The Work of Maurits Cornelis (M.C.) Escher

1. The Work of Maurits Cornelis (M.C.) Escher Presented by: Tiana Taylor Tonja Hudson Cheryll Crowe “For me it remains an open question whether [this work] pertains to the realm of mathematics or to that of art.” - M.C. Escher

2. Small Group Discussion - Guiding Questions 1) Give examples of how Escher’s work can be utilized at the following levels: elementary, middle / high school, beyond. 2) Do you believe Escher’s work is more in the realm of mathematics or art? 3) How can technology be used to create Escher-inspired art? Other technology/software? 4) Knowing there are other artists/mathematicians with Escher’s vision, how can you use this information in constructing lessons in mathematics?

3. Tessellations Isometric drawings Transformations Impossible figures Non-Euclidean geometry Symmetry Polygons Limiting Idea Infinity Polyhedra Spherical geometry Topics touched by Escher

4. History of M.C. Escher • Maurits Cornelis Escher was born in Holland in 1898. • Poor student who had to repeat two grades • Architecture to Graphic Arts to professional work • Inspired by Italian architecture • Two categories: • Geometry of Space • Tessellations • Polyhedra • Hyperbolic Space • Logic of Space • Visual Paradoxes • Impossible Drawings • MC Escher died in 1972 and Snakes was his last contribution to the world of art and mathematics. Snakes (1969)

5. Artist and/or Mathematician “The ideas that are basic to [my works] often bear witness to my amazement and wonder at the laws of nature which operate in the world around us. He who wonders discovers that this is in itself a wonder. By keenly confronting the enigmas that surround us, and by considering and analyzing the observations that I had made, I ended up in the domain of mathematics. Although I am absolutely without training or knowledge in the exact sciences, I often seem to have more in common with mathematicians than with my fellow artists.” ~ M. C. Escher

6. Escher at the Elementary Level (K-5)

7. NCTM StandardsPre-K – Grade 2 Connections • Recognize 2D and 3D shapes • Investigate and predict results of putting together and taking apart shapes • Recognize and apply slides, flips, and turns • Recognize and represent shapes from different perspectives

8. NCTM StandardsGrade 3-5 Connections • Investigate, describe, and reason about the results of combining and transforming shapes • Explore congruence and similarity • Predict and describe results of sliding, flipping, and turning 2D shapes • Identify and describe line and rotational symmetry • Recognize geometric ideas and relationships and apply them in everyday life

9. Escher-Inspired Lego Designs M.C. Escher "Relativity" http://imp-world.narod.ru/art/lego/

10. Escher-Inspired Lego Designs cont. M.C. Escher "Waterfall" http://imp-world.narod.ru/art/lego/

11. Escher for the Middle and High School (6-12)

12. NCTM StandardsGrade 6-8 Connections • Examine the congruence, similarity, and line or rotational symmetry of objects using transformations • Recognize and apply geometric ideas and relationships in areas outside the mathematics classroom, such as art and everyday life

13. NCTM StandardsGrade 9-12 Connections • Analyze properties and determine attributes of 2D and 3D objects • Explore relationships among classes of 2D and 3D geometric objects • Understand and represent translations, reflections, rotations, and dilations of objects in the plane • Draw and construct representation of 2D and 3D geometric objects using a variety of tools

14. Creating Escher’s Impossible Figures Using Computer Software Impossible Puzzle 1.10 – creates impossible figures from triangles http://imp-world.narod.ru/programs/index.html

15. Creating Escher’s Impossible FiguresUsing Computer Software Impossible Constructor 1.25 – creates impossible figures from cubes http://imp-world.narod.ru/programs/index.html

16. Escher and Beyond

17. Escher and the Droste Effect • Did Escher leave the center blank on purpose? • Is it possible to render this image via computer programming? • Hendrik Lendstra & Bart de Smit • http://www.msri.org/people/members/sara/articles/siamescher.pdf • http://escherdroste.math.leidenuniv.nl/index.php?menu=intro

18. Beyond Escher • Victor Acevedo • 1977 Escher inspired pilgrimage to Alhambra in Granada Synapse - Cuboctahedronic Periphery

19. Beyond Escher • Dr. Helaman Ferguson • Sculpture • “celebrates the remarkable achievements of mathematics as an abstract art form” Thirteenth Eigenfunction on the Helge Koch fractal snowflake curve

20. Beyond Escher • Dr. Robert Fathauer • In 1993 founded Tessellations • Researcher for NASA’s Jet Propulsion Laboratory • Fractal Iteration Fractal Knots No. 1

21. Beyond Escher • Dr. S. Jan Abas • School of Computer Science at the University of Wales • Islamic Geometric Patterns: his heritage Islamic Pattern in Hyperbolic Space

22. Small Group Discussion - Guiding Questions 1) Give examples of how Escher’s work can be utilized at the following levels: elementary, middle / high school, beyond. 2) Do you believe Escher’s work is more in the realm of mathematics or art? 3) How can technology be used to create Escher-inspired art? Other technology/software? 4) Knowing there are other artists/mathematicians with Escher’s vision, how can you use this information in constructing lessons in mathematics?

23. References Abas, S. J. (2001). Islamic geometrical patterns for the teaching of mathematics of symmetry [Special issue of Symmetry: Culture and Science]. Symmetry in Ethnomathematics, 12(1-2), 53-65. Budapest, Hungary: International Symmetry Foundation. Abas, S. J. (2000). Symmetry 2000. Retrieved November 10, 2007, from http://www.bangor.ac.uk/~mas009/pcont.htm Alexeev, V. (n.d.). Programs. Retrieved November 6, 2007, from http://imp-world.narod.ru/programs/index.html Alexeev, V. (n.d.). Impossible world: Legos. Retrieved November 6, 2007 from http://imp-world.narod.ru/art/lego/ Art-Baarn, C. (n.d.). Escher and the droste effect. Retrieved November 10, 2007, from http://escherdroste.math.leidenuniv.nl/index.php?menu=intro De Smit, B. & Lenstra Jr., H. W. (2003). The mathematical structure of Escher’s print gallery, Notices of the AMS, 50(4). Fathauer, R. (n.d.) The iteration (fractal) art of Robert Fathauer. Retrieved November 10, 2007, from http://members.cox.net/tessellations/IterationArt.html Ferguson, H. (2003). Helaman Ferguson Sculpture. Retrieved November 10, 2007, from http://www.helasculpt.com/index.html Gershon E. (2007). Escher for real. Retrieved November 6, 2007, from http://www.cs.technion.ac.il/~gershon/EscherForReal/Penrose.gif Impossible Staircase (n.d.) Retrieved November 6, 2007, from http://www.timhunkin.com/ page_pictures/a119_f3.jpg

24. References Mathematics Behind the Art of M.C. Escher (n.d.). Retrieved November 6, 2007, from http://www.math.nus.edu.sg/aslaksen/gem-projects/maa/0203-2-03-Escher/ main3.html M.C. Escher Company B.V. (2007). The official M.C. Escher website. Retrieved November 6, 2007, from http://www.mcescher.com/ NNDB (2007). M.C. Escher. Retrieved November 6, 2007, from http://www.nndb.com/people/308/000030218/ Platonic Realms (2007). The Mathematical Art of M.C. Escher. Retrieved November 10, 2007, from http://www.mathacademy.com/pr/minitext/escher/index.asp Robinson, S. (2002). M.C. Escher: More mathematics than meets the eye. SIAM News, 35(8).