A Mathematician Has Finally Solved the Infamous Goat Problem
From popular mechanics
A mixture of complex and trigonometric values has shown a long approximate math problem.
The goat problem presents important ways in which pure math does not exactly match the "real world".
The secret is a complex analysis that provides a new frame of reference for the problem.
How do you mix the irrational world of pi with the real world of. . . Goats This is the spirit of a long unsolved math problem that sounds deceptively simple.
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The problem: How much rope does a goat need to be able to eat grass in a circle with an area of exactly half an acre?
The answer has been around for centuries and provides food for thought for mathematicians who like to think about an idle headscratcher. Now there is finally a concrete equation.
The goat problem is a living example of what it means to round off your answer. Steve Nadis from Quanta explains the difference:
"To illustrate the difference, consider the equation x2 - 2 = 0. One could derive an approximate numerical answer, x = 1.4142, but that's not as accurate or satisfactory as the exact solution, x = √2."
Photo credit: Mnchnstnr, public domain, via Wikimedia Commons
Or consider Zeno's Paradox, the famous thought experiment in which a frog halves its distance across a pond - and literally never gets to the other side. How does Zeno's chicken cross the road? (Is not it.)
With a few moments of thought, the goat problem quickly becomes an exercise in many overlapping approaches. For this reason, every answer that has been offered since the 18th century has also been an approximation.
In the 1980s, mathematicians made great strides by blowing out a very hypothetical two-dimensional - simple in pure math, impractical in reality - into a 3D space with different math. If this doesn't sound intuitive, consider how much calculus is activated by switching from x to x2.
And now there is finally an exact solution for the first time. The mathematician Ingo Ullisch oriented himself on previous researchers who had made progress in this area. He introduced a complex analysis that resembles an algebra with an optional add-on for imaginary numbers.
By multiplying a range of values expressed as the tell-tale a + bi of complex numbers, he was able to reduce the problem to a still confusing but accurate expression. Quanta explains the catch:
"Ullisch's solution is not as simple as the square root of 2. It's a bit more absurd - the ratio of two so-called contour integral expressions with numerous trigonometric terms thrown into the mix - and can't say in a practical sense how long the goat leash last should. It still takes approximations to come up with a number that will be useful to anyone involved in animal husbandry. "
What is fun about the goat problem, from which mathematicians admit that it has no relation to any other question or even math domains, is that it acts as a kind of math Rosetta brick. Whatever your area of expertise, there is likely a way to approach the problem and model it using your own modeling and analysis.
Something cool about any exact equation is that, technically, it can be equated to any other exact equation and examined for similarities.
? Now look at this:
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