The Mathematics of M.C. Escher's Art

A Dutch graphic, tessellation-, print-, woodcut-, lithograph-, mezzotintmaker, Maurits Cornelis Escher (1898 - 1972), is known for his impeccably construed worlds of impossibility, infinity, and  visual paradoxes, which integrate mathematics into art. Exploring the shape and logic of space, and ultimately capturing hyperbolic space on a two dimensional plane, his works use both the mathematical perfection and distorition of some of the spatial and geometric relations. 

(Relativity)

Among other concepts, Escher employed and further developed H.S.M. Coxeter and Poincare's circle model, George Pólia's seventeen plane symmetry groups, and Roger Penrose's geometric tilings and figures. Furthermore, Escher was fascinated by the Moorish Alhambra Palace in Granada, Spain, where all symmetry patterns are present. On the other hand, his works inspired many others, such as J.F. Schouten, J.W. Wagenaar, or crystallographers concerned with polychromatic symmetry. Escher's tessellations usually feature sets of invariant tiles which interlock with copies of themselves; they represent either local or global regularity, and encompass orderly, algorythmic, self-similar repetition outside traditional symmetry groups. What is more, if colouring is employed, although it is compatible with symmetry, more often than not, there is no ballance with the four-fold rotation symmetry, or four colour theorem. 

(Lizzard Square)

Escher played with the perspective, used multiple points of view, applied shifting vanishing points, manipulated shadow and light, made background become figure and vice versa. He employed infinite loops (called strange loops by Hofstadter), stairs, reflective surfaces, windows, and geometric figures, and in order to create optical illusions, or visual paradoxes, he used different kinds of reflections and symmetries, colour and shape combinations, rotations, and translations. 

(Convex and Concave)

He employed and mixed different kinds and senses of symmetry. His best-known lithograph, Drawing Hands, for instance, does not represent a literal symmetry, but balance with respect to center point, and left and right. He used an absolute, as well a non-absolute, yet dynamic symmetry of halves, i.e. the application of a perfect point symmetry and a rotation by 180 degrees called anti-symmetry, counter-change, or half-turned symmetry. Not surprisingly, one of the most important features of Escher's works is duality, i.e. the counterbalance of two opposing notions in the broadest sense of a dualistic nature of entities: one defining the other, figure becoming ground, light dissolving into shadow, etc. 

(Day and Night)

Another aspect of Escher's art is his specific depiction of order and its disruption. His favourite shapes and figures were Penrose polygons (see for instance his use of Penrose triangle in Waterfall), Necker cube (e.g. in Belvedere), Möbius Strip (e.g. in Swans), and last but not least, polyhedra (see for instance a stellated dodecahedron used in Gravity or in Order and Chaos). 

(Order and Chaos)

To sum up, Escher's lithograph print, Raptiles, represents probably all types and senses of symmetry (many of them not explicitly mentioned in the present text) which Escher put to use. The print features symmetry as ballance, duality, order, regularity, invariance, compatibility, economy, and closure.  

(Raptiles)

For further information, see:

- a video lecture entitled Symmetry in the works of M.C. Escher given by Doris Schattschneider at Moravian College;

- an article by Doris Schattschneider entitled Escher: a mathematician in spite of himself based on a talk given in July 1986 at the Eugene Strens Memorial Conference on Intuitive and Recreational Mathematics held at the University of Calgary;

- Douglas Hofstadter, Gödel, Escher, Bach: an Eternal Golden Braid (New York: Basic Books, 1979).