  The study of the set of limit roots associated to an infinite Coxeter group
was initiated by Hohlweg, Labb\'{e} and Ripoll and further developed by Dyer,
Hohlweg, P\'eaux and Ripoll. The Davis complex associated to a finitely
generated Coxeter group $W$ is a piecewise Euclidean CAT(0) space on which $W$
acts properly, cocompactly by isometries. The one skeleton of the Davis complex
can be identified with the Cayley graph of $W$. In this paper we define a
natural map from the Davis complex into the normalised imaginary cone of a
based root system.
