Spherical Geometry: Polygons

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Objective: Discover properties of polygons on the sphere

We have seen before that a polygon in the plane is defined as follows: A polygon is a closed figure made by joining line segments, where each line segment intersects exactly two others. On the sphere we will use a similar definition:

A polygon on the sphere is a closed figure made by joining geodesic segments, where each geodesic segment intersects at most two others.

You may use [Spherical Easel] to help you answer the following questions if you like.

  1. In Euclidean (Planar) Geometry there are 3-, 4-, 5-gons etc., but there are no 1- and 2-gons. Are there 1-,2-, 3-, 4-, 5-gons etc. on the sphere? Draw some examples of those that exist.
  2. Focus on the 3-gons for a moment. Are there regular triangles? If so, what are their angle measures?
  3. Draw 4 different triangles (different sizes) and measure the sum of their angles. (see below for some comments on measuring angles and anglesums in Spherical Easel)
  4. Based on your experiment in question 3, what can we say about the sum of the angles in a triangle on the sphere? Give a convincing argument.
  5. Are there quadrilaterals on the sphere? If there are, how would you construct one?
  6. Are there any squares or rectangles on the sphere? Why or why not?
  7. What kind of tessellations can you draw on the sphere? Which of the planar tessellations may be adapted to the sphere? Which ones can definitely not be adapted (if any)?

Angle measures in Spherical Easel: After you draw a triangle you can go to the measurement menu and select triangle. This will compute sides and angles etc. To find the angle sum choose the calculator from the measurement menu and enter (M4+M5+M6)*180/PI. This adds the angle measures M4, M5, and M6 and then converts the measurements from radians to degrees.