Sangaku Optimization Problems

(All animations written by David Schultz in MAPLE (TM).  Source code available upon request: davvu41111@mesacc.edu)

 

Problem Statement

A square is constructed using the far-left endpoint of a  segment of fixed length. For what side length of the square will the area of the red triangle be a maximum? Sacred Mathematics: Japanese Temple Geometry. Fukagawa, H. & Rothman, T. 2008.

 

 

Problem Statement

A circle of varying radius is constructed from the far-right endpoint of a segment of fixed length. A right triangle is formed using the circle's center and the two endpoints of the segment. A square is constructed using the circle, the hypotenuse,  and the segment. Find the side length of the square that maximizes the square's area. Sacred Mathematics: Japanese Temple Geometry. Fukagawa, H. & Rothman, T. 2008.

 

 

Problem Statement

Two squares of fixed side length are positioned side-by-side. Allow the left-hand square to rotate counter-clockwise with its bottom-left vertex and bottom-right vertex remaining in contact with the horizontal and the far-right square respectively. A segment is constructed using the upper-right vertex of the fixed square and the upper-right vertex of the rotating square. A vertical segment can be constructed using the upper-left vertex of the rotating square, the horizontal axis, and the intersection of the previously constructed segment. What is the maximum length this vertical segment attains? Sacred Mathematics: Japanese Temple Geometry. Fukagawa, H. & Rothman, T. 2008.

 

 

Problem Statement

A square of fixed side length is constructed. If we shrink the vertical diameter of the square and keep the side lengths fixed, a rhombus is formed. Within the rhombus another square can be formed. For what side length of the inner square will the area between the rhombus and the inner square be maximized? Sacred Mathematics: Japanese Temple Geometry. Fukagawa, H. & Rothman, T. 2008.

 

 

Problem Statement

In a given sector of a circle of fixed radius, R, a smaller circle of varying radius, r, is constructed. As the smaller radius increases, a chord tangent to the inner circle with left-endpoint fixed cuts off a region of varying area. What should the radius of the inner circle be in order to maximize this area? Sacred Mathematics: Japanese Temple Geometry. Fukagawa, H. & Rothman, T. 2008.

 

 

Problem Statement

When as square piece of paper of fixed side length is folded as shown in the figure, a circle is formed in the upper-left-hand corner which is tangent at three points to the paper. First show the red segment and the red radius are equivalent for all folds. Then determine where the paper should be folded in order to maximize the area of the circle. Adapted from: Japanese Temple Geometry Problems. Fukagawa, H. & Pedoe, D. The Charles Babbage Research Center, Winnipeg, 1989.

 

 

Problem Statement

Construct a right triangle with hypotenuse of length 1 which maximizes the length of the arc within the triangle created by the circle whose center lies on the left-endpoint of the hypotenuse. A History of Japanese Mathematics. Smith, D. & Mikami, Y. 1914.

 

 

Problem Statement

A rectangular piece of paper is folded so that two opposite corners coincide. If the height of the rectangle is fixed at a given length, what dimensions of the rectangle will give the maximum area of the shaded triangle? The Sangaku in Gumma. Gumma Wasan Study Association, 1987.

 

 

 

Problem Statement

The diagonals of a trapezoid are fixed with lengths a and b with b < a. What is the horizontal length, x, which produces the trapezoid of maximal area? Sanpo-Jojutsu, pg. 151.

 

 

 

Problem Statement

What is the shortest circular arc of which the altitude above its chord is one? A History of Japanese Mathematics. Smith, D. & Mikami, Y. 1914.