How One Line in the Oldest Math Text Hinted at Hidden Universes

Discover strange new universes that turn up at the core of Einstein’s General Relativity. Head to to start your free 30-day trial, and the first 200 people get 20% off an annual premium subscription. If you’re looking for a molecular modeling kit, try Snatoms – a kit I invented where the atoms snap together magnetically – ▀▀▀ A massive thank you to Prof. Alex Kontorovich for all his help with this video. A huge thank you to Prof. Geraint Lewis and Dr. Ashmeet Singh for helping us understand the applications of Non-Euclidean geometry in astronomy/cosmology. Lastly, a big thank you to Dr. Henry Segerman and Dr. Rémi Coulon for helping us visualize what it’s like to be inside hyperbolic space and helping us understand hyperbolic geometry. ▀▀▀ Images: Euclid via Science Museum Group - Geodesy survey via ams - John Wheeler via NAS Online - ▀▀▀ References: Dunham, W. (1991). Journey through Genius: Great Theorems of Mathematics. John Wiley & Sons. Bonola, R. (1955). Non-Euclidean geometry: A critical and historical study of its development. Courier Corporation. Library of Congress. (n.d.). The Library of Congress. - Euclid’s Elements, Wikipedia - The History of Non-Euclidean Geometry, Extra History via YouTube - We (could) live on a 4D Pringle - Physics for the Birds via YouTube - Parallel Postulate, Wikipedia - Prékopa, A., & Molnár, E. (Eds.). (2006). Non-euclidean geometries: János Bolyai memorial volume (Vol. 581). Springer Science & Business Media. St Andrews, University of. (n.d.). Bolyai. MacTutor History of Mathematics. - Bolyai, J. (1896). The Science Absolute of Space.. (Vol. 3). The Neomon. Gauss, Wikipedia - Singh, U. (2022). Gauss-Bolyai-Lobachevsky: The dawn of non-euclidean geometry. Medium. - Landvermessung, D. Z. (1929). Abhandlungen ueber Gauss’ wissenschaftliche Taetigkeit auf den Gebieten der Geodaesie, Physik und Astronomie Bd. 11, Abt. - Nikolai Lobachevsky, Wikipedia - Lobachevskiĭ, N. I. (1891). Geometrical researches on the theory of parallels. University of Texas. A Problem with the Parallel Postulate, Numberphile via YouTube - Riemann, B. (2016). On the hypotheses which lie at the bases of geometry. Birkhäuser. - Einstein, A. (1905). On the electrodynamics of moving bodies. Annalen der physik, 17(10), 891-921. - ESA/Hubble. (n.d.). Hubblecast 90: The final frontier of the Frontier Fields. ESA/Hubble. - Agazie, G., et al. (2023). The NANOGrav 15 yr data set: Constraints on supermassive black hole binaries from the gravitational-wave background. - Secrets of the Cosmic Microwave Background, PBS Spacetime via YouTube - Wood, C. (2020). How Ancient Light Reveals the Universe’s Contents. Quanta Magazine. - Collaboration (2014). Planck 2013 results. XVI. Cosmological parameters. A&A, 571, A16. - WMAP Science Team, NASA. (2014). Matter in the Universe. WMAP, NASA. - What Is The Shape of Space, minutephysics via YouTube - Shape of the universe, Wikipedia - Crocheting Hyperbolic Planes: Daina Taimina by Ted, via YouTube - Hyperbolic Crochet model - ▀▀▀ Special thanks to our Patreon supporters: Adam Foreman, Amadeo Bee, Anton Ragin, Balkrishna Heroor, Bernard McGee, Bill Linder, Burt Humburg, Dave Kircher, Diffbot, Evgeny Skvortsov, Gnare, Jesse Brandsoy, John H. Austin, Jr., john kiehl, Josh Hibschman, Juan Benet, KeyWestr, Lee Redden, Marinus Kuivenhoven, Mario Bottion, Max Maladino, Meekay, meg noah, Michael Krugman, Paul Peijzel, Richard Sundvall, Sam Lutfi, Stephen Wilcox, Tj Steyn, TTST, Ubiquity Ventures ▀▀▀ Directed by Casper Mebius Written by Casper Mebius, Petr Lebedev, Emily Zhang, Derek Muller, and Alex Kontorovich Edited by Jack Saxon Animated by Fabio Albertelli, Ivy Tello, and Mike Radjabov Illustrations by Jakub Misiek and Celia Bode Filmed by Derek Muller Produced by Casper Mebius, Derek Muller, and Han Evans Additional video/photos supplied by Getty Images, Pond5, and by courtesy of: NASA, NASA’s Goddard Space Flight Center, NASA Goddard Flight Lab/ CI Lab, NASA’s WMAP science teams, ESO, and ESA/Hubble. Music from Epidemic Sound Thumbnail by Ren Hurley