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Spring 2009 Week 2

Consider any two opposite vertices (i.e. vertices which are not connected by an edge or surface) of a unit cube (i.e. a cube for which the edges are all 1 unit in length).

1.    What is the minimum distance between the two vertices if one can only travel along the edges of the cube?

2.    What is the minimum distance between the two vertices if one can travel in a straight line through the interior of the cube?

3.    What is the minimum distance between the two vertices if one can only travel along the surface of the cube?

4.    Consider now the triangle that is created by the single line segment which constitutes the path of minimum distance in part 2 and the two line segments which constitute the path of minimum distance in part 3.  What are the angles (in degrees) of this triangle?