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 introduction to quasicrystal in Zometool website

I saw a page at zomeool company website about the quasicrystal model made by the zometool. There is a paragraph about the model displayed in the 2nd floor, outside the Pan-Guan Lecture Room of the chemistry departmen, National Taiwan University. My former student, Hsin-Yu Ko, kindly introduced the model to the students from the King David School in Taipei, possibly early last summer.

http://zometool.com/goto/quasicrystals

The history of the zometool model of this quasicrystal is here.

準晶模型的前世與今生 The evolution of the 3D quasicrystal model

I wrote something in my facebook about the origin of the zometool model of quasicrystal (mainly in Chinese with a little bit English) which my students and I designed and created :

準晶模型的前世與今生 The evolution of the 3D quasicrystal model

Helen Yu, the dealer of zometool in Taiwan, also posted a lot of photos about the reconstruction of this model together with the carbon onion at the National Taiwan Science Education Center here:
Gluing& Hanging Models (Photo Credits)- keeping uploading..

Zometool: carbon onion and quasicrystal almost done

Photos by Helen Yu.

Two artworks for the Mathematical Exhibition of Joint Mathematical Meeting

Chern and I submitted two artworks to the Joint Mathematical Beading, which were accepted today: 1. Super Buckyball of Genus 31
2. Beaded Hilbert Curve, step two

In addition to the beadworks we submitted, we also noticed five Platonic bead models made by Ron Asherov. His bead models have multiple beads in an edge, which are similar to our works a few years ago. I labeled this type of bead models with Edge with Multiple Beads, where you can find all the posts. He doesn't seem to know our works along this direction, though.
He also mentioned that the Nylon string passes adjacent edges exactly once with carefully chosen path of string, which is simply the consequence of Hamiltonian path on the dual graph of the corresponding Platonic solids. We can view the whole beading process as a path through the face of polyhedron. Thus if there exists a Hamiltonian path through each face once (Hamiltonian path for the dual polyhedron), then the Nylon string will go through each beads exactly twice and only twice. Of course, you can also say the Nylon string will go through the adjacent edges exactly once. They are the same thing.

Here is a model made by Chern Chuang almost five years ago:

I was quite surprised by its rigidity when Chern showed me this model. At that time, people questioned me about the meaning of beads. I told some of my colleagues that spherical beads represent chemical bonds instead of atoms. Atoms are not shown in the bead model explicitly, instead, they are located at somewhere three beads meet. Most chemists feel uncomfortable with this connection. So Chern and I tried try to explore with the shape of beads and multiple beads and hope that they can better represent the shape of chemical bonds. So that is why we have these models in which multiple beads represent an edge.

But now, I have the valence sphere model of chemical bond as the theoretical foundation of bead models. Spherical beads are in fact the simplest possible approximation of electron pairs, in accord with the principle of Occam's razor. So to build a model of a molecule with only beads and strings is equivalent to performing a molecular analogue computation with beads. The result of computation is the approximate electron density of the corresponding molecule without referring to the Schrödinger equation or atomic orbitals (this is the comment I got from Prof. H. Bent). I have written an article on the connection between bead models and valence sphere model in Chinese for September issue of Science Monthly (科學月刊). I think I should write something about the molecular analogue computation with beads.

Of course, what I am saying above is to view bead models as molecular models. Results of mathematical beading do not need to have any connection to the molecular world. For instance, the beaded Hilbert curve accepted by the JMM mathematical art exhibition is a good example.

Update: equation of states for election

I added one more point to my equation of states for election using the result of US 2012 presidential election.

More explanation of this figure can be found in the previous post.

More explanation of this empirical relation can be found in the following draft a few years ago:

I also designed a homework problem for a course on physical chemistry four years ago:

The goal is to understand this empirical relation through the ensemble theory of statistical mechanics. Hopefully, students can have a better understanding between microstates and macrostates, and also the concept of equation of states.

Zometool: quasicrystal and carbon onion

National Taiwan Science Education Center (NTSEC, 國立臺灣科學教育館) is going to have new area for the chemistry demonstration. They also decided to hang the two zometool models, carbon onion and quasicrystal, we designed, constructed and donated (Chemistry department, National Taiwan University). However, the original structures, particularly the quasicrystal, are not stable enough to be hung under the ceiling. So we decided to strengthen the original quasicrystal model by adding a stick along the short diagonal of each face. These sticks are blue, which, I believe, can make the the whole model much more colorful. Additionally, my students have also carefully glued each part together. It takes a lot of work. So, the whole project is still unfinished yet.

From left to right: 郭岷翔，范原嘉（Yuan-Jia, Fan），詹欣穆，秦逸群，黃泓穎 (photo by Helen Yu)

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.