Matrix sequences associated with the Ducci map and the mediant construction of the rationals

The Ducci map on Rn takes a vector [x1,…,xn]T to the vector of adjacent-entry differences [|x1-x2|,…,|xn-x1|]T. This paper considers the case where n=3, with the focus being on those sequences of 3×3 matrices that can be used to carry out iterations of the Ducci map. This linear-algebraic perspective reveals an unexpected connection between the Ducci map, the process of forming mediants of rational numbers, and the Stern–Brocot tree.