This thesis presents a novel connection between a group of triangle transformations and the group of Lorentz transformations in spacetime R1,2. This connection is the third part of the thesis. The first part shows convergence to collinearity for a wide class of Markov chains of triangles (including iterated barycentric subdivision from the literature), where the triangles are generated by iterating i.i.d. random triangle transformations. The second part shows convergence to infinite time for a wide class of Markov chains in spacetime R1,2 (including relativistic versions of random walks on the hexagonal lattice), where the spacetime points are generated by iterating i.i.d. random Lorentz transformations.