Math and the Art of M.C.Escher

MT A124

Old style Geometry Courses

• Start with theory

• Often starts with polygons: triangles, squares, etc.

• Talk about SAS theorems, and similar topics

• Few applications

This Geometry Course

• Learn geometry in context

• We start with the applications and examples.

• Discover patterns, while developing our intuition

• Develop the theory last

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

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

At this point you may ask: What exactly IS geometry?

Answer:

Geometry is a systematical way of describing the world around us.

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).

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.

There are going to be a host of other topics we can look at.

• Impossible figures

• Fractals

• The fourth Dimension

• The shape of space

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.

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!

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?

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

Comment about the Reader:

• You cannot take the class without it.

• Make sure you have it by next class period at the very latest.

Your assignment:

• Go to my website http://euler.slu.edu/Dept/Faculty/ bart/bart.html

• 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.