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:

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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 – Geometry #6

Mathematics was crucial for mankind before ancient Greeks came along. Humans needed mathematics to solve their everyday problems and that is why they were learning it. But ancient Greeks changed that as they developed mathematics for joy. This is one of the reasons why they didn’t limit themselves to the daily problems.

One of the problems ancient Greeks considered is today known as the Delos Problem, or Doubling the Cube. Even the brightest philosophers were helpless against this specific problem. Now I will tell you two common told stories about how Greeks started dealing with this problem.

Surviving the Plague

According to Theon of İzmir (a city in modern Turkey), this story was inside one of the books of Eratosthenes that were lost.

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Around 430 BC a devastating plague had arisen in ancient Athens. Leaders of the city were desperate against the plague and they had no idea how to save the people of Athens. During the plague God speaks to the people through an oracle: In order to stop the plague they had to build a new altar. But this altar should have twice the volume of the previous altar.

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Plague of Athens

It was seemingly an easy task for the engineers of the Athens. Although they were unable to build the altar as God wanted them to. According to Plato, Greeks were in illusion as they claimed to know everything about geometry. And with this task God was teaching them a lesson. Plato thought God didn’t want people to build the altar. He only wanted to show people how ignorant they are.

Grave of Glaucus

Second story is being told in one of Archimedes’ books. Apparently Eratosthenes wrote a letter to the King of Greece and mentioned this story.

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King Minos

Zeus and Europa had a son named Minos. King Minos is one of the leading characters in the Greek mythology. In the story it is being told that King Minos’ son Glaucus died at an early age. King wanted his engineers to build a massive grave for his late son. Eventually King thought the grave that was built was rubbish and wasn’t suitable for a royal. He ordered his engineers to double the volume of the cube-shaped grave. In order to do that Minos told the engineers to double the sizes of the grave.

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This caused a huge problem as new volume turned out to be eight times the old volume when the sides of the cube-shaped grave were doubled. Neither Minos nor his men were unable to solve this problem.

Three Impossible Problems

I have to remind you that these men had only a compass and an unmarked ruler when they were dealing with this problem. But little they knew was that doubling the cube was one of the three problems that can’t be solved with a compass and an unmarked ruler. (I’ll be talking about the other two in the upcoming articles.) Gauss was the first person who claimed this but he didn’t back his claims with a proof. The first proof came from Pierre Wantzel in 1837! It means at least 2250 years after the problem first came out.

Let’s try to solve the problem with modern mathematics notations:

Assume that we have a cube that has 1 unit sides. Its volume is 1*1*1=1 unit. Doubling the volume of a cube makes 2 units of volume. Then we must find the cube that has volume 2. If such cube has sides a, volume of that cube become a*a*a = a3.

Thus,

a3 = 2

a = 3√2.

We solved the unsolvable… or did we?

Obviously we managed the solve it. But ancient Greeks didn’t have our modern mathematics notations. Actually they didn’t even have numbers. They had to find 3√2 length with an unmarked ruler and a compass. Even with our marked rulers, it is impossible to find how long 3√2 is.

How Long?

In order to find how long 3√2 is, we can use a method called Neusis Drawing. But I will use the power of origami and show you how to find that irrational length.

First of all I took a square paper and using origami techniques to divide the square into three equal parts.

Then I folded the paper such that point A touches the left side of the square as point B touches the line that is in the height of point C.

I called the point A touched on the left side as D. Distance from D to F is 3√2 times the distance from D to E.

Here is how Peter Messer showed this origami technique:

One wonders…

A question that was keeping even the most brilliant minds busy for more than 2000 years can be solved in the matter of seconds using origami. How can this happen? What is the missing sides of compass and ruler?

M. Serkan Kalaycıoğlu

Real Mathematics – Geometry #5

pythagoras1

I was wondering; if there was a list of hall of fame for famous ancient Greeks Pythagoras would find himself in the top ten for sure. What is striking about his fame is that it comes directly from a geometry property. Although mathematicians know that so called Pythagorean Theorem was known to other cultures at least 1000 years before he “discovered” it.

Pythagorean Theorem: In a right-angled triangle sum of the squares of the perpendicular sides gives the square of the hypotenuse that is the longest side of the triangle.

It is being told that there are 367 different proofs for this theorem. Some of them are so similar, even mathematicians have trouble seeing the difference among these proofs.

Let’s check a few of the proofs.

Proof 1

Elisha Loomis talks about a proof for the Pythagorean Theorem in his book “The Pythagorean Proposition”. This proof is special because it came from a high school student named Maurice Laisnez.

I decided to use cutting papers for the explanation. First of all I cut a random right-angled triangle and then made 3 more copies of it.

I lined these four triangles up such that it gave me a square inside a square:

pissa2

Since sides of the inner square are c, it has area c2.

Now let’s line the triangle as follows:

pissa3

Marked areas 1 and 2 are squares and their area is equal to the area of the inner square from the previous alignment. Now let’s find the areas of 1 and 2: They make a2 and b2.

Their addition will make c2. Hence:

a2 + b2 = c2

Proof 2

For the second proof I decided to go to the ancient China.

Zhoubi Suanjing is believed to be written around 500 BC to 200 BC. In the Loomis’ book you can find this proof in the page 253.

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Pythagorean Theorem’s proof in the Suanjing.

Again I will cut four right-angled triangles for the explanation of the proof. But this time I will cut the triangles such that their perpendicular sides will have length 3 and 4 units. Chinese mathematicians tried to find the third side of the triangle as follows.

pisasaads

In order to start the proof I lined the triangles up like below and a tiny square formed in the middle:

pisasa

Tiny square A has sides that have 1 unit each. This is why area of A is 1 unit as well.

We know that the area of one triangle is (3*4)/2 = 6 units. There are four of such triangles and that gives us 6*4 = 24 units of area. When I add the area of A to this result, I can find area of the whole square as 25 units.

If area of a square is 25 units, its one side is square root of the area: √25 = 5 units.

From here we found length of the third side from the triangles:

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This proof shows us that 3-4-5 triangle and Pythagorean Theorem were both known in ancient China.

One wonders…

A farmer dad wants to retire. He would like to divide three of his lands to his two sons equally. But he wants to do that without dividing the lands from each other. What should he do?

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X-Y-Z are squares as DCG is a right triangle.

M. Serkan Kalaycıoğlu

Real Mathematics – Geometry #4

Geometry has a sacred book: Elements. Author of Elements, Euclid, used only these tools when he discovered his geometry:

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A compass and an unmarked ruler… He showed what can or can’t be done with them in geometry.

Dividing a Finite Straight Line into Two Equal Parts

We already know that one can draw a straight line segment between any two points. Let’s say we have a straight line between the points A and B. Euclid found an ingenious method to divide AB into two equal parts using his only two tools. Let’s assume that AB is 6 cm.

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This way we will know that Euclid’s method works only if we find two parts that are 3 cm long.

According to Euclid one should take point A and point B as the centers of two equal circles with radius AB:

Euclid says that one should define the intersection points as C and D:

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Then he suggests one to connect C to D:

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At this point Euclid talks about two outcomes:

  1. CD is perpendicular to AB.
  2. CD divides AB into two equal parts.

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As seen in the picture CD really divides AB into two equal parts. One can use this method and see that those two lines are perpendicular with a quadrant.

Dividing a Random Angle into Two Equal Parts

Obviously Euclid didn’t stop there and tried to figure out if it was possible to divide any given angle into two equal parts. Let’s consider straight lines AB and BC intersects and form a 90-degree angle:

Euclid says that one should take B (the intersection point) as center and draw a circle with random radius. This circle will intersect AB and BC at two points: D and E.

Now Euclid tells us to use our compass and draw two equal circles that have centers D and E:

As seen above, these circles intersect at two points. Let’s choose the point F and connect it with B:

Euclid claims that the straight line BF divides the angle into two equal parts:

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We can see with our quadrant angle is divided into two equal parts of 45 degrees.

One wonders…

Is it possible to use Euclid’s methods and divide any given straight line into three equal parts?

Try to answer the same thing for an angle.

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 – Geometry #2

Mathematics without numbers

Around 2700 years ago ancient Greeks were in total control of every part of science (Philosophy, geometry and mathematics in particular.). For centuries Greek hundreds of historically important figures like Thales, Pythagoras, Eudoxus and Euclid dominated mathematics.

Ancient Greek mathematicians had a significant difference. Unlike their colleagues from other parts of the world, they choose not to use number symbols. According to them, geometry was the foundation of mathematics, and like everything in mathematics numbers arose from geometry as well.

Even though they created respectable number systems and symbols, comparing to their advanced knowledge in other branches of mathematics (particularly geometry) they were behind with numbers. It was like as if they didn’t care about number systems and symbols as much as they cared geometry and other parts of science.

It is mesmerizing to hear that founders of geometry didn’t need numbers in their works.

Ruler, Compass and Unit

In ancient Greece, philosophers (meaning scientists) used magnitude instead of numbers. They were drawing straight line segments to show a magnitude. In other words, ancient Greek mathematicians were drawing lines instead of writing number symbols. Moreover, they used unmarked ruler and compass as their only tools. (I’ll be explaining the use of them in the upcoming articles.)

Q: How did Greeks manage to make mathematics without numbers?

Assume that we have positive integers a and b.

Addition

Their addition makes a+b. Using straight line segments we can show a+b as follows:

a+b

Extraction

If a is greater than b, extraction can be written as a-b. This can be shown with line segments like the following:

a-b

Multiplication

Multiplication of them gives a.b. We can use properties of triangles in order to explain multiplication with lines. Assume that we have a triangle with side a and 1:

Now we will extend the sides of this triangle so that, the side a will become a.b while the side 1 becomes b.

Division

Let’s say that we want to find a/b with lines. This time we can use a similar approach that we used in multiplication. First we construct a triangle with sides a and b:

Then we shorten the sides so that length of the side a will become a/b while length of the side b becomes 1.

Taking the Square Root

To take the square root of the number a, first we should draw a straight line segment that has length a+1. Then we mark the segment such that left side of the mark will have length a, and right will have length 1. Finally draw a semicircle that has diameter a+1. Now draw a perpendicular from the circle’s boundary to the marked point. That perpendicular line will have length √a.

One wonders…

If ancient Greeks knew how to make calculations, does it mean that they were involved with algebra and number theory too? (Check out the name Diophantus.)

M. Serkan Kalaycıoğlu

Real Mathematics – Geometry #12

Cogito Ergo Sum*

*I think, therefore I am. You probably heard these words before. They belong to Rene Descartes. Descartes is believed to be the person who invented modern philosophy. Actually he was a groundbreaker for mathematics.

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Rene Descartes (1596 – 1650)

One of his best discoveries for mathematics has a mesmerizing story. Even though it is unknown whether this story is true or not, I find it amusing.

It is a known fact that Descartes had a rough childhood. He was constantly battling with various illnesses which were major setbacks for his early school days. Every day he was able to attend to his school around noon. This has become a habit for Descartes. For the rest of his life he spent his mornings in bed. (Except final days of his life in Sweden)

Story of Descartes’ discovery is based on this fact. Allegedly one morning as he was lying down, he saw a fly on his ceiling. He started thinking about fly’s position. There was one question in his mind: “How can I describe this fly’s position to someone who hasn’t been with me in this room?”

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Descartes began his answer with assumption. If we assign a corner of the ceiling as the starting point, it is possible to reach the fly with only two directions for movement from that point: To the width and to the length.

This is the story of how Cartesian coordinate system was first discovered. Analytic geometry was born with Descartes’ publication on the issue.

Analytic Geometry = Cartesian Geometry = Coordinate Geometry

Cartesian Coordinate System

Let’s say that left bottom corner of Descartes’ ceiling is the starting point. There are two available paths for us: Up or right.

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According to Descartes one can reach any point of the ceiling with two specific movements from the starting point: X amount to the right, Y amount to the up.

In that case the point (or the object) would have a position which could be expressed with two different numbers. If X represents amount of movement to the right, and Y to the left, then position can be shown as (X, Y).

Example: Assume that unit of movement is cm, and fly sits on the ceiling as shown:

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If we sit at the starting point and make 4 cm to the right & 3 cm to the left, then it is safe to call the position of the fly (4,3).

Catching the Thief

Materials:

  • Cartesian coordinate system that has size 6×6.
  • Dice.
  • Pencil/pen and paper.
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Coordinate system needed for the game.

Catching the thief is a multiplayer game. One player gets to be the detective as the other one gets to be the thief. Detective’s mission is the catch the thief as soon as possible.

Thief rolls the dice twice and determines its position: (First dice, Second dice).

Detective makes its first guess. If he/she is right, then the game is done.

If detective is wrong in his/her first guess, he/she receives a text message from the thief. Message reads a number. This number represents the total of the differences between these positions.

Let’s assume that the thief is at the position (2,2) and detective makes his/her guess as (4,1). Differences of the positions are: 4-2=2 and 2-1=1. Add them together and we’ll get 2+1=3. Hence thief sends the number 3 in his/her message.

Detective makes his second guess according to this information and this process continues until thief is caught.

Example

Thief rolls the dice. First try reads 3, second reads 4. Hence thief’s position is determined: (3,4).

Detective makes his/her first guess with (2,2) and misses.

Thief calculates the differences of the positions: 3-2=1 and 4-2=2. Thief adds them together: 1+2=3.

Detective receives the message: “3”. Now detective is in a slightly better situation.

Possibility 1: Detective adds 3 to the x. Guess: (5,2).

Possibility 2: Detective adds 2 to the x, and 1 to the y. Guess: (4,3).

Possibility 3: Detective adds 1 to the x, and 2 to the y. Guess: (3,4).

Possibility 4: Detective adds 3 to the y. Guess: (2,5).

Possibility 5: Detective subtracts 2 from the x, and 1 from the y. Guess: (0,1).

Possibility 6: Detective subtracts 1 from the x, and 2 from the y. Guess: (1,0).

Possibility 7: Detective subtracts 2 from the x, and adds 1 to the y. Guess: (0,3).

Possibility 8: Detective subtracts 1 from the x, and adds 2 to the y. Guess: (1,4).

Possibility 9: Detective adds 2 to the x, and subtracts 1 from the y. Guess: (4,1).

Possibility 10: Detective adds 1 to the x, and subtracts 2 from the y. Guess: (3,0).

After the message from the thief, detective knows for sure that the thief is at one of these 10 positions. Detective continues to make his guesses and uses the same strategy until he/she catches the thief.

One wonders…

  1. Determine how many possibilities can the detective have in case he/she guesses (3,0)?
  2. I am thief and you are the detective. Your first guess is (3,3) and I send you the text “2”. Where am I? Leave your answer to the comments.

M. Serkan Kalaycıoğlu

Real Mathematics – Geometry #10&11

Q: How many regular polygons are there?

In the previous articles I have shown how to construct an equilateral triangle using Euclid’s method which means using a compass and an unmarked ruler. An equilateral triangle is a regular polygon as it consists of equal sides and angles. We can even conclude that an equilateral triangle is the smallest regular polygon.

Then let me continue with increasing the number of sides to construct more polygons. A regular polygon with four equal sides and angles… Hmmm… A square!

A regular polygon with five sides: Pentagon.

A regular polygon with six sides: Hexagon.

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A regular polygon with fifty sides: Pentacontagon.

There are no limits for the number of sides of regular polygons we can create. Although after certain number of sides it is almost impossible to distinguish a regular polygon from a circle.

Pentacontagon (left) and its comparison with a regular polygon that has 200 sides.

Regular polygons exist in two dimensional worlds. What if we try to construct regular objects in three dimensions?

Q: How many regular polyhedrons are there?

There are two specific properties for regular polyhedrons: Each of their faces is the same regular polygons and there are same numbers of regular polygons meeting at each corner.

Let me give you the answer right away: There are five different regular polyhedrons. Exactly five!

The very first time I heard about it, I thought it was bizarre to have only five different regular polyhedrons while there are infinite number of regular polygons. How can coming to three dimensions from two dimensions changes so much?

School of Broad Shouldered

Around 427 BC a baby named Aristocles was born in Athens. When he grew up, Aristocles had wide shoulders which resulted with him adopting the nickname “Plato” that meant “broad” in Greek. (This story is from C.J. Rowe’s Plato.)

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Plato (427 BC. – 347 BC.)

Majority of the people have no idea that Plato was an integral individual for mathematics’ development. His persistence for clearer explanations for proof and hypothesizes evolved mathematics completely. Although his biggest contribution, not just to mathematics but to all branches of science, was the school he founded.

In 387 BC he attempted to build a school in Athens, on the land of a guy named Academas. He gave the school his name: Academy. For over 900 years Plato’s Academy was the home of countless philosophers. (Just for comparison to its worth: University of Bologna was built 1400 years after Academy was founded.)

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Above Academy’s door: “Let no man ignorant of geometry enter here.”

Platonic Objects

Polyhedrons are also called Platonic objects because he was the first to explain that there are exactly five of them.

  1. Tetrahedron: An object formed by four equilateral triangles.
  2. Cube: An object formed by six squares. I guess I didn’t need to explain it.
  3. Octahedron: An object formed by eight equilateral triangles.
  4. Dodecahedron: An object formed by twelve pentagons.
  5. Icosahedron: An object formed by twenty equilateral triangles.

platonic_solids

Why Five?

When examined carefully one might see the properties that are exclusive to the regular polyhedrons. These properties both explain and prove why there are exactly five of them.

I’ve already mentioned that regular polyhedrons are three dimensional objects. If you study a regular polyhedron, you’d see that at least three regular polygons meet at a point (corner). Let’s try it with the smallest regular polyhedron.

Take any corner on a tetrahedron. You’ll see that three equilateral triangles meet at that corner. When it is reduced to two dimensions, it would look like as follows:

IMG_6348

The corner has 360 degree around itself. One equilateral triangle has 60 degrees of internal angle on this specific point which means there are

360 – (60+60+60) = 180

degrees of empty space around the corner. By means of this empty space it is possible to construct a three dimensional object. If there were no space, then this shape would not have the flexibility and thus it would not be turned into a three dimensional object.

We are getting close. Let’s use induction and go from tetrahedron to general.

In order to construct a polyhedron we must follow these rules:

  • Draw a point on a paper.
  • Draw three regular polygons which meet at that point.
  • Add the internal angles of the regular polygons. If it doesn’t exceed 360 degrees, then it is possible to construct a regular polyhedron with these regular polygons.
  • Also one can continue adding same regular polygons as long as summation never exceeds 360 degrees.
  • If the summation of the internal angles of the regular polygons is 360 degrees or more, then this shape can’t be converted into three dimensions. This means a regular polyhedron can’t be constructed with such regular polygons.

Example 1: Tetrahedron.

Take a point and draw three equilateral triangles around it. One internal degree of equilateral triangle is 60, which make 60*3=180 degrees around the point. It is less than 360, that is why we know it is possible to construct a Platonic solid with them: Tetrahedron.

Example 2: Tetrahedron + 1 more equilateral triangle.

If we add one more equilateral triangle, internal angles around the point will make 240 degrees, which is still less than 360. Hence it is possible to construct a Platonic solid with four equilateral triangle: Octahedron.

Example 3: Tetrahedron + 2 more equilateral triangles.

We continue to add more equilateral triangles. Now we have five of them around a point, which gives 300 degrees. We are still under 360 degrees, thus we can construct a Platonic solid with them. This is our third Platonic solid and we constructed them with equilateral triangles: Icosahedron.

Example 4: Tetrahedron + 3 more equilateral triangles.

We add the sixth triangle. There is 60*6=360 degrees around the point. It is impossible to construct a Platonic solid with these triangles. Actually this shape stays only in two dimensions.

Example 5: Cube and adding a square to it.

We are done with equilateral triangles, hence we move to the next regular polygon: Square. Internal degree of a square is 90, which gives 90*3=270 degrees around the point. It is less than 360 degrees, and that is why it is possible to construct Platonic solid with three squares. This is the fourth Platonic solid: Cube.

If we add one more square, then internal angles around the point will add up to 90*4=360 degrees. It is impossible to construct a Platonic solid in this case.

Example 6: Dodecahedron and adding one more pentagon.

Since we are done with squares, we move to the next regular polygon: Pentagon. A pentagon has internal degrees 108 at its corners. Three pentagons around a point will give 108*3=324 degrees which is less than 360 degrees. So it is possible to construct a Platonic solid with three pentagons meeting at a point. This is our fifth Platonic solid.

Although if we a forth regular pentagon, internal degrees will be 108*4=432 degrees. It is more than 360, which makes the pentagons overlap. Hence it is impossible to construct a Platonic solid with these.

One wonders…

Continue to the next polygon: Hexagon. Examine why it is impossible to construct a Platonic solid with them.

M. Serkan Kalaycıoğlu

Real Mathematics – Strange Worlds #2

What shape is it?

I’ve been waiting for this day: I am finally abducted by some space aliens.

I am getting back to my senses now. Oh, they speak Turkish. That’s odd. And their favorite drink is black tea from Rize. It helped me a lot when they offered me some tea. I am relaxed now and their leader wants to have a word with me:

-Earthling! We kidnapped you so that we can conduct some experiments on you and learn about human body. But we are fair, so you’ll have a chance to prove your intelligence first. If you solve this problem, we’ll take you back to your home. Here is the problem: We’ll blindfold and leave you to a random asteroid. Your quest is to find what kind of shape that asteroid has.

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Christopher Columbus-Like

I am all alone on a strange asteroid now. Space ship is watching over me. Think Serkan, thiiiink… Oh, I think I got it! I will move in same direction just like Columbus did. If I get back to where I started, it means this asteroid is curved, like a ball.

I asked for a spray from the aliens so that I can leave a trail mark and understand when and if I get back to my starting point. I walk and walk… Finally I reach my starting point. I got my ticket back home: This asteroid is curved like a sphere, probably like Earth.

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When I am talking about a sphere, it doesn’t have to be a perfect one. It could be looking like this as well.

I am getting ready to talk with the leader of the aliens. But suddenly an idea pops in my head: What if the shape has a hole in it, like a bagel or a donut?!

Because, if I walked on the right direction, I could very well reach my starting point even though I am on a bagel/donut shaped asteroid.

What am I supposed to do now?

Am I on a ball? Or am I on a bagel?

A New Kind of Mathematics For You

This problem is actually one of the most classical problems of topology. Okay but what is topology?

Until now I’ve told you about two important discoveries of Euler: Euler’s solution to the Königsberg bridge problem and Euler’s polyhedron formula.

Euler’s Königsberg solution directly affected (more than 150 years after his solution) the birth of a new mathematics branch named graph theory. His polyhedron formula also helped mathematicians to define what topology is. I must mention that graph theory is just a sub-branch of topology.

Topology:

  • Two Greek words: Topos (space) + Lopos (science).
  • Also known as rubber sheet geometry.

Euclidean Geometry vs. Topology

  • In Euclidean geometry; it is allowed to move things around and flip them over. But you can’t stretch or bend objects without changing their properties.
  • In topology; you can bend, twist and even stretch objects without affecting their properties. But you can’t cut, add or punch a hole in topology.
  • In Euclidean geometry; angles and lengths are important.
  • In topology; they don’t matter.
  • This is why in Euclidean geometry a square and a triangle has different properties even though they are the same thing in topology.

The Most Popular Example

You might find it strange that a mathematics branch does not care about lengths and angles. Actually topology is involved in our daily lives. More than you could ever imagine.

In transportation, especially in metro and tramway systems, maps are great examples for topology. Distances and directions are distorted in the interests of simplicity which is a use of topology. Also it is neglected how big the stops are. They are all represented with dots and they are connected with lines. In short, metro maps are actually graphs.

In the following picture it is clear to see that all stops (which are shown with dots) are divided with same distance even though they are not equal in reality.

M2_Hattı
This picture shows some of the stops of Istanbul metro.

Hint: What shape is it?

You probably noticed it: That problem is actually about the topological property of the asteroid. Using something elastic (like Play-Doh) would help you to solve it.

M. Serkan Kalaycıoğlu