Mathematics 141 Opening Group Project


    We live on a sphere, and it is easy to find spheres around us (balls, etc.).  Using these models we can get an idea of how geometry on a sphere is different from geometry in the plane.  
    We do not get very far without running into differences.  A point is the same in the sphere as in the plane, but what about lines?  Discuss and write about possible ideas of what we should call a line in the sphere and why.  
    The standard definition of lines in the sphere are great circles.  These are the circles going the long way around the sphere.  In the case of the earth, the equator and meridians of longitude are great circles (note, the other “lines” of latitude are not great circles, and so we won’t be calling them lines).  One of the reasons that great circles are called lines is that they provide the shortest distance between any two points.  Note, however, that they’re pretty different from our lines in the plane.  First of all, they meet themselves, and do not go on indefinitely.  Second, any two points have two line segments connecting them (one “obvious” one, and one going the long way around the other side of the sphere).  Finally, the north and south pole (or any other so-called antipodal points opposite on the sphere) can be connected by any number of lines (think of all the meridians of longitude).  Experiment with a ball and some rubber bands to see what these lines look like, and how they interact.
    How many times do two lines on the sphere intersect?  How is this different from the plane?  What can you say about parallel lines on the sphere?  
    Now that we’ve got the basics down (we know what points and lines are), which of the basic axioms for geometry still hold on the sphere?  
    And, for serious differences, what happens with triangles?  Check out  the sum of the angles of triangles on the sphere.  Using a basketball size sphere, can you make a triangle with three right angles?  Can you make a triangle whose angle sum is even greater than 270°?  Which theorems about triangles are still true if any?  Provide counter examples for the ones that are false.  
    By now you have probably noticed that all lines intersect each other.  Therefore, if we take not intersecting as our definition of parallel lines, there are none!  What happens, if we take congruent alternating interior angles as our definition instead?  Which theorems about parallel lines remain true?  Provide counter examples for those which are false.  

    Your report should include comments and pictures for each of the above questions.  You are encouraged to turn in models if you desire.