These problems are about cubes. To fix notation, we start with a square ABCD as the base of the cube and then label the square on top as A', B', C', D', where the point A' is the point opposite (not above) A, etc.

You are asked to build two models, one inside the other. The models can be of cardboard, possibly of transparent plastic overhead material, or maybe drinking straws.

1. Build a rigid cube with sides at least 3 inches. Call this side length s. Let this cubical "box" with a lid, so that it can be opened (see C below).
2. Three of the square faces of the cube have A as a vertex. For each square face, A is one endpoint of a diagonal of the face. What are the 3 vertices of the cube that are the other endpoints of these diagonals? These 3 vertices form an equilateral triangle. What are the lengths of the sides of this triangle?
3. The 3 vertices of B, along with A, form a polyhedron with faces made of triangles. Make an accurate cardboard model of this polyhedron that will fit precisely into the cube of A, so that the vertices of this polyhedron are also vertices of the cube.
4. Bring the models to class.

In each of these problems, you should use your cube to give you a good idea of what the figure looks like. But then you should construct these figures accurately and in proportion, not just as a sketch. The ratios of distances should be as accurate as possible, using geometrical reasoning to figure out the proportions. State what the proportions are and justify your answers.

1. Imagine a cube so that the diagonal AA' is parallel to the z-axis in (x,y,z) space. Draw the image of the cube (all its vertices and edges) by a parallel projection onto the (x,y,0) plane, where the direction of the projection is parallel to the z-axis.
2. Again, imagine the cube with A at the original and AA' on the positive z-axis. Also assume that vertex B is in the (x,0,z) plane. Then draw the image of the cube by parallel projection onto the (0,y,z) plane, where the projection direction is parallel to the x-axis.