Simulate and plot particles moving under gravity in a 2D universe. This should be easy. Force should probably be 1/r , not 1/r^2 as it is in 3D. One gets circular and flower-shaped periodic orbits for fine-tuned initial conditions, otherwise non-periodic Lissajous-like paths.
Planiverse by Dewdney.
For the purposes of entertainment, other exponents can also be explored.
Simulate gravity in 4D: inverse cube law. Simulation is probably easy, but depicting it is hard. Like 2D, no stable periodic orbits, but plotting a mess could still be pretty.
Motivating observation: in 2D and 3D, a collection of particles in orbits decaying due to friction forms a 2D accretion disc. What happens in 4D?
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