You have a 3x3x3 cube. You bore a circular hole, of diameter 1, from the center of the top face to the center of the bottom face. You bore a similar hole from the left face center to the right face center, and another hole from front center to back center. How much material was removed?

(In reply to re: picture of the center by nj)

No, that's not the removal of a sphere from the unit cube. Notice on the lower left (as on all 8 pieces), there is a cusp directed toward the center of the cube--that's not the same cusp as that which is on the edge of the cube, which is visible in this perspective near it. 

The cusp toward the center is probably best visible on the farthest bottom corner of the cube, where the triangle is shown, in perspective, in three subtriangles, especially the dark, shaded one to the right, the shading caused by the direction of the cusp toward the center of the cube. Each of the three pieces (subtriangles in perspective) is a portion of a cylindrical surface.

Note the piece in the upper-right of the picture. It's top left and bottom right cusps are on edges of the cube.  The other one that's on a third edge of the cube is seen be shading difference.  The cusp on the lower left perspective edge of that piece is not on the face of the cube, but rather, directed toward the center of the cube.  Again, all 8 pieces are identical, and consist of three planar faces coincident with faces of the cube, and three segments of cylindrical surfaces.

 

Posted by Charlie on 2005-03-07 13:55:54