Circle Limit Exploration

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Objective: Explore Escher's Circle Limit prints to develop an intuition for hyperbolic geometry

Recall that in Spherical Geometry we determined that shortest distance was measured by great circles. We called the distance minimizing curves geodesics, and these geodesics play the same role as lines do in Euclidean Geometry.


In Circle Limit I, the red lines are meant to represent straight lines in this new geometry. These curves are also called geodesics. Note that the geodesics in hyperbolic geometry are either straight lines through the cented of the disk or semi-circles that meet the edgeof the disk in a right angle. Segments of these geodesics will form the sides of polygons. The polygons in hypebolic geometry will look "pinched" to our Euclidean eyes.

Keep in mind that in this geometry the fish are all the same size. They look different to us, but if we were to use a hyperbolic ruler we would find that every fish in this tessellation is exactly the same size.

  1. What type of polygons do you see in this figure? NOTE: In this picture we should interpret all fish as having the same size. A smaller figure does not mean that it is physically smaller. Think of it as being farther away.
  2. Looking at the angles in a triangle, what will the angle sum be? Equal to, greater than or less than 180 degrees? Why?
  1. What is the highest degree of rotation?
  2. What is the underlying geometric tessellation?

Circle Limit II by M.C.Escher

Copy of Circle Limit II by Doug Dunham Circle Limit IV

  1. What is the highest degree of rotation?
  2. Draw geodesics in this figure. What is the underlying geometric tessellation?

Circle Limit IV by M.C.Escher

  1. What is the highest degree of rotation? What other degrees of rotation are present?
  2. Draw geodesics in this figure. What is the underlying geometric tessellation?
  3. Draw a geodesic NOT passing through the center point.
  4. How many geodesics can you draw through the center point, so that the new geodesic does not meet the geodesic you picked? Another way to ask the same question: How many geodesics pass through the point so that the new geodesic is parallel to the first geodesic?

A Separate Note About Circle Limit III

File:Circle Limit III
Circle Limit III

This circle limit picture is quite interesting. At first we might think that the white lines represent geodesics, but they actually do not. It takes a little bit of work looking at the tessellation to see why not. You may want to print out a copy of Circle Limit III so you can do some measurements.

  1. Pick one of the triangles and see if you can determine the angles by considering the number of polygons at a vertex. Assume all angles at the vertex are equal. (They would be if all the fish were congruent.)
  2. What is the sum of the angles in this triangle? Is this possible in hyperbolic geometry?
  3. Look at where the wite lines meet the boundary of the disk. At what angle do the white lines seem to meet the boundary of the disk? Why is this a "problem"?

Handin: A sheet with answers to all questions.