Articles
Equation explainers written for search and understanding — each linking to interactive visualizations, sound, and story mode on Equation Universe.
Friedmann Equation Explained
How the expansion rate of the universe links matter density, spatial curvature, and dark energy.
Pythagorean Theorem Explained
Why a² + b² = c² is the oldest equation still taught to every student — and how it built architecture, navigation, and GPS.
Euler's Identity: The Most Beautiful Equation
e^(iπ) + 1 = 0 links five fundamental constants in one line — and powers all of modern signal processing.
Taylor Series Explained
How smooth functions become infinite polynomials — and why calculators can compute sin, cos, and e^x.
Newton's Second Law Explained
F = ma — the equation that launched classical mechanics and every engineering discipline built on motion.
Universal Gravitation Explained
Newton's F = Gm₁m₂/r² describes why planets orbit, tides rise, and satellites stay aloft.
Bernoulli's Equation Explained
Pressure, speed, and height trade off along a streamline — essential for wings, pipes, and blood flow.
The Wave Equation Explained
∂²u/∂t² = c²∇²u — the PDE behind sound, light, earthquakes, and strings.
The Heat Equation Explained
How temperature diffuses through materials — from cooking pans to CPU cooling and climate models.
Ideal Gas Law Explained
PV = nRT connects pressure, volume, temperature, and moles — the workhorse of chemistry and meteorology.
Lorentz Force Explained
How charged particles curve in electric and magnetic fields — from auroras to particle accelerators.
Heisenberg Uncertainty Principle Explained
ΔxΔp ≥ ℏ/2 — why you cannot simultaneously know position and momentum with arbitrary precision.
Shannon Entropy Explained
H = −Σ p log p measures information surprise — and sets limits on compression and communication.
SIR Epidemic Model Explained
How differential equations track susceptible, infected, and recovered populations during outbreaks.
Logistic Growth Explained
dN/dt = rN(1 − N/K) — populations that boom, then level off as resources cap growth.
Hubble's Law Explained
v = H₀d — distant galaxies recede faster, evidence that the universe is expanding.
Gradient Descent Explained
θ ← θ − η∇L — the optimization loop behind virtually all modern machine learning.
Transformer Attention Explained
How scaled dot-product attention lets models weigh which tokens matter — the core of GPT and modern AI.
Equations Every Engineer Should Know
A curated list of essential equations — from F=ma to Fourier, Bernoulli, and the heat equation — with interactive explorations.
Navier-Stokes Equation Explained
The governing equations of fluid flow — from blood in arteries to weather, aircraft, and the million-dollar millennium prize.
Bayes' Theorem for Beginners
Learn P(A|B) = P(B|A)P(A)/P(B) — the equation for updating beliefs with evidence, from medicine to AI.
Schrödinger Equation Explained
The wave equation of quantum mechanics — how ψ describes electrons, atoms, and the semiconductor revolution.
What is Entropy?
Entropy measures disorder and irreversibility — from steam engines to information theory to the arrow of time.
The Equation Behind Rocket Science
From Tsiolkovsky to staging to orbital mechanics — the math that gets payloads off Earth.
Maxwell's Equations Explained Visually
Four equations that unified electricity, magnetism, and light — and predicted radio waves before they were detected.
Why E = mc² Changed the World
Einstein's mass-energy equivalence revealed that matter is stored energy — reshaping physics, medicine, and cosmology.
What is the Fourier Transform?
Learn how the Fourier transform decomposes any signal into pure tones — and why it powers MP3, MRI, Wi-Fi, and AI.
What is the Tsiolkovsky Rocket Equation?
Understand Δv = ve ln(m0/mf) — the equation that made spaceflight mathematically possible, from Konstantin Tsiolkovsky to SpaceX.