Triangle Shortest Cut

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...
Cantor's Theorem

Infinitely many Infinities

/Posted by: MathArt
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...
Cantor Diagonal Argument Reels Uncountable

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...
cantor enumeration rationals countable

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...
Sierpinski Triangle

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 ConstructionPeano Curve Construction

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 ConstructionHilbert Curve Construction

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 pythagoras proof

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 cubed positive integers

Sum of Cubes

/Posted by: MathArt
$\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...
sum of inverse powers of four

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 threesum of inverse powers of three

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 squares visual proof

Sum of Squares

/Posted by: MathArt
$ 1^2 + 2^2 + 3^2 + 4^2 + \ldots + n^2 = \frac{1}{6} \ n (n+1) (2n+1) $ \(...
sum of positive integers visual proof

Sum of Positive Integers

/Posted by: MathArt
$\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}}...
sum of positive integers visual proof binomial

Sum of Positive Integers

/Posted by: MathArt
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...