When Mathematics and Art meet, we always discover wonderful images and concepts. The thing we love the most is when this beauty spreads also to the proofs, and they become elegant.
In this video, we tell a tale about chains of tangent spheres in the three-dimensional space, with an unexpected solution, called the Soddy's hexlet.
The summer of 2022 is the "Summer of Math Exposition", thanks to the idea of 3blue1brown. People are encouraged to put content about explanations of math online. This is our entry for this contest! https://youtu.be/hZuYICAEN9Y
Edit: we are so happy for getting a honorable mention in the SoME2 contest! Thank you everybody for supporting us! https://www.3blue1brown.com/blog/some2 and https://www.youtube.com/watch?v=cDofhN-RJqg
#SoME2 #3blue1brown #summerofmathexposition
This is a nice website with many proof about circular inversions: https://artofproblemsolving.com/wiki/index.php/Circular_Inversion
This is the paper of Frederick Soddy: "The bowl of integers and the hexlet", Nature, London, 139 (3506): 77–79, doi:10.1038/139077a0.
00:00:00 - Introduction
00:01:22 - a Tale of Tangent Spheres
00:03:29 - Spherical inversions
00:07:45 - The proof
00:12:20 - Conclusions
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