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 – 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:

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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:

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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 – Geometry #3

“There is no royal road to geometry.”

From Euclid to the king who asked Euclid if there is an easier way to learn geometry.

Up until now I have mentioned Euclid and his book Elements a few times. This masterpiece is actually a collection of 13 books and was considered as the source of only known geometry for thousands of years. Historical figures including Newton, Leibniz, Omar Khayyam and many others learned mathematics through Euclid’s Elements.

First book of Elements starts with 23 seemingly obvious and simple definitions. I will mention some of them below.

Elements Book I

Definition 1: A point is that of which has no parts. (Zero dimensions)

Definition 2: A line is length without breadth. (One dimension)

Definition 3: The extremities of a line are points.

Definition 4: A straight line is any one which lies evenly with points itself.

Definition 8: A plane angle is the inclination of the lines to one another when two lines in a plane meet one another and are not lying in a straight-line.

Definition 15: A circle is a plane figure contained by a single line such that all of the straight-lines radiating towards from one point amongst those lying inside the figure are equal to one another.

After reading these definitions for the first time, a few question marks popped up in my head.

For instance the first definition suggests that a point has no dimensions. If that’s so, how can one show a point lying on a plane?

Is it even possible to show something that has no dimensions?!

Which of these two can suggest a point to us? Obviously their sizes don’t matter and neither of them is an illustration of an actual point.

In this context, second definition is not different from the first one: One can’t draw something that has no breadth.

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Eighth definition is about angles. In order to draw an illustration for a random angle one must know how to draw lines, straight lines and dots.

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I’ve just showed you that even basic geometrical shapes are impossible to demonstrate. We can only imagine them in our minds. This means that in a way architects are selling illusions.

It is being told that mathematics has abstract and tangible parts. Whenever a student is dealing with abstract mathematics, teacher ought to give tangible examples so that student can comprehend with the subject easily. Nevertheless, we are helpless even when we want to give a full tangible explanation to a simple thing like a straight line.

Magic inside the Elements

In the first proposition of the first book of Elements given a random straight line, Euclid is showing us how to draw an equilateral triangle from that line.

Just to remind you, Euclid only used an unmarked ruler and a compass in his methods. Stop here and try to think of a way to construct an equilateral triangle from a random straight line.

Euclid’s Method

  1. Assume that we have a finite straight line AB.
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  2. Take AB as radius and draw a circle that has center A.
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  3. Now take AB as radius and draw another circle that has center B this time.
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  4. These circles will intersect at two points. Call one of them C.
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  5. Connect A to C. One can easily see that AB and AC are radii; hence they are equal in length.
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  6. Then connect B to C. One can observe that BC and BA are radii; hence they are equal in length.
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  7. AB and AC, BA and BC are equal. Since AB and BA are the same straight line one can conclude that AB=AC=BC.
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  8. These three straight lines construct an equilateral triangle.
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One wonders…

These methods are taken from a book that was written around 2300-2400 years ago. What I find fascinating about mathematics is that we are not even capable of showing what a dot is, but we can also explore other planets using the power of the language of mathematics.

Now use Euclid’s materials (an unmarked ruler and a compass) and try to draw the twin of a given random straight line. Hint: Analyze the second proposition of the book I of Elements.

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