Math

How Giving Up Reality Solved the Cubic Equation

For centuries, a general solution to the cubic equation was considered impossible—until mathematicians embraced complex numbers, abandoning the strict requirement that math reflect physical reality.

References:

• Videos:
  • 500 Years of Not Teaching the Cubic Formula – YouTube
  • Imaginary Numbers are Real – YouTube
• Books and Papers:
  • Dunham, W. (1990). Journey Through Genius: The Great Theorems of Mathematics. Link
  • Toscano, F. (2020). The Secret Formula, Princeton University Press. Link
  • Bochner, S. (1963). The Significance of Some Basic Mathematical Conceptions for Physics, Isis, 54(2), 179–205. Link
  • Muroi, K. (2019). Cubic Equations of Babylonian Mathematics, arXiv:1905.08034. Link
  • Branson, W. (2014). Solving the Cubic with Cardano. Link
  • Rothman, T. (2013). Cardano v Tartaglia: The Great Feud Goes Supernatural, arXiv:1308.2181. Link
  • Siadat, M. V., & Tholen, A. (2021). Omar Khayyam: Geometric Algebra and Cubic Equations, Math Horizons, 28(1), 12–15. Link
  • Merino, O. (2006). A Short History of Complex Numbers, University of Rhode Island. Link
• Historical Works:
  • Cardano, G. (1545). Ars Magna (Dover reprint, 1993). ISBN: 0-486-67811-3
  • Bombelli, R. (1579). L’Algebra. Link
• Software:
  • Manim Community Developers. (2021). Manim – Mathematical Animation Framework (v0.13.1). Link

Einstein was wrong about black holes — what else?

This video was inspired by conversations with leading physicists and key scientific literature. Below are the references:

References

• Thorne, K. (1995) – Black Holes & Time Warps: Einstein's Outrageous Legacy
• Hamilton, A. J. S. (2021) – General Relativity, Black Holes, and Cosmology – Link
• Relativity Playlist by ScienceClic – Link
• Black Hole Events by PBS Space Time – Watch
• Newton’s Letters via The Newton Project – Link
• Einstein, A. (1915) – Die Feldgleichungen der Gravitation – Link
• Schwarzschild, K. (1916) – Über das Gravitationsfeld eines Massenpunktes – Link
• Wali, K. C. (1982) – Chandrasekhar vs. Eddington—An Unanticipated Confrontation – Link
• How to Build a Black Hole – Watch
• Oppenheimer, J. R., & Volkoff, G. M. (1939) – On Massive Neutron Cores – Link
• Oppenheimer, J. R., & Snyder, H. (1939) – On Continued Gravitational Contraction – Link
• Schwarzschild Geometry by Andrew Hamilton – Link
• Why Time and Space Swap by ScienceClic – Watch
• Why All World Maps Are Wrong by Vox – Watch
• Hamilton, A. J., & Lisle, J. P. (2008) – The River Model of Black Holes – Link
• Mapping The Multiverse by PBS Space Time – Watch
• Rotating Black Hole via Wikipedia – Link
• Wormhole Travel by PBS Space Time – Watch
• Morris, M. S., & Thorne, K. S. (1988) – Wormholes in Spacetime and Their Use for Interstellar Travel – Link

🖼 Visuals & Simulation References

• ScienceClic – “What would we see if we fell into a black hole” – Watch
• Kazmierczak, J. et al. (2021) – NASA’s NICER Tests Matter’s Limits – Link
• Bridgman, T. et al. (2024) – M5.1 Flare at Active Regions 13559 & 13561 – Link
• Schnittman, J. et al. (2019) – Black Hole Accretion Disk Visualization – Link
• Wiessinger, S. et al. (2020) – A Decade of Sun – Link
• Skelly, C. et al. (2017) – What is a Neutron Star? – Link
• Earth texture – Link
• First image of Sgr A* – Link
• Image of M87 – Link
• Polarized light image of Sgr A* – Link

How Newton Changed the Way We Calculate Pi

For millennia, mathematicians calculated π using geometry and slow convergence. Then Newton introduced a revolutionary approach using calculus, drastically accelerating the process.

References:

• Books and Academic Sources:
  • Arndt, J., & Haenel, C. (2001). Pi-Unleashed. Springer Science & Business Media. Link
  • Dunham, W. (1990). Journey Through Genius: The Great Theorems of Mathematics. Wiley. Link
  • Borwein, J. M. (2014). The Life of π: From Archimedes to ENIAC and Beyond. In From Alexandria, Through Baghdad (pp. 531–561). Springer. Link

Special Thanks:

• Alex Kontorovich – Professor of Mathematics at Rutgers University and Distinguished Visiting Professor for the Public Dissemination of Mathematics at the National Museum of Mathematics (MoMath)

Credits:

• Written by: Derek Muller & Alex Kontorovich
• Animation by: Ivy Tello
• Filming: Derek Muller & Raquel Nuno
• Edited by: Derek Muller
• Music: Jonny Hyman & Petr Lebedev
• Additional Music: Epidemic Sound
  • "Particle Emission"
  • "Into the Forest"
  • "Stavselet"
  • "Face of the Earth"
  • "Firefly in a Fairytale"
• Thumbnail by: Gianmarco Malandra & Karri Denise