Real MATHEMATICS – Strange Worlds #13

Love of Dinosaurs

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

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

My favorite paleontologist: Ross Geller.

Dragon Curve

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


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

How to construct a dragon curve?

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

After first iteration you will end up with the following:


After second iteration:


Third and forth iterations:

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

One wonders…

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


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


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


Fold the paper second time in half:


Unfold it carefully:


Do the same things for the third time:

And finally repeat the same process for the fourth time:

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

M. Serkan Kalaycıoğlu

Real MATHEMATICS – Strange Worlds #12

“More to this than meets the eye…”

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

Maps of Norway and USA are as shown below:

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

USA: 19.924 km

Norway: 25.148 km

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

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

Measuring Device

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

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

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

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

Box Counting

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

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


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

This shape passes through exactly 14 squares.

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

This time the shape passes through 32 squares.

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


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

One wonders…

Calculate the fractal dimension of the following shape:


M. Serkan Kalaycıoğlu

Real Mathematics – Strange Worlds #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.


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.)


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

Here are the segments and all of Koch snowflake:


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:


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


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:


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:


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:


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.


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?


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

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

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


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

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


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

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

Continue cutting every segment into two halves:


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

At the point a formula breaks out:

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

Let me call dimension degree as d.

For 4 line segments:

(1/1/4)d = 4

4d  = 4

d = 1.

This concludes that line has 1 dimension.


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:


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.


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

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

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

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


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

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


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

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


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

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

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

Coastline Paradox

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

1 cm?

1 mm?

1 mm divided by 1 billion?

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

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

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

Root of the Problem

British mathematician Lewis Fry Richardson (1881-1953) had done a very interesting research in the first part of the 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.


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