Diary, December 1999

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Saturday, December 4, 1999

Staring at a painting

As usual, I walked by the gallery `Beeld en Aambeeld'. This morning I decided to go in, and ask about the painting by Billy Foley which has been in my mind lately because I realized that it would fit in our bedroom. After some questions, I found out that it was only recently sold to someone on an art fair in Hengelo. They also told that there had been a great interest for his works, and that more is coming. Strange, but somehow I had the idea that the painting already had gone back to Ierland.

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.

Friday, December 10, 1999

Goat problem

A variant of the traditional goat puzzle is stated as follows: "A farmer tethers his goat outside the fence of a circular field so that it can eat an area equal to 1/2 the field. What is the length of the robe". Note that as the goat moves within the tangent, the chain begins to wrap around the outside of the fence.

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:

pi * l²/2 + l³/3

Suprisingly, this expression is not so complicated as one whould expect. To solve the riddle we have to solve the following 3^rd degree equation:

2l³ + 3pi * l² - 3pi = 0

There is an analytical solution to this equation, but I am not going to give it here.

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:

pi * l²/2 + l³/3 - a³/3 + l - pi

The value of a is determined by the following equation:

asin(a/sqrt(a² + 1)) - a + l - pi = 0

To calculate the lenght of the robe that matches with some given surface appears to be rather hard, if the robe needs to be longer than pi.

Tuesday, December 14, 1999

The eight 'o clock news

This evening, Annabel suddenly asked me: "why do you watch the news?". I replied with "To know what happens in the rest of the world". Then she asked why I wanted to know this. A damn good question, I have to admit. Then she told me that watching the eight 'o clock news often makes her get nightmeres.

Monday, December 27, 1999

The Rules

Today, I bought the book The Rules by Ellen Fein and Sherrie Schneider for Annabel. I am going to give it to her when she is old enough to understand it, which might be only in a decade.

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