This lab will be devoted to a couple of tasks that will be turned in as Assignment 6.
On Lenart Sphere overlays construct two figures.
Construct a spherical regular octahedron and the midpoints of the edges. Connect these points to divide each fact of hte octahedron into 4 triangles. Some triangles are congruent and some are not. Give details about the distances and angles for one of each kind of triangle (i.e., one from each set of congruent triangles).
Construct the spherical cuboctahedron, whose vertices are the midpoints in the figure above (but not the vertices of the octahedron). Again, tell all distances and angles of each kind of polygon..
Construct with Sketchpad the stereographic images of the figures above. For the cube, place the north pole (the center of projection) in the center of a quadrilateral face. For the second figure, place the north pole at one of the vertices of the octahedron.
Measure the angles and distances. Hint: Angles between great circles are the usual angles. Distance between points = angle between polars. Alternatively, you can measure distance between two points with the same latitude by figuring out the Euclidean length of the segment in 3D from the definition of stereographic projection. Write down an explanation of how you measured these quantities.
For methods of constructing great circles with GSP, you can refer to Lab 6.
Save your work to be turned in as part of Assignment 6. You can work in pairs on the GSP, but if you do, turn in only one set of sketches with two signature, not two copies of the same thing.
Note: For a good visualization of spherical polyhedra, download Kaleidotile as we did in Lab 10 of Math 444.
EXTRA CREDIT: The stereographic images on the assignment are static. Make a cube or one of the other figures that can be dragged so that the vertices move on the sphere. This can be done by using the pole-polar constructions to construct a moveable 90-90-90 triangle.