- Visions of Symmetry, pg. 31-44, 77-78.
Relevant examples from Escher's work:
- Escher's sketches of Polya's 17 wallpaper patterns. Visions of Symmetry, pg. 24-26.
A wallpaper pattern is a plane figure which has more than one direction of translation symmetry. Most actual wallpaper fits the bill, because it must repeat left to right in order to hang in strips, and it must repeat vertically so that multiple strips can be cut from the same roll.
The lattice of translations.
The Seventeen Wallpaper Groups
Escher's Use of Symmetry
Escher's Regular Division of the Plane Drawings served as source material for his finished printed works. The vast majority of these fall into one of seven symmetry groups: p1, p2, p3, p4, p6, pg, and pgg. These are exactly the symmetry groups which have no reflection symmetry - only translation, rotation, and glide reflection. If two creatures meet on a line of mirror symmetry, they must have a flat edge, and recognizable figures from life rarely have perfectly straight edges. Because of this, Escher mostly avoided mirror symmetry, although he did create a few drawings where bilateral symmetry of the motif leads to overall mirror symmetry of the pattern.
Some animal motifs, generally larger animals, are usually seen from the front or side, and so look silly when viewed upside down or at an angle. Escher's was careful, at least in his later work, to avoid symmetries containing rotation when working with such animals. On the other hand, Escher writes :
When a rotation does take place, then the only animal motifs which are logically acceptable are those which show their most characteristic image when seen from above.
For example, insects and lizards occur frequently in Escher's work when rotation symmetry is present.
- Proof of the 17 wallpaper group classification by George Baloglou.
- Tilings by Dror Bar-Natan.
- The 17 wallpaper groups (animated) by Hop.
- The 17 wallpaper patterns in traditional Japanese art at Mathematics Museum (Japan).
- Wallpaper groups by David Joyce.
- Visions of Symmetry, pg. 78