Topology

Real Mathematics – Strange Worlds #18

Every year in December, each city changes drastically. Suddenly we find ourselves surrounded by decorations that remind us of the upcoming new year.

Steve the teacher starts to decorate his classrooms for the new year like he does every year. Though, Steve the teacher set his mind on using new year decorations for his mathematics lessons.

New Year Decorations Game (N.Y.D.G.)

Steve’s creation N.Y.D.G. is a multiplayer game. This is why the game is played in knockout stages/rounds. The winner of the game wins the new year decorations and gets to decorate the classroom as he/she wishes.

Content of N.Y.D.G.

In each knockout round, students are given 4 decorations as follows:

Players wind the decorations one another.
The winding procedure should be done secretly from the opponent.
Each player has at most four moves for winding.

Let’s use an example to explain what a “move” means during the winding procedure.

Assume that the first move is made with the red decoration as follows:

This counts as one move. The red one undergoes the blue and green decorations in this move. Let the next two moves are as follows:

In the second move, the yellow decoration undergoes the green and red ones, while the blue one passes over the green and yellow decorations. The illustration (up-right) shows us how the winding looks after these 3 moves.

In the end, winding gives us a braid.

The Goal of The Game

In any round, to knock your opponent out, you should solve the braid of your opponent faster than your opponent solves yours. (Solving a braid means, bringing the decorations to their first state. For instance, in the example given up, the first state is yellow-green-blue-red in order.)

Braids

Braids have a very important part in daily life. We encounter them not just in new year decorations, but also in a piece of cheese, a hairstyle, a basket or even in a bracelet:

In case you wish to understand what braids mean in mathematics; one can take a look at Austrian mathematician Emil Artin’s works from the 1920s.

Let’s call the following an identity braid from now on:

In Steve the teacher’s game, the ambition is to go back to the identity braid from a complex braid in the shortest amount of time. To do that, we can use Artin’s work on braids.

Example One: Solving two ropes.

Assume that we have two ropes tangles with each other as follows:

The inverse of this rope is:

If we combine these two ropes, when each rope to be stretched, the result will give us the identity braid:

Example Two: Solving three ropes.

Take three ropes and make a braid as follows:

There are three intersections in this braid:

1: Green over the blue.

2: Red over the green.

3: Blue over the red.

Now, you should repeat these steps, but from last to the first this time. Then, you should do these moves:

Move #1: Blue over the red.

Move #2: Red over the green.

Move #3: Green over the blue.

Finally, the combination will give you the identity braid. Try and see yourself.

Paper and Braids

Take an A4 paper and cut the paper using a knife like the following:

Then, hold the paper from its sides and rotate it 90 degrees to the left. You will end up with some kind of a braid:

One wonders…

How can you use Emil Artin’s work in the game of Steve the teacher?
In “example two”, rotate the ropes 90 degrees to the left. Start investigating the intersections from left to right. What do you notice?
Play Steve the teacher’s game with an A4 paper. (It is more than enough to use 3 or 4 cuts on the paper.)

M. Serkan Kalaycıoğlu

Real Mathematics – Graphs #7

Serkan’s System

Serkan the math teacher, hands out a specific number of problems to his students. Kids who can solve 1 or more of those problems would get a certain prize. At the beginning of each semester, Serkan and his students sit down and agree on what kind of prize is going to be distributed. For the current semester, oreo is chosen as the prize:

If Serkan the math teacher hands out 10 problems:

10 Oreos for the kids who solved 10, 9 or 8 of those problems,
5 Oreos for the kids who solved 7, 6 or 5 of those problems,
2 Oreos for the kids who solved 4, 3 or 2 of those problems,
1 oreo for the kids who solved 1 problem,
Absolutely nothing for the students who solved… well… none of those problems.

If you take a careful look at the numbers, you can see that Serkan the math teacher selected those numbers with a kind of logic: 10, 5, 2 and 1.

These are the natural numbers that can divide the number of the problems (that is 10) without any remainder.

Prize Distribution Machine (P.D.M.)

One month later…

Serkan the math teacher had faced some problems 4 weeks into the semester. He realized that it took hours to distribute the prizes since he has 10 classes in total.

Serkan the math teacher had to use almost all his free time in school to distribute the Oreos. This led him to think about a machine that would help him with the distribution:

P.D.M. will have 4 different compartments. (Because of 10, 5, 2 and 1.)
The volumes of those compartments will be measured with Oreos. They will be 10, 5, 2 and 1 Oreo-sized.
Oreos will enter the machine from the 10-Oreo-sized compartment. From there, Oreos will move to the other compartments using the connections that will be established.
Golden Rule: To establish a connection between any two compartments, the size of those compartments must be factors of one another.

Connections of the compartments for 10 problems:

For 10-Oreo-sized: 5, 2 and 1.
For 5-Oreo-sized: 10 and 1.
For 2-Oreo-sized: 10 and 1.
For 1-Oreo-sized: 10, 5, and 2.

Then, the sketch of the P.D.M. would look like the following:

Is this another graph?!

If you are familiar with graph theory (or if you read the graph section of the blog) you can recognize that the sketch of Serkan the math teacher’s machine is a planar graph:

You should connect the numbers (dots) using lines (connections) according to the golden rule.

One wonders…

What if Serkan the math teacher asks 12 problems?

For 12 problems, the numbers of prizes are going to be: 12, 6, 4, 3, 2 and 1.

In such a situation, can Serkan build his machine? In other words; is it possible to connect the dots for 12-sized P.D.M.?

Hint: First, you should consider where the lines should be. Also, you can arrange the dots in any order you’d like.

M. Serkan Kalaycıoğlu

Real Mathematics – Strange Worlds #17

Topology On Your Head

Often you see me writing about changing our perspective. For example, when you encounter a baby first thing you do is to make baby sounds and try to make the baby laugh. Whereas if you’d looked carefully at baby’s hair, you could have seen a very valuable mathematical knowledge hidden on the baby’s head:

As shown above, there is a point on each baby’s head. You can see that the hair besides that point is growing in different directions. Can you tell me which direction the hair grows at that exact point?

Hairy ball theorem can give us the answer.

Hairy Ball Theorem

Hairy ball theorem asks you to comb a hairy ball towards a specific direction. The theorem states that there is always at least one point (or one hair) that doesn’t move into that direction.

You can try yourself and see it: Each time at least one hair stands high. This hair (or point) is a sort of singularity. That hair is too stubborn to bend.

Baby’s hair is some kind of a hairy ball example. (I use the expression “some kind of”because the hairy ball has hair all over its surface. Though the baby’s head is not covered with hair completely.) This is why the point on the baby’s head is a singularity. It is the hair that gives a cowlick no matter how hard you comb the baby’s hair.

Torus

Hairy ball theorem doesn’t work on a torus that is covered with hair. In other words, it is possible to comb the hair on a torus towards a single direction.

No Wind

Hairy ball theorem can be used in meteorology. The theorem states that there is a point on earth where there is no wind whatsoever.

To prove that, you can use a hairy ball. Let’s assume that there is wind all over the earth from east to west. If you comb the ball like that, you will realize that north and south poles will have no wind at all.

On Maps

The hairy ball theorem is a kind of a fixed point theorem. Actually, it is also proven by L. E. J. Brouwer in 1912.

One of the real-life examples of the fixed point theorem uses maps. For example, print the map of the country you live in, and place it on the ground:

There is a point on the printed map that is exactly the same as the map’s geographical location.

“You are here” maps in malls or bus stops can be seen as an example of this fact.

One wonders…

Assume that all the objects below are covered with hair. Which one(s) can be combed towards the same direction at all its points? Why is that?

M. Serkan Kalaycıoğlu

Real Mathematics – Strange Worlds #16

The Walk

Select two points in the classroom.
Draw a line between them.
Send a student to one of those points.
Once the student starts his/her walk, he/she should arrive at the other point exactly 10 seconds later.
Everybody in the classroom would count to 10 to help the walker.
Ask the student to do the same walk twice while recording the walk using a camera.

The goal of the experiment

After the experiment is done, the following question is asked to the classroom:
“Is there a moment during both walks when the student stands at the exact point?”
In other words, the student walks the same distance in the same amount of time at different speeds. The goal is to find if there is a moment in both walks when the student passes the exact point on the line.
First of all, we should give time to the students for them to think and brainstorm on the problem. Then, using the video shots, the answer is given.
The most important question comes at last: Why so?

Weeding out the stone

In my childhood, one of my duties involved weeding out the stones inside a pile of rice. To be honest, I loved weeding out. Because I was having fun with the rice as I was making different shapes with it.

Years later when I was an undergrad mathematics student I heard of a theorem that made me think of my weed out days. This theorem stated that after I finish the weed out, there should be at least one rice particle that sits in the exact point where it was before the weed out started. (Assuming that the rice particles are covering the surface completely.) In other words; no matter how hard to stir the rice particles, there should be at least one rice particle that has the exact spot where it was before stirring.

This astonishing situation was explained by a Dutch mathematician named L.E.J. Brouwer. Brouwer’s fixed point theorem is a topology subject and it is known as one of the most important theorems in mathematics.

The answer to the walking problem,

The walking problem is an example of Brouwer’s fixed point theorem. This is why the answer to the question is “yes”: There is a moment in both walks when the student stands at the exact point on the line.

I will be talking about Brouwer’s fixed point in the next article.

One wonders…

A man leaves his home at 08:00 and arrives at another city at 14:00. Next morning at 08:00 he leaves that city and arrives at his home at 14:00, using the exact roads.

Conditions

Starting and finishing points are the same, as well as the time intervals of both trips.
The first condition means that the man could travel in his choice of speed as long as he sticks to the first condition.

Is there a point on these trips where the man passes at the exact time during both trips?

Hint: You could assume that the distance is 600 km and the man must finish that in 6 hours. For instance, he could have been traveling 100 km/h the first day, and the next day 80 km/h in the first 2 hours; 100 km/h in the next 2 hours, and 120 km/h in the last 2 hours of the trip.

M. Serkan Kalaycıoğlu

Real MATHEMATICS – Strange Worlds #15

Human Knot Game

Inside a classroom divide students into groups such that each group has at least 5 students. Groups should stand up and form a circle before following the upcoming instructions:

1. If there is an even number of student in a group:

Have each student extend his/her right hand and take the right hand of another student in the group who is not adjacent to him/her.
Repeat the same thing for the left hand.

2. If there is an odd number of student in a group:

Have each student (except one of them) extend his/her right hand and take the right hand of another student in the group who is not adjacent to him/her.
Then take the extra student and have him/her extend his/her right hand so that the extra student can hold the left hand of another student who is not adjacent to him/her.
Finally, repeat the process for the students whose left hands are free.

In the end, students will be knotted.

Now, each group should find a way to untangle themselves without letting their hands go. To do that, one can use Reidemeister moves.

Reidemeister Moves

Back in 1926 Kurt Reidemeister discovered something very useful in knot theory. According to him, in the knot theory one can use three moves which we call after his name. Using these three moves we can show if two (or more) knots are the same or not.
For example, using Reidemeister moves, we can see if a knot is an unknot (in other words, if it can be untangled or not).

What are these moves?

Twist

First Reidemeister move is twist. We can twist (or untwist) a part of a knot within the knot theory rules.

Poke

Second one is poke. We can poke a part of a knot as long as we don’t break (or cut) the knot.

Slide

Final one is slide. We can slide a part of a knot according to Reidemeister.

One wonders…

After you participating in a human knot game, ask yourselves which Reidemeister moves did you use during the game?

M. Serkan Kalaycıoğlu

Real MATHEMATICS – Strange Worlds #14

In my youth, I would never step outside without my cassette player. Though, I had two knotting problems with my cassette player: the First one was about the cassette itself. Rewinding cassettes was a big issue as sometimes the tape knotted itself. Whenever I was lucky, sticking a pencil would solve the whole problem.

The second knotting problem was about the headphone. Its cable would get knotted so bad, it would take me 10 minutes to untangle it. Most of the time I would bump into a friend of mine and the whole plan about listening to music would go down the drain.

The funny thing is I would get frustrated and chuck the headphone into my bag which would guarantee another frustration for the following day.

A similar tangling thing happens in our body, inside our cells, almost all the time.

DNA: A self-replicating material that is present in nearly all living organisms as the main constituent of chromosomes. It is the carrier of genetic information.

DNA has a spiral curve shape called “helix”. Inside the cell, the DNA spiral sits at almost 2-meters. Let me give another sight so that you can easily picture this length in your mind: If a cell’s nucleus was at the size of a basketball, the length of the DNA spiral would be up to 200 km!

You know what happens when you chuck one-meter long headphones into your bags. Trying to squeeze a 200 km long spiral into a basketball?! Lord; knots everywhere!

Knot Theory

This chaos itself was the reason why mathematicians got involved with knots. Although, knot-mathematics relationships existed way before DNA researches started. In the 19th century, a Scottish scientist named William Thomson (a.k.a. Lord Kelvin) suggested that all atoms are shaped like knots. Though soon enough Lord Kelvin’s idea was faulted and knot theory was put aside for nearly 100 years. (At the beginning of the 20th century, Kurt Reidemeister’s work was important in knot theory. I will get back to Reidemeister in the next post.)

Is there a difference between knots and mathematical knots?

For instance the knot we do with our shoelaces is not mathematical. Because both ends of the shoelace are open. Nevertheless a knot is mathematical if only ends are connected.

The left one is a knot, but not mathematically. The one on the right is mathematical though.

Unknot and Trefoil

In the knot theory we call different knots different names with using the number of crosses on the knot. A knot that has zero crossing is called “unknot”, and it looks like a circle:

A rubber band is an example of an unknot.

Check out the knots below:

They look different from each other, don’t they? The one on the left has 1 crossing as the one on the right has 2:

But, we can use manipulations on the knot without cutting (with turning aside and such) and turn one of the knots look exactly like the other one!

This means that these two knots are equivalent to each other. If you take a good look you will see that they are both unknots. For example, if you push the left side of the left knot, you will get an unknot:

Is there a knot that has 1 crossing but can’t be turned into an unknot?

The answer is: No! In fact, there are no such knots with 2 crossiongs either.

How about 3 crossings?

We call knots that have 3 crossings and that can’t be turned into unknots a “trefoil”.

Even though in the first glance you would think that a trefoil could be turned into an unknot, it is in fact impossible to do so. Trefoil is a special knot because (if you don’t count unknot) it has the least number of crossings (3). This is why trefoil is the basis knot for the knot theory.

One of the most important things about trefoils is that their mirror images are different knots. In the picture, knot a and knot b are not equivalent to each other: In other words, they can’t be turned into one another.

Möbius Strip and Trefoil

I talked about Möbius strips and its properties in an old post. Just to summarize what it is; take a strip of paper and tape their ends together. You will get a circle. But if you do it with twisting one of the end 180 degrees, you will get a Möbius band.

Let’s twist one end 3 times:

Then cut from the middle of the strip parallel to its length:

We will get a shape like the following:

After fixing the strip, you can see that it is a trefoil knot:

To be continued…

One wonders…

1. In order to make a trefoil knot out of a paper strip, we twist one end 3 times. Is there a difference between twisting inwards and outwards? Try and observe what you end up with.
2. If you twist the end 5 times instead of 3, what would you get? (Answer is in the next post.)

M. Serkan Kalaycıoğlu

Real MATHEMATICS – Graphs #6

Bequeath Problem

King Serkan I decides to allocate his lands to his children. Obviously he had set up some ground rules for the allocation:

Each child will get at least one land.
Same child can’t have adjacent lands.

Problem: At least how many children should Serkan I has so that allocation can be done without a problem?

Map #1

Let’s start from a simple map:

In this case Serkan I can have two children:

As you can remember from the previous article a map and a graph is irreversible. If we represent lands with dots, and let two dots be connected with a line if they are adjacent, we can show maps as graphs:

Let’s add another land to this map:

Adding a land on a map is the same thing as adding a dot on a graph:

Map #2

Let’s assume there are three lands on a map:

We can convert this map into graph as follows:

As seen above, three children are needed in order to fulfill Serkan I’s rules:

Map #3

Let the third map be the following:

According to Serkan I’s rules, we will need four children for such map:

Map #3 can be shown as a graph like the following:

Map #4

For the final map, let’s assume Serkan I left a map that looks like USA’s map:

Surprisingly four children are enough in order to allocate the lands on the map of USA:

What is going on?

Careful readers already noticed that adding a dot that connects the other dots in the second map’s graph gives the graph of the third map. Same thing is true for the first and second maps:

Hence, adding a new dot to the graph means adding a new child.

Q: Is it possible to create a map that requires at least five children?

In other words: Is it possible to add a fifth dot to the graph so that it has connection to all existing four dots? (Ps: There can be no crossing in a graph as our maps are planar.)

Then, all we have to do is to add that fifth dot… Nevertheless I can’t seem to do it. When I add the fifth dot outside of the following graph:

It is impossible to connect 1 and 5 without crossing another line. No matter what I try, I can’t do it:

Four Color Theorem

About 160 years ago Francis Guthrie was thinking about coloring maps:

“Can the areas on any map be colored with at most four colors such that no pair of neighboring areas get the same color?”

Incredibly the answer is yes.

This simple problem was introduced for the first time by Francis Guthrie in 1852. Not until 1976 there was no proof for Guthrie’s conjecture. Only then with the help of computers the conjecture was proved. This proof is crucial for mathematics world as it is known as the mathematical theorem that was proven with the help of computers.

One wonders…

Add a fourth dot to the graph you see above and connect that dot to the existing dots. (You are free to place the fourth dot wherever you want on the graph.)

Now check your graph: One of those four dots is trapped inside the lines, isn’t it?

Can you fix that?

Explain how you can/can’t do it.

M. Serkan Kalaycıoğlu

Real MATHEMATICS – Graphs #5

Cruel Traffic Light

I have to drive past the same crossroad almost every single day. Obviously I stop at the longest-lasting traffic light of the crossroad. Within time I started loving these moments because it gave me a chance to think my life over. Though, it doesn’t take too long for me to start thinking about mathematics.

One of those days I found myself questioning the traffic lights and their relationship with mathematics. (After all mathematics is everywhere; isn’t it?) Soon after I realized that there was my beloved graph theory behind traffic lights.

Light #1

Let’s assume a one-way street with two traffic lights: One for the vehicles (we’ll call it A), other for the pedestrians (we’ll call it B):

In such situation we have to avoid accidents. This means whenever A has green light, B must have red light and vice versa. We will not take account of the situation when both lights are red. Because even though there won’t be any accident, neither of the sides will be standing still (which is nonsense):

All these can be shown using graph theory: Lights will be represented by dots. Dots in a graph will be connected with lines if they are not in the same color:

Same thing can be shown with maps: If A and B are two neighboring countries, they should be colored in different colors to avoid confusion:

Light #2

This time we will assume a two-way street with three traffic lights: Two for the vehicles from opposite sides (we’ll call them A and B), and one for the pedestrians (we’ll call that C):

In such a situation when C is red, either or both of A and B should be green. When C is green, then both A and B should be red:

We’ll skip the situation where all three of them are red as no one would move in such situation.

We can show these using graph theory and map coloring as follows:

Since we set all the rules graphs make it much easier and clearer to understand the situations.

Light #3

Finally we have a two-way street (A and B), a right turn (C) and two pedestrian lights (D and E) as follows:

This time it is a much complex situation:

Although using graph theory makes it all easier for us to understand:

How Many Colors?

We will color the following graphs using the same rules we just established above: If two dots are connected with a line, then those dots must have different colors.

Chromatic Numbers: Whenever a graph is being colored, ambition is the use the least number of colors. This number is also known as the chromatic number of a graph.

Here we need 3 colors. Hence chromatic number of the graph is 3. Let’s add one more dot and line to the graph:

This time chromatic number of the graph becomes 2. We’ll add another dot and line:

Now chromatic number becomes 3 again. Let’s add a dot and a line for the last time:

Chromatic number is back to 2.

To be continued…

One Wonders…

What did just happen? What did you notice? Why is it happening?

How can you increase the chromatic number?

M. Serkan Kalaycıoğlu

Real MATHEMATICS – Graphs #4

I visited the world famous Hermitage museum in Saint Petersburg (Russia) back in 2014. Hermitage is so huge that it has 1057 rooms and one would have to walk around 22 km to see all the rooms. And numbers of artistic & historical items are not that shallow either: It is believe that it would take a little over 11 years if one would spare 1 minute for each piece of art.

hermitage-map-st-petersburg — Map of Hermitage.

Now you can relate why I had trouble when I wanted to visit Hermitage for a few hours. I didn’t have enough time and I wanted to see important art works such as the Dessert: Harmony in Red by Henri Matisse. Eventually I realized that this is a problem I had faced before.

Actually, each and every one of you must have faced such problems in your daily lives. Most common one: “Which route you should take between home and work during rush hour traffic?”

Postman’s Path

Facing such problem in Saint Petersburg is a pleasing coincidence as this magnificent city once was home to one of the giants of mathematics: Leonhard Euler. If you take a look at the first article of graphs you can see that Euler is the person who initiated the discovery of Graph Theory with his solution to the famous Seven Bridges of Königsberg problem.

In 1960 (almost 230 years after the solution of Königsberg) a Chinese mathematician named Mei-Ko Kwan took a similar problem into his hands:

A postman has to deliver letters to a given neighborhood. He needs to walk through all the streets in the neighborhood and back to the post-office. How can he design his route so that he walks the shortest distance?

This problem is given the name Chinese Postman Problem as an appreciation to Kwan. There are a few things you should know about the postman problem:

Postman must walk each street once.
Start and finish point must be at the post-office.
Postman must fulfill these two conditions in the shortest possible time.

Power of Graphs

Mei-Ko Kwan turned to Euler’s Königsberg solution in order to solve the postman problem. According to Kwan postman problem could have been shown as a graph: Lines represent the streets and letters represent the houses.

Example Graph:

20190324_210039 — A, B, C, D and E are the houses as the lines are the streets. The numbers above the lines show how much time it takes to walk from one house to another.

Reminder (You can read the first three articles of Graphs and learn the detail of the following.)

What is an Euler circuit?

Euler circuit is a walk through a graph which uses every edge (line) exactly once. In an Euler circuit walk must start and finish at the same vertex (point).

How do we spot an Euler circuit?

A graph has an Euler circuit if and only if the degree of every vertex is even. In other words, for each vertex (point) count the number of edges (lines) it has. If that number is even for each vertex, then it is safe to say that our graph has an Euler circuit.

Solution

As you can see above, if the degree of every vertex in a graph is even, then we can conclude that the graph has an Euler circuit. Let’s assume that the following is our graph:

The numbers above each line represents the distances (e.g. in km) between the points.

First of all we must determine the degree of each vertex:

As seen above all the vertices have even degrees. This is how we can conclude that the graph has an Euler circuit even without trying to find the path itself. Hence the postman can use the existent roads and finish his route in the shortest time:

When a graph has vertices with odd degrees, then we must add new line(s) to the graph in order to create an Euler circuit. These are the steps you can follow:

Find the vertices that have odd degrees.
Split these vertices into pairs.
Find the distance for each pair and compare them. The shortest pair(s) shows where to add new line(s).
Add the line(s) to the graph.

Let’s use an example and test Kwan’s algorithm for the solution. Assume that A is the starting point. First we must check the degrees of the vertices:

Unfortunately A and D have odd degrees which is why the postman can’t finish his walk:

Here we have only one pair that is A-D. There are three routes between A-D. If we find the shortest one and add that to our graph, we will have an Euler circuit:

The shortest one is the line between A-D as seen above. If we add that line, we will have the shortest route for the postman:

One wonders…

If the post office is at A, what is the route for the postman to take so that he will finish his day in the shortest way?

M. Serkan Kalaycıoğlu

Real MATHEMATICS – Strange Worlds #13

Love of Dinosaurs

I loved reading weekly television guides when I was a child. Thanks to these booklets I knew when to catch my favorite cartoons and movies. This is why I was able to watch some great movies such as Jurassic Park more than once.

Jurassic Park (along with the famous cartoon Flintstones) was the reason why so many kids from my generation interested in genetics, paleontology and obviously dinosaurs. After 1993 when Jurassic Park was a big success, majority of the kids (including me) started learning names of the dinosaurs starting with Tyrannosaurus Rex.

My favorite paleontologist: Ross Geller.

Dragon Curve

In my mid 20s I was doing research about fractal geometry and I eventually found myself with Jurassic Park. Apparently in 1990 Jurassic Park novel was first published. There were strange shapes just before every chapter named as “iterations”. These iterations were actually showing some stages of a special fractal: