Thanks to Noether, we now know that in classical mechanics, conserved quantities come from symmetries. For a planet orbiting the Sun, conservation of energy comes from time translation symmetry. Conservation of the angular momentum comes from rotation symmetry. But there's another, more mysterious conserved quantity: the eccentricity vector, which points along the long axis of the planet's elliptical orbit.
What symmetry gives *this* conserved quantity?
Symmetry under rotations in 4-dimensional space! These include the obvious rotations in 3-dimensional space which give angular momentum. The other 4-dimensional rotations act in a much less obvious way, and give the eccentricity vector.
But how does the 4th dimension get into the game?
In this article, I explain how a planet orbiting the Sun is secretly isomorphic to a particle moving around on a sphere in 4-dimensional space. This is a nice explanation of the 4-dimensional rotation symmetry.
But the planet is also isomorphic to a *massless* particle moving at the speed of light on a sphere in 4-dimensional space! This brings relativity into the story, and sets the stage for still more shocking developments.
https://johncarlosbaez.wordpress.com/2025/07/20/the-kepler-problem-part-4/
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