“For me it remains an open question whether [this work] pertains to the realm of mathematics or to that of art.” - M.C. Escher Contents Page 1 Introduction The Art of Alhambra Our Area of Focus Our Aim Page 2 The Principals behind Tessellations - Translation - Rotation - Reflection - Glide Reflection Page 3 Mathematics in Escher's Art - Translation - Rotation - Glide Reflection - Combination Page 4 Our Original Tessellations - The Catch (Translation) - Under the Sea (Rotation) - The Herd (Glide Reflection) - Dumbo & Butterfly (Combination) Page 5 Possible links with Architecture Conclusion References Principals of Tessellations What are tessellations? It is an arrangement of closed shapes that completely cover the plane without overlapping or leaving gaps. To comprehend how Escher creates his seemingly impossible to understand tessellations, we must first understand certain principles of tessellations. It is however important to remember to differentiate between a tiling an a tessellation. By definition, tilings require the use of regular polygons put together such that it completely covers the plane without overlapping or leaving gaps. Tessellations however, do not need the use of regular polygons, below is an animated example. To illustrate the principals behind a simple tessellation pattern, a tiling consisting of equilateral triangles of degree 6 at each vertex is used as an example to illustrate these principles. There are basically 4 ways of how a diagram can be “mapped onto” itself, namely , by translation, rotation, reflection and glide reflection.We will discover in the following diagrams, how this design can be “mapped onto” itself, which is the fundamental idea of how tessellations work TRANSLATION Translation is the moving of a pattern over a certain distance, such that it coincides and cover the underlying pattern again. ROTATION Rotation similarly is the rotation of a pattern at a fixed origin and fixed angle such that it covers the underlying pattern also.   REFLECTION Reflection is the mapping of a pattern by “mirroring” the image with respect to an axis of reflection. The image is therefore a mirror image of the original pattern. GLIDE REFLECTION Glide reflection is basically a combination of translation and reflection of the diagram. The abovementioned ways are the possible shifts whereby a pattern can be made to mapped onto itself. There are some patterns that only allow for translation and others that may allow for a combination of other types of symmetry. Escher has discovered that there are as many as 17 different patterns that could be subjected to such “mapping” onto itself. However, we will only study simple patterns such as squares, triangles and hexagons for easy understanding of how his artworks are created. Nonetheless, such patterns still offers us endless possiblities of tessellation artworks that could be created by the 4 simple mathematical principles. next page>