A physicist, an engineer and a mathematician were all challenged to build the shortest possible fence around a small herd of resting cattle.
The physicist went first. He took out a piece of graph paper and plotted the position of each cow, giving each cow a pair of x-y coordinates. Then he determined the lines connecting all the points. Finally he constructed a fence based on his diagram. When he finished he turned to the others and said “I’m done. And since the interior region bounded by line segments connecting the cattle-points is convex, it follows that the boundary is minimal. Q.E.D.”
Then it was the engineer’s turn. First he secured a strong fence-pole near the cattle. Next he attached one end of a six-foot-high roll of wire fence to the pole and walked around the cows slowly letting out the roll of wire fence until he came back to the post. Then he gave the roll to the physicist and told him to start pulling. As he the physicist pulled, the engineer ran around the outside of the fence kicking the cows, flailing his arms, and screaming at them to make them get up and move into the middle; meanwhile while he was yelling “Pull the fence tighter! Pull the fence tighter!” Finally the cows were shoved so close together that they couldn’t move and the fence was wrapped around them so tightly that it was leaving marks on their hides. The engineer nailed the other end of the fence to the post, cut away the roll and said “There, that is the shortest fence.”
Finally it was the mathematician’s turn. He walked over to the roll of wire fence, cut off a small piece, wrapped it around himself and declared: “I’m on the outside.”
One day a farmer called up an engineer, a physicist, and a mathematician and asked them to fence of the largest possible area with the least amount of fence.
The engineer made the fence in a circle and proclaimed that he had the most efficient design.
The physicist pointed out that fencing off half of the Earth was certainly a more efficient way to do it.
The mathematician just laughed at them. He built a tiny fence around himself and said “I declare myself to be on the outside.”