Which theorems from classical theory of continued fractions have 3(or multi) dimesional analogs?
Of cause classical one is a periodicity of Klein polyhedra. Probably there are some more...
Which theorems from classical theory of continued fractions have 3(or multi) dimesional analogs? Of cause classical one is a periodicity of Klein polyhedra. Probably there are some more... 


(1) There is 3D analog of Vahlen's theorem, see http://link.springer.com/article/10.1007%2Fs1100600600186?LI=true (2) 3D isolation theorems and extemal Davenport forms (see Cassels "An Introduction to the Geometry of Numbers" and SwinnertonDyer, "On the product of three homogeneous linear forms" Acta Arith., 1971, 18, 371385). The key role here play numbers $2\cos\frac{2\pi}7$, $2\cos\frac{4\pi}7$, $2\cos\frac{6\pi}7$ (3D Golden Ratios). 

