# 平面座標上長方形沙發旋轉問題之解的存在性

2004年

## 摘要或動機

1.當長與寬比值為無理數時，此問題無解

2.當長與寬比值是最簡分數時；若分子為奇數，此問題無解

3.當長與寬比值為偶數時，此問題有解

In this paper we discuss the solution of rotating sofa problem as follows : The
condition is : Merely allow to rotate the sofa several times by rotating 90 degrees
clockwise or counterclockwise around the vertex. (maybe A, B, C, or D in Fig. 1)
The question is : What’s the relationship between the length and the width of the
sofa, if we request the sofa translated next to the original position with direction
unchanged. (as shown in Fig. 1 with A’B’C’D’).

We take this problem as a mathematical one of rotating a rectangle in plane coordinates.
Then we derive the desired equations by using the tools of plane coordinates, trigonometric
functions, complex number, polar form of complex number, and vector. Finally, we
prove that：

1. When the ratio of length and width is irrational, the problem has no solution.

2. When the length of sofa is odd in the ratio of length and width, the problem
has no solution.

3. When the ratio of length and width is even, the problem has solutions.

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