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

Hyperbolic buckyball

拍攝時間：2015/4/28; 地點：北一女
作品完成時間：2015/2/8

Another surface with negative Gaussian curvature

2015/4/4

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

Hyperbolic soccerball

I posted many hyperbolic bead models before. But most of them are periodic surface structures in 1- to 3-D dimensional spaces. Examples are various periodic minimal surfaces. In these models, one needs to pay attention to the subtle periodic conditions in the course of beading. Sometimes, it makes the beading quite difficult.
Here, I show a simple construction of hyperbolic soccerball (truncated order-7 triangular tiling) consisting of infinitely many heptagons (blue beads), each of them are connected to seven neighboring heptagons by only one carbon-carbon bond, which is represented by a yellow bead in the model shown below.

Following the spiral beading path by adding hexagons and heptagons, eventually one obtains the hyperbolic soccerball, or more exactly a hyperbolic graphitic snowflake. There is no need to worry about the periodic conditions among different parts of the structure.

From wiki: Truncated order-7 triangular tiling

In principle, one can also use kirigami (paper cutting) to make a model of the hyperbolic soccerball. But I found that the beading technique is much easier for making robust structure of this object due to the nature of mathematical beading. Also the bead hyperbolic soccerball should be able to model the local force field of hyperbolic soccerball to certain extent because the bead model not just gives the connectivity of the molecular graph right, but also mimics the microscopic repulsions among chemical bonds.

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

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

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.

Hyperbolic bead model of lettuce

I took this picture of hyperbolic lettuce made of beads in Yan-Ping North road, where you can find many craft stores, a few months ago. It is amazing that people have made so many interesting geometric models with beads for a long time, possibly without knowing the underlying mathematics.

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)

Hyperbolic "buckyball" with octagons

I played with carbon tetrapods described in the Diudea and Nagy's book last weekend and discovered quite accidentally that it is possible to make another type of hyperbolic "buckyball" with neighbored nonhexagons separated by only one CC bond. Particularly, all nonhexagons in this structure are octagons. So this one is different from another hyperbolic "buckyball" made of heptagons and hexagons, or simply heptagonal buckyball. The skeleton of this structure is the same as the heptagonal "buckyball". But each unit cell, which consists of two connected tetrapods, has 96 carbon atoms. I am not sure if people have mentioned this kind of hyperbolic "buckyball" or not.

Heptagonal "buckyball", C168 (This model was given to Prof. Sonoda as a gift when I was invited to Japan for a series of talks and workshops.):

(photo by Dancecology)

Octagonal "buckyball", C96: