Home The Atlas of Advanced Mathematics

Reader's Guide

Mathematics and Physics

Euler's Identity The Hyperbolic Plane Euler's Polyhedron Formula Infinitely Many Primes The Prime Number Theorem The Irrationality of √2 The Principle of Stationary Action The Euler–Lagrange Equation Noether's Theorem Bayes' Theorem Maxwell's Equations Einstein's Field Equations Schrödinger's Equation The Dirac Equation

Subatomic Matter Mass in the Quantum World The Hierarchy of Cosmic Structure Cosmology and Astrophysics Glossary

Quantum Technologies

Quantum Foundations and Theory Quantum Information Quantum Algorithms Quantum Communications Quantum Networking Quantum AI Building a Quantum Computer Quantum Hardware Engineering Glossary

Connected Systems

Information Theory Coding Theory Digital Signal Processing Cryptography Graph Theory Network Science 6G: The Mathematical Frontier

Brain

Regions of the Brain

Neuroscience

Computational Neuroscience

Cells and Development

How Cells Find Their Role

Musculoskeletal Atlas

Head and Neck The Spine The Shoulder The Arms The Hands The Abdomen The Hip The Legs The Knee The Foot and Ankle The Feet

Euler's Polyhedron Formula

Vertices minus edges plus faces always makes two: why the count never fails, what bends it, and how one number grew into topology

Euler's Polyhedron Formula — eight solids counted from their coordinates, the two-trees proof, the five regular solids, twelve pentagons on every football, angle defects, a torus, a tesseract and the hairy-ball theorem

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