A Dual Approach to Triangle Sequences: A Multidimensional Continued   Fraction Algorithm
  A dual approach to defining the triangle sequence (a type of multidimensional continued fraction algorithm, initially developed in NT/9906016) for a pair of real numbers is presented, providing a new, clean geometric interpretation of the triangle sequence. We give a new criterion for when a triangle sequence uniquely describes a pair of numbers and give the first explicit examples of triangle sequences that do not uniquely describe a pair of reals. Finally, this dual approach yields that the triangle sequence is topologically strongly mixing, meaning in particular that it is topologically ergodic. 
S. Assaf, L. Chen, T. Cheslack-Postava, B. Cooper, A. Diesl, T.   Garrity, M. Lepinski, A. Schuyler