5. Classification

5.1. Glide Rotation. For the classification of tilings we make use of the symmetry operations that are needed to map one tile of the tiling onto another tile. In flat tilings these operations are called translation, rotation, reflection and glide reflection. When we want to map a tile onto another tile in one of the cylindrical tilings shown in Figure 31 to 34, none of the flat operations will give the result we want. Neither translation, nor rotation, nor reflection, nor glide-reflection will map one tile onto another. What we need here is a combination of two operations, which are translation and rotation. While glide reflection is also a combination of two operations (translation and reflection), the most logical solution seems to be that we introduce a new operation: Glide Rotation, in literature often referred to as “screw rotation”.
Figure 36: Glide Rotation.
Figure 37: Pentagonal tiles.
5.2. Polygons. In non-flat tilings all the tiles are polygons. Curved edges are not possible because of the use of flat tiles. An edge between two flat tiles, not laying in a plane is part of the intersecting line of the two planes in which the tiles are laying, and therefore the edge is always a straight line. So another property that we could use for classification might be the number of sides of the polygon. And because convex as well as non-convex polygons can be used, the position of the non-convex angle can also be used for classification. Figure 38 shows some of the main types of tiles for non-flat cylindrical tilings. The notation is according to the way it’s being described in Heesch’s and Kienzle’s book Flächenschluss [3] in which they present the types for normal flat tilings.
Figure 38: Classification.
5.3. Type B. In the pictures 39 to 41 you can see examples of the use of tiles B-pentagonal tiles (Figure 39 and 40) and a B-hexagonal tile (Figure 41). The concept of non-planar tilings leads to new interesting structures. The method to create cylindrical tilings with flat tiles described in Section 5.1 is one of the methods I found. To create the tilings shown in Figure 37 and Figures 39 to 41 other methods had to be used.
Figure 39: Cylindrical tiling - pentagons.
Figure 40: Top view.
Figure 41: Cylindrical tiling - hexagons.
Figure 42: Convex pentagons - Type C.
We will use the classification ‘Type C’ for tilings in which the tiles are convex polygons. Figure 42 shows a cylindrical tiling of Type C with the use of convex pentagons.