Japanese Temple Geometry: SketchPad Files


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My first glimpse of Japanese Temple Geometry was through an article in the Scientific American Magazine by Tony Rothman, with the co-operation of Hidetoshi Fukagawa (Unfortunately the article is no longer kept on their site). The article's introduction explains:-


"Of the world's countless customs and traditions, perhaps none is as elegant, nor as beautiful, as the tradition of sangaku, Japanese temple geometry. From 1639 to 1854, Japan lived in strict, self-imposed isolation from the West. Access to all forms of occidental culture was suppressed, and the influx of Western scientific ideas was effectively curtailed. During this period of seclusion, a kind of native mathematics flourished. Devotees of math, evidently samurai, merchants and farmers, would solve a wide variety of geometry problems, inscribe their efforts in delicately colored wooden tablets and hang the works under the roofs of religious buildings. These sangaku, a word that literally means mathematical tablet, may have been acts of homage--a thanks to a guiding spirit--or they may have been brazen challenges to other worshipers: Solve this one if you can! For the most part, sangaku deal with ordinary Euclidean geometry. But the problems are strikingly different from those found in a typical high school geometry course. Circles and ellipses play a far more prominent role than in Western problems: circles within ellipses, ellipses within circles. Some of the exercises are quite simple and could be solved by first-year students. Others are nearly impossible, and modern geometers invariably tackle them with advanced methods, including calculus and affine transformations."


This set me on a quest to locate the book from which most of the article was based upon:

JAPANESE TEMPLE GEOMETRY PROBLEMS. H. Fukagawa and D. Pedoe. Charles Babbage Research Foundation, Winnipeg, Canada, 1989.

It was NOT easy to locate for someone no longer in academe but after 6 months or so, and with the help of a very nice librarian from Cambridge University, I was able to get a copy on loan from the National Library of Canada - for two weeks only and not to be removed from my local library during this period!!! Phew...
Well, it was truly worth it. The book contains some of the most inspirational geometric problems that I have ever seen. For me, their beauty lies both in their apparent simplicity and accessibility to all levels. However, this often underlies some very demanding mathematics, needing a great deal of insight that (for me at least!) comes from lots of hard work!

The Book can actually still be purchased by writing to the Office Manager at:

Charles Babbage Research Center
P.O. Box 272, St. Norbert Postal Station
Winnipeg, MB
Canada R3V 1L6

My thanks go to Jessica for this address.

Any files below that are prefixed by a "J" come directly from the above book and refer directly to problems given within the text. More will be added along the way...this may take some time because their are over 250 true Japanese Temple Geometry problems and a further 100 problems based upon a square that are of a similar nature! None-Japanese Temple files may be here because the results/Theories contained within may be useful background material.

Notes:
(1). All of these filenames in this page are LONG! - Therefore, if you want to preserve these (Because Version 3.* of SketchPad does not use long filenames) you must either RIGHT-CLICK on the file links within the above directories and then choose "Save Target As...". This allows you to save the file directly to a disk/drive or after using a LEFT-CLICK select the "Save this file to disk" Option Box rather than selecting the "Open this file from its current location". If you choose this last option then Geometer's SketchPad will automatically open - if you have it! - but then after viewing the file if you then try to save it you will not be able to use a long filename.

Sketches

D - Stewart's Theorem.gsp
A very useful but lesser known result connecting a Cevian and Triangle sides.

D - Carnot's Theorem.gsp


D - J1-2-3.gsp
Find the distance between the two parallel lines in terms of the radii of the given Circles


D - J1-2-7.gsp

D - J1-2-8.gsp
This is a most useful result that can be used in many guises in other problems.
Prove that


where a, b and c are respective radii, and d is the length of the exterior tangent

D - J2-1.gsp
The triangle is equilateral. If the internal circles have the same radii, what is it?

D - J2-3-1.gsp
Find the radii of the internal circles in terms of the hemisphere of radius r


D - J3-1.gsp Solution: PDF (87KB)

D - J3-1-2
Find the side length of the internal square and the radii of the internal circles.

D - J3-1-3.gsp
Find the radius of the incircle shown


D - J3-1-6.gsp

D - J3-2-3.gsp
Show that the following relation is true



where r is the radius of the central circle. Solution: PDF (174KB)

D - J3-2-4.gsp
Prove that the radius of the middle circle C2 is the Geometric Mean of the radii of the other two circles C1 and C3.
That is .

D - J3-2-5.gsp
Show that the ratio of the radii

D - J3-3-1.gsp
Find the radii of the internal circles.

D - J3-3-2.gsp
Find the radii of the internal circles.


D - J4-2-1.gsp


D - Own (1).gsp


D - Own (2).gsp
Find the radius of the small circle in terms of r, h and x if both outer circles have the same radius, r.


D - Japanese Temple Theorem - Radii Sum in Cyclic n-gon.gsp
A well-known Theorem - use Carnot's Theorem above to help with proof.

Link http://www.wasan.jp/tokyo/tobuti.html


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Circles | Conics | Gothic Arches |Higher Curves | Inversions | Japanese Temple Geometry | Mechanical Linkages | Miscellaneous | Miscellaneous Geometry Conditions | n-gons | Phi | Quadrilaterals | Scripting Tools | Triangles