these pictures with the bead model of D surface Wei-Chi made:

(http://www.ams.org/mathimagery/displayimage.php?album=32&pid=418#top_display_media, AMS Math Imagery)

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.

Chern Chuang and Paul Hildebrandt with G and D surfaces

A photo of Chern (with G surface) and Paul (with D surface) in the Bridges conference held in the Towson university, Baltimore, USA:

(Photo by Helen Yu)

MATH imagery of American Mathematical Society

AMS MATH imagery: Beaded Fullerene of Schwarz's D Surface.

Two hyperbolic graphitic surfaces

I made a bead model of hyperbolic graphitic surface by connecting 10 tetrahedral C₈₄ units (the model on the right, the one on the left is the C168). The skeleton of this model is similar to that of an adamantane of the smallest unit of diamond.

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.

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:

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.

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:

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.

D-type triply periodic minimal surface

From Dec 25, 2008

D surface on my mac