  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.
