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

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

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.

20190730_143315.jpg

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 – 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?

indir
Mei-Ko Kwan

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:

20190325_123338.jpg
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:

20190325_123326.jpg

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:

20190325_123311.jpg
The shortest path takes 11 km.

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:

  1. Find the vertices that have odd degrees.
  2. Split these vertices into pairs.
  3. Find the distance for each pair and compare them. The shortest pair(s) shows where to add new line(s).
  4. 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:

20190324_210039

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

20190324_210102

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:

20190324_210157

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?

20190324_210225

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.

a

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:

hjfhjfhj

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:

20190217_005201

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:

20190217_005002

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.

20190214_203034

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:

20190214_184605

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:

20190214_185221

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:

20190214_185857

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.

ispa

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 – What are the chances?! #4

Coffee of Serkan

Certain days of the week (okay; at least six days a week) I visit a specific coffee shop. Almost all the baristas know what I drink because of my frequent visits… Or do they?

My preferences change every six months. In the period of October-March, I only drink either latte or filter coffee, while in the period of April-September I prefer iced latte or berry.

October-March: In case I drink latte today, there are 80% of chances that I will be drinking latte in the next day. If I drink filter coffee today, chances of me drinking filter coffee tomorrow are 60%.

April-September: If I drink iced latte today, tomorrow I will be drinking iced latte with %80 of probability. For berry that probability is 90%.

20190205_195230
Diagram of my coffee selections in October-March period.

Question: If I drank filter coffee this morning, what are the chances that I will drink a latte 2 days later? (We are in February.)

This question resides one of the most crucial findings of mathematics with itself: Markov Chain.

20190205_194842
A Markov Chain example. I will be explaining what it is in details inside the article.

It is clear to see that there are two different possibilities to drink latte two days from now. Sum of their probabilities will give us our answer:

Probability of drinking filter coffee (0,6) the next day, and latte (0,4) two days later: 0,6*0,4 = 0,24.

Probability of drinking latter (0,4) the next day, and again latte (0,8) two days later: 0,4*0,8 = 0,32.

Probability of drinking latte two days from now: 0,24 + 0,32 = 0,56.

It means the chance is 56%.

One wonders…

  1. Does it matter which day of February it is today?
  2. Would it change the answer if you learn that I drank filter coffee yesterday? Please elaborate your answer.
  3. If I drink iced latte on June 11th, what is the probability of me drinking berry on June 14th?

Driverless Cars

In case you make a simple web search you will see that there are thousands of pages of articles that question where flying cars are. A few generations including mine have been dreaming about flying cars whenever we were just kids. “Back to the Future” was one of the main reasons why we had such dreams. And it is not like we expect time travel. We just want flying cars!

gelecege-donus-hakkinda-bilmedikleriniz,Z6sLULxzT02ta5L2MlbZvA

It is 2019 and there are still no flying cars around. Technology developed as much as making driverless cars only. (Only?!)

Decision making systems are among the key technologies needed for building driverless cars. Because a self-driving car will make hundreds of decisions even if it travels short distances.

Gist of decision making systems is the Markov Chain I mentioned in the Serkan’s Coffee. A concept known as Markov Decision Process is the powerful tool that is being used for driverless cars.

Markov Decision Process (MDP): It is a mathematically formulation for decision and control problems with uncertain behavior.

Memory-less Probability

Markov Chain: If there is Markov Chain inside an event or system, future of that system depends only at the current state of the system; not to its past. And it is possible to predict the future of that system.

One of the examples of Markov Chain is Drunkard’s Walk. Reminder: A drunkard makes random decisions while he/she is trying to find his/her home. Assume that the drunkard had made these moves:

rand1

Drunkard’s next move will not depend on the previous moves he/she had made. This only depends on his/her current state and probabilities of the possible moves.

rand2
If drunkard will move from the point F, there are four possibilities and none of them depend on previous steps the drunkard took.

It goes the same for driverless cars: Decisions will not depend on the previous ones. For example if a driverless car is heading towards traffic lights its decision will depend on the color of the traffic light; not the left turn it made 200 meters behind.

Markov Chain was found more than 100 years ago and it is being used in economy, meteorology, biology, game theory and even modern technologies like driverless cars and voice recognition systems.

Mathematician Family

The person who gave Markov Chain its name is a Russian mathematician called Andrei Markov. His little brother, Vladimir Markov, was also internationally recognized mathematician. Vladimir died because of tuberculosis at the age of 25. Andrei’s son Andrei Markov Jr. was also a mathematician.

Politics and Andrei

Andrei Markov was involved with politics too. He was not in favor of Romanov dynasty which ruled Russia between 1613 and 1917. He showed his opposition with not participating in the 300th year celebration of Romanov dynasty in 1913. Instead he celebrated 200th year anniversary of the Law of Large Numbers! (I’ll get back to Law of Large Numbers later.)

M. Serkan Kalaycıoğlu