Do you remember the time I was kidnapped by the aliens? I saved my life thanks to Euler’s formula. Today I will show another method for solving that problem for those who don’t like memorizing formulas.
This method is so remarkable; it actually comes from a theorem called “The Remarkable Theorem”. “Theorema Egregium” belongs to one of the two names I often mention in my articles: Gauss. Well, a remarkable theorem would only fit to the Prince of mathematics anyway.
Gauss’ remarkable theorem shows us the degree of the mathematics internalization in our daily lives. I will use an example in order to explain the theorem:
Example: The (Remarkable) Way of Holding Papers
Take an A4 paper, hold it from left side and try to read what is written on the paper. If you don’t apply any kind of force on your paper through your fingers it will stand like the following:
Paper got bent like this because of gravity. This situation is so internalized in our minds; we solve it with a reflex:
Bending the paper into an inward position helps us beat the gravity and get the paper into a more readable position.
Why?
Why do we have to bend the paper in such situation?
Gauss’ Remarkable Theorem: Gauss says that a plane or an object will have the same Gaussian curvature even after it gets bent, twisted, rotated etc.
Let’s calculate Gaussian curvature of the paper while it lies on the floor. To do that we must select a random point A on the paper and draw two perpendicular lines that go through A:
Now let us analyze the curvatures of these two lines. For each line there will be three possibilities:
- If the line is straight, it has zero curvature.
- If the line is bent outward, it has positive (+) curvature.
- If the line is bent inward, it has negative (-) curvature.
Gaussian curvature is calculated with the multiplication of the curvatures of those two lines. A paper sitting on the floor has zero curvature for both lines which means the paper has zero Gaussian curvature:
Holding the Paper
In the end according to the Gauss’ remarkable theorem Gaussian curvature of a paper will always be zero. No matter what we do to a paper its Gaussian curvature will be zero.
Now let’s go back a bit and examine the Gaussian curvature of the paper when I was holding it. At first paper was curved outward. This means that the paper has positive curvature horizontally:
But vertically paper has zero curvature:
Multiplying positive with zero makes zero, which means the paper has zero Gaussian curvature. Gauss’ remarkable theorem holds on.
Next, we will be analyzing the Gaussian curvature of the paper when my hand forced it on an inward position. Vertically inward position means a negative curvature as horizontally paper has zero curvature. Multiplication gives a zero Gaussian curvature for the paper. Gauss’ theorem is indeed remarkable!
Gaussian Curvatures
- Zero Gaussian Curvature:
Both horizontally and vertically it has zero curvature. - Positive Gaussian Curvature:
Positive(horizontally) times positive(vertically) makes positive Gaussian curvature. - Negative Gaussian Curvature:
Negative(horizontally) times positive(vertically) makes negative.
Running Away From the Aliens
Now you can solve the puzzle of the aliens without needing of a formula. All you need is to pick a starting point and determine the Gaussian curvature.
Zero Gaussian curvature means a paper-like, positive means a sphere-like and negative means a saddle-like shape.
One wonders…
The (Remarkable) Way We Hold a Pizza Slice
How do you hold your pizza slice while you are eating it?
Why?
M. Serkan Kalaycıoğlu
kmatematikatolyesi.com 