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:

Red undergoes the green.

The inverse of this rope is:

Green undergoes the red.

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

The Amateur Mathematician

Pierre de Fermat

Birth: 1601, France

Death: 1665, France

Two names lead the first half of 17th-century mathematics: Rene Descartes and Pierre de Fermat. It is rather unusual since neither of them saw themselves as mathematicians as first.

Pierre de Fermat worked as a lawyer and a government official from 1631 until his death. He is regarded as the most important amateur mathematician of all time.

Entry to math circles

Fermat started working on mathematics in the late 1620s. He loves creating problems in the number theory and proving them.

Fermat became famous among mathematicians with his letter to Mersenne* dated back to April 26th, 1636. In his letter, Fermat worked for subjects such as Galileo’s free-fall experiment and Apollonius’ conics which led him to correspond with many mathematicians.

Even though his famous letter involved mathematical physics, he was interested in number theory. This is why, whatever the subject was in his letters, he somehow brought it to a problem in number theory. He wanted mathematicians to prove the problems he created and solved. But these problems were incredibly hard, and soon mathematicians started irritated by him. For instance, Frenicle de Bessy thought that Fermat was teasing him with his difficult problems.

Last Theorem

Today, Fermat is best known for his famous last theorem: Fermat’s Last Theorem. Fermat didn’t consider himself as a mathematician. Hence, he didn’t publish any of his works. In fact, sometimes he was writing his theorems and proofs in the blank parts of books.

One of those theorems was Fermat’s Last Theorem*.

* Fermat’s Last Theorem

“No three positive integers a, b, and c satisfy the equation an + bn = cn for any integer value of n greater than 2.”

Fermat also noted the following:

“I have discovered a truly remarkable proof which this margin is too small to contain.”

Fermat’s Last Theorem was finally proved by British mathematician Andrew Wiles in 1994. That means the theorem was unproven for almost 358 years!

Alienation

Between 1643 and 1654, Fermat was alienated by the science community. There were a few reasons for that such as the civil war and the plague that affected where he lived. But one of the main reasons was his dispute with Descartes.

In his time, Descartes was influential among French scientists. So, when Fermat made a negative comment about his beloved work La Geometrie, Descartes went after him. He (wrongly) criticized Fermat’s work for maxima-minima-tangent*.

Even after Fermat proved that his work was complete, Descartes continued the argument by claiming that Fermat wasn’t a sufficient mathematician.

*Maxima-Minima-Tangent

You probably heard derivatives and integral if you took a math course in university. Actually, the first developments of those two subjects belonged to a 17th-century French lawyer.

Using a simple example, I will show you how Fermat’s method worked:

Assume that we have a straight line with length a:

My ambition is to find a point on it such that it separates the line into two parts, and multiplication of the length of those two parts is maximum.

Assume that one of those parts has length x. Then the other part would have length a-x:

Then, the maximum becomes:

x . (a-x)

ax – x2

So, how can we find this length x?

At this point, Fermat discovered an ingenious method. He says that we should add length e to x:

But this length e should be so small, we might as well consider it as zero. In other words, e is infinitely small. Now we can substitute every x in our solution with x+e:

a(x+e) – (x+e)2

Use basic algebra to simplify the equation:

ax + ea – x2 – 2xe – e2

From this we can reach;

ax – x2 = ax + ea – x2 – 2xe – e2

Again simplify the equations:

ea = 2xe – e2

Now, divide everything by e:

a = 2x – e

Remember that Fermat said e is so close to zero, it is in fact zero? Then:

a = 2x

Our line a has length 2x. This means that if we split a straight line into two equal parts, their multiplication becomes maximum.

Final Years

After his dispute with Descartes ended, Fermat got in touch with mathematicians. His correspondence with Blaise Pascal led to the set up of the probability theory.

Fermat was far ahead of his time in number theory. He almost carried the whole branch by himself in the 17th-century. His findings and problems were appreciated much after his death in the modern number theory. For me, his title will always be the greatest amateur mathematician of all time.

M. Serkan Kalaycıoğlu

A New, Strange World

Janos Bolyai

Birth: 1802 – Romania

Death: 1860 – Romania

Most people think of Count Dracula whenever Transylvania is mentioned. However, I think of another name at first: Janos Bolyai.

Failed dreams of a young mathematician

Janos Bolyai (from now on I will mention him as Bolyai) is the son of Hungarian mathematician Farkas Bolyai. He showed great potential even when he was just 5-6 years old, and mastered calculus* when he was 13.

*Calculus

The branch of mathematics that deals with the finding and properties of derivatives and integrals of functions, by methods originally based on the summation of infinitesimal differences.

In 1816, Farkas asked his old friend and mathematics teacher Gauss* to take his son as a pupil so that young Bolyai can have the best possible mathematics education. Although, Gauss rejected Farkas’ offer. This wasn’t going to be the only bad new Bolyai gets from Gauss.

*Gauss

A German mathematician who is also known as the prince of mathematics.

For young Bolyai, the best possible choice was to go to Vienna and study military engineering. He was an outstanding student there and finished 7 years of study in just 4 years and joined the army in 1823. He earned a living there until 1834.

A new geometry

Farkas Bolyai spent most of his career for finding a proof (or disproof) for Euclid’s parallel postulate* but failed in the end. It is not a surprise that young Bolyai took the matter in his own hands. He started working on the subject in the early 1820s.

Euclid’s parallel postulate

Also known as the fifth postulate that is given by the Greek mathematician Euclid in the first book of his masterpiece Elements.

Basically, it states that no two infinite parallel lines meet at a point.
L1 and L2 will meet at a point if they are not parallel to each other. Euclid’s fifth states that in essence.

Bolyai spent almost all of his spare time in his army duty for mathematics. In November 3rd, 1823 he wrote a letter to his father, mentioning his findings for the first time with these words:

“… out of nothing, I created a new and strange world.”

A year after this letter Bolyai completed his idea for non-Euclidean geometry. At first, Farkas was distant to his son’s findings, but at 1830 he realized how important his discovery was. This is why he convinced his son to write down his idea. In 1831, Bolyai wrote 24 pages-long appendix in his father’s book.

Farkas sent the appendix to his old friend Gauss to evaluate his son’s work. After reading the appendix, Gauss made two important comments to two separate people…

The idea

Bolyai’s idea at its core: Imagine a new geometry where Euclid’s fifth postulate isn’t true. In other words; a new geometry where parallel lines can meet.

In Euclidean geometry, the shortest path between two points is a straight line. However, according to Bolyai’s non-Euclidean geometry shortest path is a curve. You can read about this subject here.

One could explain Bolyai’s idea like this:

In Euclidean geometry, internal angles of any given triangle add up to 180 degrees. But if we draw a triangle on a sphere (e.g. on Earth), angles will exceed 180 degrees:

Bolyai’s geometry (today is known as hyperbolic or non-Euclidean geometry) was a brand new geometry.

The breakdown

Gauss told one of his friends “I regard this young geometer Bolyai as a genius of the first order.”. But, at the same time he wrote a letter to Farkas and showed a much different attitude:

“… to praise this work would be praising myself, as I’ve had the same ideas some 30-35 years ago.”

Today we know that Gauss, in fact, held similar ideas with Bolyai thanks to a letter of his dated back to 1824. But, this happened much after that he told. It is believed that Gauss wasn’t feeling comfortable about publishing his ideas publicly.

Gauss’ comments on his appendix hit Bolyai hard. His idea stayed unknown for the mathematics community and he was deeply affected after Gauss’ remarks. Soon after, his health had gone bad and he was forced to leave the army in 1834, and he started living an isolated life from his beloved mathematics.

Paranoid

Even after all this, Bolyai kept working on mathematics. In 1848, he received a work that is written by a Russian mathematician named Lobachevsky. Lobachevsky’s work was published in 1829 (before his appendix) and covered almost the same ideas as he had held for non-Euclidean geometry. Gauss knew about this work as well. In fact, he praised Lobachevsky for his work.

After investigating the work deeply, Bolyai believed that Lobachevsky was not a real person, and it was all Gauss behind this work. Unfortunately, Bolyai was slowly losing his mind.

Bolyai stopped his mathematics in his last years and died in poverty in 1860. It is known that he left around 20.000 pages of unpublished mathematical work behind. They can be found in the Bolyai-Teleki library located in Targu Mures.

Today, we honor him by calling non-Euclidean geometry as Bolyai-Lobachevsky geometry.

M. Serkan Kalaycıoğlu

A Mathematician Inside Seven Sages

Thales

Birth: Around 624 BC, Miletus

Death: Around 547 BC, Miletus

Thales Theorem: The diameter of a circle always subtends a right angle to any point on the circle.

In high school math, Thales is known with this theorem even though he is one of the most important names in the history of science. The funny thing is that he most likely didn’t even find this theorem. Then, who is this Thales guy we ought to know?

Thales is known as one of the Seven Sages of Greece as well as the first known natural philosopher.

Seven Sages of Greece are the seven important figures of 7th and 6th century BC Greece including Thales, Pittacus, Bias, Solon, Cleobulus, Myson and Chilan.

Natural philosopher title comes from Thales’ diversity as it is said that he worked in mathematics (especially geometry), engineering, astronomy, and philosophy. There is not even a single written work of his left today. Everything we know about him was written by others centuries after his death. This had led many legends after his name.

Historians believe that Thales visited Egypt where he learned mathematics and engineering. He is known as the first person who introduced geometry to Greeks. It is said that while he was in Egypt, he calculated the length of pyramids by just looking at their shadow.

Whenever the sun makes 45 degrees with Earth, the shadow of a particular object becomes equal to its own length. This is known as one of the methods Thales used in his calculation.

According to another legend, he guessed the time of the solar eclipse in 585 BC. During his time, people were able to guess the lunar (moon’s) eclipse. Although, it was impossible to guess where and when the solar eclipse was gonna occur using 6th BC’s knowledge and technology. Today we believe that even if this happened, it was just an astonishing guess for Thales.

Still, some believe that Thales really guessed the solar eclipse thanks to his brilliance. After all, he was one of the Seven Sages. Actually, as Socrates stated, he was the only natural philosopher inside that prestigious group.

Thales believed that everything comes from water. According to him, the Earth was shaped like a disk and it floats on an infinite ocean. He suggested that the earthquakes were the result of the Earth’s movement on the ocean. This was a first in the history of science as Thales’ ideas were based on logic instead of supernatural phenomena.  

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.

person holding black cassette tape

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.

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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_main_001

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?

maxresdefault (4)

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.

180px-Example_of_Knots.svg

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:

20190730_135245.jpg
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:

lanaa

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”.

Blue_Trefoil_Knot
Trefoil knot.

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.

trefoilandmirror

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:

20190730_131333.jpg

We will get a shape like the following:

20190730_134158-1.jpg

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:

20190423_000256.jpg

In this case Serkan I can have two children:

20190423_000310.jpg

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:

20190423_000334.jpg

Let’s add another land to this map:

20190423_000402.jpg

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

20190423_000507.jpg

Map #2

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

20190423_000523.jpg

We can convert this map into graph as follows:

20190423_000537.jpg

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

20190423_000553.jpg

Map #3

Let the third map be the following:

20190423_000627.jpg

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

20190423_000642.jpg

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

20190423_000728.jpg

Map #4

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

480271690e1e0485f71988e273730559

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

amarikaaa

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…

20190423_003916.jpg

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 – 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:

20190226_152316

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.

loga

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

One wonders…

Calculate the fractal dimension of the following shape:

20190228_005941

M. Serkan Kalaycıoğlu

Real Mathematics – Strange Worlds #11

Fractional Dimension

When you try to measure the length of a coastline, your finding will increase as your measuring device decreases. It means that there is a proportion between these magnitudes. This is why it is possible to find different (even infinite) lengths for a random coastline.

Mathematician Mandelbrot named this proportion as “fractal dimension”.

In the Euclidean geometry a dot has 0, a line has 1, a plane has 2 and a cube has 3 dimensions. But, in the nature shapes of objects are not regular as shown in the Euclidean geometry. In the early 20th century a mathematician named Felix Hausdorff discovered that some shapes have non-integer dimensions. Later on we started calling this non-integer dimension idea as Hausdorff-Besicovitch dimension. This idea was basis for fractal geometry’s development.

In the previous article I showed how one can calculate dimension of a shape in the Euclidean geometry. Same formula can be used in order to calculate objects that don’t have regular shapes. For that, I will be talking about a couple special fractals.

Snowflake

Swedish mathematician Helge von Koch created a geometrical shaped named after him: Koch snowflake.

To create a Koch snowflake, one can start drawing a straight line. Then that line should be divided into thirds as the middle part gets erased:

Draw sides of an equilateral triangle above the removed segment: (In other words, add a peak where there is a gap.)

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Continue the same process forever and you will get Koch fractal:

Here are the segments and all of Koch snowflake:

karr.jpg

Now let’s use the dimension formula to the Koch snowflake. We only need the number of parts and their lengths in each step of the construction of the Koch snowflake.

In the first step, we had a straight line that was divided into 1/3s:

20190224_161654

In the second step we ended up with 4 of those 1/3s:

Adsızmbm

If we examine each step of the Koch snowflake we will end up with 4 parts that have 1/3 lengths. Therefore fractal dimension of Koch snowflake (which I call d) can be found as follows:

(1/3)d = 4

d ≈ 1,26.

Koch Curve

Let’s try a variant of the Koch snowflake, which we call Koch curve. This time we will draw sides of a square instead of an equilateral triangle.

So, we will start with a straight line that is divided into thirds. Then we will remove the middle part and draw sides of a square that has no bottom line:

Next few stages of the Koch curve will look like the following:

Here we see that in each step, we end up with 5 parts that have length 1/3:

555

Apply this to the dimension formula and this fractal’s dimension will be as follows:

(1/3)d = 5

d ≈ 1,4649.

What does this difference in dimensions mean?

Between the curve and the snowflake, curve has more roughness and it takes up more area than the snowflake. Hence one can conclude that higher dimension means more roughness and more area for Koch fractals:

To be continued…

One wonders…

Another handmade fractal is Sierpinski triangle. This famous fractal was first discovered more than 100 years ago and named after a mathematician named Waclaw Sierpinski.

To construct Sierpinski triangle, one must start with an equilateral triangle:

20190224_214207

Then mark middle of each side and connect those points to form a new triangle:

At this point, there are four smaller versions of the original triangle. Cut the middle one out and you will have three equilateral triangles that have half of the side lengths of the original triangle:

20190224_214247

Repeat the steps forever and you will get Sierpinski triangle:

  1. Show that Sierpinski triangle is a fractal.
  2. Calculate the dimension of the Sierpinski triangle and compare your result with Koch snowflake.

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.

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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:

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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:

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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

4d  = 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

2d  = 4

d = 2.

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

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

(1/1/4)d = 16

4d = 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.

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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:

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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:

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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:

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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 20th 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.

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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