Topology – Page 3

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 7^th 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!

Shortest Path between Two Points

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:

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

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.

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.

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

Real Mathematics – Strange Worlds #1

Euler Characteristic

Take a piece of paper and draw dots on it.
Draw as many lines as you want between those dots.
Lines may not cross.
Every dot must be connected with a path of lines.
Closed lines make a “face”.
Every plane has at least one face. It is the area of the plane and it lies on outside of the dots and lines.
In each time the formula
Number of Dots – Number of Lines + Number of Faces = 2
will justify itself. It is called Euler’s Polyhedron Formula.

Example 1: Three Dots

Put three dots on a paper and draw lines between them like following.

There are two faces as shown in the following photo.

Apply Euler’s formula and you will get two!

Example 2: Four Dots

Put down four dots on a paper and draw lines as following.

In total there are three face, four dots and five lines. Euler’s formula gives us the same result which is two!

Example 3: Five Dots

Five dots, seven lines and four faces are shown in the following photo which justifies Euler’s formula.

How about if you add a new dot?

Newly added dot F could give us three new lines and two new faces. At first it looks like Euler’s formula won’t add up this time.

However it works out just fine. Euler’s formula gives us the same result under these conditions.

Example 4: Five Dots and No Faces

You might be wondering what would happen if we don’t enclose the lines and create a face. We all know from the initial conditions that every plane has at least one face that is outside of the dots and lines.

Therefore we’ll draw four lines between our five dots. We can travel from one dot to another and no lines are crossed. Hence all conditions are justified.

In the end we realize that five dots, four lines and a face give us the result two when we apply them into the Euler’s formula. Euler is always right!

For Those Who Love History

Until now, Euclid and Euler’s names appeared in my articles frequently.

Euclid is the author of a book called Elements which has all the information that you learn in your geometry lessons until you graduate from high school. This is the reason why the geometry that is being taught in schools is called Euclidean Geometry.

On the other hand, Euler is known as one of the most productive scientists of all times with his archive of over 800 works.

Elements, what consists of 13 books about the foundations of mathematics, was written BC. 200s. For thousands of years countless scientists and philosophers learned mathematics with this magnificent work. Isaac Newton is known as the person who started modern science (and also labelled as the father of modern physics) had started learning mathematics with Euclid’s Elements.

Even 2000 years after it was written, Elements were known as the only source of geometry. In fact Immanuel Kant, who is one of the most influential philosophers of all time, claimed that it was illogical to think of geometry besides Euclid’s. This is one of the many reasons why geometry was believed to be completed and there was nothing else to be found in it. Kant had such a powerful influence over scientists even Gauss, who is called as the prince of mathematics and often cited as the greatest mathematician of all time, was afraid to express his thoughts about a new kind of geometry. I’ll be talking about Gauss and new geometry in another article.

Euler had a rare gift and an unprecedented will for solving problems which were thought as “impossible to solve”. His name is given to so many formulas thanks to his outstanding works. But there is a formula he found, most of the mathematicians in the world agree that it gives the most elegant result of mathematics. It is a formula that works for Euclidean geometry too. It is called Euler’s Characteristic.

One wonders…

How about three dimensions: Would Euler’s formula work for it too? For example, would it work for a sphere? Take a sphere, put random two dots on it and connect those dots. What do you see?

Take a third dot and connect it to the other two dots. What do you see now?

M. Serkan Kalaycıoğlu

Real Mathematics: Graphs #2

You’ve been sent for scouting a forest nearby your village. You are done with your duty and on your way back home you are bringing a wolf, a goat and some cabbage with you. Wolf would like a piece of goat and goat is looking hostile against the cabbage, but they can’t do anything with your presence. Then you come across with a deep river. Luckily for you, there is a small boat on coast.

The Catch: Boat is so small it allows you to take only either one of wolf, goat or cabbage. Is there any way you can take everything to the other side of the river safely?

Solving Puzzles with Graphs

In the previous article I told you how Euler used a brand new approach to a seemingly insignificant puzzle and how his method became the basis of graph theory. Drawing graphs in certain cases really improve the chances of reaching an answer. Graphs also help us understand why there is not a solution, in case problem has no solution. Graphs are so powerful at times they can even show us what kinds of conditions are needed to solve a problem that has no solution.

In the river problem, let’s use Euler’s methods. You are all on the left side of the river and you have to get across the right side of it. Now assume that left side of the river is assigned as position 1, and right side as position 2.

There are wolf, goat and cabbage. In the puzzle, those three stand on the position 1 which might be shown as a point named 111 (wolf-goat-cabbage on the positions 1-1-1). Ultimate goal of the puzzle is to find a way to get them all to the point 222.

There are eight different positions for the wolf, goat and cabbage: 111, 112, 121, 122, 211, 212, 221 and 222. For instance 112 means wolf is at position 1 (left side of the river), goat is at position 1 (left side of the river) and cabbage is at position 2 (right side of the river).

Drawing the Graph

We assumed that all these 3-digit numbers represent a point. To get from one point to another, we could only change exactly one digit of it, as we are allowed to take exactly one of wolf, goat or cabbage into our boat. Then there are limited ways to go from one point to another. Let’s assume that if you can go from one point to another, then there is a line connecting those two points.

For instance 111 and 112 has a connection as there is exactly one digit that differ those numbers. Although 111 and 212 has no connection as there are more than one digit that differ those numbers.

Finally we found our points and lines and how they could be drawn as a graph. I think Euler would be proud of us.

Our starting point 111 has three options: 112, 121, 211. Our graph looks like following when all relationships between points are shown:

As our ambition is to reach point 222 from the point 111, it is easy to see the ways to the solution in the graph. However we can’t move between points as we’d like to. We can only use paths which satisfy puzzle’s rules: If they are left alone, wolf eats goat and goat eats cabbage.

Solution

The point 111 has three options: 112, 121 and 211. If we select 112 (taking cabbage to the right side) wolf and goat would be alone, so this path is not good for us. The point 211 would mean leaving goat with cabbage which is forbidden as well. Hence there is only one way to select from the point 111: The path to the point 121.

The point 121 has three options: 111, 122 and 221. We can’t go back, so the point 111 is out. We should choose either 122 or 221.

Choosing 122

This would mean that we are taking cabbage to the right side of the river, near the goat. There are again three options: 121, 222 or 112. We can’t go backwards, so the point 121 is out.

222: Choosing this path would give us the answer. But, in order to do that we must leave goat and cabbage alone and go get wolf. Until we are back to the right side of the river, goat would be doing its final chewing. Hence, we can’t choose this path.

112: This is the only real option we can choose.

From 112, there are again three options: 122, 212 and 111. We can’t go backwards, so the points 122 and 111 are out. Path to the point 212 is the only choice.

At the point 212, wolf and cabbage are on the right side of the river and goat is on the left side. Here, we can go to the left side, pick goat with us and reach the point 222. This would give us the solution we were looking for.

Choosing 221

Now go back and choose the path to the point 221. There are again three options from 221: 222, 211 and 121. We can’t choose the point 121 as it would mean going backwards. Going to the point 222 would solve all our problems, but if we try to do that, we would have to leave wolf and goat alone at the position 2. This gives us the option 211.

The point 211 also gives us three options: 111, 221 and 212. Choosing 111 and 221 would mean going backwards. Then we must follow the path to 212. From the point 212, we can directly choose the path to the point 222. And we would again find the solution within the lines of the rules of the puzzle.

Check It Out

Add other animals and vegetables to these three and try to draw the graph. Using the graph, see if you can find a solution to your puzzle.

M. Serkan Kalaycıoğlu

Real Mathematics: Graphs #1

Puzzles and mathematics have a deeper relationship than one might assume. When you take a look at history of mathematics, it is not that rare to see a puzzle causing mathematics to change.

Without a doubt, Seven Bridges of Königsberg (1736) is the most famous one of those stories.

Leonhard Euler, one of the most influential mathematicians of all times, was given this puzzle/problem in the 18^th century. Euler had a great reputation for solving “unsolvable” questions, and major of Königsberg asked for his help. Euler was not pleased to receive this problem at first as he said “… I don’t believe this question has anything to do with mathematics.” Although after some time he believed that the problem was interesting enough to deal with.

Euler’s solution to Seven Bridges of Königsberg problem culminated a couple brand new mathematics branches to emerge: Topology and graph theory. What fascinates me about this story is that these branches came out almost 150 years after Euler’s solution.

Seven Bridges of Königsberg

Take any book on graph theory. Open first chapter’s first title. You’ll see Euler’s solution for this simple problem:

The old town of Königsberg has seven bridges as shown on the map. Is it possible to take a walk through the town, visiting each part of the town and crossing each bridge once and only once?

I encountered with this problem for the first time in the junior year of my math major. I first thought that the question was dull and I could solve it with ease. After spending a night with full of disappointments and a lot of cursing, I gave up. I knew that it was impossible to take that walk and I knew that it was possible when I removed one bridge. But I didn’t know why.

Yeah… You can’t solve it that way Serkan…

Soon enough, I realized that I needed fundamentals of the graph theory to understand the answer. Euler’s solution to this simple puzzle is the basis of a whole new mathematics branch: Graph theory.

What Euler Did

First thing Euler did was to imagine that bridges and lands as two different groups. According to him, solution of Königsberg problem didn’t depend on the physical appearances of them. Euler realized that it didn’t matter how long or curvy a bridge is, or if a land is an island or not.

He just thought that bridges could be represented as edges numbered from 1 to 7, and lands could be represented as vertices named A,B,C and D. When all Königsberg problem map is put into a piece of paper with using on edges and vertices, we would end up with a graph.

Euler’s handwriting for the solution of the Königsberg problem.

Degree of a vertex is the number of edges it has.

According to Euler, in order to find a solution for Königsberg problem either of these should be found in the graph:

Exactly two vertices should have odd number of degrees. That would mean one of the vertices is the start of the tour as the other is the finish.
All of the vertices should have even number of degrees. That would mean one could start the tour from any vertex.

Why?

If a vertex is neither start nor finish of the tour, then in case one edge comes to that vertex another one should leave so that the vertex would be neither the start nor the finish. (Because if an edge comes to a vertex and another doesn’t leave, it means that vertex is the finish of the tour. And if an edge leaves from a vertex but another doesn’t come back to it, then it means that vertex is the start of the tour.) This would mean that the vertex should have even number of degree. Although, if vertex has odd number of edges, in order to keep odd degree we shouldn’t let another edge to leave in case there is an edge coming to that vertex. This would mean that the so called vertex is either start or finish of the tour.

Also, if there are no vertices in the graph with an odd degree, then a chosen vertex could well be both start and finish of the tour.

Eulerian Path: If a path/tour in a graph contains each edge of the graph once and only once, then that path is called Eulerian Path.

I am aware that the paragraph above is a bit confusing. Let me give you a few examples to clear your mind about degree of a vertex.

Examples:

One Bridge: Two lands and one bridge graph.
Two Bridges: Two lands and two bridges graph.
Three Lands: Add land C into the previous graph.

In this case there are two vertices with odd number of degrees. Hence, there is a solution for this graph but vertices B and C should be the start and finish of the path.

Solution to Königsberg

Euler saw the Seven Bridges of Königsberg as this graph.

How Königsberg problem looks like as a graph.

Numbers of degrees for the vertices are 5, 3, 3 and 3 in order. This means that the Königsberg graph has more than two odd numbers of vertices which is a violation for the rule. This is concludes that there is no such tour/path for the seven bridges of Königsberg.

Topology is the mathematical study of the properties that are preserved through twisting and stretching of objects. (Tearing, however, is not included.) Topologically a circle and an ellipse are equivalent to each other.

A simit is topologically equivalent to a coffee cup.

Check it out

Which objects are topologically equivalent to each other ?