Category: YouTube Shorts

The Mandelbrot set, named after French-American mathematician Benoît B. Mandelbrot, arises in the study of complex numbers. How It Works A Fractal of Extraordinary Detail How It Works We begin by picking some number c in the complex plane. Using c, we define a function: take a number, multiply it by read more

A unit square is a square with a side length of one unit. Suppose you have a collection of some number of unit squares, and we will call this number n. You want to pack all these squares into a large square enclosure with no overlap. For a given number of unit squares n, what read more

Topology studies the properties of spaces and figures that remain invariant under continuous deformations such as stretching, twisting, or bending, as long as there are no cuts or new attachments. A circle and an ellipse are topologically equivalent, whereas a circle and a figure eight are not. The Foundations The central concept is the topological read more

Zero dimension corresponds to the point, defined since antiquity as the entity without length, width, or depth. Euclid described it in his work Elements around 300 BC as “that which has no parts,” establishing the foundation upon which geometry is built. What Is a Point? A point has no measurable magnitude but serves the essential read more

Imagine being lost in a forest. You do not know where you are or what direction you are facing, but you do know the exact shape and size of the forest. What is the best path to take to ensure the lowest escape time in the worst case scenario? An Unsolved read more

For a long time, mathematicians assumed that no function could be continuous everywhere and differentiable nowhere. The Weierstrass function proved them wrong. Discovery A Fractal Before Fractals Why Did This Matter? Discovery The Weierstrass function was discovered by German mathematician Karl Weierstrass and first published in 1872. Weierstrass defined it using read more

Goldbach’s Conjecture A Problem Born from Letters Why Can’t We Prove It? Goldbach’s Conjecture Goldbach’s conjecture, one of the greatest unsolved mysteries in math, is also very easy to write. Every even number greater than two is the sum of two primes. You can check this in your head for small read more

Set theory is the fundamental language of mathematics, describing and organizing collections of objects called sets. A set is usually denoted with braces. For example: A = {1, 2, 3} Its elements can be numbers, points, objects, functions, or even other sets. Basic Operations Why Is Set Theory So Important? Basic read more

Imagine you are working in 2D and you have some kind of convex shape that can contain any shape with a diameter of one. Let’s call this shape the cover, since it can be used to cover up any of the diameter-1 shapes. The diameter-1 shapes can be translated (slid), rotated (spun around), and reflected read more