3.3 Relation between Glide Reflection and Swing Translation. Let's go back to the Escher tiling with the Glide Reflection for a while. There is an alternative way to make the transformation from the light horseman to the dark one. I will illustrate this using another tiling of M.C. Escher, regular tiling No. 17 [1].
Figure 8: M.C. Escher, regular tiling nr. 17.
Another way to make the transformation from one white bird to the white bird which is in the row above is the use of Swing Translation (Figure 9, 10).
Figure 9: Swing translation.
Applying Swing Translation in a continuous way to the white bird tile will result in the weird spiral shape of Figure 10. In fact the 2D tiling is then an intersection with a plane and these spiral shapes. Note that this kind of helix can be made with the bird looking outwards and with the bird looking inwards. And the helical shapes can be entwined!
Figure 10: M.C. Escher - No. 17 helix.
Figure 11: Leonardo da Vinci - Double Helix