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Overlapping Cubes - reasoning about rates

Please explore the applet "Overlapping Squares - reasoning about rates" before you use this applet. Two identical cubes can be made to overlap in many ways. Here are two methods of varying the degree of overlap of the cubes - one in which face diagonals on the cubes are aligned and one in which the cubes have parallel faces. What conjectures do you have about the nature of the common volume for shapes other than cubes? In what ways is this applet similar to the Overlapping Squares applet? In what ways is it different? Yet another method of overlapping two cubes is the following - align a body diagonal of each cube along a line in space. How does this method of overlapping the cubes compare with the two methods explored in this applet? What questions could/would you ask your students based on this applet? What conjectures do you have about the common volume if the two cubes are not related by a simple translation along a line of symmetry?