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:

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 – Geometry #17

Why do we have round wheels in cars and bicycles?

Square Wheels

Let’s construct a square wheel like the following and test if it can be used as an efficient wheel:

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After turning the squared-wheel 45 degrees the square will look like the one on the right:

At this position it is clear even to the naked eye that the height of the wheel is taller comparing to its original state. If we continue turning the wheel for another 45 degrees it goes back to its original position.

Squared-wheel has big disadvantages. A car or a bicycle with squared-wheels will be doomed as height of the vehicle constantly changes.

Triangle Wheels

Among the triangles equilateral is the best one for constructing a wheel as all of its sides have equal length:

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After turning the triangular-wheel 60 degrees to the left, it will have the same height:

Although let’s go 30 degrees back and examine the height of the wheel:

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Here height of the wheel is clearly taller than the height of the original state. This proves that it is not appropriate to construct a triangular-wheel on a vehicle. Otherwise you might have problems with your spine.

Power of the Circle

The reason why circle is the most powerful and convenient shape for our vehicles is that neither of its height nor width changes while rotating. This separates circle from all the other polygons and makes it the best shape for wheels.

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Nonetheless, one wonders if circle is the only possible shape for wheels.

Reuleaux Triangle

Leonardo da Vinci is one of those names that appear in your mind when someone mentions Renaissance. The relationship between this magnificent figure and Reuleaux triangle comes from a world map that was found inside his pupil Francesco Melzi’s notes:

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This is known as one of the very first maps that included America. It is believe that this 1514-dated drawing was due to Leonardo da Vinci. If this is true, then it is safe to assume that da Vinci was the first person ever who used the Reuleaux triangle.

It was Leonhard Euler whom discovered the shape and explained it mathematically almost 200 years after da Vinci’s map. You probably realized this from my articles: “It is either Euler or Gauss.”

How come it is called Reuleaux? Because a century after Euler, a German engineer named Franz Reuleaux discovered a machine using Reuleaux triangle. In 1861 Franz Reuleaux wrote a book that made him famous and today he is known as the father of kinematics.

How to Construct Reuleaux Triangle?

I will show you my favorite method for its construction using three identical circles. First draw a circle that has radius r:

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Then pick a point on that circle as a center and draw a second circle again with radius r:

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At last, pick one of the crossings of the circles as center and draw a third circle with radius r:

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The central area of these three circles is a Reuleaux triangle:

When a Reuleaux triangle is rotating, it will have the same height at all times just like the circle:

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A bike that has a Reuleaux-triangled wheel.

One wonders…

  1. Using Euclid’s tools (a compass and an unmarked ruler) draw an equilateral triangle.
  2. Try to construct Reuleaux triangle with on the equilateral triangle.
  3. Can you contruct Reuleaux polygon(s) other than the triangle?

M. Serkan Kalaycıoğlu

Real Mathematics – Geometry #15

Drawing a square

I am dealing with geometry and I imagine that I am in ancient Greece again. Aegean sea is in front of me and I am sitting on a marble between two huge white columns while holding an unmarked ruler and a compass.

First I draw a circle that has center at A and has radius r:

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Then I draw the same circle but taking its center at B this time:

çember2

I connect the points A and B with a straight line. Then I draw two perpendiculars from the endpoints of the line AB:

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I connect the point E to the point F and end up with the ABEF square which has side lengths r:

Biggest circle that can be drawn into ABEF will have diameter r and touch the square at exactly four points:

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Area

In order to find the area of a square one can take the square of one side that gives r2.

To find a circle’s area one should multiply the square of the radius with π. In our inscribed circle we calculate the area as πr2/4.

Ratio of these areas would give π/4.

Weight

Now let’s make an experiment. For that all you need is some kind of cardboard cut as a square and a precision scale. Using the scale find the weight of the square-shaped cardboard.

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Then draw the biggest possible circle inside this square. Cut that circle out and find its weight with the scale.

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Since we are using the same material ratio of the weights should be equal to the ratio of the areas. From here one can easily find an approximation for the number π:

0,76/0,97 = π/4

3,1340… = π

One of the main reasons why we only found an approximation is that the cardboard might not be homogeneous. In other words the cardboard might not have equal amount of material on every point of itself.

Another reason for finding an approximation is that I didn’t cut the square and the circle perfectly.

One wonders…

Draw a circle and then draw the biggest-possible square inside that circle. Find their areas and measure their weights. See if you found an approximation.

M. Serkan Kalaycıoğlu

Real Mathematics – Killer Numbers #2

In the previous article I was talking about the numbers which put an end to Hippasus’ life. These numbers are not only fatal; they are also incommensurable as well. On top of these, it is impossible to write these killer numbers as ratios of two other numbers.

I believe that there are more than enough reasons to choose a name such as “irrational” for these numbers. For me, it is astonishing to accept that there are some lengths which we can’t measure although they are just in front of us.

√2: One of the most famous irrational numbers.

Whether we realize it or not we can easily spot these lengths in everything that has square shape. Just divide a square diagonally into two equal parts and you will get two right-angled equilateral triangles.

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Assume that the square had side lengths 12. This gave a right-angled equilateral triangle with perpendicular sides with length 12. If we apply the Pythagorean Theorem:

This is an irrational number.

In case you’d like to measure this length, you will see a number that has infinite decimals: 16,97056…

I wonder what would happen if I call this number 17.

√2 is Finally Rational

If 12√2=17, we would get:

pisag5

We did it! √2 can be written as a ratio of two other numbers. It means √2 is rational. From now on we can write 17/12 wherever we see √2.

Although let’s stick to geometry a little bit more and see if we really got something or not.

Proof by Contradiction

First we divide the triangle as follows:

We can see that there are two identical right-angled triangles (A and B) that have perpendicular sides with length 5 and 12, and another right-angled triangle (C) that is equilateral.

Let’s analyze the triangle C from close. It has perpendicular sides with length 5 and a hypotenuse that has length 7. Using Pythagorean Theorem we can conclude:

img_45541

25 + 25 = 49.

50 = 49.

This is a contradiction.

√2 is not rational.

One Wonders…

Check and see what would happen if we used a square that has side lengths 10.

Real Mathematics – Killer Numbers #1

Hippasus: First Victim of the Science Mob

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Pythagoras is a very well known historic figure. Even though most of the people know him through the geometry theorem attributed to him, he had accomplished more than just a theorem. He was also the head of the first known science mob in the history.

Pythagorean Theorem: In a right-angled triangle square of the perpendicular sides add up to the square of the third side of the triangle that is also known as the hypotenuse.

Pythagoras was born in the island of Samos. He had an enormous reputation as a mathematician throughout the ancient Greece. His followers (Pythagoreans) chose to live as their leader. They were a tight and closed group that ate neither meat nor beans and isolated themselves from having any kind of possession.

According to Pythagoras universe was built on the numbers. Every number had a character and everything that is happening around us could be explained with numbers. He believed that numbers have categories such as beautiful, ugly, masculine, feminine, perfect and such. For instance 10 was the best number because it contained the summation of the first four numbers: 1+2+3+4=10.

Pythagoreans also believe that every number is rational: Meaning that each number can be represented as a division of two other numbers. (E.g. 10/2 = 5)

Oath Breaker

One day one of Pythagoras’ followers broke his oath and asked the forbidden question: What is the length of the hypotenuse of an equilateral right-angled triangle?

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Geogebra shows that the hypotenuse is around 1,41 units. This is not the exact value of the length as this length can never be measured.

Hippasus was a devoted Pythagorean. One day he sailed away with his brothers. When he was in the open sea, he started thinking about the problem of the right-angled equilateral triangle. In the end he claimed that he found irrational numbers. This was an oath breaker as it was forbidden to question Pythagoras’ words. Hippasus never came back from that trip, and Pythagoreans continued to keep the existence of the irrational numbers as secret.

Incommensurables: Do they exist?

According to the Pythagorean Theorem: Length of hypotenuse on a right-angled equilateral triangle.

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If Hippasus was wrong, √2 was a rational number which means √2 can be written as the division of two other numbers. Let’s say that this is true and a/b is equal to √2.

Ps: a and b are relatively prime. This means that a/b can’t be simplified; they are the smallest numbers for that ratio.

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Let’s square both sides so that we are free from the square root.

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Now send the denominator to the left side of the equality.

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This actually means that two squares that have side b add up to another square that has side a.

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Hence, we just need to show that when we add two identical squares, we can get another square.

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Since the little squares add up to the large square, let’s try to put them inside the large one.

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As seen above, little squares intersect in the middle and leave gaps on the corners. If we stick to our initial assertion, this intersection must have same area as the gaps. But there is something absurd here, because this intersection is a square. Also the gaps are identical squares that add up to the intersection.

If I call sides of the little squares d, and the big square c:

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This result is the same as our starting point. We just found ourselves in a loop which means that our initial assertion was wrong. √2 can’t be shows as a ratio of a/b. Hence, √2 is not a rational number.

One Wonders…

  1. Try to prove that √2 is an irrational number, using Euclid’s tools which are compass and an unmarked ruler.
  2. How can we understand if √3 is rational or not? (Hint: Try to prove geometrically like I did in the article.)

M. Serkan Kalaycıoğlu

Real Mathematics – Killer Numbers#6

Socrates’ Lesson

In the previous articles I have talked about Plato and his effect on science; particularly geometry. Thanks to his book named Meno, we know about one of the most influential philosophers of all times: Socrates.

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Meno was another book of Plato that was written as dialogues. In this book there were two main characters: Meno and Socrates.

In the beginning of the book Meno asks Socrates if virtue is teachable or not. Even though Meno is crucial for understanding Socrates’ philosophy, there is one part of the book that interests me the most.

Problem

The book gets interesting when Socrates starts asking “the boy” who was raised near Meno. At first, Socrates is asking the boy to describe shape of a square and its properties. After a series of questions Socrates asks his main problem: How can one double the area of a given square?

This is an ancient problem that is also known as “doubling the square”. The boy answers Socrates’ questions and eventually finds the area of a square with side length of 2 units. The boy also concludes that since this area has 4 units, double of such square should have 8 units. But when asked to find one side of such square, the boy gives the answer of 4 units. However after his answer the boy realizes that a square with sides of 4 units has 16 units of area, not 8.

Classical Greek Mathematics

After this point the boy follows Socrates’ descriptions in order to draw a square that has 8 units of area. At first Socrates commands the boy to draw a square that has sides 2:

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This square’s area is 4 units. Then Socrates tells him to draw three identical squares:

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Now Socrates tells the boy to unite these squares as follows:

tekkare

Socrates asks the boy to draw the diagonals in each square. They both know the fact that a diagonal divides a square into two equal areas:

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It is easy to see that the inner square has a total area of 8 units:

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One side of the inner square is the diagonal from small squares. In order to find that diagonal the boy uses Pythagorean Theorem:

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Conclusion

Even though he only uses a compass and an unmarked ruler, the boy found a length that is irrational thanks to Socrates’ instructions. Back in ancient Greece numbers were imagined as lengths/magnitudes. This is why as long as they constructed it neither Socrates nor the boy cared about irrationality of a length.

Pythagoras and his cult claimed that all numbers are rational and they tried to hide the facts that irrational numbers exist. But in the end philosophers like Socrates won the debate and helped mathematics to flourish into many branches.

M. Serkan Kalaycıoğlu

Real Mathematics – Killer Numbers #4

Lattice Points

Imagine a page from a squared notebook. Take all the edges of little squares and leave the vertices. If you assume that horizontally and vertically distance between two consecutive points is exactly 1 unit. I name this plane as the system of lattice points.

In the system of lattice points every point is represented by a number duo:

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Pick’s Theorem: An alternative method in order to find the area of a polygon.

In the system of lattice points one can construct as many polygons as wanted with joining points together. Pick’s theorem is handy for finding the areas of these polygons.

According to the theorem only two things are needed to find the area of a given polygon: Number of points polygon has on its edges (let’s call it e) and the number of points staying inside the polygon (let’s call it i). Pick’s theorem gives the following formula for the area of a polygon sitting on the system of lattice points:

Area = i + (e/2) – 1

Example 1: Triangle.

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Assume that there is a triangle as shown above. Number of points on its edges e is equal to 4 as the number of points inside the triangle i is equal to 0. Hence Pick’s theorem says the area of this triangle is:

Area = 0 + (4/2) – 1

          = 0 + 2 – 1

          = 1 unit.

We know from basic geometry that the area of a triangle is the half of height times base: (2*1)/2 = 1 unit.

Example 2: Square.

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In the picture we can see that e=12 and i=4. Thus the area is:

4 + (12/2) – 1 = 4 + 6 – 1 = 9 units.

We know that area of a square is the square of the lenght of its edge which makes 3*3=9 units.

Square and triangle are easy examples and perhaps you are thinking that Pick’s theorem is redundant. Then let’s continue drawing a more complex polygon and find its area.

Example 3: Polygon.

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Now Pick’s theorem shows its strength. Without the theorem we’d have to divide this polygon into various polygons and then calculate areas one by one. But with Pick’s theorem it is just counting points:

e=12 and i=72. Hence the area of the polygon can be found with:

Area = 72 + (12/2) – 1 = 72 + 6 – 1 = 77 units.

Equilateral Triangle

Up to this point it looks like we are dealing with geometry but the headlines said that it is an article about numbers. Let me change the course of the article with a question.

Q: Is it possible to draw an equilateral triangle on our points system while the corners of the triangle sits on the lattice points?

For instance let me draw the base of an equilateral triangle that has 2 units of length:

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Third corner (Q) sits between lattice points.

As seen above third corner of the triangle won’t be on a lattice point.

One wonders…

Can you prove that this is the case for each and every equilateral triangle on the system of lattice points?

To be continued…

M. Serkan Kalaycıoğlu