Real Mathematics – Strange Worlds #2

What shape is it?

I’ve been waiting for this day: I am finally abducted by some space aliens.

I am getting back to my senses now. Oh, they speak Turkish. That’s odd. And their favorite drink is black tea from Rize. It helped me a lot when they offered me some tea. I am relaxed now and their leader wants to have a word with me:

-Earthling! We kidnapped you so that we can conduct some experiments on you and learn about human body. But we are fair, so you’ll have a chance to prove your intelligence first. If you solve this problem, we’ll take you back to your home. Here is the problem: We’ll blindfold and leave you to a random asteroid. Your quest is to find what kind of shape that asteroid has.

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Christopher Columbus-Like

I am all alone on a strange asteroid now. Space ship is watching over me. Think Serkan, thiiiink… Oh, I think I got it! I will move in same direction just like Columbus did. If I get back to where I started, it means this asteroid is curved, like a ball.

I asked for a spray from the aliens so that I can leave a trail mark and understand when and if I get back to my starting point. I walk and walk… Finally I reach my starting point. I got my ticket back home: This asteroid is curved like a sphere, probably like Earth.

IMG_5900
When I am talking about a sphere, it doesn’t have to be a perfect one. It could be looking like this as well.

I am getting ready to talk with the leader of the aliens. But suddenly an idea pops in my head: What if the shape has a hole in it, like a bagel or a donut?!

Because, if I walked on the right direction, I could very well reach my starting point even though I am on a bagel/donut shaped asteroid.

What am I supposed to do now?

Am I on a ball? Or am I on a bagel?

A New Kind of Mathematics For You

This problem is actually one of the most classical problems of topology. Okay but what is topology?

Until now I’ve told you about two important discoveries of Euler: Euler’s solution to the Königsberg bridge problem and Euler’s polyhedron formula.

Euler’s Königsberg solution directly affected (more than 150 years after his solution) the birth of a new mathematics branch named graph theory. His polyhedron formula also helped mathematicians to define what topology is. I must mention that graph theory is just a sub-branch of topology.

Topology:

  • Two Greek words: Topos (space) + Lopos (science).
  • Also known as rubber sheet geometry.

Euclidean Geometry vs. Topology

  • In Euclidean geometry; it is allowed to move things around and flip them over. But you can’t stretch or bend objects without changing their properties.
  • In topology; you can bend, twist and even stretch objects without affecting their properties. But you can’t cut, add or punch a hole in topology.
  • In Euclidean geometry; angles and lengths are important.
  • In topology; they don’t matter.
  • This is why in Euclidean geometry a square and a triangle has different properties even though they are the same thing in topology.

The Most Popular Example

You might find it strange that a mathematics branch does not care about lengths and angles. Actually topology is involved in our daily lives. More than you could ever imagine.

In transportation, especially in metro and tramway systems, maps are great examples for topology. Distances and directions are distorted in the interests of simplicity which is a use of topology. Also it is neglected how big the stops are. They are all represented with dots and they are connected with lines. In short, metro maps are actually graphs.

In the following picture it is clear to see that all stops (which are shown with dots) are divided with same distance even though they are not equal in reality.

M2_Hattı
This picture shows some of the stops of Istanbul metro.

Hint: What shape is it?

You probably noticed it: That problem is actually about the topological property of the asteroid. Using something elastic (like Play-Doh) would help you to solve it.

M. Serkan Kalaycıoğlu

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