In the present paper, we study the (1,1,1)-discrete surfaces introduced in {Jam04]. In \[Jam04], the (1,1,1)-discrete surfaces are not assumed to be connected. In this paper, we prove that assuming connectedness is not restrictive, in the sense that, any two-dimensional coding of a [1,1,1]-discrete surface is the two-dimensional coding of both connected and simply connected ones. In the second part of this paper, we investigate a particular class of discrete surfaces: those generated by infinite smooth words. We prove that the only smooth words generating such surfaces are K_{\{1,3\}}, 1K_(1,3) and 2K_(1,3)$, where K_(1,3)=33311133313133311133313331 is the generalized Kolakoski's word on the alphabet {1,3}.