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

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Published 2023-10-21

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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.

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Images:

Euclid via Science Museum Group - ve42.co/Euclid

Geodesy survey via ams - ve42.co/Geodesy

John Wheeler via NAS Online - ve42.co/Wheeler

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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. - ve42.co/LibofCongress

Euclid’s Elements, Wikipedia - ve42.co/Elements

The History of Non-Euclidean Geometry, Extra History via YouTube - ve42.co/ExtraHistory

We (could) live on a 4D Pringle - Physics for the Birds via YouTube - ve42.co/4DPringle

Parallel Postulate, Wikipedia - ve42.co/Parallel

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. - ve42.co/Bolyai

Bolyai, J. (1896). The Science Absolute of Space.. (Vol. 3). The Neomon.

Gauss, Wikipedia - ve42.co/Gauss

Singh, U. (2022). Gauss-Bolyai-Lobachevsky: The dawn of non-euclidean geometry. Medium. - ve42.co/CPNonEuclidean

Landvermessung, D. Z. (1929). Abhandlungen ueber Gauss' wissenschaftliche Taetigkeit auf den Gebieten der Geodaesie, Physik und Astronomie Bd. 11, Abt. - ve42.co/Landvermessung

Nikolai Lobachevsky, Wikipedia - ve42.co/Lobachevsky

Lobachevskiĭ, N. I. (1891). Geometrical researches on the theory of parallels. University of Texas.

A Problem with the Parallel Postulate, Numberphile via YouTube - ve42.co/NumberphileParallel

Riemann, B. (2016). On the hypotheses which lie at the bases of geometry. Birkhäuser. - ve42.co/Riemann

Einstein, A. (1905). On the electrodynamics of moving bodies. Annalen der physik, 17(10), 891-921. - ve42.co/Einstein1905

ESA/Hubble. (n.d.). Hubblecast 90: The final frontier of the Frontier Fields. ESA/Hubble. - ve42.co/Einstein1905

Agazie, G., et al. (2023). The NANOGrav 15 yr data set: Constraints on supermassive black hole binaries from the gravitational-wave background. - ve42.co/NANOGrav

Secrets of the Cosmic Microwave Background, PBS Spacetime via YouTube - ve42.co/PBSCMB

Wood, C. (2020). How Ancient Light Reveals the Universe's Contents. Quanta Magazine. - ve42.co/AncientLight

Collaboration (2014). Planck 2013 results. XVI. Cosmological parameters. A&A, 571, A16. - ve42.co/Planck2013

WMAP Science Team, NASA. (2014). Matter in the Universe. WMAP, NASA. - ve42.co/WMAP2014

What Is The Shape of Space, minutephysics via YouTube - ve42.co/SpaceShape

Shape of the universe, Wikipedia - ve42.co/UniverseShape

Crocheting Hyperbolic Planes: Daina Taimina by Ted, via YouTube - ve42.co/Hyperbolic

Hyperbolic Crochet model - ve42.co/Crochet

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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.

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Thumbnail by Ren Hurley