Ghhm
Japan Temple Geometry
Abstract
In this science project the problems, which were written at Japanese temple boards are considered. These problems are differing from the European geometry by their solutions. Translated chapters from the book of Fukagawa and Pedoe were devoted to ellipses and n-gons, different combinations of the ellipses, circumferences and quadrilaterals, spheres, spheres and ellipsoids, different combination of the spheres and n-gons. In this science project the most interesting and original solutions of geometric problems were considered, in the base of using affine transformation.
Aim: Our aim is to show solutions of Japanese problems, which are new for Western mathematics; and to reveal these types of geometrical questions. In our opinion, solving different types of problems can help everyone to enlarge the outlook in mathematics.
CONTENTS
Chapter 1. Ellipses and triangles ……..……………….………..…...……. 3
Chapter 2. Ellipses and tetragons..……………………………….……..... 4
Chapter 3. Ellipses, circumferences and rhombuses …………………... 5
Chapter 4. Spheres …………………………………………...……………. 6
Chapter 5. Spheres and Ellipsoids ………………………………………...7
Chapter 6. Spheres, pyramids and prisms …………….…………………8
Conclusion …………………………………………………………………...9
History of sangaku.……………………………………..……………………10
Literature ………………………………………………..……………………11
Problem 1.
On ellipse O(a,b) three points (A, B and C) are chosen so that areas (S1, S2 and S3) of curvilinear triangle, are equal. Show that area of triangle ABC is [pic][pic]ab.
Solution.
The Ellipse shall be converted into circle b by using affine transformations, triangle ABC becomes some triangle A'B'C'. We shall show that this triangle is equilateral. Since ratio of affine
Abstract
In this science project the problems, which were written at Japanese temple boards are considered. These problems are differing from the European geometry by their solutions. Translated chapters from the book of Fukagawa and Pedoe were devoted to ellipses and n-gons, different combinations of the ellipses, circumferences and quadrilaterals, spheres, spheres and ellipsoids, different combination of the spheres and n-gons. In this science project the most interesting and original solutions of geometric problems were considered, in the base of using affine transformation.
Aim: Our aim is to show solutions of Japanese problems, which are new for Western mathematics; and to reveal these types of geometrical questions. In our opinion, solving different types of problems can help everyone to enlarge the outlook in mathematics.
CONTENTS
Chapter 1. Ellipses and triangles ……..……………….………..…...……. 3
Chapter 2. Ellipses and tetragons..……………………………….……..... 4
Chapter 3. Ellipses, circumferences and rhombuses …………………... 5
Chapter 4. Spheres …………………………………………...……………. 6
Chapter 5. Spheres and Ellipsoids ………………………………………...7
Chapter 6. Spheres, pyramids and prisms …………….…………………8
Conclusion …………………………………………………………………...9
History of sangaku.……………………………………..……………………10
Literature ………………………………………………..……………………11
Problem 1.
On ellipse O(a,b) three points (A, B and C) are chosen so that areas (S1, S2 and S3) of curvilinear triangle, are equal. Show that area of triangle ABC is [pic][pic]ab.
Solution.
The Ellipse shall be converted into circle b by using affine transformations, triangle ABC becomes some triangle A'B'C'. We shall show that this triangle is equilateral. Since ratio of affine