Spherical Geometry: Polygons
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: Definition: A polygon on the sphere is a closed figure made by joining geodesic segments, where each geodesic segment intersects at most two others.
- 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.
- Focus on the 3-gons for a moment. Are there regular triangles? If so, what are their angle measures?
- Draw 4 different triangles (different sizes) and measure the sum of their angles.
- 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.
- Are there quadrilaterals on the sphere? If there are, how would you construct one?
- Are there any squares or rectangles on the sphere? Why or why not?
- 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)?