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.

Hyp-circle-limit-i-tess.png

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 edge of 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

  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

Circle-limit-IV.jpg
  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?

Handin: A sheet with answers to all questions.