Real Mathematics – Strange Worlds #3

Inheritance

We all have those relatives whom we only see at weddings or religious holidays. Inheritance is my favorite thing about those relatives.

I’d accepted that I would never win any kind of lotto due to my bad luck. This is why I sometimes dream that a lawyer is handing me a fat banking check from my mother’s father’s 7th cousin’s son-in-law. That is the kind of distant relative I love! I used to imagine how I would spend that money in Moscow, Cannes or LA. But after some time, I got bored of imagining these things.

You already know the fact that when you play a game on your computer or whatever, you try to finish that game at its hardest level. Because only then you can enjoy the game you play. Therefore I started imagining conditions to get the inheritance.

In one of those imaginations, lawyer hands me an envelope and says:

“Mr. Serkan; you’ll find a map inside that envelope, and that map show the whereabouts of an Ottoman treasure. You have only 15 hours to get to the treasure. If you fail to do so, that money will be giving to the rival of your favorite football team.

There is a plane and a crew waiting for you at the airport. What they need is you to find where to go and using which route. Good luck sir.”

Shortest Path

I found the coordinates using google maps (helloooo, it is easy) and I found the destination: Los Angeles (LAX) airport. Now all I have to do is to find the shortest path between Istanbul and Los Angeles.

In 3-dimension, shortest path between two points is a straight line. I assigned dots to Istanbul and Los Angeles which meant finding the straight line between them will give me the solution. Oh, my poor imagination; it was an easy one!

Serkan! What is the shape of the Earth?

If Earth was flat, I would say that I created and solved the dullest problem in the world of imagination. Although, it is known that the Earth is round. What? You are a flat-Earth believer huh? Leave this site please!

I checked web site of Turkish Airlines. It shows the routes between Istanbul and other cities all over the world. The one between Istanbul and Los Angeles is shown as follows:

thy rota

Okay but how come it is that shape? Another example between İstanbul and New York.

projections6
On the map, shortest distance is the straight line. But airplanes follow that curve which might seem odd here.
projections7
This is how Earth would look from above. The straight line from the flat map turns out to be a curve and airplane’s route looks like a straight line.

Because Earth is round, you might say like a ball. And when you pick two points on round objects and then connect those points, you’d get a curve! You can try it with a rope or a play-doh and see it yourself.

Then it is not always possible to find the shortest path on a flat map. There is my favorite question: Why?

Because round objects just like Earth can’t be scaled down into 2-dimension from 3-dimension 100% correctly. There are really successful maps but none of them are showing the Earth in correct scales. Try to make a triangle on a ball, cut it out. It will have curved sides.

We can back this fact with another simple example. Take an orange and draw a triangle on it. Peel that triangle and you will see that the triangle’s sides are not straight lines even though you draw them straight. More importantly, whatever you try, you just can’t straighten the sides of that triangle. When you push down one side, others will be up.

However, we can straighten this triangle using topology. Again take a play-doh. You can stretch the sides and make a triangle with sides that are straight. Although stretching ruins the true scale of the triangle. In the end, what you get would look like the real triangle, but never the same of it.

Next: Eratosthenes and his magnificent method.

M. Serkan Kalaycıoğlu

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