  Representations of planar triangulations as contact graphs of a set of
internally disjoint homothetic triangles or of a set of internally disjoint
homothetic squares have received quite some attention in recent years. In this
paper we investigate representations of planar triangulations as contact graphs
of a set of internally disjoint homothetic pentagons. Surprisingly such a
representation exists for every triangulation whose outer face is a 5-gon. We
relate these representations to five color forests. These combinatorial
structures resemble Schnyder woods and transversal structures, respectively. In
particular there is a bijection to certain alpha-orientations and consequently
a lattice structure on the set of five color forests of a given graph. This
lattice structure plays a role in an algorithm that is supposed to compute a
contact representation with pentagons for a given graph. Based on a five color
forest the algorithm builds a system of linear equations and solves it, if the
solution is non-negative, it encodes distances between corners of a pentagon
representation. In this case the representation is constructed and the
algorithm terminates. Otherwise negative variables guide a change of the five
color forest and the procedure is restarted with the new five color forest.
Similar algorithms have been proposed for contact representations with
homothetic triangles and with squares.
