Escher’s Tessellations: Symmetry and Groups Expressed as Art

Artist M. C. Escher’s work, “Regular Division of the Plane,” extensively uses different tessellations that are altered into various drawings.  These tessellations, or shapes that tile a plane, tended to be constructed out of altering various polygons, such as triangles and squares.  In one technique, Escher would find polygons that tile a plane, and then alter them to give them forms of animals or patterns, which would result in a form of art.

Figure 1. A method of creating tiles that cover the plane. The base form is a square, and any changes that were done to one edge are done opposite on the adjacent edge [1]

On its surface, this may just be a technique an artist used to create his works. However, these works exhibit various mathematical concepts within.  The first of these concepts is symmetry.  Each of Escher’s tile drawings exhibit at least one of the four symmetries: reflection, rotation, glide and translation.  He began studying the seventeen different plane symmetry groups extensively, as introduced to him by a paper from the mathematician Pólya, in order to produce many of these tilings [2, p. 708]. A symmetry group is the set of all isometries, or a translation that preserves distance, that map a pattern onto itself.  The seventeen groups that Escher studied and emulated are also known as the two-dimensional crystallographic groups, as periodic patterns are classified by their symmetry groups in the same way crystallographers classify crystals [3, p. 441].  Escher’s artwork clearly displays various concepts of the symmetry groups, and as a result it has used to teach symmetry and translation concepts, as well as a source of discovery, such as when beginning to look at color symmetry groups [2, pp. 714, 716].  The deep rooted concept of symmetry shows the technical mathematical nature of the various pieces within “Regular Divisions of the Plane,”  despite Escher’s personal investigations tending to focus on creating art, which is arguably a form of mathematics in its own right [2, p. 706].

Figure 2. The seventeen plane symmetry groups [3]

Beyond these symmetry groups, more general sub groups can be analyzed within Escher’s artwork.  Using a fixed prototile, the symmetry group is used to produce the whole tiling.  Because a symmetry group includes a number of isometries in order to produce the entire tiling, one can deduce that subgroups of this symmetry group fix certain edges or vertices of the tilings [4, p. 36].  This also applies to the colorings of the tiles, as different symmetries produce different permutations, or arrangements of a set, for a given tessellation.

The use of these subgroups is how mathematicians came up with fractal salamanders to tile a plane.  Using one of Escher’s symmetry drawings that made use of salamanders, mathematicians recreated the design with self-similar fractal “salamanders” [5]. Though they may not be as aesthetically pleasing, these salamanders were systematically created by different subgroups and properties of symmetry, mathematically producing the same form Escher was working with.  Even further, these fractal salamanders are self-similar, which means they are made up of themselves, as subdivisions of these sets produce the same set.  Using Escher’s work as a guide, mathematicians were able to emphasize concepts of group theory to a greater extent, while preserving the artistic nature of the various tilings.  Although perhaps not the original intention, Escher’s artwork has presented numerous concepts in mathematics, showing his contributions as not only an artist, but as a mathematician as well.

Figure 3. A fractal salamander, along with a depiction of its self-similarity [5]

References:

[1] Haak, Sheila. “Transformation geometry and the artwork of MC Escher.” Mathematics Teacher 69.8 (1976): 647-652.

[2] Schattschneider, Doris. “The mathematical side of MC Escher.” Notices of the AMS 57.6 (2010): 706-718.

[3] Schattschneider, Doris. “The plane symmetry groups: their recognition and notation.” The American Mathematical Monthly 85.6 (1978): 439-450.

[4] Senechal, Marjorie. “The algebraic escher.” Structural Topology 1988 núm 15(1988).

[5] Gelbrich, Götz, and Katja Giesche. “Fractal escher salamanders and other animals.” The Mathematical Intelligencer 20.2 (1998): 31-35.