A poem from the 3rd century BC, attributed to Archimedes, about counting the number of cattle in a Sicilian herd wasn’t fully solved until the 20th century.
**Errata** At 03:13, there is a typo in the second equation from the top (the black cows). It should say 9,717/10 + 143/2800*{sigma}, not 9,717/10 + 143/1,260*{sigma}. The subsequent steps are correct. **
Time Stamps:
00:00 - Intro
00:51 - Part I: The Bulls
02:36 - Part II: The Cows
04:31 - Part III: The White and the Black Bulls
05:38 - Part IV: The Yellow and the Dappled Bulls
07:33 - History of Pell Equations
09:12 - Path to a Solution
10:36 - Archimedes
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Some useful resources about the Cattle Problem of Archimedes:
Archimedes related web site maintained by Chris Rorres: https://math.nyu.edu/Archimedes/Cattle/Statement.html
@numberphile video on the Cattle Problem: https://youtu.be/dNxyFtqcNss?si=Nj9uMMpSHL5ZlA43
Video by @OliverThomas1 explaining methods to solve the problem: https://youtu.be/qn_mJzm5QII?si=y24GZyl5f0bKqqWc
“Archimedes’ Cattle Problem” - Ilan Vardi - https://math.nyu.edu/Archimedes/Cattle/cattle_vardi.pdf
“A Simple Solution to Archimedes’ Cattle Problem” - Antti Nygren - https://math.nyu.edu/Archimedes/Cattle/isbn9514259327.pdf
Presentation by Alex Walker about the Cattle Problem, which includes the hypothesis about Apollonius of Perga: https://awwalker.com/wp-content/uploads/2018/02/homslides_7_cattle_problem.pdf
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Music from Epidemic Sound
Images of mathematicians are from Wikimedia Commons
**A Note about how I use AI generated images in my videos**
The emerging ability of artificial intelligence to generate compelling images from text prompts opens new possibilities for compelling storytelling. However, when mixed with real historical imagery, it has the potential to create confusion, or worse, if not handled properly. Additionally, there is important discourse about issues of ethics, intellectual property, etc surrounding AI imagery.
I have used Midjourney AI to create “stock” image elements including backgrounds, illustrations, and objects. I do this because it allows much greater specificity than is possible with stock image services (you need a white cow and bull, a yellow cow and bull, a spotted cow and bull, and a black cow and bull, and you want them to be in an "old" painted style? You got it!).
Additionally, in this video I used AI generated images of what we popularly imagine Archimedes to look like. Not only are there (obviously) no photographs of Archimedes, there are no surviving images of Archimedes whatsoever. Most of the modern imagery of him is based on a statue, located in the Naples, Italy Archeological Museum. The problem is that scholars eventually figured out that this was actually a statue of ArchiDAMUS, a 4th century king of Sparta, not ArchiMEDES.... All of this being the case, I see no risk of misinforming or creating historically inaccurate images in generating his likeness using AI (after all, it is just as "inaccurate" as any other drawing or painting that exists).