Enantimorphs for compounds of five tetrahedra

Left- and right-handed compounds of five tetrahedra built by a student, 資工系 徐衍新, in my class, Molecular Aesthetics 2020.

Workshop for students from Kanagawa University

I gave one more workshop for students from Kanagawa university, Japan. Today, I tried something new, instead of working on C₆₀ for each student, I asked them work on two zeolite structures, zeolite A and Faujasite, together after making the famous 30-ball Sangaku problem, which just gave them enough beading experience to move on. Both of these two zeolite structures consist of the same structural unit (Secondary Building Units, SBUs), namely truncated octahedrons. It seems to be easy for them to work together, then combine them into these two framework types. Here are a few pictures from the workshop.

Students were very happy when they succeeded in making the zeolite A.

A few photos from the NTU Chemistry Camp

I gave a workshop for the summer chemistry camp of the chemistry department at the National Taiwan University early this month. This is a summer camp for the local high school students. There are about 60 participants this year. Here a few photos from the workshop:

The title and the first slide of my workshop for the chemistry summer camp:

C20 vs C56

Workshop in Seoul

Mr. Horibe and I will give a workshop for the Bridges conference this coming August. Basically, we will follow the format we had in our joint workshop given in Taiwan this March. Horibe-San will first give a half-hour talk on the Sangaku in general and a special Sangaku problem in particular. And he will further describe the math about this Sangaku problem, particularly its connection to the continued fraction and then proceed to the construction of a physical model of this Sangaku problem before all participants make their own models that consist of 30 small wooden balls and a central large Ping-Pong ball.

Here are two photos of Mr. Horibe from workshop held in the math department of Academia Sinica (in the NTU main campus, Taipei) on Mar. 15.

The workshop paper, From Sangaku Problems to Mathematical Beading: A Hands-on Workshop for Designing Molecular Sculptures with Beads, can be found here (pdf file).

High-Genus Fullerene

Another dodecahedrane

I made one more valence sphere model (VSM) of dodecadrane (C₂₀H₂₀) yesterday. Although this model looks simple compared to other bead models I have made, I still feel satisfied every time I made a bead model of this molecule.

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.

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?

Sangaku and mathematical beading

Mr. Horibe, as a math teacher in Japan, is particularly interested in the connection between mathematical beading and the traditional Japanese temple mathematics, namely Sangaku （算額）. His teacher and colleague, Hidetoshi Fukagawa ( 深川英俊), is famous for his work on Sangaku and published a book, Sacred Mathematics: Japanese Temple Geometry, with Princeton's physicist Tony Rothman.

There are a few questions of Sangaku which are related to tangent spheres. For instance, the following problem appears in Fujita Kagen’s 1796 edition of Shinpeki Sanpo. (Collection of Fukagawa Hidetoshi.)



The translation of this problem given in the "Sacred mathematics" is
"Twenty small balls of radius r cover one big ball of radius R where each small ball touches three other small balls. Find R in terms of r."


Another problem related to the bead model of C₂₀ appeared in 1830 book Sanpo Kisho, or Enjoy Mathematics Tablets, by Baba Seitoku (1801–1860)



In this problem, small spheres correspond to beads that represent 30 edges of a dodecahedron.

To illustrate these two problems, Mr. Horibe made a beautiful, but non-standard bead model consisting of 20 ping-pong balls punctured with three holes at suitable places in order to connect them with an elastic rubber string. Of course, this model is a dodecahedron with ping-Pong balls located at the 20 vertices. Using the similar technique, he also made a bead model of an icosahedron consisting 12 Ping-Pong balls located at the vertices too.



To illustrate the second problem with 30 small spheres in touch with a large central sphere, Mr. Horibe did something even more amazing. He tried to make a composite bead model for the problem 2. This model consists of 30 small spheres outside (i.e. a dodecahedron) and a large sphere inside with small and large spheres satisfying the correct ratio of their radii, i.e. R=sqrt{5} r. This was a difficult task. Mr. Horibe quickly realized that it was very expensive to ask people to made two wooden beads exactly with this ratio. So he could only look around for different kind of balls in many stores in Japan, particularly he always brought a calculator and a ruler with him to measure if he was lucky enough to find out just the right beads that satisfied this condition. Quite fortunately, he found out a right size of metal ball for the central large sphere. So he made two nice models, one for Fukagawa and another for himself, for the famous Sangaku problem. In the following pictures I took at Nagoya's Children and Family Center, you can see how Mr. Horibe stretched the small spheres apart and pull the inner large sphere out. It was quite a show.

High-genus fullerenes

I made two more high-genus fullerenes with octagonal necks using 8mm beads and 0.6mm Nylon threads. If I remember everything correctly, Chern and I should have made four of this molecule (one in China and three in U.S.) before.

HG-Fullerenes with octagonal necks No. 5 & No. 6

Since this model contains less than one thousand beads, so I managed to finish one of them in a day.

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.

"Super" dodecahedron consisting of 20 punctured C60s

This is the final bead model of a dodecahedron by connecting 20 C60s with three holes described in previous post. The C60 units in this structure is distorted quite significantly.

Four face-sharing pentagonal dodecahedra

E. A. Lord, A. Mackay, and S. Ranganathan described in their book, "New geometries for new materials", a simple cluster consisting of four face-sharing pentagonal dodecahedra arranged in a tetrahedral configuration (pp.48). Here is a bead model of this cluster.

In their book, there are more clathrate structures that one might be able to construct with beads.

Clathrate cluster

I bought some more green rice-shape beads last week and managed to finish this interesting clathrate cluster of 60 dodecahedra in the last weekend. One can still see deformation of many dodecahedra in this clathrate cluster though.

Another set of beaded models of platonic solids

Here is another set of beaded Platonic solids made with the same type of beads.

two more pictures of an sp3 structure

Ping-Pong ball in a beaded dodecahedron

From beaded fullerene

Chuang made it.

Platonic solids with cylindrical bonds

Cylindrical beads that have capsule shape are perfect pedagogical materials for creating physical models of fullerenes since they can look like the chemists' intuition of chemical bonds and, at the same time, effectively mimic the steric repulsion among different beads. To demonstrate these points, here, we use these beads to create the five Platonic solids. 

Platonic Solids

I bought these elongated beads last week, on sale, of course. The shape of these beads are closer to that of chemical bonds. I have used these beads to create five Platonic solids. They look great.

Tetrahedron


Cube



Octahedron




Dodecahedron



Icosahedron