Lagrange's theorem says that a real number has an eventually periodic continued fraction expansion if and only if it's a quadratic irrational — a root of a quadratic equation with integer coefficients. The classical proof is algebraic: manipulate the recurrence relations of continued fraction convergents until periodicity falls out. It works, but it treats the continued fraction as a sequence of numbers rather than a geometric object.
Broise-Alamichel & Parkkonen (arXiv:2603.12425) prove Lagrange's theorem geometrically, and in doing so extend it to continued fractions over the complex numbers, quaternions, and octonions. The key insight: finite continued fraction expansions correspond to fixed points of parabolic transformations in the appropriate modular group, while eventually periodic expansions correspond to fixed points of loxodromic elements. The connection to quadratic equations follows from the Clifford algebra formalism that unifies all these cases.
The geometric perspective reveals why Lagrange's theorem should generalize. The theorem isn't really about real numbers or continued fractions — it's about the dynamics of Möbius transformations acting on hyperbolic space. A periodic continued fraction traces a closed orbit under a discrete group action. The orbit closes precisely when the transformation is loxodromic, which happens precisely when the fixed point satisfies a quadratic equation over the appropriate integer ring.
Moving from real to complex continued fractions (Hurwitz's version) to quaternionic ones doesn't change the underlying hyperbolic geometry — it changes the dimension of the hyperbolic space. The same dynamical argument applies in each dimension because the group structure is preserved.
The original algebraic proof obscured this. By proving Lagrange's theorem with algebra, you get the result for reals. By proving it with geometry, you get it for every division algebra simultaneously.