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(4): picture of the center
Actually, what is depicted there, between equations 16 and 17 for the 3-cylinder case, is the intersection of all three cylinders, that is, all points that are part of all of the three cylinders.
What I depicted was the complement of the union of all three cylinders, that is all points that are part of none of the three cylinders.
Points within the cube that are part of only one or two of the cylinders are found in neither my solid set (the 8 pieces), nor in the solid depicted on Wolfram, so the pieces are not complementary with regard to the central cube. Looked at another way, their object, placed within the central void of my object, would still leave empty volumes, consisting of those points that are in exactly one or two of the cylinders.
Consideration of both shapes could be useful for someone who wants to do calculus on both, and use them in combinatorial analysis (inclusion/exclusion) to get the whole desired volume (the void in my depiction).
Posted by Charlie
on 2005-03-07 15:55:58