(5) Combined use of symmetry operations

Shapes of human footprints and orders of leaves are created by the operation "advancing reflection" which simultaneously uses parallel movement and mirror reflection (The "advancing reflection" is sometimes called "slipping reflection" or "glide reflection" in Mathematics).

The shape in the left of Figure 293 will be bilateral symmetry if it is cut vertically at the center and shift slightly as shown in the right of the same figure. In other words, it is known that "advancing reflection" consists of two operation of "translation" (shift) and "mirror reflection".

This way, new symmetries can be created by combining "translation", "rotation", "mirror reflection", and "extension". The combination will be 13 in the case of Wolf (Figs. 281-a, b).

In summary,
           Independent use of the basic operations
                      "translation" (t), "mirror reflection" (s), "rotation" (d), "expansion" (h).
          Combined use of the two operations
                      "translation + rotation" (td), "translation + mirror reflection" (ts), "rotation + 
                        mirror reflection" (ds), "translation + expansion" (th), "rotation + expansion" (dh), 
                      "mirror reflection + expansion" (sh).
          Combined use of the three operations
                      "translation + rotation + expansion" (tdh), "translation + mirror reflection + 
                       expansion" (tsh), "rotation + mirror reflection + expansion" (dsh).
These will give us a strong hint to create shapes.
 
 

191   193   292   294   Figure 291: Symmetry of "advancing reflection" by Mitsuko Shiratori. Figure 292: Symmetry of "rotation + translation". Figure 293: Explanation of operations of mirror reflection and translation included                         in the "advancing reflection".  Figure 294: Symmetries including "rotation", "expansion",  and  "mirror reflection".

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