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Within an hour, I came across a book about a man that claimed to have become enlighted after having stared at a wall for nine years. What does enlightenment mean if appearently you could afford the luxury of only staring at a wall. I wonder if this man did any work during these nine years. His family, or his vilage, must have provided this "saint", or should I say "mad man", with food and housing.
But, I could not stop thinking about the idea of a man becoming obsessed by a painting, and ending up staring at it for nine years. Might be an interesting idea for a novel.
At first sight this looks like a very complex problem. What is the area cover by a robe sweeping around a circle? Lets start with a unit circle, and first assume that the robe is shorter than pi (that is half the circumference of the unit circle). Say that the length of the robe equals l, and that the part of the robe that is free from the circle while it wraps around the circle equals x. For each x the end of the robe will move about x * dx when x is changed for a value of dx. The surface that is covered equals to approxamily x2/2 * dx. If we integrate this area where x varies from 0 to l we get the expression l3/6 for the area of which the robe wraps on one side of the circle. From this follows that formulea for the surface covered by the goat if the robe has length l is:
But what if the robe is longer than pi? Because then the robe will go more than half around the circle, and then there is a small area that the goat can reach from going around the circle from both sides. An important point is the furtherst point opposite the point where the robe is attached to the fence, which can be reached by the goat. This is the furtherst point that the goat can reach from going round the circular fence from both sides. When the goat reaches this point, some part of the robe is wrapped around the fence, and some part is not. Lets assume that the length of this part equals a. Then the total surface (after some calculations) appears to be: