Fibonacci numbers and continued fractions

Fibonacci is probably the most well known sequence in mathematics. Today, we will see how Fibonacci numbers can be used to construct beautiful patterns called continued fractions.

Divide a line segment by compass

Using compass and straightedge, we can construct the midpoint of a line segment, and we can easily divide a line segment into, say three equal parts. The question is, is it possible to do these constructions with just the compass.

The answer is "yes"! Indeed, it is possible to construct the midpoint of a line segment, and it is possible divide a line segment into a number of equal parts by using the compass alone. How amazing is that?! Today, we will look at these constructions!

Construction by compass alone

Normally in geometric construction problems, we use straightedge and compass. Today, we will look at an unusual type of construction problems where we are only allowed to use compass.

Star of David theorem

Today we will learn about a very beautiful theorem in geometry -- the Star of David theorem. This theorem is a consequence of the Pascal hexagon theorem and the Pappus theorem.

Chessboard and pyramid

Have you heard of the legend of the chessboard? The story goes like this. There was a wise man who invented the chess game and introduced it to a king. The king loved the game so much that he told the wise man that he could choose anything for a reward. The wise man then pointed to the chessboard and asked for 1 grain of wheat for the first chess square, 2 grains of wheat for the second square, 4 grains of wheat for the third square, 8 grains of wheat for the fourth square, and repeat this doubling pattern. This sounds like a very little reward but at the end the king didn't have enough wheat to reward the wise man.

Today, we will calculate to see how much wheat the wise man actually requested. Is it a lot? Is it little? I'll give you a clue, it's HUGE!

To see how big this reward is, we will calculate the number of pyramids that we could build up using all this wheat.

Sum of reciprocal squares

Today we will look at a very fascinating proof due to Euler for the following identity: $$\frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \frac{1}{5^2} + \dots = \frac{\pi^2}{6}$$
The mathematician Euler formulated this proof in 1734 when he was 28 year old.

Taylor series

To celebrate the $\pi$ day, in our previous post, we were introduced to a very beautiful identity due to Euler
$$\frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \frac{1}{5^2} + \dots = \frac{\pi^2}{6}$$

The mathematician Euler had an intriguing method to derive this identity. Euler's method employed the Taylor series, so today we will learn about Taylor series, and in the next post, we will look at Euler's technique.

When I look in your eyes

When people say to each other something like "there is a universe hidden in your eyes" or "I'm lost in your eyes, my spirits rise to the skies", they are not exaggerating. There is actually a mathematical truth in these sayings!

Today, we will prove the following mathematical fact:
there exists a polygon in the human eye whose perimeter is equal to the circumference of the earth
In a circle of diameter 1 cm (about the size of the iris of the human eye), there exists a polygon whose perimeter is equal to the circumference of the earth (about 40,000 km)!