Lab 5

This lab has some hands-on activity and some Sketchpad activity. The instructions for the Lenart spheres say DRAW, which means you can measure. But it should be an accurate figure. For methods of constructing great circles with GSP, you can refer to Lab 4.

You can work in pairs. Show an instructor your work when done. It is suggested you save this and/or print it out. This work will be part of Assignment 6.

Part 1: Octahedron and Cube in the Sphere

• On the Lenart Spheres, draw 3 mutually orthogonal great circles that cut up the sphere into 8 congruent triangles. The intersection points of the circles are the vertices of a regular octahedron inscribed in the sphere.
• For one of these triangles ABC , draw the midpoints A', B', C' of the sides and draw the medians, which are also the perpendicular bisectors of the sides. These great circles are concurrent at a point inside the triangle. This point is the circumcenter and center of symmetry of the triangle. It is also a vertex of a cube inscribed in the sphere.
• Draw accurately all the vertices of the inscribed cube and then connect the vertices by drawing the arcs that lie over the edges of the cube. We will call the figure made up of the 8 vertices and 12 arcs a spherical cube.
• Make a note (use your homework also) of the (x,y,z) coordinates but also the spherical distance from a cube vertex and an octahedron vertex.
• Finally, if E and F are adjacent vertices of the cube, and E* and F* are the antipodal points, sketch the rectangle EFE*F* inscribed in a great circle and tell its dimensions if the radius of the circle is 1.

Part 2: Stereo Image of Part 1

• Make an accurate Sketchpad construction that shows the stereographic image of the spherical cube. You should take the center of projection (north pole) as one of the vertices of the octahedron.

Part 3. Midpoint triangle of octahedral triangles

• Returning to the 90-90-90 triangle ABC from part 1, construct the spherical midpoint triangle A'B'C'.
• In the figure, explore by measuring the lengths of the triangle A'B'C' and the corner triangle such as A'B'C. Also figure out the approximate areas and the angles by measuring.
• See whether you can figure some exact answers by reasoning.
• View the figure of all the midpoint triangles by attaching 4 corner triangles to form quadrilaterals. What does the polyhedral version of this figure look like?

Part 4. GSP Midpoint triangle of octahedral triangles

• Continue with your stereo figure and add the stereographic image of the sides of triangle A'B'C' and the other midpoint triangles to your figure. Do some of the great circles coincide?
• How many great circles are needed to give all the sides of all the midpoint triangles?

Part 5. Completion of Lab 4.

Part of Assignment 6 will be to turn in two careful figures of a sphere with latitude circles and longitude lines, each drawn every 15 degrees. Also, given the latitude and longitude of a point you should be able to do a straigthtedge and compass construction of the stereographic image of the point (e.g, in time for the midterm). The cubical work above is a good example.

The material about Wulff diagrams in Lab 4 may help.

• Construct (not draw) a figure inside a circle that shows the image of the southern hemisphere of this sphere by stereographic projection from the North Pole.
• Construct (not draw) a figure inside a circle that shows the image of the "western" hemisphere of this sphere by stereographic projection from the point on the equator and at 90 degrees east longitude..