Spherical Easel Exploration
From EscherMathJump to navigationJump to search
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? 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.
- Take a triangle and compute 1/2 bh in the three possible ways. Do we get the same value no matter what side we choose as a base? Do the values for 1/2 bh 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.