A Mathematician Inside Seven Sages

Thales

Birth: Around 624 BC, Miletus

Death: Around 547 BC, Miletus

Thales Theorem: The diameter of a circle always subtends a right angle to any point on the circle.

In high school math, Thales is known with this theorem even though he is one of the most important names in the history of science. The funny thing is that he most likely didn’t even find this theorem. Then, who is this Thales guy we ought to know?

Thales is known as one of the Seven Sages of Greece as well as the first known natural philosopher.

Seven Sages of Greece are the seven important figures of 7th and 6th century BC Greece including Thales, Pittacus, Bias, Solon, Cleobulus, Myson and Chilan.

Natural philosopher title comes from Thales’ diversity as it is said that he worked in mathematics (especially geometry), engineering, astronomy, and philosophy. There is not even a single written work of his left today. Everything we know about him was written by others centuries after his death. This had led many legends after his name.

Historians believe that Thales visited Egypt where he learned mathematics and engineering. He is known as the first person who introduced geometry to Greeks. It is said that while he was in Egypt, he calculated the length of pyramids by just looking at their shadow.

Whenever the sun makes 45 degrees with Earth, the shadow of a particular object becomes equal to its own length. This is known as one of the methods Thales used in his calculation.

According to another legend, he guessed the time of the solar eclipse in 585 BC. During his time, people were able to guess the lunar (moon’s) eclipse. Although, it was impossible to guess where and when the solar eclipse was gonna occur using 6th BC’s knowledge and technology. Today we believe that even if this happened, it was just an astonishing guess for Thales.

Still, some believe that Thales really guessed the solar eclipse thanks to his brilliance. After all, he was one of the Seven Sages. Actually, as Socrates stated, he was the only natural philosopher inside that prestigious group.

Thales believed that everything comes from water. According to him, the Earth was shaped like a disk and it floats on an infinite ocean. He suggested that the earthquakes were the result of the Earth’s movement on the ocean. This was a first in the history of science as Thales’ ideas were based on logic instead of supernatural phenomena.  

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.

img_4552

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.

kc3b6k2

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.

kc3b6k21

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

kc3b6k22

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

kc3b6k23

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

kc3b6k24

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

karekc3b6k1

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

karekc3b6k2

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:

kc3b6k25

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.

philosophydiscourse-cropped-425x259

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:

1kare

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

4kare

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:

kosekare

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

kosekare2

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

karepis

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 – 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.

eratosthenes-1-259x300

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.

plague-of-athens
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.

minos
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.

kc3bcp2-e1537876946556

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:

pasdpfsdf

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 #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.

çizgi

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.

açı

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.
    çem
  2. Take AB as radius and draw a circle that has center A.
    çem1
  3. Now take AB as radius and draw another circle that has center B this time.
    çem2
  4. These circles will intersect at two points. Call one of them C.
    çem3
  5. Connect A to C. One can easily see that AB and AC are radii; hence they are equal in length.
    çem4
  6. Then connect B to C. One can observe that BC and BA are radii; hence they are equal in length.
    çem5
  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.
    çem6
  8. These three straight lines construct an equilateral triangle.
    çem8

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 #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.

regular-polygons-6-638

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.)

plato
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.)

geometri-bilmeyen-giremez
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 – Geometry #9

Finding Pi

Humans discovered a connection between a circle’s circumference and its diameter around 4000 years ago. This connection, what we call the number pi now, affected everyone who was involved in mathematics and/or engineering in the early times. Countless people, including historical figures, worked in order to find what this number is, but all of them failed to find such answer.

Some of them tried to find an approximation and got really close. When it was their time to shine for geometry, one ancient Greek whom we are familiar with had taken approximation methods to a different level. This famous Greek dude was Archimedes.

Archimedes is believed to live between 287 BCE and 211 BCE. If you take a look at the history of mathematics it was common to see approximations for the number pi, especially in Babylon, Sumer, ancient Egypt and China civilizations.

Although, Archimedes was different in an important way: Before him, method to find the number pi was not that complicated. A certain circle would be inspected and compared to a similar regular polygon. Their areas or perimeters were thought to be same which would give an approximation for pi.

Let’s try to find an approximation for the number pi using this method.

Draw a circle and divide it into 12 equal parts, like a pizza.

Cut the 12th slice into two equal parts and lay down all the slices side to side.

Slices will seem like a rectangle which is a regular polygon. At this point circle and rectangle are believed to have same shapes.

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Circumference of circle is found with pi*R, where R is the diameter of the circle. It is obvious to the naked eye that the long sides of the rectangle would give the circumference of the circle. Hence, we can find an approximation for the number pi.

Try this for another circle that has different diameter length.

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And take the arithmetic mean of both results. There is the number pi… Well, it is just an approximation.

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Archimedes’ Method

Even though there were so many engineering marvels as well as important geometry knowledge, Archimedes was not happy with the methods used to find the number pi. He developed his own method that required inscribing and circumscribing the unit circle with regular polygons. Archimedes did something different than the methods which were used before him: He continued using polygons with more sides and compared those results until he found the best polygon.

His method today is known as “the method of exhaustion”. It is the earliest known version of calculus.

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You could draw a unit circle yourself. Try to inscribe and circumscribe a square. You’d get pi between 2,8 and 4, which is not bad for the first try. If you continue with polygons that have more sides, you’d get even better intervals.

The thing is, Archimedes didn’t try to find an exact value. Instead he found an interval which was 99,9% accurate!

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Archimedes found this interval with starting a hexagon, and he finished his method at a 96-sided polygon. You could also see that 22/7, which is given to students for pi in school, is the upper limit that Archimedes found nearly 2200 years ago!

M. Serkan Kalaycıoğlu