Abstract:
Mathematics is everywhere in life. Even within the short dinner time, it helps me solve two big problems.
Scene 1: I have confidence in saying that the four legs of my kitchen table have the same length, since it cost me a lot of money. Unfortunately, it wobbles because of my old floor, which I cannot afford to fix right now. Fortunately, the Dyson-Livesay Theorem gives me a cheaper solution. It tells me that I can fix this by just rotating the table by some angle.
Connect the four feet of our rectangular table diagonally with two line segments. Then these two segments intersect at some angle α and form two diameters of some sphere S2. (See Figure 1(a), 1(b).) Imagine lifting the table, along with the sphere, high above the ground and let f(x) denote the vertical distance from a point x on that sphere to the floor. This function is clearly continuous on our sphere. The Dyson-Livesay Theorem states that we can find two points p and q on the sphere S2 such that f(p) = f(−p) = f(q) = f(−q) and (p, q) = α. That means, if we rotated the table in space so that the four table feet fit into the locations p,−p,q and −q and lowered it to the floor it would rest firmly. Therefore, the same result can be accomplished by simply turning the table on the ground, while keeping the intersection of the diagonals on the same vertical line.
