"Conway's Game of Life" is an incredible cellular automaton that produces life-like complex behaviors from just four simple rules. This article deeply explores its mathematical background, detailed rules, famous patterns, and implementation examples.
Can you find Pi (π) just by dropping a needle on the floor? We explain in detail the astonishing classic problem of probability theory proposed in the 18th century, Buffon's Needle, from its mathematical proof to simulation.
A detailed explanation of the Central Limit Theorem, one of the most important theorems in statistics, from intuitive understanding to mathematical proof and Python simulation.
A comprehensive guide to chaos theory and the butterfly effect, nonlinear dynamical systems with sensitive dependence on initial conditions, covering their historical background, mathematical foundations, fractals, and real-world applications.
What happens if you infinitely add 1+2+3+4+...? Normally, it diverges to infinity, but in a specific field of mathematics, it is known to yield a surprising result of -1/12. In this article, we explain in detail the Ramanujan summation, which derives this counterintuitive result, along with its mathematical background and applications in physics.
An in-depth look, using formulas and diagrams, into 'Zipf's Law', the amazing mathematical rule governing the world from natural language to city scale and economics.
A deep dive into the Axiom of Choice and Zorn's Lemma, from their intuitive meaning to mathematical formalization and their importance in modern mathematics.
What is the Secretary Problem? Through this classic example of the optimal stopping problem, we explain in detail the '1/e law (approx. 37% rule)', a mathematical approach for making the best choices under uncertainty. We explore how to apply it to hiring and everyday decision-making, with mathematical proofs and simulations.
There are differences in 'size' among infinities. We explain in detail the process of proving that the cardinality of real numbers is strictly greater than the cardinality of natural or rational numbers, using the 'Diagonal Argument' proposed by Georg Cantor.
A detailed explanation of the Chinese Remainder Theorem (CRT), originating from ancient Chinese arithmetic texts, covering its mathematical principles to its important applications in modern public-key cryptography (RSA cryptography).
A detailed explanation of the famous topological theorem, the 'Hairy Ball Theorem', from intuitive understanding to rigorous mathematical proof, and its real-world applications in meteorology and computer graphics.
The 'Seven Bridges of Königsberg' problem solved by the 18th-century mathematician Leonhard Euler. This article delves deeply into the history and mathematical background of how this problem pioneered new fields of modern mathematics, such as graph theory and topology.
From the doubts about Euclid's parallel postulate, to the establishment of non-Euclidean geometry by Gauss, Bolyai, Lobachevsky, and Riemann, and leading to Einstein's general theory of relativity, we explain the magnificent history of mathematics and physics.
An easy-to-understand explanation of the physical background and mathematical difficulties of the 'Yang-Mills Equation and Mass Gap Problem', one of the Millennium Prize Problems.
A deep dive into the existence and smoothness of the Navier-Stokes equations, one of the Millennium Prize Problems of the Clay Mathematics Institute.
A deep dive into the 'Remarkable Theorem' discovered by Carl Friedrich Gauss, from its mathematical background to its everyday application when eating pizza.
A deep explanation of the Prime Number Theorem, a monumental achievement in mathematics regarding the regularity of prime number appearances. From the prime-counting function to its relationship with the Riemann Hypothesis.
A detailed mathematical explanation of the Birch and Swinnerton-Dyer Conjecture (BSD Conjecture), one of the Millennium Prize Problems, from the basics of elliptic curves to its deep connection with L-functions.
Fermat's Last Theorem, the most famous unsolved problem in the history of mathematics. How was this difficult problem, which rejected the challenges of many genius mathematicians for 360 years after the death of Pierre de Fermat, proven by Andrew Wiles? We explain its dramatic history and mathematical background in detail.
What is Galois Theory? Why is there no formula for solving equations of degree 5 or higher? We will explain the foundations of group theory left by Évariste Galois and the beauty of symmetry in mathematics in an easy-to-understand way.
