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!

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?

Some photos from the 3rd International workshop between College of Science, National Taiwan University and Faculty of Science, Okayama University

Early this month, I gave another workshop for the 3rd International workshop between College of Science, National Taiwan University and Faculty of Science, Okayama University, which is a one-week workshop for undergraduate students from both universities. Most of courses offered in the workshop were advanced chemistry courses such as quantum chemistry, biophysical chemistry, inorganic chemistry, etc. Mine is the simplest one, though. No any background in science or chemistry is required. I gave a half-hour background introduction before participants started to make their own buckyballs. Below are some photos from the workshop:

First page of my slides:

One more picture of super buckyball

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: