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Klein Bottle – The 4D Shape With No Inside or Outside
The Klein Bottle: A Surface with No Inside or Outside Constructing a Klein Bottle The Klein Bottle: A Surface with No Inside or Outside The Klein bottle, named after German mathematician Felix Christian Klein, is another example of a non-orientable surface. However, unlike the Möbius strip, it has no boundary, meaning read more
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The Complex Equations of The Mandelbrot Set Explained
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
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Unsolved Geometry Puzzle – The Square Packing Problem Explained
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
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Topology Explained in 45 Seconds
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
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Zero Dimensional Space Explained
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
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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 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 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 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 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 read more
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