Tag: mathematics

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

A Seifert surface, named after German mathematician Herbert Seifert, is an orientable surface whose boundary is a knot or link. The Simplest Example Seifert Surfaces of Links Why Are Seifert Surfaces Important? The Simplest Example The simplest example is a disc, which is a surface whose boundary is a circle, and read more

The fourth dimension arises by adding time to the three spatial dimensions. In 1905, Albert Einstein formulated the special theory of relativity and proposed that time should not be treated as an external parameter but as a dimension intertwined with space. Minkowski’s Spacetime The Light Cone A New Kind of Geometry read more

Here’s your formatted article: Hilbert’s Hotel: The Hotel That Is Always Full but Never Out of Room One New Guest Infinitely Many New Guests Hilbert’s Paradox What Does This Tell Us About Infinity? Hilbert’s Hotel: The Hotel That Is Always Full but Never Out of Room Hilbert’s Hotel is a thought read more

Ulam’s Packing Conjecture: Why Spheres Are the Worst Packers Packing Density The Conjecture Why Is This So Hard to Prove? Ulam’s Packing Conjecture: Why Spheres Are the Worst Packers Ulam’s packing conjecture concerns the packing of identical shapes in three-dimensional space. Imagine taking a bunch of identical convex solids and using read more

Here’s your formatted article: Discrete Mathematics: The Math Behind Computation The Pillars of Discrete Mathematics Why Does Discrete Mathematics Matter? Discrete mathematics studies structures whose values are separate and not continuous. Unlike mathematical analysis, which focuses on smooth variations, it deals with countable elements, whether finite or countably infinite, that can read more