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.

Euclidean and Non-Euclidean Geometry

• Introduction to Mathematics and M.C. Escher
  • M.C. Escher: Life. Work.
  • Escher on Display A collection of photographs of Escher art that can be found in the Netherlands. The art represented ranges from items in the Escher museum to pillars, facades, etc in buildings
  • Fundamental Concepts: Classifications. Plane Geometry.

• Symmetry and Isometries
  • Introduction to Symmetry: Reflections, rotations, and rosette patterns. Color symmetry.
  • Frieze Patterns: Translations, glide reflections, and frieze patterns.
  • Wallpaper Patterns: Lattices. The 17 groups. Classification flow chart. Escher's use of symmetry.
  • Isometry Groups: Isometries. Composition. Multiplication tables. Groups of isometries.
    • Fieldtrip: Seeking Symmetry

• Tessellations
  • Introduction to Tessellations
  • Tessellations by Polygons: Regular tessellations. Triangle tessellations. Quadrilateral tessellations. Convex polygon tessellations.
  • Tessellations by Recognizable Figures: Escher's polygon systems. Techniques for tessellation.
  • Aperiodic Tessellations Random tessellations and Penrose tessellations.
    • Project: Tessellation Art Project

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

Further topics in Geometry and Mathematics

• Similarity and Fractals
  • Similarity Transformations: Similarity. Dilation. Iteration.
  • Fractals: Self-similarity. Fractals.

• Art and Perception
  • Depth and Perspective: Depth in art. Linear Perspective. 2D vs. 3D. Impossible Figures.
  • The Fourth Dimension: Dimension. 4D as space. 4D as time. Fourth dimension in art and literature.
    • Project: Flatland and the Fourth Dimension
    • Project: Art and Mathematics Project
    • Fieldtrip: The Saint Louis Art Museum

• Topology
  • The Mobius Band and Other Surfaces
  • Knot Theory

• History of Mathematics
  • Egyptian Mathematics: Introduction A discussion of the history of mathematics as well as an explanation of the Egyptian number system.
  • Egyptian Mathematics: The mathematical papyri
  • Egyptian Mathematics: The Pyramids

• Education
  • Elementary Education (K-8) NCTM standards and some resources.
  • Secondary Education (9-12) NCTM standards and some resources.

Resources

• Index of Escher Artwork

• Index of Explorations

• Index of Exercises

• Index of Fieldtrips

• Index of Projects

• References

Course Information

• SLU MATH 124: Math and Escher - Fall 2008 - Dr. Anneke Bart

• SLU MATH 124: Math and Escher - Fall 2008 - Dr. Steve Harris

• SLU MATH 124: Math and Escher - Spring 2008 - Dr. Bryan Clair

Instructor Resources

Instructor Resources

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. Drs. Bart and Clair created most of the materials for the Escher Wiki. Further important contributions have been made by Dr. Steve Harris - a colleague from Saint Louis University.

Contributions by others are welcomed. We are always interested in further development of this online text.

• Policies and Guidelines for Editors

Copyright

This work © 2008 Anneke Bart and Bryan Clair. Others may use these materials in the classroom without asking for permission. If materials are used in the classroom or for other educational purposes, we would appreciate a link to our page from your website.

Many images in this work are free of copyright and are available at Wikimedia Commons. The image page for these images will contain the appropriate link.

All M.C. Escher works © Cordon Art BV - Baarn - the Netherlands. All M.C. Escher works (c) 2007 The M.C. Escher Company - the Netherlands. All rights reserved. Used by permission. www.mcescher.com