Sangaku (one example above, carved into a wooden tablet) are japanese geometrical problems originating in Japanese temples of the 17th century. The problems usually consist of a group of shapes that share one surrounding tangent shape. Sangaku were presented in temples in honor of the gods, and this is the most fascinating thing about Sangaku; the fact that geometric contemplations could have religious properties. Any combination of art math and faith is an enigma, but also seemingly natural, why sacrifice an animal or fast for the purpose of appeasing a higher power, when one could create mathematical conjecture and theorize about perfect shapes in honor of any ethereal forces or traditions you observe. Doesn’t geometrical theory seem so much purer and theological then a hymn or diet? My surprise and confusion at the utility of geometry as a sacred persuit compelled me to share this idea with any who don’t know about these types of problems and to open one of the problems I found as open discussion for the class.
After researching some existing sangaku problems and their solutions, I found a problem I would like to explore and dissect (pictured above) but I haven’t been able to piece together the construction of the problem after assigning the value of diameter of the largest circle (tangent) as one, so obviously there is more progress to be made. Anyone who is interested in solving or analyzing this problem, please contribute, I’m posing this as a problem and a challenge.
This Sangaku looks like it can't be solved because the big inner circle looks like it totally arbitrary.
ReplyDeleteyou have to assign a value to the diameter of the outer circle, 0 is the center point of the tangent circle and 0' is the center point of the largest inner circle. should have clarified that in the original post.
DeleteThere's a stained glass window at my church that's a fantastic sangaku. I've solved it many times; doing so feels similar to (and perhaps in fact is) praying. I'll have to add a photo of it to our map of Found Geometry.
ReplyDelete@Angus, could you provide a link to where you found the sangaku you posted? Thanks!
http://www.cut-the-knot.org/pythagoras/1331Sangaku.shtml
DeleteThanks for sharing that link, Angus.
Delete@All: It should be noted that in the sangaku Angus posted, the sizes of the circles are not fixed. Rather, we're to prove a relationship among the radii of the smaller circles. Follow the link Angus posted for details.
Finally, I've added by church sangaku to the Found Geometry map.