A Modern-Day Sangaku

Prof. George Hart

An interdisciplinary activity and exhibit, that blends art,
mathematics,
computer science, and traditional Asian culture.

On monday, April 24, 2006 (note the date change) Stony Brook students, faculty, and staff are invited to help me create a modern-day sangaku in the Wang center. This will be a large geometric sculpture---a sphere 6.5 feet in diameter---assembled from ten thousand eight hundred small plastic components.  The image above is a computer-generated rendering which gives a hint of how complex it will be. I will need many helpers during the day to get the parts together. We will start at 10:00AM, but feel free to come and go whenever you are free during the day. We might finish around 2:00 or 3:00, depending on how many people show up to participate.

Sangaku (also written san gaku) is a Japanese tradition of celebrating geometrical beauty, popular during the Edo period (1603-1867) by members of all social classes. The traditional sangaku is a shaped wooden tablet on which a geometric problem or theorem is written with colorful paint. The tablet was displayed in a temple or shrine as an offering, as a puzzle for others to think over and understand, and as a way for the creator to show off the geometric discovery. Many are so beautiful that they are considered works of art. Although most of these sangaku were lost, almost nine hundred survive, scattered all throughout Japan.

I propose to create a modern-day celebration of geometric beauty by leading members of the Stony Brook community in a large colorful geometric construction. Our temple or shrine-like environment for displaying the result will be the Wang center, which provides an ideal atmosphere for people to contemplate the beauty of the geometric form. The form we will make is a three-dimensional shadow of a four-dimensional polytope in the 120-cell family. Although we are updating to a three-dimensional construction from the traditional two-dimensional tablet, its content like many sangaku contains numerous nested and tangent circles and spheres. A mathematical understanding of the derivation is not necessary for participants to help assemble it or for viewers to appreciate the wonderful patterns and relationships within its complex structure. I will be able to give assembly directions that anyone can follow, and will explain key mathematical ideas so participants get an intellectual sense of its richness in addition to the fun hands-on experience.

A model from this four-dimensional family is especially appropriate for this context because Japan is currently a leading center for research in this area of geometry. I will bring along a copy of Koji Miyazaki’s recent book (in Japanese) on Hyper-Structures, which includes computer-drawn images of the form that we will make. Miyazaki, of Kyoto University, also is the editor of a journal (in Japanese and English) devoted to four-dimensional geometry.

I believe no one has ever made a physical model of this structure, so this is a world premier event. As with traditional sangaku, it will be a work of devotees colorfully celebrating the beauty of geometry. It will remain on display for over two weeks, until the end of the semester. I will have a small disassembly event on friday May 5 when we take it apart and pack up the pieces.

Here is another view of the identical structure.
From this angle, one sees six "spokes" around a hexagon.