A little bridge between M.C. Escher and Leonardo da Vinci

The regular division drawings of M.C. Escher are considered an important part of his artistic work. He made about 150 basic drawings of regular divisions, some of which were used later in his prints. In almost all of these drawings, it is the tile, the motif, that plays the leading role. However, there are a few exceptions. In his own definition of regular division of the plane, given in Regelmatige vlakverdeling [2, p.94] Escher says that the tiles should fit tightly together on all sides, so that there is no space between them. In other words, the joint, the grout, the layer of mortar used by bricklayers to cement each stone to an adjacent stone, separates them in practice, but can theoretically be reduced to nothing. Mathematicians would call these joints "edges" of the tiling; edges are never considered to have any width.

We can say that this is the mathematical point of view. From an artistic point of view, the separating-lines between tiles will always be there; we can’t ignore them. We can give these boundaries more attention and even go so far as to omit the tiles. What we have then is just a grid of joints, connected in some regular way, or a latticework: a plane with a lot of carefully outlined holes. Chinese windows and screens often display such latticework; a nice collection of such designs can be found in reference [5].

In a few of Escher’s sketches these lines that separate the tiles indeed seem to take over the leading role. For example, this can be seen in his regular division drawing number 11 (Baarn ‘42) in his Abstract Motif Notebook [4, p. 87] and in his regular division drawing number 133 (Baarn ‘67) [4, p. 226], which has been redrawn by computer in Figure 6. At first glance, this focus on the space between the tiles may seem to be only a slight shift of attention, but it opens up a vast area of artistic possibilities for which Escher might not had have the time to investigate. If we use Escher's own metaphor about wandering in his beautiful garden of regular division, it’s like discovering the gate to another garden adjoining it.
And like a real garden, no discription can substitute for seeing the blooms; these explorations are shared primarily through pictures.