昨天到科教館參加「探索化學世界」的開幕，碰到不少記者，也是第一次跟那麼多記者講話。國語日報記者陳祥麟與陳壁銘替我跟zometool準晶模型拍了幾張照片，寫了一個有關串珠的報導：

分子結構像串珠 教授動手做模型 (2012/12/30)

I met a few local reporters in the opening ceremony of "The wonderland of chemistry" held in the National Center for Science and Education yesterday. This was the first time I talked to so many reporters. A photographer from the Mandarin Daily News (國語日報), a traditional Chinese children's newspaper published daily in Taiwan (wiki), took a picture of me with the giant Zometool model of quasicrystal hung under the ceiling in the exhibition room. The reporters of the Mandarin Daily News also wrote a report about my bead models in Chinese.

## Sunday, December 30, 2012

## Saturday, December 29, 2012

## Thursday, December 27, 2012

### High-geneus fullerene by Mr. Horibe

I just got an email from Mr. Horibe with a link to his page, which gives many photos on the different stages of making a dodecahedral high-genus fullerene. It should be useful for anyone who wish to make this model.
Mr. Horibe also emailed me a beautiful picture of
the backyard of his house in Tajimi. We took a picture together
at the same place this June, without snow though.

## Wednesday, December 26, 2012

### Beaded trefoil knot using two beads per edge

## Sunday, December 23, 2012

### Oval shaped beads

I bought some plastic oval shaped beads a few months ago. I only made a buckyball with this kind of beads and never used them again. I saw these beads this afternoon accidentally and decided to make some more simple models with these oval shaped beads.
So, here are five Platonic solids and a rhombic triacontahedron I made.

## Thursday, December 13, 2012

### Giant beaded buckyball

I went to the 109th anniversary of TFGH yesterday and took a photo of this interesting giant buckyball.

## Tuesday, December 11, 2012

### 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.

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.

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.

## Monday, December 10, 2012

### 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.

## Sunday, December 2, 2012

### Bead models and Omnitruncated dodecaplex for Math Education meeting

I had an exhibition for a local meeting on math education held in the math-astronomy building (數學天文館) of the National Taiwan University this weekend. Math-Astronomy building is close to the chemistry building, 積學館, where I am working. But I only have two tables for my beadworks. So I chose a few larger ones for this exhibition. The following pictures of my exhibition were taken from the Facebook of Ms. Helen Yu.
Helen also prepared a giant Zometool model, a 3-meter omni-truncated dodecaplex, for this meeting. More than 10 grad students from the theory group of chemistry department and a few students from the math department participated the final stage of this project one day before the meeting. We faced a number of problems during the construction due to the huge size of the model. It is hard for the model to hold its own weight especially when one tried to build the top first. But eventually, we solved almost all of the problems, except the final piece located at the North Pole of this model. Even though one of students is about 197 cm high, still we couldn't fix that one. Fortunately, it is not easy to notice it. So, we decided to leave it slightly incomplete because the model will be dismantled after a week of display in the mathematics/astronomy building.

## Wednesday, November 21, 2012

### 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.

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.

## Friday, November 16, 2012

### 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.

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

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

## Thursday, November 15, 2012

### 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..

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..

## Monday, November 12, 2012

## Friday, November 9, 2012

### 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.

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.

## Thursday, November 8, 2012

### 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.

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.

## Tuesday, November 6, 2012

### 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)

## Thursday, November 1, 2012

### 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.

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.

## Friday, October 26, 2012

### 珠璣科學(Zhu-Ji Science)

In the November issue of Science Monthly (科學月刊), I wrote the sixth article of the Zhu-Ji Science series (珠璣科學). This one is about the particular problem of "30 small balls cover a big ball" from the Japanese temple geometry.

金必耀, 左家靜, 珠璣科學─日本寺廟幾何與正十二面體串珠模型 (Zhuji Science - Japanese temple geometry and the bead model of regular dodecahedron), 科學月刊 2012, 33(11).

金必耀, 左家靜, 珠璣科學─日本寺廟幾何與正十二面體串珠模型 (Zhuji Science - Japanese temple geometry and the bead model of regular dodecahedron), 科學月刊 2012, 33(11).

### NTU newsletter: Chemist Builds Intricate Nanomolecular Models with Beads

I was told by a secretary of Science school of the NTU (National Taiwan University) early this year that they wanted to write about the beaded molecules. But I didn't know that they really wrote something in the June issue of NTU newsletter until now.

## Saturday, October 20, 2012

### 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)

## Friday, October 19, 2012

### 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.

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.

## Tuesday, October 16, 2012

### Beaded Hilbert Curve (Step Two)

A couple of years ago I made a (half of a) Hilbert curve, as I recall it was inspired by a conversation with Bih-Yaw. And soon I forgot about this and went on to other beaded molecules. It was at the Bridges this year that I came across the 3D-printed sculptures of Dr. Henry Segermen. Then I decided to make another beaded model for this amazing mathematical figure.

I finished submitting this, together with a short introduction on both the novelty of the beading technique and the curve itself, to the Joint Mathematics Meeting 2013, which will be held in San Diego next January. I don't think I could physically be there at the meeting, though.

I finished submitting this, together with a short introduction on both the novelty of the beading technique and the curve itself, to the Joint Mathematics Meeting 2013, which will be held in San Diego next January. I don't think I could physically be there at the meeting, though.

Labels:
Exhibition,
Hilbert curve,
Space Filling

## Wednesday, September 26, 2012

### Animated dodecahedral beaded structure by Horibe-San

Mr. Horibe (Horibe-San) just informed me a beautiful beaded structure he made with a single string! The address is here:
http://horibe.jp/JPGBOX/3540BOX/Beads3540_ss.htm.

BTW, he also made an animated sequence of the whole beading procedure using Javascript for this structure. We can see how he managed to make each node sequentially along a Hamiltonian path on 12 vertices of a dodecahedron and a Hamiltonian path through each face within a node. It looks great!

BTW, he also made an animated sequence of the whole beading procedure using Javascript for this structure. We can see how he managed to make each node sequentially along a Hamiltonian path on 12 vertices of a dodecahedron and a Hamiltonian path through each face within a node. It looks great!

## Saturday, September 22, 2012

### Two pictures of C20 Sangaku problem

I wrote a simple matlab script to generate the 3D structure of the C20 Sangaku problem.
The first one is viewed from a 3-fold rotational axis and the second one is grom the 5-fold axis. Following the original problem as shown in the previous post, I used blue color to denote ten small spheres located on a great circle.

## Friday, September 21, 2012

### Ancient proof of R/r=sqrt{5}

In the Meiji era, mathematicians of Japan didn't use trigonometry to prove R/r=sqrt{5} for the problem about a big ball covered by 30 small balls. I doubt that they knew the trigonometry as we know today.
Instead, a regular pentagon as shown in the following figure was recognized. Then the answer follows naturally. It is a smart proof, isn't it?

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