The Fun Side of Mathematics: Theorema Egregium
Today let’s talk about the fundament of every natural and technical science — mathematics. As the noted mathematician Carl Friedrich Gauss once said, mathematics is the “queen of the sciences”. Undoubtfully, it has been successfully used as a key to uncovering the laws of physical reality.
Personally, I’m a huge fan of mathematics. I admire how various and beautiful this science is. I often spend time on reading about fun and curious math theorems and unsolved problems. As a computer science student, I see so many mathematical ideas and applications used in programming. I am convinced that understanding mathematics and logic is crucial for a computer science specialist.
Unfortunately, many students face difficulties when studying mathematics and acquire a biased attitude towards the subject. That’s why my today’s goal is to show a different approach to learn about mathematics — from its fun side!
I will show you the beauty of mathematics on the example of one of my favorite theorems — Theorema Egregium (Latin for excellent or remarkable theorem). Remember who called mathematics the “queen of the sciences”? Carl Friedrich Gauss, that same 19th-century math genius came up with this theorem.
However, let’s start with a common situation — you decide to take a slice of pizza and almost take a bite but suddenly it flops over and dangles limply from your fingers. The crust isn’t stiff enough to support the weight of your pizza slice. Don’t worry, Carl Gauss will teach you the best way to hold your pizza providing mathematical reasoning for that. Simply fold the pizza slice into a U shape. This keeps the slice from flopping over, and you can proceed to enjoy your meal.
As you might have already realized, behind this pizza trick lies Theorema Egregium, an important finding in the mathematical field of curved surfaces.
Let’s proceed to the mathematical foundation of this idea. Imagine rolling a sheet of paper into a cylinder. It seems obvious that the paper is flat, while the cylinder is curved. However, Gauss thought about this in a different way. He came up with the definition of the curvature of a surface which doesn’t change when you bend the surface.
Let’s say an ant lives on the surface of the cylinder. It can move in multiple different ways: follow a curved path when moving around the cylinder, follow a flat path when moving along out a straight line, or choose a random path.
The idea of Theorema Egregium is to define the curvature of a surface as it takes all these possibilities into account.
Here’s how to find the curvature at the point according to the theorem. Starting at any point, find the two most extreme paths: the most concave path and the most convex path. To find the curvature at the point, multiply the curvature of those two paths together (curvature is positive for concave paths, zero for flat paths, and negative for convex paths).
In the case of our ant at the point on the cylinder, the two extreme paths available to it are the curved, circle-shaped path, and the flat, straight-line path. Since the curvature of the flat path is zero, the result for the surface curvature is also zero. Therefore, according to Gauss, the cylinder is flat which reflects the fact that you can roll one out of the paper.
If instead, the ant lived on a sphere, there would be no flat paths available to it. Now every available path is a curve, and so the surface curvature represents a positive number. Thus the spheres are curved which, similarly, reflects the fact that you can never bend a sheet of paper into a ball.
Now let’s return to our pizza problem. I hope my explanation didn’t take too much time for your slice to get cold.
The slice is obviously flat as it lays in the pizza box. Theorema Egregium tells us that after bending it still remains flat and therefore one direction of the slice must always remain flat.
When the slice flops over, the flat direction is pointed sideways, which isn’t convenient for eating it. However, if you force it to become flat in the other direction (just like in the picture below), you can successfully transfer it!
This theorem has many other fascinating and much more complicated applications. That’s the beauty of mathematics — you can find identical patterns that can be expressed mathematically in different phenomena.
The world of mathematics is amazing and even entertaining. I hope that you had fun reading about the basic theory of Theorema Egregium.
Resources used:
https://en.wikipedia.org/wiki/Theorema_Egregium
https://www.wired.com/2014/09/curvature-and-strength-empzeal/
https://insidetheperimeter.ca/physics-eating-pizza/
