In January 2008, I spent two wonderful weeks in Kyoto, as a guest of
the Japan Society for the
Promotion of Science
. I am very grateful to
the Society and to Professor Koji Miyazaki and to Professor Hideki
Tsuiki for inviting me to come and for making many excellent
I am also grateful to Professor
Terufumi Ohno of the Kyoto University
who did many preparations for the activities shown here, which I did at
the museum during the visit. Here is an announcement
The image above shows a six-foot diameter projection model of the
truncated 600-cell, in the lobby of the Kyoto University Museum, Kyoto,
Japan. This is the first of two workshops I led there on January 6,
Above, you can see participants starting the workshop, getting familiar
with the Zometool components. Initially, everyone makes a regular
icosahedron for practice, even though only one is needed in the final
Now we are making more complex
icosahedra which are compressed relative to regular icosahedra. Ninety
modules are required all together, so it is great to have many people
Above, we have built many of the modules. Five
different shapes are required. We count them and put them aside until
it is time to assemble them.
I had initially tried to start at the center and work outwards,
but the structure was very fragile. So we replanned and started
assembling from the bottom up.
As it grows, people can work from all sides. It can be understood as a
central core surrounded by four layers of icosahedra. Each layer is
more flattened. The outermost layer of thirty icosahedra is completely
The final result is interesting to study.
Mathematically the form is a three-dimensional projection of a uniform
four-dimensional polytope called The
Five truncated tetrahedra and one
icosahedron meet at each vertex. It is one
polytopes in the H4 family that can be made in
projection with Zometool parts
. We thank Image Mission Inc.
parts used in the
workshop. There are more photos (with Japanese explanation) here
The second workshop at the museum was to make a sculpture out of CDs
connected with tie wraps. The shape is based on a truncated
icosahedron. We made two copies of it and hung them up on display. We
cut off the protruding tie wraps of the one on the left, but the one on
the right we kept "hairy".
This workshop begins by making a ring of ten CDs Five are colored and
face outwards. They correspond to the vertices of the truncated
icosahedron. Five others, which correspond to the edges of the
truncated icosahedron, face inwards. Twelve of these modules are
required for one sculpture.
We start to assemble the modules and
hang them up so we can work around
them from all sides.
We continue adding modules and connecting them to each other, working
downwards from the top. There are six colors used for this sculpture,
with opposite modules being the same color. In the background above,
you can see a drawing
which we made before the assembly, to learn its structure.
At the same time, we work on a second sculpture which has the same
form. There are 150 CDs and 180 tie wraps in each sculpture.
The final result is very shiny and attractive, but it will not last
long. The CDs will soon crack from the bending. So this is a fun
educational activity, but it does not make a permanent sculpture. For
more information about this construction, see my paper on p. 205 in the
Some of the photos above were taken by Kenji Nagai of the Japan Zome Club
. There is a report (in
Japanese) with more photos here
article about the CD
workshop was printed
next day in the Kyoto section of the newspaper Asahi Shimbun
. You can see it on my citations
I gave a talk at Kyoto University in the clock tower that is behind
this tree, which symbolizes the university.
I had wonderful meals with wonderful people.
I saw this interesting sangaku.
Thank you Koji Miyazaki.
Thank you Hideki Tsuiki.