This set of Constructions has an A part on the Lenart Spheres and a B part on paper (can be done with GSP) and a Part C which consists of a description of your method, plus a couple of short questions about the figure.
For Constructions 4-1A and 4-2A, you should do the actual work for the portfolio on the overlays in the big plastic box with the red lid in the Math Study Center, as before. Then sign your work.
For Constructions 4-1B and 4-2B, you should start with a circle on a piece of paper and construct the stereographic image of the spherical figures of the A-constructions as described below.
Background: Both of these constructions are about the spherical tessellations defined by the vertices of some inscribed Platonic solids. For a bit more background, check this link.
Construct 8 points on the sphere that are the vertices of an inscribed cube. Then for each edge of the cube, connect the endpoints with great circle segments (arcs) so that the sphere is covered by 6 congruent quadrilatrals.
Hint for the construction: Remember that some of the Platonic solids are dual to each other, fit inside each other, etc. Your task is to figure out how to locate the vertices by a precise straightedge and compass construction.
Construction 4-1A: Do this construction on a Lenart Sphere.
Construction 4-1B. Carry out the construction of the stereographic image of the same figure, starting with the equatorial circle e. Make the center S of e the center of a face of the cube.
Writing 4-1C: (1) Write down clearly the steps and strategy that you used for this construction. (2) Explain what are the angles at the vertex of the quadrilateral (3) Check that this angle measure is consistent with the area formula.
Then as a second phase, investigate the perpendicular bisectors of the other
sides. Figure out
how many distinct perpendicular bisectors there are.
Recall that eight of the vertices of a a dodecahedron form a cube (actually this can be done in 5 ways). From the opposite point of view, you can start with a cube to get 8 vertices and construct the others by adding a "tent" to each cubical face. You can do the same on the sphere, but the golden ratio no longer is the key idea. Start with a cube (a duplicate of your work in 4-1 and figure out how to add the remaining vertices to get the spherical tessellation by pentagons that comes from the dodecahedron.
Notes: If you need to brush up on this construction, go to this link from 444 for several references.
Construction 4-2A: Do this construction on a Lenart Sphere.
Construction 4-2B. Carry out the construction of the stereographic image of the same figure, starting with the equatorial circle e. Make the center S of e the center of a face of the cube.
Writing 4-2C: (1) Write down clearly the steps and strategy that you used for this construction. (2) Explain what are the angles at the vertex of the pentagon (3) Check that this angle measure is consistent with the area formula.
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Leave your spherical "A" work in the Math Study Center. Turn in the rest no later than Wednesday, March 9.