  This paper is about turedos, which are Turing machine whose head can move in
the plane (or in a higher-dimensional space) but only in a selfavoiding way, by
putting marks (letters) on visited positions and moving only to unmarked,
therefore unvisited, positions. The key parameter of turedos is their lookup
radius: the distance up to which the head can look around in order to make its
decision of where to move to and what mark to write. In this paper we study the
hierarchy of turedos according to their lookup radius and the dimension of
space using notions of simulation up to spatio-temporal rescaling (a standard
approach in cellular automata or self-assembly systems). We establish that
there is a rich interplay between the turedo parameters and the notion of
simulation considered. We show in particular, for the most liberal simulations,
the existence of 3D turedos of radius 1 that are intrinsically universal for
all radii, but that this is impossible in dimension 2, where some radius 2
turedo are impossible to simulate at radius 1. Using stricter notions of
simulation, intrinsic universality becomes impossible, even in dimension 3, and
there is a strict radius hierarchy. Finally, when restricting to radius 1,
universality is again possible in dimension 3, but not in dimension 2, where we
show however that a radius 3 turedo can simulate all radius 1 turedos.
