Have you ever wondered how to cut a ham sandwich into two equal parts with one slice? Or how to find a point on a sphere that has the same temperature as its opposite point? Or how to fold a protein or control a robot using polygons? If you are curious about these questions, you might be interested in learning about the Borsuk-Ulam theorem, a mathematical theorem that has many surprising and fascinating applications.
The Borsuk-Ulam theorem states that for any continuous function from a sphere to a Euclidean space of the same dimension, there is a point on the sphere that maps to the same point as its antipodal point. For example, if you map the surface of the Earth to a plane, there is a point on the Earth that has the same coordinates as its opposite point. This means that you can always find a pair of antipodal points on the Earth that have the same temperature, pressure, humidity, or any other property.
One of the applications of the Borsuk-Ulam theorem is the ham sandwich theorem, which states that for any number of objects in a space with the same dimension, there is a way to cut all of them in half with a single slice. For example, in three-dimensional space, a ham sandwich with two slices of bread and a piece of ham can be cut into two equal parts with one plane. The ham sandwich theorem takes its name from this case, but it can also be applied to other objects, such as cakes, pizzas, or pancakes.
Another application of the Borsuk-Ulam theorem is in planar polygon spaces, which are spaces of polygons with fixed side lengths in the plane. They can be used to model various physical systems, such as protein folding and robotics. The Borsuk-Ulam theorem can be applied to planar polygon spaces to prove some results about their topology and geometry. For example, it can be used to show that any planar polygon space has an even number of vertices.
The Borsuk-Ulam theorem is one of the most applied theorems in topology. It was conjectured by Ulam at the Scottish Caf´e in Lvov and proven by Borsuk in 1933. It has many equivalent formulations and generalizations for different types of functions and spaces. It has also been used in different areas of mathematics, such as combinatorics, differential equations, and economics.
As one mathematician said, “The Borsuk-Ulam theorem with various generalizations and many proofs is oneof the most useful theorems in algebraic topology.” If you want to learn more about this theorem and its applications, you can check out some online resources or books on topology and geometry. You might discover some new ways to look at the world and have fun with math.
– Ham sandwich theorem – Wikipedia, Wikipedia, 2021-10-25
– BORSUK-ULAM THEOREM AND APPLICATIONS, University of California, Los Angeles, 2021-01-17
– Some applications of the Borsuk-Ulam Theorem, University of Chicago, 2017-09-25
– The Borsuk-Ulam theorem for planar polygon spaces, ScienceDirect, 2021-01-01