Ants and bracket generating distributions in dimensions 5 and 6.

We consider a mechanical system of three ants on the floor, in two situations. In the first situation ants move according to Rule A, which forces the velocity of any given ant to always point at a neighboring ant; in the second situation ants move according to Rule B, which forces the velocity of every ant to be parallel to the line defined by the two other ants. We observe that Rule A equips the 6-dimensional configuration space of the ants with a structure of a homogeneous distribution, and that Rule B foliates this 6-dimensional configuration space onto 5-dimensional leaves, each of which is equipped with a homogeneous distribution. The symmetry properties and the local invariants of these distributions are determined. In the case of Rule B we study and determine the singular trajectories (abnormal extremals) of the corresponding distributions. We show that these satisfy an interesting system of two ODEs of Fuchsian type.