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 – I

What kind of relationship can soap bubble and mathematics have?

Obviously Michael Dorff, a professor of mathematics from Brigham Young University, is interested in playing with soap bubbles. Click here for Dr. Dorff’s video.

We, mathematicians, love talking about how mathematics is inside of everything around us. There are some of us who discover the mathematics in a thing even though it seems to have nothing to do with mathematics. Dr. Dorff is one of those special scientists who found mathematics when he dunked a cube into a bucket full of soapy water twice.

When Dr. Dorff dunked his cube twice, he realized that there was a soap bubble inside the cube which is in the shape of a cube. In other words, dunking made it possible for us to see a hypercube.

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A hypercube

What is a hypercube?

Unfortunately human mind is unable to understand more than three dimensions. Although you might have heard that higher dimensions exist. We can show that through physics and mathematics. Hypercube is the name of the cube shape we know in the fourth dimension. Hence what Dr. Dorff discovered was that when he dunked a cube inside a soapy water twice, you get to see a fourth dimension cube.

There is another beautiful thing hidden inside Dr. Dorff’s experiment. Let’s go back in time before I tell you about it.

2400 year ago

Greek names dominate when we talk about ancient times and science. Majority of people heard at least one or two Greek names from this magnificent era. There are so many ancient Greek philosophers who influenced world history even though there have been more than 2 millennia since their works were written.

 There is an ancient Greek mathematician named Euclid who influenced many scientists for more than 2000 years. His work Elements is known as the second most printed book in the history, right after Bible. The Elements which was consisted of 13 books is considered among one of the most striking books of all times. Until 19th century, Elements were the only source of geometry for scientists such as Newton, Leibniz, Euler etc.

A comparison: Elements were written almost 2400 years ago while the number zero (0) was explained mathematically only around 1350 years ago.

What is inside Elements?

For now I will be talking about the first book of Elements. Book I starts with 23 definitions.

Definition #1: A point is that of which there is no part.

Euclid started writing down what geometry is from scratch. He started with the definitions of point, line, straight-line, surface… and so on. Then he gave 5 axioms/postulates.

Postulate #1: A straight-line can be drawn from any point to any point.

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Let us consider we have two specific points on a plane. We can draw infinite number of lines between them even though we can draw only one straight-line. A straight-line is also known as the shortest path between any two points.

Soap Bubble

Euclid says that the shortest path between two points is a straight-line. But what if we are looking for the shortest path between four points?

Dr. Dorff’s soapy water experiment gives us the answer. I’ve told you that when you dunk a cube inside a bucket full of soapy water, you get a soap bubble inside the cube which gives hypercube. If you dunk the cube only once, you will get a bubble which is in a shape that is a combination of letters H and X. This bizarre bubble shows the shortest path between four points.

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Shortest path between four points.

Changing your perspective

Especially in the 90s, 3-D paintings were very popular. These paintings were meaningless until you see what is really hidden inside them. And when you see it, it is impossible not to see it again.

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Mathematicians are like those who look and see beyond 3-D paintings. When they see where mathematics is hidden around them, they can’t help but look for new examples. This is one of the most fundamental things needed in order to understand mathematics: Changing your perspective.

M. Serkan Kalaycıoğlu

 

Real Mathematics: Geometry – #7

Ancient Greek philosophers used only compass and an unmarked ruler to come up with incredible results in geometry. In this article, I’ll be talking about what kinds of methods were used in geometry’s holy book Euclid’s Elements in order to cut a segment into as many equal parts as we’d like.

So far, I’ve showed in my previous articles how to bisect any segment and any angle with just using a compass and an unmarked ruler.

How about trisecting a given segment?

Trisecting a segment is shown in Euclid’s Elements, Book 6, Proposition 9. Let me try to show you Euclid’s method in a nutshell.

Define any two points A and B on a plane and construct a line segment between them. Choose a point C which is not on the line segment AB.

Draw a line from the point A passing through C.

Use compass to mark the points D, E and F on the line AC such that AD, DE and EF are equal line segments.

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Connect F and B. Then draw line segments from points D and E which should be parallel to BF.

Thus, line segment AB is cut into three equal parts. Actually, using Euclid’s method, we can divide AB into as many equal parts as we’d like.

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It might seem easy at first sight but when you think about it, there are a few information we need in order to use Euclid’s method for trisecting a line segment. If you are careful enough, you might have realized that one should know how to draw a parallel to any given line. In order to do that, we should know the method from Elements Book I, Proposition 31, which requires how to create a specific angle on a random point (Elements Book I, Proposition 23). And it doesn’t end here: Proposition 23 requires knowledge of proposition 22 from the same book.

Suddenly Euclid’s ingenious methods seem a bit complicated.

Muslim Ingenious

Elements were written more than 2400 years ago and humanity should thank a few Muslim scholars for its existence today!

Al-Nayrizi (865-922) was one of the first Muslim philosophers who read and commented on Elements. Actually you could still buy his commentary on Elements even today on amazon.com even though it costs a little over 200 USD.

Al-Nayrizi has come up with a truly magnificent method for cutting a segment into equal parts.

Let’s start with drawing a straight line AB. Al-Nayrizi’s method requires only the following knowledge: Drawing lines through A and B respectively which are perpendicular to the line AB.

If we’d like to cut AB into n equal parts, we should mark n-1 equal segments on these perpendicular lines. We could easily do that using a compass. Let’s assume that we’d like to cut AB into four equal parts. That means we need 4-1=3 equal segments on the perpendicular lines.

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Now all we have to do is to connect the dots as shown below.

In the end we managed to cut AB into four equal parts.

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M. Serkan Kalaycıoğlu