Escher s Circle Limit Anneke Bart Saint Louis University http://math.slu.edu/escher/ Introduction What are some of the most fundamental things we do in geometry? In the beginning we mainly look at lines, segments, polygons and circles. This is even reflected in the axioms ( ground rules ) of geometry. Euclidean Geometry (Planar Geometry) We look at polygons and tessellations we can create with these polygons. A tessellation is a covering of the plane by shapes, called tiles, so that there are no empty spaces and no overlapped tiles. Tessellations are also called tilings. Spherical Geometry What happens when we try to draw polygons on a sphere, or if we try to tile a sphere? We would soon realize that the axioms of planar geometry do not work. We actually get a different geometry.
This begs the question: Are there any other geometries? If so, what do they look like? Hyperbolic Geometry What does it look like? This is a drawing of hyperbolic space. This space is an infinite space. There is some distortion present. We need to interpret all the fish as being the same size. The ones near the edge are not smaller. They are far away (compare to perspective drawing!) To travel fast in this geometry we travel along semi-circles perpendicular to the edge.
On the left we see examples of geodesics (hyperbolic straight lines) On the right we see a stick figure walking off into the distance. We will now explore hyperbolic geometry by looking at Escher s Circle Limits. All M.C. Escher works Cordon Art BV - Baarn - the Netherlands. All M.C. Escher works (c) 2007 The M.C. Escher Company - the Netherlands. All rights reserved. Used by permission. www.mcescher.com
Escher's Circle Limit Exploration - EscherMath http://math.slu.edu/escher/index.php?title=escher%27s_circle_limit_explor... Escher's Circle Limit Exploration From EscherMath Objective: Explore Escher's Circle Limit prints to develop an intuition for hyperbolic geometry Materials Printed copies of Circle Limit I, Circle Limit II, Circle Limit III, and Circle Limit IV (Heaven and Hell). All four Circle Limit prints, dimmed: Image:Four-dim-circle-limits.pdf Circle Limit I Recall that in Spherical Geometry, the shortest path between two points is along a great circle. These shortest paths are called geodesics, and the geodesics play the same role as do straight lines in Euclidean geometry. Escher's Circle Limit prints are based on a new kind of geometry, Hyperbolic Geometry. The red lines shown on Circle Limit I are the geodesics in this new geometry. These curves will play the role of straight lines. Each red line follows the spines of a line of fish. 1. There are two types of red line marked in the Circle Limit I figure. Describe them. Draw more geodesics by following the spines of other rows of fish. Describe the curves that result. In these pictures of hyperbolic geometry, geodesics come in two forms, either straight lines through the center of the disk, or arcs of circles that meet the disk's edge at 90. Segments of geodesics form the sides of polygons. Polygons in hypebolic geometry will look "pinched" to our Euclidean eyes. 2. What type of polygons do you see in this figure? 3. Compare the angle sum of one of these polygons to the corresponding angle sum for Euclidean geometry. Circle Limit I is a picture of a surface called "hyperbolic space", but it is a distorted picture. In actual hyperbolic space, these fish would all have the same size and shape. M.C. Escher, Circle Limit I (1958) with geodesics in red. 4. 5. What is the highest order of rotation symmetry for this print? Describe the geometric tessellation underlying Circle Limit I. 1 of 3 4/21/08 11:14 AM
Escher's Circle Limit Exploration - EscherMath http://math.slu.edu/escher/index.php?title=escher%27s_circle_limit_explor... Circle Limits II and IV M.C. Escher, Circle Limit II (1959) M.C. Escher, Circle Limit IV (Heaven and Hell) (1960) For Circle Limit II: 6. 7. What is the highest order of rotation? Draw geodesics in this figure. Describe the underlying geometric tessellation. For Circle Limit IV: 8. What is the highest order of rotation? What other orders of rotation are present? 9. Draw geodesics in this figure. Describe the underlying geometric tessellation. 10. Draw a geodesic NOT passing through the center point. 11. How many geodesics can you draw through the center point, so that the new geodesic does not meet the geodesic you picked in the previous question? 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? Circle Limit III 2 of 3 4/21/08 11:14 AM
Escher's Circle Limit Exploration - EscherMath http://math.slu.edu/escher/index.php?title=escher%27s_circle_limit_explor... This Circle Limit is the most subtle. The white lines look like the geodesics in the other Circle Limit prints, but they are not the same. A closer look shows that these white lines are not geodesics at all. 12. Pick a triangle and determine its corner angles by considering the number of polygons at a vertex. Assume all angles at each vertex are equal (they are, though the distortion makes this harder to believe). 13. What is the sum of the angles in the triangle? Is this possible in hyperbolic geometry? 14. Look at where the white 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"? M.C. Escher, Circle Limit III, 1959. Handin: Marked up Circle Limit prints and a sheet with answers to all questions. Retrieved from "http://math.slu.edu/escher/index.php/escher%27s_circle_limit_exploration" Category: Exploration This page was last modified 16:32, 17 April 2008. 3 of 3 4/21/08 11:14 AM
Circle Limit I Circle Limit II Circle Limit III Circle Limit IV