Geometry In Escher

The work of Maurits Carnelis Escher (M.C. Escher) is widely considered the most popular example of the mathematical influence in art. Though never formally trained in math, Escher's initial interest in decorative art sparked a curiosity in certain mathematical areas such as geometric shapes, tessellations and spatial planes/demensions. His interest in both aesthetic and logic resulted in provoking visual representations of multiple dimension. Escher's understanding of mathematics in combination with his artistic skill provides a rare translation between the seemingly separate languages of math and art. "In mathematical quarters, the regular division of the plane has been considered theoretically . . . Does this mean that it is an exclusively mathematical question? In my opinion, it does not. [Mathematicians] have opened the gate leading to an extensive domain, but they have not entered this domain themselves. By their very nature thay are more interested in the way in which the gate is opened than in the garden lying behind it." (M.C. Escher on tessellations viewed at Alhambra www.mathacedemy.com)

The variety of math used in his body work extends from basic geometric shape to hyperbolic geometry; though there is no need to cover such a wide subject range to explain mathematic influence. Escher was first inspired by the gridded tile patterns, designed in the 14th Century by the Moors, at the Alhambra castle in Granada. Escher was fascinated by the idea of dividing the plane with geometric shapes. Tassellations, the arrangement of closed shapes that do not overlap or allow gaps, became a staple in his work. Typically tassellations are created with regular shapes, such as polygons. Escher was inspired by these patterns and the richness they added to a two dimensional surface, though also understood the geometric concept of three dimensions. Escher was interested in translating the concepts, and line drawings, of space beyond two dimensions to a believable visual depiction. He used many shapes, regular and irregular, that along with a use of certain isometries such as translation and rotation. On the Euclidian plane, N-fold rotational symmetry states that with a particular point (2-D) or axis (3-D) rotation of 360/n the shape/object will not change. Briefly considering some basic variables:

the notation for n-fold symmetry is Cn, or solely "n"

the actual symmetry group is specified by the point or access of symmetry, together with the "n"

For each point or axis of symmetry the abstract group type is cyclic group Zn of order n,

Some groups have the same geometric and abstract notations, in which case distinguish the difference

The fundamental domain (smallest part of an object/pattern, which based on symmetry, determines the whole object/pattern)

is a sector of 360/n.

*Example without additional reflection symmetry: n = 2, 180 (an example of this would be, yin and yang)

*Cn is the rotation group of a regular n-sided polygon in 2-D

Note: 1-fold is no symmetry and 2-fold is the most basic symmetry.

This isometries determined in this equation alone allowed for a wide variety of patterns; combining additional isometries, such as reflection and gliding reflection, is how Escher created such complex patterns. Going even one step further, Escher created what he called a metamorphosis. He would progressively alter the shapes, letting them slowly distort an interact with each other, and eventually transform/reform into new shapes. His distortions obeyed the 3, 4 or 6-fold symmetry of the original pattern in order to maintain the tassellation. As seen in 'Development 1' , using a number of calculated isometries he progressively transformed a regular polygon into a lizard while maintaining the tassellation. After this technique developed, he added a third dimension; doubling the equations used as well as adding additional geometric and abstract group types. 'Reptiles' is an amazing example of Escher's perfected calculation and technique. With only just partial comprehension of the mathematical genius behind this lithograph, the ease of the lizards crawling in and out of separate dimensions is even more astonishing. This image proves that a three dimensional plane can appear to exist in a two dimensional plane accurately illustrated geometry.

Geometric shapes were a major interest for Escher appearing in many of his works. The most popular were the Platonic solids, the only five convex regular polyhedrons. These shapes are marveled for unique dimensions: all points (plus every point along the straight line segments connecting them) are within the objects, as well as identical faces with same number of faces around each vortex. In translucent wood work,'Four Regular Solids,' Escher intersected four of the five Platonic solids with all their symmetries perfectly aligned allowing four solids to be seen at once. In 'Order and Chaos' he connected the shapes of the solids once again only once connected stellated the shape, replacing the faces with solid triangles, which making a strong three dimensional star shape. Another famous work with connected solids is 'Stars,' in which the solids are connected in various ways without their faces, showing an amazing series of complex connected lines. The influence of math is quite apparent in Escher's work, as it is the foundation of his subject matter and also anchors his aesthetic. However, his addition of familiar subjects, such as lizards or birds, bring a unique sense of identification and curiosity. Instead of depicting only mathematical structures or influenced patterns, Escher brings common life elements into his calculated planes. Fellow mathematicians enjoyed this aspect of the work as it gave a deeper point of reference in defining the spacial