My name is Grant Sanderson. Videos here cover a variety of topics in math, or adjacent fields like physics and CS, all with an emphasis on visualizing the core ideas. The goal is to use animation to help elucidate and motivate otherwise tricky topics, and for difficult problems to be made simple with changes in perspective. For more information, other projects, FAQs, and inquiries see the website: https://www.3blue1brown.com
This video unpacks M.C. Escher's "Print Gallery" lithograph, revealing how its paradoxical infinite loop relies on a conformal grid derived from complex analysis to transform a linear Droste effect into a continuous circular zoom, mathematically resolving the mysterious blank center.
Grant Sanderson demonstrates why high-dimensional geometry—essential for modern AI—defies human intuition through counterintuitive sphere packing puzzles, revealing that high-dimensional cubes (not spheres) behave bizarrely as their corners stretch to distance √n while edges remain fixed, ultimately building toward the elegant but underappreciated formula for the volume of n-dimensional balls.
The Hairy Ball Theorem establishes that every continuous tangent vector field on a sphere must contain at least one zero vector, creating unavoidable constraints in systems ranging from video game physics to meteorology.
Laplace transforms convert differential equations into algebraic expressions on the complex s-plane, enabling analysis of dynamic systems—such as driven harmonic oscillators—by examining pole locations to distinguish transient decay from steady-state behavior without solving full time-domain equations.
The Laplace transform decomposes functions into their constituent exponential components by integrating f(t)*e^(-st) from zero to infinity; when the complex frequency s matches an exponential hidden within f(t), the integrand becomes constant causing the integral to diverge into a pole, simultaneously converting differential equations into algebraic problems by transforming derivatives into multiplications by s.