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> Unfortunately, there’s a catch. Ullisch’s solution ... can’t tell you, in a practical sense, how long to make the goat’s leash. Approximations are still requi

by bryan0 6y ago

> Unfortunately, there’s a catch. Ullisch’s solution ... can’t tell you, in a practical sense, how long to make the goat’s leash. Approximations are still required... Ok, then in what sense is this an exact solution?

whatshisface 6y ago
In the sense that sqrt(2) can be an exact solution, even though you can't write it down as a decimal.
- gowld 6y ago
  No, sqrt(2) is exact and no approximation is needed -- it's the diagnoal of a 1x1 square. Goats and grass don't care about decimal number systems. As the article says, the solution to the goat problem isn't practically constructible: "It’s a bit more abstruse — the ratio of two so-called contour integral expressions, with numerous trigonometric terms thrown into the mix"
  - whatshisface 6y ago
    >No, sqrt(2) is exact and no approximation is needed -- it's the diagonal of a 1x1 square. That's like saying that the goat-grass system as described is exact, and no approximation is needed. I can write 1 x sqrt(2) just as easily as I can write 1 x (goat-grass constant). We arbitrarily choose what constants and symbols are allowed when we use the phrase "exact solution." Philosophically, every solution is at most breaking down an answer into other solutions.
  - Koshkin 6y ago
    Problem is, if you make sqrt(2) exact - say, by an appropriate choice of the number system - then 1 will not be exact. This is because the ratio sqrt(2) / 1 is irrational in any number system.
    - thaumasiotes 6y ago
      But this isn't true. As gowld points out, you can easily represent an exact 1 and an exact sqrt(2) simultaneously; all you need is compass and straightedge. Interestingly, that's as far as it goes; if you start with a line of length 1, compass and straightedge will let you construct a line with length equal to the square root of any rational number. You can do addition, multiplication, subtraction, division... and square roots, and there things come to a stop.
      - whatshisface 6y ago
        But what singles out a compass and straightedge as a "the" construction system? There are infinity other ways to construct numbers, each one with a potentially different set of constructable numbers.
        thaumasiotes 6y ago
        You can use them to draw lines which you can then use to cut leather. They don't need to be "the" construction system; they are a construction system, and a very simple one.
        whatshisface 6y ago
        A pencil tied to the end of a string which is itself tied to the outside of a circle is also a construction system, one that happens to be able to construct the goat-grass area exactly. :)
        thaumasiotes 6y ago
        > one that happens to be able to construct the goat-grass area exactly. How? How did you get a string of the correct length? Note that the tool you described is a compass.
      - Koshkin 6y ago
        > compass and straightedge But those are not symbols. We're talking symbols (digits).
- deleted 6y ago
  [deleted]
castratikron 6y ago
It's in closed form
svat 6y ago
It's clearer from the Wikipedia article (https://en.wikipedia.org/w/index.php?title=Goat_problem&oldid=993368252 https://en.wikipedia.org/w/index.php?title=Goat_problem&oldi...): earlier we only had expressions for r like https://wikimedia.org/api/rest_v1/media/math/render/svg/9758becd77e7b4102ccdf50ff0845a749d236dc2 https://wikimedia.org/api/rest_v1/media/math/render/svg/9758... and https://wikimedia.org/api/rest_v1/media/math/render/svg/a63430b98b5e138a3d73831a2b7ecb6825be7a5b https://wikimedia.org/api/rest_v1/media/math/render/svg/a634... (from which of course one could numerically compute the answer r=1.1587…), but now thanks to this paper we have the expression https://wikimedia.org/api/rest_v1/media/math/render/svg/5b7451ea92f107e56651328ad15bfcc3276f87f2 https://wikimedia.org/api/rest_v1/media/math/render/svg/5b74... that does not involve r on the right-hand side.
- Sniffnoy 6y ago
  Hm, if it's based on this ratio of contour integrals, shouldn't it be possible to do better than this? Like why would it be so hard to find the residues for these poles? Shouldn't that be just a bit of formal Laurent series manipulation? What am I missing here?
  - Someone 6y ago
    If the series don’t cancel out nicely (likely, I would guess), wouldn’t you end up with some infinite sum? If so, as this example shows (contour integrals in a closed form?) “closed form” is loosely defined, but I think most would say something with an infinite sum wouldn’t be one (but then, https://en.wikipedia.org/wiki/Closed-form_expression#Analytic_expression https://en.wikipedia.org/wiki/Closed-form_expression#Analyti... says: “In particular, special functions such as the Bessel functions and the gamma function are usually allowed, and often so are infinite series and continued fractions. On the other hand, limits in general, and integrals in particular, are typically excluded.[citation needed]”
    - Sniffnoy 6y ago
      No, you shouldn't end up with an infinite sum; if you only want to know the series finitely far out, you only need to know finitely many terms. However, it seems the problem is hard for other reasons. I had assumed, without checking, that they'd put the center of the circle at 3pi/8 because it's a pole. Nope! As best I can tell from some graphing tools, there is indeed precisely one pole in the circle, and it's on the real line, but it's close to the right endpoint; I don't know that it has any nice form. So I imagine that getting any sort of exact series expansion around there -- or even just getting exactly the first few terms, i.e. the first few derivatives there, which would be all you'd need -- would be difficult for that reason. (Although, the higher the order of the zero, the more initial terms you'd need...) Edit: Actually, I guess it looks like a zero of order 1? Except that doesn't make sense, because then the top contour integral would be zero...
6gvONxR4sf7o 6y ago
Imagine an exact solution like this: sin(cos(log(98234/123)+tan(exp(pi/5))-1)+pi + integral(complicated function(x) dx from 0 to 1) It doesn't tell you in a practical sense how long to make the goat's leash. So you find the numerical version to however many decimals you want to get a useful approximation.
pfortuny 6y ago
The unknown is isolated: r=very-difficult-to-compute-expression
elil17 6y ago
The word exact here is an inexact description of what sort of solution this really is - it's a closed form explicit solution. A closed form solution means that the equation is limited to to certain common mathematical operations and is finite in length. An explicit solution means that the quantity we are solving for is isolated on one side of the equation.

by bryan0 6y ago

whatshisface 6y ago

In the sense that sqrt(2) can be an exact solution, even though you can't write it down as a decimal.

gowld 6y ago
No, sqrt(2) is exact and no approximation is needed -- it's the diagnoal of a 1x1 square. Goats and grass don't care about decimal number systems. As the article says, the solution to the goat problem isn't practically constructible: "It’s a bit more abstruse — the ratio of two so-called contour integral expressions, with numerous trigonometric terms thrown into the mix"
- whatshisface 6y ago
  >No, sqrt(2) is exact and no approximation is needed -- it's the diagonal of a 1x1 square. That's like saying that the goat-grass system as described is exact, and no approximation is needed. I can write 1 x sqrt(2) just as easily as I can write 1 x (goat-grass constant). We arbitrarily choose what constants and symbols are allowed when we use the phrase "exact solution." Philosophically, every solution is at most breaking down an answer into other solutions.
- Koshkin 6y ago
  Problem is, if you make sqrt(2) exact - say, by an appropriate choice of the number system - then 1 will not be exact. This is because the ratio sqrt(2) / 1 is irrational in any number system.
  - thaumasiotes 6y ago
    But this isn't true. As gowld points out, you can easily represent an exact 1 and an exact sqrt(2) simultaneously; all you need is compass and straightedge. Interestingly, that's as far as it goes; if you start with a line of length 1, compass and straightedge will let you construct a line with length equal to the square root of any rational number. You can do addition, multiplication, subtraction, division... and square roots, and there things come to a stop.
    - whatshisface 6y ago
      But what singles out a compass and straightedge as a "the" construction system? There are infinity other ways to construct numbers, each one with a potentially different set of constructable numbers.
      - thaumasiotes 6y ago
        You can use them to draw lines which you can then use to cut leather. They don't need to be "the" construction system; they are a construction system, and a very simple one.
        whatshisface 6y ago
        A pencil tied to the end of a string which is itself tied to the outside of a circle is also a construction system, one that happens to be able to construct the goat-grass area exactly. :)
        thaumasiotes 6y ago
        > one that happens to be able to construct the goat-grass area exactly. How? How did you get a string of the correct length? Note that the tool you described is a compass.
    - Koshkin 6y ago
      > compass and straightedge But those are not symbols. We're talking symbols (digits).
deleted 6y ago
[deleted]

castratikron 6y ago

It's in closed form

svat 6y ago

It's clearer from the Wikipedia article (https://en.wikipedia.org/w/index.php?title=Goat_problem&oldid=993368252 https://en.wikipedia.org/w/index.php?title=Goat_problem&oldi...): earlier we only had expressions for r like https://wikimedia.org/api/rest_v1/media/math/render/svg/9758becd77e7b4102ccdf50ff0845a749d236dc2 https://wikimedia.org/api/rest_v1/media/math/render/svg/9758... and https://wikimedia.org/api/rest_v1/media/math/render/svg/a63430b98b5e138a3d73831a2b7ecb6825be7a5b https://wikimedia.org/api/rest_v1/media/math/render/svg/a634... (from which of course one could numerically compute the answer r=1.1587…), but now thanks to this paper we have the expression https://wikimedia.org/api/rest_v1/media/math/render/svg/5b7451ea92f107e56651328ad15bfcc3276f87f2 https://wikimedia.org/api/rest_v1/media/math/render/svg/5b74... that does not involve r on the right-hand side.

Sniffnoy 6y ago
Hm, if it's based on this ratio of contour integrals, shouldn't it be possible to do better than this? Like why would it be so hard to find the residues for these poles? Shouldn't that be just a bit of formal Laurent series manipulation? What am I missing here?
- Someone 6y ago
  If the series don’t cancel out nicely (likely, I would guess), wouldn’t you end up with some infinite sum? If so, as this example shows (contour integrals in a closed form?) “closed form” is loosely defined, but I think most would say something with an infinite sum wouldn’t be one (but then, https://en.wikipedia.org/wiki/Closed-form_expression#Analytic_expression https://en.wikipedia.org/wiki/Closed-form_expression#Analyti... says: “In particular, special functions such as the Bessel functions and the gamma function are usually allowed, and often so are infinite series and continued fractions. On the other hand, limits in general, and integrals in particular, are typically excluded.[citation needed]”
  - Sniffnoy 6y ago
    No, you shouldn't end up with an infinite sum; if you only want to know the series finitely far out, you only need to know finitely many terms. However, it seems the problem is hard for other reasons. I had assumed, without checking, that they'd put the center of the circle at 3pi/8 because it's a pole. Nope! As best I can tell from some graphing tools, there is indeed precisely one pole in the circle, and it's on the real line, but it's close to the right endpoint; I don't know that it has any nice form. So I imagine that getting any sort of exact series expansion around there -- or even just getting exactly the first few terms, i.e. the first few derivatives there, which would be all you'd need -- would be difficult for that reason. (Although, the higher the order of the zero, the more initial terms you'd need...) Edit: Actually, I guess it looks like a zero of order 1? Except that doesn't make sense, because then the top contour integral would be zero...

6gvONxR4sf7o 6y ago

Imagine an exact solution like this: sin(cos(log(98234/123)+tan(exp(pi/5))-1)+pi + integral(complicated function(x) dx from 0 to 1) It doesn't tell you in a practical sense how long to make the goat's leash. So you find the numerical version to however many decimals you want to get a useful approximation.

pfortuny 6y ago

The unknown is isolated: r=very-difficult-to-compute-expression

elil17 6y ago

The word exact here is an inexact description of what sort of solution this really is - it's a closed form explicit solution. A closed form solution means that the equation is limited to to certain common mathematical operations and is finite in length. An explicit solution means that the quantity we are solving for is isolated on one side of the equation.