Real Mathematics: Killer Numbers #3

Theodorus of Cyrene

City of Cyrene was one of the ancient Greek cities that were located in the Northern Africa at around 5th century BC. We are certain that Theodorus (465 BC – 398 BC) was born in Cyrene and he was a tutor of Plato. Theodorus probably met Socrates and lived in Athens for some time. Unfortunately we don’t know much about his life.

Although we know through his student Plato that Theodorus had made significant works on irrational numbers. During ancient times, even though they had number symbols called Attic (also  known as Herodianic symbols) Greek philosophers did not care about number symbols. According to their thinking numbers were just magnitudes of lines. Hence they used line segments to represent numbers.

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Plato

Pythagoreans thought that every number can be shown as a ratio of any other two rational numbers, and that is why they claimed that numbers are rational. Oddly enough a geometric property what we call Pythagoras Theorem contradicted Pythagoreans’ claims as an isosceles right triangle with 1 unit length sides would have a hypotenuse that is √2 unit length. Unfortunately for them √2 is not a rational number. In other words, it can’t be shown as a ratio of two rational numbers.

Theodorus was also a Pythagorean, but he proved that √2, √3, √5, … are all irrational. For some reason he stopped at √17.

In this article I’ll be talking about a work of Theodorus now called Spiral of Theodorus.

Construction of the Spiral of Theodorus

This lovely geometric shape, which is also called “Einstein Spiral”, “π Spiral” and “Square Root Spiral”, could be constructed by anyone who knows Pythagoras Theorem.

At first construct an isosceles right triangle with 1 unit length sides. Through Pythagoras Theorem we can calculate the length of the hypotenuse as √2.

IMG_5384

In the next step, continue drawing a base with length 1 unit that is perpendicular to the hypotenuse of the previous triangle and construct a second triangle which would have hypotenuse with length √3.

As long as you continue the same process, you will be ending up with hypotenuses with lengths √4, √5, √6, √7 … Theodorus stopped at √17. Probably he ended his structure at 17 because that is the final triangle before overlapping starts.

IMG_5379

It is obvious to the naked eyes that these triangles form a beautiful spiral, which is called Spiral of Theodorus.

Nothing like a spiral

What is so special about the Spiral of Theodorus?

  • All the lengths of the hypotenuses of the spiral, except the perfect square number lengths, are irrational.

    IMG_5390
    √1, √4, √9, √16, √25… are the only numbers that are rational.
  • If one continues to add triangles which would mean that the spiral is going to the infinity, no two hypotenuses overlap.

    IMG_5380
    Even if it looks too close, no two hypotenuses overlap in the Spiral of Theodorus.
  • The windings of the spiral would have length π between themselves as one adds infinitely many triangles.

    IMG_5383
    I have found 3,1 in the first 30 triangles. If I kept going for infinity, this number would approach to π.
  • The angle between two consecutive perfect square number hypotenuses would approach to 360/π as the spiral goes to infinity.

    360/π = 114,591559026… which is very close to what I draw. I was certainly lucky but if you draw this perfectly, as you approach to infinity, you would find exactly. 360/π.

Killer Curve

This is my favorite property of the Spiral of Theodorus: Cut off every single triangle from the spiral and align them on the x-y coordinate system.

If you connect the tipping points of every triangle, you would end up with the y=√x curve. I like calling irrational numbers as “killer numbers” because of the story of poor Hippasus. I think it would be suitable to call this curve The Killer Curve.

I also used a program called GeoGebra and find the following result.

teodorusgeo

One might wonder…

What would happen if you take the first triangle like the following?

IMG_5394

What kinds of changes do you observe?

M. Serkan Kalaycıoğlu

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