Tag: mathematics
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The Kissing Number Problem Explained
A broad category of problems in math are called sphere packing problems. They range from pure math to practical questions like figuring out how to stack fruit at the grocery store. Some of these problems have solutions, but others, like the kissing number problem, are still tricky. What Is a Kissing Number? What Do We… read more
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Escaping a Forest with Math – Bellman's Lost in a Forest Problem
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 Problem Fat Forests Why… read more
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Every Proof That There Are Infinitely Many Primes Explained
What Is a Prime Number? Euclid’s Proof Factorial Proof Erdős’s Proof What Is a Prime Number? Think of a natural number. That is, a number used for counting, like six. Next, think of another natural number, like two. If we calculate 6 / 2, the result is 3. Since 3 is also a natural number,… read more
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The Most Jagged Function in Math – The Weierstrass Function
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 an infinite sum called… read more
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The Conjecture Unsolved Since 1742
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 numbers. 18 = 13… read more
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The Basics of Set Theory Explained
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 Operations The basic operations… read more
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Unsolved Geometry – The Lebesgue Universal Covering Problem Explained
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
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The Surface Hidden Inside Every Knot – Seifert Surfaces Explained
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 a circle is just… read more
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Spacetime – The 4th Dimension
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 Why Does This Matter?… read more
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The Infinity Paradox of Hilbert’s Hotel
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 experiment proposed by German… read more
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