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.

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.

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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 #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