Transformations & Symmetry

This blog will be dedicated to two examples of transformations with symmetrical isohedral tilings or tessellations. Polya describes 17 groups that tessellate the plane and Geoffery Shepard in Structural Topology 15 (1988) describes 93 isohedral patterns and groupings for coloring tiles. My goal is to show how a symmetrical isohedral tiling can be easily transformed from one design to another. Figure 1 below depicts Jets & Pterodactlys in the same shape that is a mirror image of each tile crossing the design. The pterodactyls go right to left and the jets from left to right.  If you are able to create a design with a same shape and it tessellates the plane it just takes some creativity to transform it into another recognizable figure.

Transformation I - Jets & Pterodactyls (2016)

Figure 2 depicts a different type of tessellation that is a p4 rotation. The bird and the fish tiling is excatly the same shape but rotates 90 degrees to tile the plane.

Transformation 2 - Birds & Fish