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.

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:

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.

Visiting Horibe and workshop at Nagoya Child and Family Center

Under the great arrangement of prof. Sonoda, I have this opportunity to visit Mr. Horibe in his place in the Aichi prefecture last Saturday (6/9/2012), two days before I returned back to Taiwan. It was really nice to meet him and saw many of his amazing mathematical beadworks. But, he doesn't speak English, so I can only communicate with him mainly through the translation of Prof. Sonoda. Most often, they spoke to each other in Japanese. But I still had plenty of time taking pictures and inspected his beadworks. I hope my understanding about his work is correct. I will post some of his beadworks and also compare his works with mine in the next few posts. But what I wrote could be my misunderstanding.


I knew Mr. Horibe and his beadworks through Prof. Sonoda two years ago. As I mentioned before that Prof. Sonoda is a missionary of cube kaleidoscope and has the passion to share this activity to kids all over the world. He visited Taiwan two years ago to give not only his own chemistry research, but also a workshop on the cube kaleidoscope. I participated both events and also shown him my works on beaded molecules. After he returned to Japan, he knew quite accidentally from a secretary of Prof Ono that a local high school teacher had set up a website which contains a lot of similar mathematical beadworks. So prof. Sonoda informed me about Horibe's beadworks immediately. Finally we have the chance to meet each other.

On Sunday, Mr. Horibe and his wife also participated the workshop held at Nagoya Child and Family Center near B's Hotel hosted by Ms Hiroe Takeuchi (6/10/2012).

Workshops in Japan

Prof. Sonoda of Kyushu university invited me to Japan next month to have a few lectures and workshops in a few different places of Japan. Prof. Sonoda also invited Mr. Kazunori Horibe when I give a workshop in Nagoya Child and Family Center.

形狀美麗的分子（Chemistry & Chemical Industry, Vol. 62, 2012）

Prof. Sonoda (園田高明) of Kyushu University sent me a special issue on "Beautifully Shaped Molecules" of "Chemistry & Chemical Industry" published by Japanese Chemical Society. There are five articles by famous Japanese chemists in this issue. The photos I showed here are from an article on "Self-Assembled Polyhedra" (my translation based on the Kanji in the title, I don't know Japanese, though) by Makato Fujita (藤田 誠) of Tokyo University.

Photos of Workshop for Exchange Students from Okayama Univeristy

Mr. Hwang took many photos for my workshop last Tuesday.

As usual, I gave a 30-min short talk before we started to make beaded C₂₀ and C₆₀.

While I was talking, students were free to examine the beadworks I brought to the workshop.

Prof. Sonoda (Kyushu University) also visited Taiwan in those few days. So I invited him to this workshop. He seemed to have fun making a beaded C₂₀.

Almost all of students succeeded in making their own C₂₀ and C₆₀ in this 3-hour workshop. You can see how happy they were when they finally made their own buckyballs.

After the workshop, I took a picture with my colleagues.

Cube kaleidoscope (Wonder Mirror Box, or CUMOS cubic cosmos scope)

N asked me about the cube kaleidoscope Chern posted before. If you are interested in getting instructions and materials for making your own cube kaleidoscope, you should contact Prof. Takaaki Sonoda (園田 高明) of Kyushu University (九州大學). (reports about him in Czechish??? and in Japanese).

Here I just want to show the other two cube kaleidoscopes, kaleidostereochemistry and kaleidogarden, I made when Prof. Sonoda visited Taiwan about one year ago.

It is really fun to make your own kaleidoscope. Basically, one needs to have six mirrors to make a cube kaleidoscope. Once you have the necessary materials, you need to make patterns on two or three mirrors by scratching part of mirror away by using suitable stick. For details, please contact Prof. Sonoda.

In addition to the two kaleidoscopes I made by myself, I also got another professionally made cube kaleidoscope from Prof. Sonoda as a gift.

You can also find extra information at this site (CUMOS cubic cosmos scope). Here is one I found at that site:

More images is in this gallery.