### Triangle Shortest Cut

/Posted by: MathArt
Let’s say you want to cut an equilateral triangle into two pieces such that both of them have the same size. The goal is to minimize the length of the cut (which doesn’t have...

### Real numbers are uncountable

/Posted by: MathArt
We already showed that the rationals are countable, now we will show that the reels $\mathbb{R}$ are uncountable, meaning there is no one-to-one correspondence between the natural numbers $\mathbb{N}$ and the reels $\mathbb{R}$. The...

### Rationals are Countable

/Posted by: MathArt
Cantor defined a set to be countable if you can find a way to enumerate all of its elements, even if the enumeration never finishes. The natural numbers $\mathbb{N}$ are the standard example of...

### Fractals

/Posted by: MathArt
Previously, to draw the Hilbert Curve and the Peano Curve, we designed an explicit algorithm that recursively defines the shape of the curve. If we want to generalize this recursive construction, we end up...

### Peano Curve

/Posted by: MathArt
The Peano Curve is another space-filling curve, meaning if you do enough iterations it will come arbitrary close to any point. Source Code The algorithm to construct a Peano curve is given in this...

### Hilbert Curve

/Posted by: MathArt
The Hilbert Curve is one of the most known space-filling curves, meaning if you do enough iterations it will come arbitrary close to any point. This way you can get a mapping between 1D...

### Pythagorean Tiling

/Posted by: MathArt
A pythagorean tiling is another beautiful way to prove Pythagoras’ theorem. It also shows that every plane can be tiled by two squares of arbitrary size. $\definecolor{colorRed}{RGB}{255,50,50}\definecolor{colorGreen}{RGB}{50,205,50}\definecolor{colorBlue}{RGB}{50,50,255}\definecolor{colorOrange}{RGB}{255,165,0} a^2 + \textcolor{colorBlue}{b^2} = \textcolor{red}{c^2}$ Explanation: If...

### Sum of Squared Fibonacci Numbers

/Posted by: MathArt
Fibonacci numbers are defined as $F_n = F_{n-1} + F_{n-2}$. $\sum_{i=1}^{n} F_i^2 = 1^2 + 1^2 + 2^2 + 3^2 + 5^2 + 8^2 + \ldots + F_n^2 = F_{n}F_{n+1}$ \(...

### Sum of Inverse Powers of Four

/Posted by: MathArt
$\frac{1}{4} + \frac{1}{16} + \frac{1}{64} + \ldots = \frac{1}{3}$ Explanation: The full area of the square is $\square$. The sum of the black squares is thus $\sum_n \blacksquare = \sum_n \frac{1}{4^n} \square$. We...

### Sum of Inverse Powers of Three

/Posted by: MathArt
$\sum_{n=1}^\infty \frac{1}{3^n} = \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \ldots = \frac{1}{2}$ Explanation The square gets divided into three parts. One of them is marked black (its area is $\frac{1}{3}$), and then in...

### Sum of Inverse Powers of Two

/Posted by: MathArt

### Pythagoras

/Posted by: MathArt
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