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:

The weight of mathematics

Is there a weight for a mathematical problem? The answer is Yes, if you talk about the Sangaku problem from the Edo period of Japan. Under the kind arrangement of Prof. Sonoda and Mr. Horibe during my visit to Nagoya this May (May 11, 2014), I was fortunate enough to see a few wooden Sangaku tablets, replica and original one, and really saw that mathematical problems can be really heavy. The next day after Horibe-San gave a workshop, Dr. Fukugawa Hidetoshi and I gave the other two talks in the Nagoya City University, we visited two temples in the Nagoya area.

Dr. Fukagawa is the premier authority on the Sangaku tablets in Japan. But he had a serious cold in that weekend, but still insisted to go with us to visit these places. In addition to him and me, we were also jointed with Mr. Horibe, Prof. Sonoda, one local high school math teacher to see these Sangaku wooden tablets.

The first temple we went is the Atsuta Shrine(Atsuta Jingu/熱田神宮). Right after we arrived, we were guided to a special room in the second floor by people from the Shrine, where the replica of two Sangaku tablets (dated 1841 and 1844, respectively) are carefully stored and not on display usually. Because we were special guests of Dr. Fukagawa, so we were lucky enough to have the privilege to examine these two beautifully-made replica.

After we had a brief lunch at Atsuta Shrine, Mr. Horibe brought us to a beautiful temple, Yourinzi temple (明星輪寺), in a nearby mountain area, Mount Ikeda (Ikeda-yama, 金生山) of Ogaki city, Gifu prefecture (岐阜縣大垣市). In this Buddhist temple, there is a well preserved Sangaku tablet made in 1865. Most importantly, many mathematicians and physicists, including Freeman Dyson, have been invited by Dr. Fukagawa to visit this temple to see the Sangaku tablet before.

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).

A few photos from workshop in the Nagoya City University

During my visit to Japan this May, Prof. Sonoda and Horibe kindly arranged a special workshop in the Nagoya City University on May 10, 2014. Here are a few photos from the workshop which connecting the Sangaku problems and mathematica beading.

In addition to this workshop, Horibe and I will also give a similar workshop in the Bridges meeting (Seoul) this year.

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?

Proof of R/r=sqrt{5} for the geometry of bead C20

Even though the original drawing for this particular sangaku problem is done by the black-and-white style, it may not be a bad idea to make a colorful version for the cross section:

or may be like this one:

Sangaku problem and C20

In the book "算法助述" I got from Mr. Horibe, there is an interesting problem in pp. 51-52 as shown in the following picture. As I mentioned before, Mr. Horibe made two beautiful bead models with a large metal sphere inside. And more importantly, large and small balls have the correct ratio of radii, R/r=sqrt{5}. He told me that he gave one of these two models to Hidetoshi Fukagawa, the author of the book "Sacred Mathematica - Japanese Temple Geometry", because, through Dr. Fukagawa, he knew so much about these interesting Japanese sangaku problems.

This Sangaku problem was originally proposed by Ishikawa Nagamasa and written on a tablet hung in 1798 in Tokyo’s Gyuto Tennosha shrine. I was quite surprised by that I could understand this ancient problem without any translation. It was written in Chinese (or Kanji):

"今有以小球三十個如圖圍大球，小球者各切臨球四個與大球，小球徑三百零五寸，問大球徑幾何？

答曰：大球徑六百八十二寸。"

In English:

Thirty small balls cover one big ball where each small ball touches four other small balls and big balls.
The radius of small ball is 305 inches. Find the radius of large ball.

Answer: The radius of large ball is 682 inches.

In the page 52 of the book, there are some explanations on how to reach the answer by noting that thirty balls are located on the thirty vertices of an icosidodecahedron. It was written with some Japanese characters which I don't understand. But it is not hard to guess the rationale. One can see that if one starts from an arbitrarily chosen small ball, then moves to next ball along a fixed direction, and then the next one along the same direction. Eventually, one would trace over exactly 10 balls which are located on the large circle of a sphere with the radius equal to the sum of radii of small and large balls. Using some trigonometry, one can get the answer.

Mr. Horibe's story

Yesterday, Mr. Horibe informed me with email that he made the first regular polyhedron with duralumin tubes around 1975. After that he was busy on teaching and nothing was done for a while.

However, around 1995, he made the first bead model of dodecahedron which was consisting of 30 balls based on the problem described in the famous math book (Sangaku book, 算法助術) from the Edo period (江戶時代) of Japan. Since then, he started to make many different kinds of mathematical beadworks alone for about 17 years until we met last week. He said he will keep on making more bead models.

Two Sangaku books from Mr. Horibe

I also got two nice books on Sangaku (算額) from Mr. Horibe.

The first one is a reprint of a famous Japanese math book (算法助術) originally published in the early 19th century.



A page from this book:


The second one is the catalog for an exhibition on Sangaku in Nagoya a few years ago (2005).



A page from this beautiful catalog:



Mr. Horibe said, however, very few people came to this exhibition.

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.