Difference between revisions of "Circle Limit Exploration"

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(→‎Circle Limit III: Changed wording of the last question to be more directive. Not sure it's going to be an improvement, though.)
 
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{{objective|Explore Escher's Circle Limit prints to develop an intuition for hyperbolic geometry}}
 
{{objective|Explore Escher's Circle Limit prints to develop an intuition for hyperbolic geometry}}
  
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==Materials==
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{{printable|All four Circle Limit prints, dimmed: [[Image:four-dim-circle-limits.pdf]]}}
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* Printed copies of [[Circle Limit I]], [[Circle Limit II]], [[Circle Limit III]], and [[Circle Limit IV (Heaven and Hell)]].
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{{clear}}
  
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.
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==Circle Limit I==
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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]].
  
[[Image:Hyp-circle-limit-i-tess.png]]
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[[Image:Hyp-circle-limit-i-tess.png|thumb|right|300px|M.C. Escher, <cite>[[Circle Limit I]]</cite> (1958) with geodesics in red.]]
  
In Circle Limit I, the red lines are meant to represent straight lines in this new geometry. These curves are also called geodesics.
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The red lines shown on [[Circle Limit I]] are 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.
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.
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<ol>
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<li>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.</li>
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</ol>
  
<ol>
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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.
  
<li> What type of polygons do you see in this figure?
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<ol start="2">
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.
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<li>What type of polygons do you see in [[Circle Limit I]]?
<li> Looking at the angles in a triangle, what will the angle sum be? Equal to, greater than or less than 180 degrees? Why?
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<li>Compare the angle sum of one of these polygons to the corresponding angle sum for Euclidean geometry.
 
</ol>
 
</ol>
 
   
 
   
<ol start="3">
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[[Circle Limit I]] is a picture of a surface called "hyperbolic space", but it is a distorted picture.
<li> What is the highest degree of rotation?  
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In actual hyperbolic space, these fish would all have the same size and shape.
<li> What is the underlying geometric tessellation?
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<ol start="4">
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<li> What is the highest order of rotation symmetry for this print?
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<li> Describe the geometric tessellation underlying [[Circle Limit I]].
 
</ol>
 
</ol>
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{{clear}}
  
'''Circle Limit II by M.C.Escher'''
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==Circle Limits II and IV==
 
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[[Image:Circle-limit-II.jpg|thumb|left|300px|M.C. Escher, <cite>[[Circle Limit II]]</cite> (1959)]]
[[Image:Circle-Limit-II.jpg|300 px|Copy of Circle Limit II by Doug Dunham]] [[Image:Circle-limit-IV.jpg|300 px|Circle Limit IV]]
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[[Image:Circle-limit-IV.jpg|thumb|right|300px|M.C. Escher, <cite>[[Circle Limit IV (Heaven and Hell)]]</cite> (1960)]]
 
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{{clear}}
 
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For [[Circle Limit II]]:
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<ol start="6">
<ol start="5">
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<li> What is the highest order of rotation?
<li> What is the highest degree of rotation?
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<li> Draw geodesics in this figure. Describe the underlying geometric tessellation.
<li> Draw geodesics in this figure. What is the underlying geometric tessellation?
 
 
</ol>
 
</ol>
 
'''Circle Limit IV by M.C.Escher'''
 
  
  
<ol start="7">
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For [[Circle Limit IV (Heaven and Hell)|Circle Limit IV]]:
<li> What is the highest degree of rotation? What other degrees of rotation are present?
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<ol start="8">
<li> Draw geodesics in this figure. What is the underlying geometric tessellation?
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<li> What is the highest order of rotation? What other orders of rotation are present?
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<li> Draw geodesics in this figure. Describe the underlying geometric tessellation.
 
<li> Draw a geodesic NOT passing through the center point.
 
<li> Draw a geodesic NOT passing through the center point.
<li> 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?
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<li> 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?
 
</ol>
 
</ol>
  
== A Separate Note About Circle Limit III==
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==Circle Limit III==
  
[[Image:Circle_Limit_III|right|thumb|200 px|Circle Limit III]]
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[[Image:Circle-Limit-III.jpg|300px|thumb|right|M.C. Escher, <cite>[[Circle Limit III]]</cite>, 1959.]]
  
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.
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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.
  
<ol start="11">
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<ol start="12">
<li> Pick one of the triangles and see if you can determine the angles by considering the number of polygons at a vertex.
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<li> Pick a triangle and determine its corner 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.)
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Assume all angles at each vertex are equal (they are, though the distortion makes this harder to believe).
<li> What is the sum of the angles in this triangle? Is this possible in hyperbolic geometry?
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<li> What is the sum of the angles in the triangle? Is this possible in hyperbolic geometry?
<li> 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"?
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<li> Choose a white line and trace it to the point where it meets the boundary of the disk.
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        Carefully measure the angle it makes with the edge of the disk (you may want to draw tangent lines to the disk and the white line).
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        What angle did you get?
 
</ol>
 
</ol>
  
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{{clear}}
  
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{{handin|Marked up Circle Limit prints and a sheet with answers to all questions.}}
  
  
{{handin}}
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[[category:Non-Euclidean Geometry Explorations]]

Latest revision as of 10:12, 20 November 2013


Time-40.svg

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

Materials

Printer.svg All four Circle Limit prints, dimmed: File: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.

M.C. Escher, Circle Limit I (1958) with geodesics in red.

The red lines shown on Circle Limit I are 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.

  1. What type of polygons do you see in Circle Limit I?
  2. 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.

  1. What is the highest order of rotation symmetry for this print?
  2. Describe the geometric tessellation underlying Circle Limit I.

Circle Limits II and IV

M.C. Escher, Circle Limit II (1959)

For Circle Limit II:

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


For Circle Limit IV:

  1. What is the highest order of rotation? What other orders of rotation are present?
  2. Draw geodesics in this figure. Describe 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 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

M.C. Escher, Circle Limit III, 1959.

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.

  1. 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).
  2. What is the sum of the angles in the triangle? Is this possible in hyperbolic geometry?
  3. Choose a white line and trace it to the point where it meets the boundary of the disk. Carefully measure the angle it makes with the edge of the disk (you may want to draw tangent lines to the disk and the white line). What angle did you get?

Handin: Marked up Circle Limit prints and a sheet with answers to all questions.