Scholar Sauce | What's the Deal with Euclid's Vth Postulate? (Part 2) 500 Subscribers Special! @scholarsauce | Uploaded December 2023 | Updated October 2024, 1 hour ago.
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In his book, the Elements written in 300 BCE, Euclid described planar geometry using five postulates or axioms. One of these axioms was considerably more complicated than the others and seemed to some to be likely provable as a theorem from the other four more natural postulates. But no ancient Greek was ever able to prove Euclid's fifth postulate and this question persisted for over 2000 years until it was finally resolved in the 19th century CE. It took 2000 years to confirm that indeed you cannot prove Euclid's fifth postulate from the other four, but this proof resulted in us learning of another type of plane geometry that is completely consistent and satisfies the first four of Euclid's postulates but not the fifth. This type of geometry is called hyperbolic geometry and weird stuff happens there: triangles have less than 180 degrees, similar triangles are congruent, and rectangles are nowhere to be found. This discovery inspired further advancements in geometry including Riemannian and semi-Riemannian geometry which is the mathematical framework that Albert Einstein used to develop general relativity and advance our understanding of gravity and the universe.
Link to Part 1: youtu.be/pmGDea9ZQ4U
*Images*
_The Acropolis at Athens_
Artist: Leo von Klenze
License: Public Domain
Acquired: commons.wikimedia.org/wiki/File:Akropolis_by_Leo_von_Klenze.jpg
_Papyrus Fragment of Euclid's Elements_
Artist: Euclid
License: Public Domain
Acquired: math.ubc.ca/~cass/Euclid/papyrus/tha.jpg
_Euclid of Megara_
Artist: Justus van Gent
License: Public Domain
Acquired: commons.wikimedia.org/wiki/File:Euklid2.jpg
_Nikolai Lobachevsky_
Artist: Lev Kryukov
License: Public Domain
Acquired: commons.wikimedia.org/wiki/File:Lobachevsky.jpg
_Bust of Janos Bolyai_
Artist: Szilas (photographer)
License: Creative Commons CC0 1.0 Universal Public Domain Dedication
Acquired: commons.wikimedia.org/wiki/File:Bust_of_J%C3%A1nos_Bolyai_at_the_Mikl%C3%B3s_Zr%C3%ADnyi_Military_Academy.jpg
_Johann Carl Friedrich Gauss_
Artist: Christian Albrecht Jensen
License: Public Domain
Acquired: commons.wikimedia.org/wiki/File:Carl_Friedrich_Gauss_1840_by_Jensen.jpg
_Eugenio Beltrami_
Artist: Unknown
License: Public Domain
Acquired: commons.wikimedia.org/wiki/File:Beltrami.jpg
_Felix Klein_
Artist: Unknown
License: Public Domain
Acquired: commons.wikimedia.org/wiki/File:Felix_Christian_Klein.jpg
_Bernhard Riemann_
Artist: Unknown
License: Public Domain
Acquired: commons.wikimedia.org/wiki/File:Bernhard_Riemann_3.jpg
_Henri Poincare_
Artist: Unknown
License: Public Domain
Acquired: commons.wikimedia.org/wiki/File:Henri_Poincar%C3%A9-2.jpg
_Hendrik Lorentz_
Artist: Museum Boerhaave
License: Pbulic Domain
Acquired: commons.wikimedia.org/wiki/File:Hendrik_Antoon_Lorentz.jpg
_Hermann Minkowski_
Artist: Raum und Zeit
License: Public Domain
Acquired: commons.wikimedia.org/wiki/File:Hermann_Minkowski_Portrait.jpg
_Albert Einstein_
Artist: Ferdinand Schmutzer
License: Public Domain
Acquired: commons.wikimedia.org/wiki/File:Einstein_1921_by_F_Schmutzer_-_restoration.jpg
*Videos*
All video clips are short and used under Fair Use as this work is transformative in each case.
_Star Trek IV: The Voyage Home_
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_The Flash (TV Series)_
"_The Darkness and the Light_"
Copyright Warner Bros. Entertainment, Inc.
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_Justice League_
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_Avengers: Infinity War_
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_Despicable Me 2_
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_Star Trek: The Next Generation_
"_Sarek_"
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Acquired: yarn.co/yarn-clip/02c82a09-7335-4f8d-92ce-4dd07f821acf
_The BEST FALLS on Runway by Fashion Channel_
Acquired: youtube.com/watch?v=kYLE_kp6coQ
*Music*
Various Tracks from the YouTube Audio Library
Thank you to our first 500 subscribers! Your support makes this channel possible. Please share this video with your friends and help us get to 1000!
In his book, the Elements written in 300 BCE, Euclid described planar geometry using five postulates or axioms. One of these axioms was considerably more complicated than the others and seemed to some to be likely provable as a theorem from the other four more natural postulates. But no ancient Greek was ever able to prove Euclid's fifth postulate and this question persisted for over 2000 years until it was finally resolved in the 19th century CE. It took 2000 years to confirm that indeed you cannot prove Euclid's fifth postulate from the other four, but this proof resulted in us learning of another type of plane geometry that is completely consistent and satisfies the first four of Euclid's postulates but not the fifth. This type of geometry is called hyperbolic geometry and weird stuff happens there: triangles have less than 180 degrees, similar triangles are congruent, and rectangles are nowhere to be found. This discovery inspired further advancements in geometry including Riemannian and semi-Riemannian geometry which is the mathematical framework that Albert Einstein used to develop general relativity and advance our understanding of gravity and the universe.
Link to Part 1: youtu.be/pmGDea9ZQ4U
*Images*
_The Acropolis at Athens_
Artist: Leo von Klenze
License: Public Domain
Acquired: commons.wikimedia.org/wiki/File:Akropolis_by_Leo_von_Klenze.jpg
_Papyrus Fragment of Euclid's Elements_
Artist: Euclid
License: Public Domain
Acquired: math.ubc.ca/~cass/Euclid/papyrus/tha.jpg
_Euclid of Megara_
Artist: Justus van Gent
License: Public Domain
Acquired: commons.wikimedia.org/wiki/File:Euklid2.jpg
_Nikolai Lobachevsky_
Artist: Lev Kryukov
License: Public Domain
Acquired: commons.wikimedia.org/wiki/File:Lobachevsky.jpg
_Bust of Janos Bolyai_
Artist: Szilas (photographer)
License: Creative Commons CC0 1.0 Universal Public Domain Dedication
Acquired: commons.wikimedia.org/wiki/File:Bust_of_J%C3%A1nos_Bolyai_at_the_Mikl%C3%B3s_Zr%C3%ADnyi_Military_Academy.jpg
_Johann Carl Friedrich Gauss_
Artist: Christian Albrecht Jensen
License: Public Domain
Acquired: commons.wikimedia.org/wiki/File:Carl_Friedrich_Gauss_1840_by_Jensen.jpg
_Eugenio Beltrami_
Artist: Unknown
License: Public Domain
Acquired: commons.wikimedia.org/wiki/File:Beltrami.jpg
_Felix Klein_
Artist: Unknown
License: Public Domain
Acquired: commons.wikimedia.org/wiki/File:Felix_Christian_Klein.jpg
_Bernhard Riemann_
Artist: Unknown
License: Public Domain
Acquired: commons.wikimedia.org/wiki/File:Bernhard_Riemann_3.jpg
_Henri Poincare_
Artist: Unknown
License: Public Domain
Acquired: commons.wikimedia.org/wiki/File:Henri_Poincar%C3%A9-2.jpg
_Hendrik Lorentz_
Artist: Museum Boerhaave
License: Pbulic Domain
Acquired: commons.wikimedia.org/wiki/File:Hendrik_Antoon_Lorentz.jpg
_Hermann Minkowski_
Artist: Raum und Zeit
License: Public Domain
Acquired: commons.wikimedia.org/wiki/File:Hermann_Minkowski_Portrait.jpg
_Albert Einstein_
Artist: Ferdinand Schmutzer
License: Public Domain
Acquired: commons.wikimedia.org/wiki/File:Einstein_1921_by_F_Schmutzer_-_restoration.jpg
*Videos*
All video clips are short and used under Fair Use as this work is transformative in each case.
_Star Trek IV: The Voyage Home_
Copyright Paramount Pictures Corporation
Acquired: yarn.co/yarn-clip/fd1c3db4-8966-4c1a-b2d0-12f2987f7851 and yarn.co/yarn-clip/242688c2-6dee-48e2-89fd-29485f7526b3
_The Flash (TV Series)_
"_The Darkness and the Light_"
Copyright Warner Bros. Entertainment, Inc.
Acquired: yarn.co/yarn-clip/3cf6cd52-945b-4934-8b8b-acf80bdfb2c7
_Justice League_
Copyright Warner Bros. Entertainment, Inc.
Acquired: yarn.co/yarn-clip/a70a67af-8dc5-46bc-bb17-c57ce2813636
_Avengers: Infinity War_
Copyright Marvel
Acquired: yarn.co/yarn-clip/e8dba66a-db48-49dd-a172-5d4507822d4a
_Despicable Me 2_
Copyright Universal Pictures
Acquired: yarn.co/yarn-clip/fc26c065-61cf-4844-a951-033164bffa6b
_Star Trek: The Next Generation_
"_Sarek_"
Copyright CBS Studio Inc.
Acquired: yarn.co/yarn-clip/02c82a09-7335-4f8d-92ce-4dd07f821acf
_The BEST FALLS on Runway by Fashion Channel_
Acquired: youtube.com/watch?v=kYLE_kp6coQ
*Music*
Various Tracks from the YouTube Audio Library
