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 – Geometry #20

Escape From Alcatraz

Imagine a classroom that has 5 meters between its walls in length. Tie a 6-meter long rope between these walls. Let the rope be 2 cm high off the ground. Since the rope strained to its limits, its 1-meter long part hangs from either side of the rope.

The ultimate goal is to escape from the classroom from under this rope, without touching the rope.

Rules

  • Escape should be from the middle point of the rope.
  • One should use the extra part of the rope to extend it.
  • One of the students will help you during the escape. He/she will strain the rope for you so that you can avoid touching the rope.
  • Each student has exactly one try for his/her escape.

Winning Condition: Using the least amount of rope for your escape.

Football Field

Legal-size for a football field is between 90 and 120 meters in length. Assume that we strain a rope on a football field that is 100 meters long. We fixed this rope right in the middle of both goals while the rope is touching the pitch.

The middle of the rope sits right on the starting point of the field. This is also known as the kick-off point.

Let us add 1 meter to the existing rope. Now, the rope sits flexed, not strained, on the field.

Question: If we try to pick the rope up at the kick-off point, how high will the rope go?

Solution

We can express the question also as follows:

“Two ropes which have length 100m and 101m are tied between two points sitting 100m apart from each other. One picks the 101m-long rope up from its middle point. How high the rope can go?”

If we examine the situation carefully, we can realize that there are two equal right-angled triangles in the drawing:

Using Pythagorean Theorem, we can find the length h:

(50,5)2 = 502 + h2

h ≈ 7,089 meters.

Conclusion

Adding only 1 meter to a 100-meter long rope helps the rope to go as high as 7 meters in its middle point. This means that a 1-meter addition could let an 18-wheeler truck pass under the rope with ease.

M. Serkan Kalaycıoğlu

Real MATHEMATICS – Strange Worlds #14

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

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?

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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:

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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:

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We will get a shape like the following:

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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 – Geometry #19

How Much Chocolate?

It is midnight and my stomach is talking to me. I hope to find something to eat in the kitchen and I see a chocolate bar:

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Immediately made myself a cup of coffee and broke a piece of the chocolate bar:

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After I “killed” the broken piece I started having second thoughts about my decision: Oh God; Did I eat too much chocolate?

I placed the leftover on a grid. This way I found where both whole and the broken piece lies on the grid:

çiko1

The broken piece is shaped like a simple polygon. My goal is to calculate the area of that piece. There are several ways I could calculate the area. Although, the first thing comes to my mind is a theorem called “Gauss’ shoelace theorem”.

Gauss’ Shoelace Theorem

The shoelace theorem can only be applied to simple polygons. In order to use the theorem, I have to find where the edges of the simple polygon lie on the grid:

çiko2

Theorem uses these points just like shoelaces. But first we have to define the edges and make a list of them:

çiko3

lak1

 

 

Do not forget to add the first edge to the bottom of the list.

Now you can multiply the numbers diagonally; from right to left and left to right. Then add left to right and subtract it from right to left ones:

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{(0*0) + (5*1) + (4*3) + (5*3) + (0*0)} – {(0*5) + (0*4) + (1*5) + (3*0) + (3*0)}

{32} – {5}

27.

The area of that simple polygon can be found by dividing the result above:

27/2

13,5

M. Serkan Kalaycıoğlu

Real MATHEMATICS – Geometry #18

Protect the Cake

Holy cake is coming to the school today. It will be open for visit in one of the classrooms. You are responsible for the protection of the holy cake. But you have limited people under your command which is why you should use minimum number of guards to protect the holy cake.

About the guards: A museum guard stays at a fixed point and observes the art piece(s) from that point. Obviously guard can see every angle around him/herself by simply rotating.

Floor plan of the classroom is like the following:

20190328_130114.jpg

Find the minimum number of guards needed in order to protect the holy cake in this room.

Polygonal Rooms

First we will be examining the simplest polygon.

Example 1: Triangle room.

Assume that holy cake is inside a triangle room. One guard is enough for protecting the cake as he/she will be able to see every part of the room:

Example 2: Rectangle room.

Again one guard will be enough:

Q: Is one guard enough for all polygon-shaped rooms?

Convex-Concave: A polygon is convex if each of its internal angles is less than 180 degrees. Otherwise that polygon is called concave.

As we can see above, all convex polygons can be protected by a single guard. Although it can’t be said for all concave polygons.

Now we know that the room of holy cake is a concave polygon. Let’s examine a simple concave polygon first:

20190328_123117.jpg

It is possible to protect the room if we place the guard right on the vertex of the angle that is greater than 180 degrees:

20190328_123216.jpg

If we have a room plan as following:

20190328_123606.jpg

In such room, one guard doesn’t give enough coverage:

20190328_123732.jpg
Guard can’t see the blackened area.

Art Gallery Problem

Before moving on the more complex room plans, we have to check if there is any algorithm for finding the necessary number of guards in concave polygons. This problem was first posed by a mathematician named Victor Klee in 1973 and is called art gallery problem. In order to find the necessary number of guards in a room, we should use a method called triangulation.

Triangulation: Basically it is a process that enables us to divide any polygon into triangles.

Let’s take a concave polygon as an example. First we should use triangulation:

Then we should color the vertices of the triangles. In any triangle vertices must have different colors:

20190328_124639.jpg
We used 3 colors: 1, 2 and 3.

Number of guards is the least used vertex color. If guards take positions at those vertices, they will be able to see the room completely:

20190328_124931.jpg

Least numbered colors are 2 and 3. Placing guards on either of them is enough to cover the whole room. If, let’s say, we choose to place two guards on the color 3, room will be protected.

Solution

We are finally ready to solve the holy cake problem. First we should triangulate the floor plan of the classroom:

20190328_130612.jpg

Next, we should color the vertices of the triangles:

20190328_130833.jpg

Three colors were used as follows:

20190328_131241.jpg

The least used colors are 1 and 3. This gives us the number of guards needed (that is 6) in order to protect the holy cake. If, let’s say we choose to place the guards on the vertices with color 1, room will be covered completely.

One wonders…

  1. What would the answer be if there were columns inside the classroom as shown below?
    20190328_131619.jpg
  2. Can you find a relationship between the number of vertices of the polygon and number of guards?

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.

Jhkbi8QKyPml.jpg
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:

OGvqP9l

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:

20190318_123050

After second iteration:

20190318_123404

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:

20190226_123508

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

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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:

20190226_123649

Fold the paper second time in half:

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Unfold it carefully:

20190226_123749

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:

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:

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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.

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:

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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:

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.

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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