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Sangaku

In my exploration of the Internet in search of non-Western approaches to mathematics, I was delighted to discover a tradition in medieval Japan known as Sangaku, based on a form of traditional Japanese mathematics known as wa-san.

As I discovered from a number of sources, including here at Wikipedia , because Japan was cut off from Western societies during the period known as the Edo Period, the Japanese developed their own system to deal with the analysis of shapes and numbers. Citizens ranging from samurai, farmers and laborers would devise difficult mathematical problems and represent them in colorful wood block cuttings that they would hang from the temple walls. These served as both offerings to the gods, as well as challenges to other temple goers. An example is the picture above from 1847 appearing in the city of Uchiko.

There is apparently an excellent book, called the Sacred Geometry of Japanese Temples, which discusses the history of Sangaku and traditional Japanese mathematics in greater detail. I found this excellent set of geometry problems here (but note that the links to the answers no longer work!).

Let's start with the most straightforward one:

Q1) In the diagram below, three orange squares are drawn inside of a green triangle. If the radius of the blue circle in the upper left of the triangle is equal to 3, what are the radii of the other two blue circles?

Q2) In the diagram below, the three circles are tangent to each other and to the line beneath them. If the radius of the orange circle is 10, and the radius of the blue circle is 6, what is the radius of the red circle?

The Virtual Bead Loom

Although mathematics is often taught to be and thought to be solely the product of European culture, nothing could be farther from the truth. It IS true that Western European culture is responsible for much of the symbolic and logical structure of modern mathematics, and being fluent in this often arcane tongue is what is sometimes as thought of as being GOOD at mathematics. But the beauty of mathematics is that it tends to deal with concepts that are UNIVERSAL to the human experience, such as our understanding of shapes, patterns and quantities.

As mentioned briefly in the post Induction from Patterns, when I was in Guatemala recently I discovered that geometric patterns were an important part of Mayan civilization. I later learned that Guatemalan women have actually passed down their understanding of geometry to their daughters through the act of weaving. The patterns below are a picture I took in a street market in the town of Santiago de Atitlan along the shores of Lago Atitlan.




There is an excellent resource here called Culturally Situated Design Tools from the work of Ron Eglash at RPI which provide interactive environments to explore the connection between mathematics and art in indigenous cultures. There is a great overview of these tools in his paper here. The persistent theme across cultures is the presence of repeating geometrical patterns with different types of symmetry and regularity. There are tools to explore cornrow patterns in African American culture, beadwork in Native American cultures, and pyramid design and adornment in Mayan culture. There is little information on how to incorporate more advanced mathematics, particularly high school concepts, to the use of these tools, however.