How Does Topology Help Solve the Inscribed Rectangle Problem by Proving that Every Jordan Curve Has 4 Vertices that Form a Rectangle?  

作　　者：Heidi Hassan Heidi Hassan(Independent Researcher, Cairo, Egypt)

机构地区：[1]Independent Researcher, Cairo, Egypt

出　　处：《Journal of Applied Mathematics and Physics》2023年第4期859-873,共15页应用数学与应用物理（英文）

摘　　要：When we stare into our complex surroundings, we see objects of different shapes and sizes. However, the shape that is always present, regardless of the complexity of the object, is the circle. The circle is arguably the most fascinating shape in the universe. A circle is defined as the set of all points equidistant from a given point, which, therefore, lies at the center of the circle. One of the various properties of circles is that it has infinite inscribed squares. This is because it is a continuous function, therefore if any point in the circle is transitioned by a specific factor, the other related points in the square would be shifted by the same factor. An ellipse is a similar shape with several inscribed squares. But does any closed curve have an inscribed square? This question was proposed by Otto Toeplitz in 1911 and to this day it is not answered. Another version of this problem is the inscribed rectangle problem which will be discussed in this paper.When we stare into our complex surroundings, we see objects of different shapes and sizes. However, the shape that is always present, regardless of the complexity of the object, is the circle. The circle is arguably the most fascinating shape in the universe. A circle is defined as the set of all points equidistant from a given point, which, therefore, lies at the center of the circle. One of the various properties of circles is that it has infinite inscribed squares. This is because it is a continuous function, therefore if any point in the circle is transitioned by a specific factor, the other related points in the square would be shifted by the same factor. An ellipse is a similar shape with several inscribed squares. But does any closed curve have an inscribed square? This question was proposed by Otto Toeplitz in 1911 and to this day it is not answered. Another version of this problem is the inscribed rectangle problem which will be discussed in this paper.

关 键 词：Inscribed Rectangle Problem TOPOLOGY Jordan Curve Mobius Strip 

分 类 号：O15[理学—数学]