Closed geodesics in X₃

A periodic orbit of the diagonal flow on the space of unimodular 3-lattices is the Minkowski lattice of a totally real cubic field, deformed by a totally-positive unit: Lₜ = Λ·diag(μᵢᵗ) returns to Λ at t=1. Watch the lattice stretch along the three real embeddings and snap back.

Cubic field (closed orbit)
Phase t (scrub the orbit)
00.001
Playback speed
Show
This orbit
Drag to rotate · scroll to zoom. The orbit is closed (period 1); the stretch rates μᵢ=rᵢ² are the squared real conjugates, ∏μᵢ=1 so the covolume (and the Voronoi-cell volume) stay 1. Compact diagonal orbits in X₃ ↔ totally real cubic fields + ideal classes.
closed orbit · period 1 · cell volume ≡ 1