Response from Prof. B.

In n-space the endpoints of the vectors with one 1 and all the rest 0 will all have distance sqrt(2) from one another and in this way you get n points that form the vertices of an (n-1) dimensional simplex. For example the points (1,0,0), (0,1,0),(0,0,1) form the vertices of an equilateral triangle and the points (1,0,0,0), (0,1,0,0),(0,0,1,0), (0,0,0,1) are the vertices of a regular tetrahedron situated in four-space. If you use -1 instead of 1 in each case, what you get is related to the dual of the original figure, as you suspected.