To try everything Brilliant has to offer—free—for a full 30 days, visit . You’ll also get 20% off an annual premium subscription. Banach spaces were invented by the Polish mathematician, Stefan Banach, as part of his Ph.D. thesis. These math spaces lead to weirdly interesting geometries. Geometries where spheres becomes cubes, circles become squares, and pi = 4. _____ Join my Patreon community: Support the channel with a one-time donation: _____ Video where "Completeness of a Space" is explained in depth: Music: Epidemic Sound Animations created using Manim: Illustrations and Thumbnails: Christine Kosakowski This video was sponsored by Brilliant