Periodic lattice flow

A closed orbit of the diagonal flow on unimodular 3-lattices: the Minkowski lattice of a totally real cubic field, stretched along the three real embeddings by a totally-positive unit. Lₜ = Λ·diag(μᵢᵗ) returns to Λ at t = 1.

Lattice — closed orbit (cubic field)
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Phase t (scrub the orbit)
00.001
Flow speed
Radius
Orb size
Color-loop speed
Color band density
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This orbit
The orbit is closed (period 1); μᵢ = rᵢ² are the squared real conjugates of a unit, ∏μᵢ = 1, so the covolume stays 1. Each glowing orb is a lattice point; the color loops as the flow plays. Origin view puts you at the center looking out; external view orbits the lattice. Drag to rotate · scroll or pinch to zoom.
closed orbit · period 1 · covolume ≡ 1