Tag: mathematics
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Why You Can’t Draw These 4 Shapes Exactly
With only an unmarked straightedge and a compass, ancient geometers could bisect angles, build equilateral triangles and draw regular pentagons. Yet four simple-sounding tasks resisted every attempt for millennia. The reason turned out to be algebraic: once constructions are translated into equations, it becomes clear why squaring the circle, doubling the cube, trisecting an angle… read more
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Every Perfect 4D Shape Explained
The tesseract is the best-known four-dimensional shape, yet it is only one of six perfectly regular convex shapes in four dimensions. Each is built from identical three-dimensional solids, called cells, fitted together with complete symmetry. This article walks through all six: the 120-cell, 600-cell, 5-cell, tesseract, 16-cell and 24-cell, and shows how they relate through… read more
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We Know Everything About This Number Except Whether It Exists
Some numbers are defined so precisely that we know almost everything about them, except whether a single example exists. This article explores quasiperfect numbers, almost perfect numbers, Descartes numbers, and superperfect numbers, all built from one simple idea: adding up the divisors of a number. Perfect, Abundant, and the Sum of Divisors The divisors of… read more
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Every Knot From Simplest to Impossible Explained
Take a loop of rope, tie a knot, glue the ends together, and hold it to a mirror: for some knots, the reflection is a genuinely different knot. This article walks from the plain circle to the trefoil, the figure-eight, the cinquefoil, and finally the Perko pair, a famous case where one knot sat in… read more
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Every Shape That Can Roll Without Being Round Explained
A wheel does not have to be a circle. Some strange shapes can roll smoothly between two parallel surfaces, and others can roll across a plane along winding paths. This article tours the Reuleaux triangle, the Meissner bodies, the sphericon, and the oloid. Along the way it uses Gaussian curvature to explain why some surfaces… read more
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Every Weird Shape That Shouldn’t Exist Explained
Some mathematical objects seem to break the rules of geometry. A curve that has area, a sponge with no volume but endless surface, a one-sided surface that fits in space, and a cloud of dust that can snag a rope: each is rigorously defined and fully consistent. This article tours six of them and explains… read more
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The Most Controversial Proof in Mathematics Ever
Mathematics rests on a promise: if a proof is correct, a trained mathematician should be able to check it. Shinichi Mochizuki's claimed proof of the ABC conjecture has tested that promise, dividing proponents of his theory from prominent critics. This article explains the conjecture, why it matters, and the disagreement about whether the proposed argument… read more
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Every Particle in Physics Explained
Every Fundamental Particle in the Standard Model Explained What is everything made of? The Standard Model of particle physics describes the smallest known building blocks of matter and the forces that govern their interactions. It includes particles such as electrons, quarks, and neutrinos, along with the particles that carry the electromagnetic, strong, and weak… read more
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Every Weird Number That Shouldn't Exist Explained
Thanks for watching! Chapters: 0:00 Zero 1:40 Negative Numbers 2:40 Imaginary Numbers 3:39 Infinitesimals 4:44 Surreal Numbers 5:55 Quaternions 7:00 Octonions 8:32 Transfinite Ordinals 9:28 Irrational Numbers One useful way to organize the major number systems is as a ladder with seven levels: Natural numbers, N Integers, Z Rational numbers, Q Real numbers, R Complex… read more
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Every Proof of the Pythagorean Theorem Explained
The Pythagorean theorem is one of the oldest and most famous results in mathematics. More than 370 proofs of the theorem are known, and mathematicians have continued finding new ones for over 2,000 years. Some proofs use nothing more than geometry and rearrangement. Others use algebra, trapezoids, or even calculus. In this article, we will… read more
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