Math and the Art of M. C. Escher

Preface

The 20th century Dutch artist Maurits Cornelis Escher was a master printmaker, whose works were heavily infused with ideas from mathematics. This textbook is intended to support a mathematics course at the level of college algebra, with topics taken from the mathematics implied by Escher's artwork.

This textbook is intended to be supplemented by one or more printed books about Escher. In particular, Visions of Symmetry contains essential complementary material on Escher's life and mathematical approach to symmetry, and Magic of M.C. Escher is an excellent art book that provides a much better view of Escher's artwork and technique. It is important to remember that artwork viewed on the web is usually small, grainy, and has poor color reproduction.

Throughout the text you will find Explorations, which are short investigative projects to be done in groups in a class setting. Explorations are intended to introduce material in a way that motivates and prepares the student for the more rigorous approach in the body of this text.

|- valign="top" style="height:80px;" | width="150" | I. Introduction

1. M.C. Escher: Life. Work.
2. Fundamental Concepts: Classifications. Plane Geometry.

|- valign="top" style="height:80px;" | width="150" | II. Symmetry

1. Introduction to Symmetry: Reflections, rotations, and rosette patterns. Color symmetry.
2. Frieze Patterns: Translations, glide reflections, and frieze patterns. Isometries.
3. Wallpaper Patterns: Lattices. The 17 groups. Classification flow chart. Escher's use of symmetry.

|- valign="top" style="height:80px;" | width="150" | III. Tessellations

1. Introduction to Tessellations
2. Tessellations by Polygons: Regular tessellations. Triangle tessellations. Quadrilateral tessellations. Convex polygon tessellations.
3. Tessellations by Recognizable Figures: Escher's polygon systems. Techniques for tessellation.

|- valign="top" style="height:80px;" | width="150" | IV. Similarity and Fractals

1. Similarity Transformations: Similarity. Dilation. Iteration.
2. Fractals: Self-similarity. Fractals.

|- valign="top" style="height:80px;" | width="150" | V. Non-Euclidean Geometry

1. Introduction to Non-Euclidean Geometry
2. Spherical Geometry: Geodesics. Angle sum and area. Tessellations. Platonic solids. Duality. Euler characteristic.
3. Hyperbolic Geometry: The Poincaré Disk. Hyperbolic tessellations.
4. The Three Geometries: The classification of regular tessellations. Axioms in geometry. The shape of the universe.

|- valign="top" style="height:80px;" | width="150" | VI. Topology

|- valign="top" style="height:80px;" | width="150" | VII. Art and Perception

1. Depth and Perspective: Depth in art. Linear Perspective. 2D vs. 3D. Impossible Figures.
2. The Fourth Dimension: Dimension. 4D as space. 4D as time. Fourth dimension in art and literature.

|- valign="top" style="height:80px;" | width="150" | Index of Escher Artwork

|- valign="top" style="height:80px;" | width="150" | Index of Explorations

|- valign="top" style="height:80px;" | width="150" | Index of Exercises

|- valign="top" style="height:80px;" | width="150" | Index of Fieldtrips

|- valign="top" style="height:80px;" | width="150" | References

Authors

Anneke Bart and Bryan Clair are professors in the Department of Mathematics and Computer Science at Saint Louis University.

They have been teaching the Math and the Art of M.C. Escher course since its inception in 2000.