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.

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

Tetrahedral C28 and related structures

There are only three tetrahedral fullerenes with number of carbon atoms less than that of buckyball. They are C₂₈, C₄₀, and C₄₄, respectively. The spiral code for the smallest tetrahedral fullerene, C₂₈, is [1 2 3 5 7 9 10 11 12 13 14 15]. Following this code, we can easily make its bead model using the standard figure-eight stitch. We can see that, in this molecule, there are 12 pentagons, 3 in a group located at a vertex, and 4 hexagons located on the four faces of the tetrahedron. If we replace these pentagons by heptagons, we get a tetrapod-like structure, in which tri-pentagon vertices become tri-heptagon necks as shown in the following figure.

Using these tetrapods as building blocks, we can get the following diamond-like structure. In fact, this is exactly the structure Mr. Horibe put in the postcard. OK, if we start from other tetrahedral fullerenes such as C₄₀ and C₄₄, we can find out a lot more diamond-like structures.

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.

Gyroidal Invinciball

A graphitic gyroid is a hyperbolic object. To make it, we need to introduce octagons at suitable positions on a graphitic sheet, which is similar to the pentagons in the spherical space such as buckyball. In some sense, we can view graphitic gyroid as a kind of "ball" in the hyperbolic space.

Students from the TFGH created this gyroidal invinciball in the hyperbolic space. Unfortunately, they used 0.6mm Nylon strings for the 12mm faceted beads. The structure is too soft to stand on its own.

Gyroidal National Flag of Republic of China (Taiwan)

I went to a special ceremony for the beading competition held in the Tapei First Girls High School this afternoon. I saw this amazing 3D flag model of my country, Republic of China (i.e. Taiwan), which is made by a suitable color code of octagons in a gyroidal graphene.

BTW, you can also interpret this flag as that of US.

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.

Gyroid: simulation vs bead model

I carefully recalculated the region of Gyroidal surface and got a better comparison between the calculated surface and the bead model. The agreement is quite well. We can see the helical strips we used have made the whole structure a little bit longer than 2 unit cells along the z direction.

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)

From carbon cube to Schoen's I-WP surface

As I mentioned in the previous post, we can create necks by replacing every pentagon around corners by a heptagon. The structure thus obtained is a single unit cell for the Schoen's I-WP surface! The eight necks in this particular case are, however, too thin to be chemically stable.

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:

Dodecahedral Carbon Schwarzite

Previously, I discussed how to construct a locally hyperbolic tetrahedral building block by puncturing four holes on a tetrahedral C84 along its four tetrahedral axes. This basic unit consists of 12 heptagons, which make the local curvature negative. We can create many different kinds of interesting structures by using this kind of building blocks. One possibility is a dodecahedron. The angle between two axes emanating from the center to two any two vertices (the "tetrahedral angle") of a tetrahedron is 109.47°, which is slightly larger than the inner angle of a dodecahedron (108°). This means that we can use them to construct a dodecahedron without introducing too much local strain. Below is a dodecahedral bead model consisting of 20 such units I just made today. It took me about one week to bead all twenty units and connect them together. More than 2500 8mm faceted beads are used to make this model.

German mathematician Herman Schwarz first proposed P- and D-types triply periodic minimal surfaces (TPMS) in the 19th century. Later, about twenty years ago, Lenosky et al. theoretically suggested graphitic structures with suitable arrangement of seven-membered rings decorated in P- and D-TPMS as possible model structures of sponge carbon. Now, chemists and physicists call this kind of graphitic structures with negative Gaussian curvatures as Schwarzites. The one I have here satisfies this criteria, so we might call it the dodecahedral carbon Schwarzite.

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.

Gyroidal surface on top of a Macbook air

There are 8 helical strips right now. I still need to make the other half.

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: