View source for Course:Homework 4: Hyperbolic Geometry -Bart07

* Read [[Hyperbolic Geometry]]
* Write up solutions to the following questions:


<ol>
<li>Draw a Poincaré disk, and draw four geodesics that don't cross.</li>
<li>Draw a Poincaré disk, and draw four geodesics through the center point.</li>
<li>Copy the Poincaré disk shown below, and draw three geodesics through the point that don't cross the line shown.
<br>
[[Image:hyp-ex-3.svg]]
</li>
<li>Draw a Poincaré disk, and draw a triangle with three 5° angles.</li>
<li>Draw a Poincaré disk, and draw a 90°-5°-5° triangle. (Hint: Put the 90° angle at the center point.)</li>
<li>Draw a Poincaré disk, and draw a pentagon with five right angles</li>
<li>Escher used angels and devils in three artworks: {{sketch|45|Angels and Devils}}, [[Sphere with Angels and Devils]], and [[Circle Limit IV (Heaven and Hell)]].<br>
 Answer each question for all three artworks:
<ol style="list-style-type:lower-alpha">
<li>How many creatures touch at one wingtip?</li>
<li>How many creatures touch at their feet?</li>
<li>Each creature is basically a triangle formed by the two wingtips and the feet.  From parts a & b, you know how many of these triangles come together at each vertex.  What are the corner angles of these triangles?</li>
<li>What are the angle  sums of these triangles?</li>
<li>What geometry is the artwork based on?</li>
</ol>
</li>

<li>Which of these Escher artworks are based on hyperbolic geometry?</li>
<ol style="list-style-type:lower-alpha">
<li>[[Path of Life I]]</li>
<li>[[Circle Limit I]]</li>
<li>[[Circle Limit II]]</li>
<li>[[Circle Limit III]]</li>
<li>[[Verbum]]</li>
<li>[[Development II]]</li>
<li>[[Square Limit]]</li>
<li>[[Snakes]]</li>
<li>[[Rippled Surface]]</li>
</ol>
</li>

<li>Find the defect and area of the following hyperbolic polygons:
<ol style="list-style-type:lower-alpha">
<li>A triangle with three 5° angles.
<li>A 90°-5°-5° triangle.
<li>A pentagon with five right angles.
<li>A 90°-0°-0° triangle. What does this triangle look like?
<li>A 20°-20°-20°-20° quadrilateral.
</ol>
</li>

<li> Compare Hyperbolic geometry to spherical and Euclidean geometry.

* What can we say about parallel lines in hyperbolic geometry?
* What can we say about the sum of the angles in a triangle?
* Do n-gons exist for all possible values of n?
* Do we have squares and rectangles in hyperbolic geometry? Why or why not?
* Do parallellograms exist in hyperbolic geometry? If so, what do they look like?
* Give an example (sketch) of the isometries: translation, rotation, and reflection.
* How do we measure the defect of a triangle?
* How do we measure the area of a triangle?
* How do we measure the defect of a general polygon?
* How do we measure the area of a general polygon?
  </li>
</ol>