Paper Folding And Cutting Sangaku
The following three problems are from the 1743 book Kanyjia Otogi Zoshi by Nakane Genjun (17011761) [Fukagawa and Rothman, pp. 7677, pp. 8587]. In an English translation, Collection of Interesting Results in Mathematics, it sounds almost like one of D. Wells' books (The Penguin Dictionary of Curious and Interesting Geometry, The Penguin Dictionary of Curious and Interesting Numbers and The Penguin Dictionary of Curious and Interesting Puzzles). And indeed it gives us samples of the Japanese recreational mathematics of the 18^{th} century. (Fukagawa and Rothman's book is devoted to a peculiar mathematical art of sangaku but they also took pains to highlight the historical background of Japan during the Edo period and along the way outline the biographies and work of many contemporary mathematicians, Nakane Genjun in particular.

References
 H. Fukagawa, A. Rothman, Sacred Mathematics: Japanese Temple Geometry, Princeton University Press, 2008
Up Contact Front page Contents Geometry
Copyright © 19962018 Alexander Bogomolny
The first problem comes with two solutions which the diagram below make quite obvious.
A solution to the second problem which is the least trivial of the three is presented by the following diagram where the light dotted lines indicate folds and the dashed lines indicate the cuts.
The diagram begs for an explanation. First fold the rectangle along the long axis LM. Then fold the corner D into J on LM. This will generate point K and divide angle at A into three equal angles of 30°. Indeed, in the right ΔAJM,
Let's check the lengths. From ΔAKD, DK = √3/3 = JK.
Here is the same decomposition with colored pieces.
Solutions to the problems with five squares are shown by the diagrams and require little explanation if at all:
(a) 

(b) 
(c) 
The latter is nothing but a proof of the Pythagorean theorem.
 An Interesting Example of Angle Trisection by Paperfolding
 Angle Trisection by Paper Folding
 Angles in Triangle Add to 180^{o}
 Broken Chord Theorem by Paper Folding
 Dividing a Segment into Equal Parts by Paper Folding
 Egyptian Triangle By Paper Folding
 Egyptian Triangle By Paper Folding II
 Egyptian Triangle By Paper Folding III
 My Logo
 Paper Folding And Cutting Sangaku
 Parabola by Paper Folding
 Radius of a Circle by Paper Folding
 Regular Pentagon Inscribed in Circle by Paper Folding
 Trigonometry by Paper Folding
 Folding Square in a Line through the Center
 Tangent of 22.5^{o}  Proof Without Words
 Regular Octagon by Paper Folding
 The Shortest Crease
 Fold Square into Equilateral Triangle
 Circle Center by Paperfolding
 Folding and Cutting a Square
Up Contact Front page Contents Geometry
Copyright © 19962018 Alexander Bogomolny
65607297