strange worlds

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 – 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 #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:

This fractal is known as Jurassic Park fractal or Dragon curve. I prefer using Dragon curve because let’s face it; dragons are cool!

How to construct a dragon curve?

Draw a horizontal line.
Take that line, spin it 90 degree clockwise. This will be the second line.
Add second line to the first one.
Repeat the same processes forever.

After first iteration you will end up with the following:

After second iteration:

Third and forth iterations:

Just before the first chapter of the Jurassic Park novel you can see the forth iteration named as “first iteration”:

One wonders…

You might find these ordinary. Then let me try to surprise you a bit. First of all cut a long piece of paper as shown below:

Did you do it? Well done! Now unite the right end of the paper with left end:

In other words the paper is folded in half. Now slowly unfold the paper such that two halves construct a 90-degree angle between them:

Fold the paper second time in half:

Unfold it carefully:

Do the same things for the third time:

And finally repeat the same process for the fourth time:

Conclusion: Whenever a piece of paper is folded four times in half, one would end up with the fourth iteration of the dragon curve.

M. Serkan Kalaycıoğlu

Real MATHEMATICS – Strange Worlds #12

“More to this than meets the eye…”

I will be using a real life example in order to explain Mandelbrot’s answer to the coastline paradox.

Maps of Norway and USA are as shown below:

It is clear to the naked eye that total coastline of USA is enormous comparing to Norway’s coastline. Nevertheless there is more to this than meets the eyes. Norway’s coastline is a lot longer than USA’s:

USA: 19.924 km

Norway: 25.148 km

There are more than 5000 km between the coastlines which is a really surprising result. But when you zoom into the maps it is easy to see that Norway’s coastline is way more irregular than USA’s. In other words Norway’s coastline has more roughness. Mandelbrot expresses this in his fractal geometry as follows: Norway coastline has a bigger fractal dimension than USA coastline.

But this doesn’t necessarily mean that a bigger fractal dimension has more length. Length and fractal dimension are incomparable.

Measuring Device

In the coastline paradox we learned that one decreases the length of his/her measuring device, then length of the coastline will increase. This information brings an important question with itself: How did they decide the length of the measuring device for Norway-USA comparison?

This is where fractal dimension works perfectly: Finding the appropriate length for the measuring device.

Q: This is all very well how can a coastline length be measured exactly?

Unfortunately it can’t be done. Today, none of the coastline or border lengths are 100% accurate. Although we are certain about one thing: We can make comparisons between coastlines and borders with the help of fractal dimension. In short, today we are able to compare two coastlines or borders even though we are not sure about their exact length.

Box Counting

Finding fractal dimension is easier than you’d think. All you need to do is to count boxes and know how to use a calculator.

Let’s say I want to calculate the fractal dimension of the following shape:

Assume that this shape is inside a unit square. First I divide the square into little squares with side length ¼ units. Then I count the number of boxes which the shape passes through:

This shape passes through exactly 14 squares.

Up next, I divided the unit square into even smaller squares which have side length 1/8 units. And again I count the number of boxes which the shape passes through:

This time the shape passes through 32 squares.

Then I use a calculator. In order to find the fractal dimension of the shape, I must find the logarithms of the number of boxes (32/14) and length of the squares ({1/8}/{1/4}). Then I must divide them multiply the answer with -1.

This random shape I drew on my notebook has around 1,19 fractal dimension.

One wonders…

Calculate the fractal dimension of the following shape:

M. Serkan Kalaycıoğlu

Real Mathematics – Strange Worlds #10

Tearing Papers Up

Use a stationery knife and cut an A4 paper. You will end up with pieces that have smooth sides. Hence you can use Euclidean geometry and its properties in order to find the perimeters of those pieces of papers:

However, when you tear a paper up using nothing but your hands you will be getting pieces that have rough sides as follows:

If you pay enough attention you will that the paper on the right looks just like map of an island.

In the previous article I used properties of Euclidean geometry to measure the length of a coastline and found infinity as an answer. Since those two torn papers look like an island or a border of a country, perimeters of those pieces of papers will also tend to infinity. This is indeed a paradox.

This paradox shows that Euclidean geometry is not useful when it comes to measure things/shapes that have roughness.

Birth of Fractals

“Clouds are not spheres, mountains are not cones, and coastlines are not circles…”

Benoit B. Mandelbrot

There are only shapes consist of dots, lines, curves and such in Euclidean geometry. On the other hand shapes of some natural phenomena can’t be described with Euclidean geometry. They have complex and irregular (rough) shapes. This is where we need a new kind of geometry.

In 1967 mathematician Benoit Mandelbrot (1924-2010) published a short article with the title “How long is the coast of Britain?” where he gave an ingenious explanation to coastline paradox. Furthermore, this article is accepted as the birth of a brand new mathematics branch: Fractal geometry.

The word fractal comes from “fractus” which means broken and/or fractured in Latin. There is still no official definition of what a fractal is. One of their most definitive specialties is “self-similarity”. Fractals look the same at different scales. It means, you can take a small extract of a fractal object and it will look the same as the entire object. This is why fractals are self-similar.

Romanesco broccoli has a fractal shape as you zoom in you will see shapes that look the same as the whole broccoli.

Fractals have infinitely long perimeters. Therefore measuring a fractal’s perimeter or area has no meaning whatsoever.

Coastlines are fractals also: As you zoom in on a coastline or a border line, it is possible to see small versions of the whole shape. (Click here for a magnificent example.) This means that if we had a way to measure a fractal’s perimeter, we can solve the coastline paradox.

So the big question is: What can one do to measure a fractal?

Dimension

According to Mandelbrot, it is possible to measure a fractal’s roughness degree; not its perimeter nor its area. Mandelbrot called this degree fractal dimension. He realized that fractals have different dimensions than the shapes we know from Euclidean geometry.

I will be talking about dimensions in Euclidean geometry before moving on to explaining how one can calculate the dimension of a fractal.

In the Euclidean geometry line has 1, plane has 2, space has 3 dimensions and so on. Teachers use specific examples when this is taught in schools. For space, a teacher asks his/her students to observe any corner of a ceiling:

There are 3 different directions one can choose to walk from that corner. While this is a valid example, it only shows what 3 dimensions look like.

Q: How can one calculate the dimension degree of an object/shape?

Line

Let’s draw a straight line and cut it into two halves:

If straight line had length 1, each segment now has length 1/2. And there are 2 segments in total.

Continue cutting every segment into two halves:

Now each segment has length ¼ and there are 4 line segments in total.

At the point a formula breaks out:

(1/Length of the line segments)^{Dimension Degree} = Number of total line segments

Let me call dimension degree as d.

For 4 line segments:

(1/1/4)^d = 4

4^d = 4

d = 1.

This concludes that line has 1 dimension.

Square

Let’s draw a square that has unit side lengths. At first cut each side into two halves:

There will be 4 little replicas of the original square. And each of those replicas has side length ½.

Apply the formula:

(1/1/2)^d = 4

2^d = 4

d = 2.

Let’s continue and divide each side into two halves. There will be 16 new squares:

Each side of these new squares has length ¼. Apply the formula:

(1/1/4)^d = 16

4^d = 16

d = 2.

This means that square has 2 dimensions.

To be continued…

One wonders…

Apply the same method and formula to find the dimension of a cube.

M. Serkan Kalaycıoğlu

Real Mathematics – Strange Worlds #9

Island S

I own a private island near New Zealand. (In my dreams) Unfortunately I put it on the market due to the economical crisis. If I can’t sell my island I will have to use charter flights to Nice instead of using my private jet.

I created an ad on Ebay. But I choose a different approach when I set a price for my island:

“A slightly used island on sale for 100.000 dollars times the coastline length of the island.

Note: Buyer must calculate the length of the coastline.”

Soon enough I got an offer from a potential buyer. Buyer said he calculated the coastline length as follows:

Buyer used three straight lines in order to measure the length of the coastline. He took each line 8 km long which gave 8*3=24 km. Hence his offer was 24*100.000 = 2.400.000 dollars.

I thought my island worth more than that. Hence I asked the buyer to evaluate his bid again. He came up with a new bid:

This time buyer used seven 5-km-long lines: 5*7=35 km. Thus his second offer was 3.500.000 dollars.

Even though new offer is higher, I thought the buyer can do better. This is why I asked the buyer to measure the length of the coastline one more time:

At last buyer used sixteen 3-km-long lines: 3*16=48 km. Therefore buyer’s final last offer was 4.800.000 dollars.

Q: What is the highest bid I can get from a buyer?

Give yourself a second and think about the answer before continuing the article.

Coastline Paradox

As the buyer decreases the length of the ruler, length of the coastline will get bigger. What is the smallest length for the ruler?

1 cm?

1 mm?

1 mm divided by 1 billion?

There isn’t any answer for the smallest length of a ruler; it can be decreased up to a point where it is infinitely small.

Since there is a disproportion between the length of the ruler and the length of the coastline, coastline can have infinity length.

This is a paradox. Because it is a known fact that there isn’t any land on earth which has infinitely long coastline. Although using buyer’s measurement method, one can’t find an upper limit for the coastline of Island S.

Root of the Problem

British mathematician Lewis Fry Richardson (1881-1953) had done a very interesting research in the first part of the 20^th century. He wanted to know what factors would reduce the frequency of wars between any two country. One of the questions he asked was the following:

“Is there any correlation between the probability of war and the shared border length among two neighbor countries?”

Richardson took Spain and Portugal as an example. Therefore he wanted to know the border length between them. Richardson was really surprised when two countries reported their measurements. Even though they measured the same length, there was a difference of 200 km between two values.

This huge difference led Richardson to pursue the topic and he eventually came up with the coastline paradox.

Is there a sensible explanation for this paradox?

To be continued…

One wonders…

How can my island’s coastline be measured if I want to sell my island for more than 6.000.000 dollars?

M. Serkan Kalaycıoğlu

Real Mathematics – Strange Worlds #8

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.

20190131_185431 — Left to right: Zero, positive and negative Gaussian-curvatured shapes.

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

Real Mathematics – Strange Worlds #7

Mathematics is tough

Most of the people I meet tell me that they have never been good at mathematics. It is upsetting to hear such thing especially when they come from adults. Because even though they don’t realize it, these adults are having negative effects on kids.

Understanding mathematics is not easy. Doing it is not easy either. And as if these are not enough, kids hear how unnecessary it is to learn mathematics. It is not hard to find memes about the redundancy of trigonometry or algebra. I love memes by the way.

What should a 10-year old think when he/she hears that the thing you can’t do at school is not even useful after school? This student would distance him/herself from not only mathematics but all sciences until the end of time.

Worst thing is that this student (future adult) wouldn’t even have any idea about what he/she lost.

In almost every country there are huge masses who don’t know what mathematics is and how useful it would have been for them.

Perspective

One of the most crucial things one can gain from mathematics is having the ability of perspective. Whenever you are facing with a problem in life it would help you to solve that problem more quickly if you are capable of looking at thing from different angles.

Perspective is not only useful for solving problems. It helps one to understand other people’s feelings, thoughts, and state. Having perspective helps you to become an empathic person.

Actually having perspective relates to a habit. Mathematics can be very handy if you’d like to build a new habit for your mind. Personally I believe topology is cut out for it.

Engineer’s Problem

Points A-A’, B-B’ and C-C’ should be connected.

A-A’: Electric system for the house.

B-B’: Gas system for the house.

C-C’: Water system for the house.

These three connections should not intersect with one another to avoid accidents such as fire. This is when the engineer is needed: Finding the right paths as electric system must be built first.

Engineer knows that there are 3 different paths for A-A’: Between B’-C’, B-C’ and C-B’. But there seems to be a big problem here. When engineer tries to connect any of those lines, it is impossible to connect B-B’ and C-C’.

How can the engineer solve this problem?

Beauty of Topology

Topology is very useful in case you can’t see a solution right away. In this amazing branch of mathematics we are allowed to deform objects. If so, then let us deform the shape of the problem in our imagination.

There would be no problem if B’ and C’ were replaced. This is the point where we’ll be using topology: If whole area of the shape was made of a kind of fluid, points B’ and C’ could be pushed away from their original places. And if we are allowed to push away these points, everything in between would have been pushed as well. This is why the connection between A-A’ would have been deformed as shown below:

You could imagine A-A’ as a string.

This deformed A-A’ path is engineer’s solution to the three connections problems.

Then, if A-A’ connection is done first, B-B’ and C-C’ connections can be done as follows:

One wonders…

In the following picture you will be seeing a famous problem called “three utilities problem”.

Now take a pen and a cup in your hands and try to find a way to connect three utilities to three houses.

M. Serkan Kalaycıoğlu

Real Mathematics – Strange Worlds #5

WHERE AM I?

Let’s assume that you have an object shaped like a sphere. Take a pen and draw a starting point on the object. Then, select a direction and move in that direction from your point. When your tour is completed, you’ll be ending up at your starting point in your original direction.

This is an obvious fact which would never change no matter which starting point or direction you choose.

Orientability: Mathematical definition of orientability is really complicated, even for a mathematics graduate. Which is why I will use an example to define it: Imagine that you are travelling to east from Hamburg with a zeppelin. If you continue travelling, and if you are lucky enough to survive, you would eventually end up at Hamburg while your zeppelin faces east. (Please leave you flat-earthers) Actually, it doesn’t matter which direction or city you choose, result will always be the same. If an object or a surface has this specialty, then we say that it has orientability.

It is absurd to think that you finish travelling as a mirror image of your starting state. If you move to the east from a point, you won’t accidently end up at the same point with your direction flipped to west. In other words, you won’t finish taking a tour on a sphere as your mirror image.

Two types of mirror images. Obviously I am talking about the one on the right.

Is it possible to have an object that doesn’t have orientability? Is it possible to go east and when you arrive at your starting point you realize that you are standing upside down?

YES!

Möbius Strip

Cut a rectangle shaped paper. This paper has two faces.

There is no interchanging from one face to another. It means, if you start moving on one face, no matter where or in which direction you started, you won’t end up in the other face. We can call these faces “roads”. So a paper has two different roads.

A paper can have two faces (roads). You can’t go from I to II.

Now glue the ends of this paper together. You will have a cylinder. A cylinder has two faces like a flat paper. It means that a cylinder has two different roads. It is not possible to use one road and end up in the other road. This is why a cylinder has orientability.

Use a transparent paper and make a cylinder. Mark a starting point on your cylinder and move it around your finger until you are at your starting point. You would end up in the same conditions.

Let’s go back: Take your flat rectangle paper and while bringing their ends together twist one end 180 degrees. Mathematicians call this shape a Möbius strip.

While constructing, that 180 degree twist on one end is the only difference between a cylinder and a Möbius strip. But, this difference has absolutely mesmerizing results. First thing to notice is that there is only one face in a Möbius strip. That means a Möbius strip has only one road.

Let’s use a transparent paper again and construct a Möbius strip. Select a starting point and move the strip around your finger. When you end up at your starting point, you’ll see that “up” became “down”, “down” became “up”. In other words, you ended up as your mirror image on a Möbius strip.

Up-right became up-left as if there is a mirror between them.

Then, there is no orientability for Möbius strips. Also, if you make a second tour on it, you would end up at the starting point as your original state.

A little bit of history

It is surprising to learn that Möbius strip was first discovered around 160 years ago. This simple but mysterious shape took its name from the German mathematician August Möbius. Although, Johann Listing, who is another German mathematician, was the first person who published about Möbius strip.

August Möbius (left) and Johann Listing (right).

That is because August Möbius and Johann Listing discovered about Möbius strip independently almost at the same time. After looking through their personal notes we understood that August Möbius discovered the strip 2 months before Listing did. Even though Listing was the first person to use the word “topology”, 2 months gave August Möbius a better mortality.

One wonders…

Construct two Möbius strips. While twisting them, twist one of them to the right and the other one to the left 180 degrees. Do you see any difference?
Construct a cylinder and cut it into its middle in the direction parallel to its longest edge. You will end up with two cylinders which are little replicas of the original cylinder.
Try the same thing on a Möbius strip. What is the result? Why did you have that result?

M. Serkan Kalaycıoğlu