Rotation and Translation of 60 degree (P3) and 90 degree (P4) Tessellations

This blog shows two examples of the translation of 60 degree rotational (P3) tessellations to 90 degree (P4) tessellations. Many symmetrical tessellation designs, that is a tessellation with mirror reflection about a central axis, are capable of being transformed from the 60 degree (P3) state to a 90 degree ( P4) form. This doesn't mean that the design will be able to be simply copied just that a form of the design can be modified from one state to another. I thought I would show a couple examples of how this works and provide some commentary on the modifications.

​Figure 1 depicts Horned Lizards as a 60 degree rotational design or a P3. The tessellation is rotated 60 degrees and repeated. Figure 2 depicts Horned Lizards as a 90 degree mirror reflection design or a P4.  The design repeats at 90 degrees as a mirror refection. Figure 3 shows Jets and Pterodactyls as a 60 degree rotational design or a P3. Figure 4 is a 90 degree reflection of the same design. 

There are many symmetrical tessellations that can manipulated in this way. Normally the approach I take is to start with a P3 (60 degree) design and overlay a it on a square grid with tracing paper. I then  stretch and create a 90 degree grid with  "mirror symmetry" and design around the center axis to complete the tessellation.

Figure 1- Horned Lizards as a 60 degree (P3) Rotational

Figure 3 - Jets and Pterodactyls as a 60 degree (P3) Rotational

Figure 2 - Horned Lizards as a 90 degree (P4) Translational

Figure 4 - Jets and Pterodactyls as a 90 degree (P4) Translation