Friday, February 3, 2023

After centuries, a simple mathematical problem obtains an exact solution

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Here is a simple sound problem: imagine a circular fence that encloses an acre of grass. If you tie a goat inside the fence, how long do you need a rope to allow the animal to access exactly half an acre?

It sounds like high school geometry, but mathematicians and math enthusiasts have been thinking about this problem in various forms for over 270 years. And although they managed to solve some versions, the goat in a circle puzzle refused to give anything other than fuzzy and incomplete answers.

Even after all this time, “no one knows an exact answer to the initial basic problem,” said Mark Meyerson, mathematician emeritus at the US Naval Academy. “The solution is only given approximately.”

But earlier this year, a German mathematician named Ingo Ullisch finally made progress, find what is considered the first exact solution to the problem – although even that comes in an unwieldy and unfriendly form for the reader.

“This is the first explicit expression that I know [for the length of the rope]Said Michael Harrison, mathematician at Carnegie Mellon University. “It is certainly a step forward.”

Of course, this won’t upset textbooks or revolutionize mathematical research, concedes Ullisch, because this problem is isolated. “It is not related to other problems or integrated into a mathematical theory.” But it’s possible that even fun puzzles like this give rise to new mathematical ideas and help researchers find new approaches to other problems.

In (and out of) the farmyard

The first such issue appeared in the 1748 issue of the London periodical The Women’s Journal: or the Women’s Almanac—A publication that promised to feature “further improvements in the arts and sciences, and many devious details.”

The original storyline involves “a horse tied for food in a Gentlemen’s Park”. In this case, the horse is tied outside of a circular fence. If the length of the rope is the same as the circumference of the fence, what is the maximum area the horse can feed on? This version was then classified as an “outside issue” as it was about grazing outside rather than inside the circle.

An answer appeared in the Personal diaryedition of 1749. It was provided by “Mr. Heath, ”who relied on“ an essay and a table of logarithms, ”among other resources, to reach his conclusion.

Heath’s answer – 76,257.86 square meters for a 160-meter cord – was an approximation rather than an exact solution. To illustrate the difference, consider the equation X2 – 2 = 0. We could derive an approximate numerical answer, X = 1.4142, but it is not as precise or satisfactory as the exact solution, X = √2.

The problem reappeared in 1894 in the first issue of the American Mathematical Monthly, recast as the original fence grazer problem (this time without any reference to farm animals). This type is classified as an interior problem and tends to be more difficult than its exterior counterpart, Ullisch explained. In the outer problem, you start with the radius of the circle and the length of the chord and calculate the area. You can solve it through integration.

“Reversing this process – starting with a given domain and asking which entries lead to that domain – is much more complex,” Ullisch said.

In the decades that followed, the Monthly published variations on the interior problem, which primarily involved horses (and in at least one mule) rather than goats, with fences that were circular, square, and elliptical in shape. But in the 1960s, for some mysterious reason, goats began to move horses in the literature on grazing issues – this despite the fact that goats, according to mathematician Marshall Fraser, are perhaps “too independent to rely on. submit to the tie ”.

Goats in higher dimensions

In 1984 Fraser got creative, eliminating the problem of the flat, pastoral estate and placing it in a larger area. he elaborate how long does a rope take to allow a goat to graze in exactly half the volume of a not-Dimensional sphere like not goes to infinity. Meyerson spotted a logical flaw in the argument and fixed Fraser’s error later that year, but reached the same conclusion: as n approaches infinity, the ratio of the tether cord to the radius of the sphere approaches √2.


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