Does anyone here know of any theorems that relate the curve of a cycloid [1] to the curve of the horopter [2] (in particular, the empirical horopter)?
In my continued quest to connect curious properties of the cycloid, I noticed hints of potential correspondences between the two curves beyond just their shape, particularly in the ways the curves of the cycloid and horopter both relate to the path of light.
A few years back, Grant did a 3Blue1Brown video with with Steven Strogatz on the cycloid Brachistochrone curve : https://www.youtube.com/watch?v=Cld0p3a43fU
And Vsauce did one with Adam Savage on the Brachistochrone [3] where they build a mechanical model of one that shows it's the fastest/optimal path among different curves, and their experiment also shows the cycloid Tautochrone [4] invariant property where objects begin up the curve at different distances apart and yet all arrive together simultaneously in constant time. https://www.youtube.com/watch?v=skvnj67YGmw
Some other interesting properties and places the cycloid shows up...
* The arclength of the cycloid curve is 8R, a rational value given a rational radius.
* The shape of the closed universe [5]. While we don't yet know if the shape of the universe is open or closed, we do know that if the universe is closed the shape of its evolution is precisely the shape of a cycloid.
* Spinors [6], octonions, and the epicycloid [7]. Electrons, protons, neutrinos, and quarks are spinors. The Rolling Spinor is like a ball, but "thanks to the 'double' in the double cover SU(2)→SO(3), a 360∘ rotation does not act like the identity. Instead, we need to rotate by 720∘ degrees to get back where we started" [8]. The two balls have a 3-to-1 ratio, and the path traced out around the larger ball is an epicycloid.
[1] Cycloid https://en.wikipedia.org/wiki/Cycloid
[2] Horopter https://en.wikipedia.org/wiki/Horopter , Hering–Hillebrand deviationn https://en.wikipedia.org/wiki/Hering-Hillebrand_deviation
[3] Brachistochrone curve https://en.wikipedia.org/wiki/Brachistochrone_curve
[4] Tautochrone curve https://en.wikipedia.org/wiki/Tautochrone_curve
[5] MIT 8.286 The Early Universe: Introduction to Non-Euclidean Space [video] https://www.youtube.com/watch?v=YfbXB_MSkSY
[6] Spinor https://en.wikipedia.org/wiki/Spinor
[7] Epicycloid https://en.wikipedia.org/wiki/Epicycloid
[8] G2 and the Rolling Ball https://golem.ph.utexas.edu/category/2013/06/g2_and_the_roll...
Split Octonions and the Rolling Ball, Dr. John Baez [video] https://www.youtube.com/watch?v=xvflQcHT5C4
Eric Weinstein explains Gauge Symmetry [video] https://www.youtube.com/watch?v=2xiEEtoa-_4