### Triangle Shortest Cut

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...

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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...

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Having seen that the rationals are countably infinite and the reels are uncountably infinite, the natural questions to ask is: How many types of infinities are there? It turns out there are infinitely many...

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...

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...

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...

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...

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...

$\definecolor{colorRed}{RGB}{255,50,50}\definecolor{colorGreen}{RGB}{50,205,50}\definecolor{colorBlue}{RGB}{50,50,255}\definecolor{colorOrange}{RGB}{255,165,0}\sum_{i=1}^{n} i^3 = \textcolor{red}{1^3} + \textcolor{colorGreen}{2^3} + \textcolor{colorBlue}{3^3} + \ldots + \textcolor{colorOrange}{n^3} = (\textcolor{red}{1} + \textcolor{colorGreen}{2} + \textcolor{colorBlue}{3} + \ldots + \textcolor{colorOrange}{n})^2 = (\sum_{i=1}^n i)^2...

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} $ \(...

$\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_{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...

$ \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \ldots = 1...

$ 1^2 + 2^2 + 3^2 + 4^2 + \ldots + n^2 = \frac{1}{3} \ n (n+1) (n+\frac{1}{2}) $ \(...

$\definecolor{colorRed}{RGB}{255,50,50}\definecolor{colorBlue}{RGB}{50,50,255} 1 + 2 + 3 + 4 + \ldots + n = \textcolor{colorRed}{\frac{n^2}{2}} + \textcolor{colorBlue}{\frac{n}{2}}...

The yellow dots represent the value of $1 + 2 + 3 + 4 + \ldots + n$ and the blue dots represent the number $n+1$. If you hover over a yellow dot you...