1. Math and the Art of M.C.Escher MT A124

2. Old style Geometry Courses Start with theory Often starts with polygons: triangles, squares, etc. Talk about SAS theorems, and similar topics Few applications

3. This Geometry Course Learn geometry in context We start with the applications and examples. Discover patterns, while developing our intuition Develop the theory last

4. First topic we will cover How did Escher create these patterns? You will learn to make similar patterns, using geometric techniques These patterns are called tessellations, wallpaper patterns or sometimes tilings

5. Another topic we will discuss Here you see decorated spheres. One by Escher, the other by Jos Leys. Can we adapt what we have learned about the geometry of a flat object to the geometry of the sphere? (YES) This leads to Spherical Geometry

6. At this point you may ask: What exactly IS geometry? Answer: Geometry is a systematical way of describing the world around us.

7. Interesting fact: This means that there are different geometries (depending on what we’re describing)! We have seen: The geometry of the plane (we use this to describe the rules that govern tessellations) The geometry of the Sphere (we use this to describe what happens on a spherical object).

8. Are there any other geometries? Yes, there is a geometry, called Hyperbolic geometry. This is a picture of it. Details will have to wait until later in the course.

9. There are going to be a host of other topics we can look at. Impossible figures Fractals The fourth Dimension The shape of space

10. Impossible figures Belvedere (top) and Waterfall (bottom) are among Escher’s most famous impossible figures. After exploring the math behind it, you will be able to see how MCE made these prints.

11. Fractals Notice how each picture is made up of the same shapes, just different sizes? This is the characteristic of a fractal. You will learn how to make these yourself. And yes, this is still mathematics, believe it or not!

12. The fourth dimension Einstein came up with the idea of space-time. We live in some sense in a 4-dimensional world. Any event takes place somewhere in space (3D) and some point in time (1D more) Thinking about 4 dimensional space is a rather fun exercise. What does a 4D being see when it looks at us? What would we see if we we moved to a higher dimension?

13. Comment about the book There are 2 editions, and either one will do. Amazon.com sells the one at the top for $21 Some used copies under $20 ! Amazon.com may have the old edition as a used book. The price ranges: $8-25

14. Comment about the Reader: You cannot take the class without it. Make sure you have it by next class period at the very latest.

15. Your assignment: Go to my website http://euler.slu.edu/Dept/Faculty/ bart/bart.html http://euler.slu.edu/Dept/Faculty/ Link to the Escher page. Go to the lower end of the page and explore the links provided for the different topics. Give me a list of the 3 topics you most want to learn about. Give some idea why.