Difference between revisions of "Spherical Easel Exploration"
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Revision as of 07:58, 24 January 2009
Go to Spherical Easel. This program will allow you to explore the geometry of the sphere. Answer the following questions as completely as possible.
- What do “lines” look like on the sphere?
- What do polygons look like on the sphere?
- Create some triangles on the sphere and measure their angle sums. Do the angles always add up to 180 (= pi radians)? Do they ever add up to 180? What is the smallest angle sum? What is the largest angle sum?
- Can we create rectangles and squares on the sphere? Why or why not?
- Can we create parallel lines? Why or why not? Can we create parallelograms on the sphere? Explain why or why not.
- Show that we can create a rhombus. Show that we also get a new type of polygon, namely a 2-gon.
- Draw a triangle and compute <math>1/2 bh</math> in the three possible ways. Do we get the same value no matter what side we choose as a base? Do the values for <math>1/2 bh</math> correspond with the area that the software computes for us? Do you think that the area of a triangle on the sphere could be given by the same formula we use in the Euclidean plane?
- On the plane we had four types of isometries we looked at: translations, rotations, reflections and glide-reflection (where the last one is of course a composite of the translation and a reflection). Do these same isometries work on the sphere? Can we translate by any distance? Can we always reflect over a mirror line? Can we rotate through any angle? How are translations and reflections related on the sphere? Are all translations reflections and are all reflections translations? Explain.