Research Question: What would Maurits Cornelis Escher’s Regular Division of the Plane with Birds look like on the torus?
Maurits Cornelis Escher was born in Leeuwarden, Holland in 1898. He showed an interest in design and drawing, and this led him to a career in graphic art. His work was not given much recognition until 1956 when he had his first important exhibition which led him to worldwide fame. He was inspired by the math he read about and his work related to those mathematical principles. This is interesting because he only had formal mathematical training through secondary school. He worked with non-Euclidean geometry and “impossible” figures. His work covered two main areas: geometry of space and logic of space. They included tessellations, polyhedras, and images relating to the shape of space, the logic of space, science, and artificial intelligence (Smith, B. Sidney). Although Escher worked with a wide variety of art, the main focus of this paper will be tessellations. This brings me to my research question: how does Maurits Cornelis Escher’s Regular Division of the Plane with Birds relate to the tiling view of the torus?
Tessellations and the torus are related to mathematics in the areas of geometry, topology, and the geometry of space. “A regular tiling of polygons (in two dimensions), polyhedras (three dimensions), or polytopes (n dimensions) is called a tessellation.” (Weisstein, Eric W.). Tessellations, or regular divisions of the plane, cover the entire plane without leaving any gaps or overlapping (http://www.mathacademy.com/pr/minitext/escher/). The word “tessellate” comes from the Greek word “tesseres” which means four in English. This relates to tessellations
because the first ones were made of square tiles which have four sides (Alejandre, Suzanne and South, Michael). A torus is a finite figure with no boundaries. It looks much like a 3 dimensional ring, or donut. A torus can be made from a rectangular paper, by first forming a cylinder and then connecting the two ends. The shapes of the ends being connected are circles. The torus is related to the mathematical study of topology and geometry. “Topologists are mathematicians who study qualitative questions about geometrical structures.” (http://www.shef.ac.uk/nps/Wurble.html). The first topological work was done by Euler. He wrote The Solution of a Problem Relating to the Geometry of Position which was the solution to the Königsberg Bridge Problem in 1736. Today, there are several different kinds of topology, including point set topology, algebraic topology and differential topology (O’Connor, J. J. and Robertson, E. F.). I will be exploring the view of the tessellation Regular Division of the Plane with Birds when made into a torus.
Fundamental domain is the piece tfrom. The fundamental domain of the tessellation, Regular Division of the Plane with Birds (picture: http://www.mathacademy.com/pr/min/escher/bigp?IMAGE=reg_div_birds) is one pairing of the black and...