[HN Gopher] After centuries, a seemingly simple math problem get...
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After centuries, a seemingly simple math problem gets an exact
solution
Author : techlover14159
Score : 170 points
Date : 2020-12-10 16:49 UTC (6 hours ago)
(HTM) web link (www.quantamagazine.org)
(TXT) w3m dump (www.quantamagazine.org)
| atsmyles wrote:
| Hey Terence Tao, you might be a great mathematician but Ingo
| Ullisch is the GOAT!
| bryan0 wrote:
| > 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?
| elil17 wrote:
| 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.
| pfortuny wrote:
| The unknown is isolated:
|
| r=very-difficult-to-compute-expression
| 6gvONxR4sf7o wrote:
| 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.
| whatshisface wrote:
| In the sense that sqrt(2) can be an exact solution, even though
| you can't write it down as a decimal.
| gowld wrote:
| 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 wrote:
| > _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.
| [deleted]
| castratikron wrote:
| It's in closed form
| svat wrote:
| It's clearer from the Wikipedia article (https://en.wikipedia.o
| rg/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/9758...
| and
| 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/5b74...
| that does not involve _r_ on the right-hand side.
| Sniffnoy wrote:
| 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 wrote:
| 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#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]"_
| IshKebab wrote:
| Ha if only the article about mathematical equations had
| actually included any mathematical equations.
|
| This is like those articles that are about a picture but
| don't include the picture.
| crazygringo wrote:
| It's fascinating how a problem that looks at first glance like it
| belongs as a basic homework problem in a calculus textbook turns
| out to be so difficult.
|
| I'm curious if there are lists of other problems that are
| similarly easy to understand in a few seconds, that seem like
| they'd be similarly easy to solve, but which turn out to be
| fiendishly hard like this one?
|
| Especially ones that can be visualized easily geometrically like
| this one.
| twic wrote:
| Kepler's equation is a bit like this [1]. A body is moving in a
| known elliptical orbit, and you would like to know where it is
| at any given time - eg a quarter of the way through its orbit
| (so going from 'mean anomaly' to 'eccentric anomaly'). The
| starting point is a handful of simple equations, of motion and
| gravitation. But there is no analytical expression that answers
| the question - you have to solve it numerically, or approximate
| it with a sufficiently accurate Taylor series.
|
| https://en.wikipedia.org/wiki/Kepler's_equation
| hrydgard wrote:
| There's always Fermat's Last Theorem.
| FartyMcFarter wrote:
| I'm struggling to come up with examples.
|
| Maybe the 3n+1 problem? It's definitely easy to understand in a
| few seconds, and definitely fiendishly hard. I'm not sure if it
| seems easy to solve though :)
|
| Another one that came to mind is about efficiently computing
| the permanent of a matrix[1]. Maybe understandable in seconds
| if you know how determinants work.
|
| [1] https://en.wikipedia.org/wiki/Computing_the_permanent
| DataDaoDe wrote:
| 3n + 1 is interesting because it seems so tantalizingly easy
| and beautiful at first glance. Its like looking at a stream
| where you can clearly see the stony bed and it appears to be
| ankle deep but once you take a step you realize the lighting
| has fooled you and you fall head over heels into icy waters.
| pas wrote:
| It's so much more interesting if you happen to know about
| the "hydra game" (the Goodstein theorem), which is also a
| question about sequences and their termination at 0.
|
| https://en.wikipedia.org/wiki/Goodstein%27s_theorem
|
| But this is easily proved (using infinite ordinals, but it
| seems it could be proved just by coming up with a similar
| concept like the infinite ordinal arithmetic, basically
| providing an upper bound for each step of the algorithm and
| showing that there's an eventual maximum to these and then
| there's a monotonic decrease, and that the number of steps
| are always finite).
| qwerty1793 wrote:
| Mathoverflow has their "long-open problems which anyone can
| understand". This includes things like the integer brick
| problem: Is there a brick where all its dimensions (width,
| height, breadth, face diagonals and main diagonal) are
| integers? And Singmaster's conjecture: How many times can a
| number (other than 1) appear in Pascal's triangle?
|
| https://mathoverflow.net/questions/100265/not-especially-fam...
| galkk wrote:
| "I have discovered a truly remarkable proof of this theorem
| which this margin is too small to contain."
|
| I can't formulate proper google search query, but I once read
| funny story where spies were planting a papers with a math
| problem like it written by a kid, very simple looking, but the
| answer had actually crazy numbers.
| darkhorse13 wrote:
| >I'm curious if there are lists of other problems that are
| similarly easy to understand in a few seconds
|
| I too would love such a list. Can anyone point me if one does
| exist?
| [deleted]
| johncolanduoni wrote:
| There's a lot of simple geometric problems like this that don't
| have known closed form solutions. A lot of simple geometric
| time to intersection problems (comes up a lot in game physics)
| don't have known explicit solutions for example. Finding them
| is interesting but once you have something you can use an
| iterative algorithm on to find a solution at arbitrary
| precision the pressure/incentive to solve them is reduced quite
| a lot.
|
| Generally mathematicians would consider the number that answers
| this problem to be "known" in the colloquial sense, even before
| this better form.
| skinkestek wrote:
| This is probably written by people who don't have real-life
| experience with goats I guess.
|
| I cannot even concentrate on the problem since I keep coming back
| to what the goat will do:
|
| - jump the fence
|
| - chew the rope
|
| - probably more
| ufo wrote:
| The original statement of the problem talks about a horse. I
| wonder when it changed into a goat.
| yetihehe wrote:
| In such cases, it's best to assume it's a special zero
| dimensional point goat, which only chews grass.
| cgriswald wrote:
| How do you tie a zero dimensional point goat? How do you know
| he's in the fence at all? How does he chew the grass? Where
| does the grass go?
| morei wrote:
| ... are you serious? You use an infinitely thin rope of
| course!
|
| Slightly more seriously, you take the limit of a alpha-
| sized goat tied with an alpha-sized rope as alpha goes to
| zero from above.
| thechao wrote:
| Literally every one of these questions pertains to real
| goats, too.
| tuatoru wrote:
| And having discovered the goat, how do you put a tether on
| something with no size?
| WJW wrote:
| You'd need a special rope to tie it as well, in addition to
| low-dimensional fencing and specialized grass.
| srathi wrote:
| First, imagine a spherical cow!
| adwn wrote:
| ...in a vacuum.
| cgriswald wrote:
| When I was a small child my parents took me to a petting zoo
| inside an amusement park. While an employee watched, the goat I
| had just fed started eating the shirt I was wearing. I was
| terrified. My parents, not wanting to hurt the goat looked to
| the employee for help. The employee said, "Yeah, they'll do
| that" and turned around, not concerned for me or for the goat.
| My parents eventually extricated my shirt and all was well. I
| have no idea why I was never afraid of goats (especially since
| I was afraid of _trees_ for a period of about six months).
| throwaway889900 wrote:
| The better question is what farmer has a circular fence and is
| actively tying livestock to said fence. The fence should be
| good enough!
| kneedeepat wrote:
| As a Naval Academy grad, it still took me a second to figure out
| why it seemed to be a thing at the Naval Academy... But then Bill
| the Goat. FIN. Beat Army.
| 2sk21 wrote:
| Is this in any way to the problem of quadrature of the lune?
| https://en.wikipedia.org/wiki/Lune_of_Hippocrates
| tunesmith wrote:
| I first misread "goat tied to the inside of the fence" as meaning
| tied to a post in the center of the circle, and was confused at
| why everyone thought this question was so hard.
| fastaguy88 wrote:
| I initially assumed that the goat was tied to the fence in such
| a way that the non-goat rope-end could move completely around
| the circumference of the fence and not be fixed to a single
| point.
| acqq wrote:
| I was also at first confused, and I find the problem as
| published in 1894 (i.e. almost 130 years ago) much clearer:
|
| "A circle containing one acre is cut by another whose centre
| is on the circumference of the given circle, and the area
| common to both is one-half acre. Find that radius of the
| cutting circle."
|
| I think that formulation had to be at least mentioned near
| the beginning of the text, when not used in the first
| sentence.
| shmageggy wrote:
| Your goat eats the "donut" and GP's goat eats the "hole",
| each eating half of the area. Not a blade of grass was
| wasted; how satisfying!
| mrlala wrote:
| I was just drawing what you said on the whiteboard, and I was
| like.. uh, am I missing something or am I a super genius? I am
| positive the latter is incorrect.
|
| I had to re-read the first paragraph multiple times until I
| understood the goat was not tethered to the center.
| InitialLastName wrote:
| There's a picture of the problem like halfway down the
| article.
| r-bryan wrote:
| Cool. Now solve it for a circular fence on the surface of a
| sphere. In fact, solve all four cases {exterior, interior of
| sphere} X {exterior, interior of fence}. Spherical trig can only
| make the solution(s) even more heroic, right?
| antiquark wrote:
| You forgot about the hypergoats.
| Akronymus wrote:
| Why not try hyperbolic space too?
| scatters wrote:
| Surface of a sphere depends on how large the fence is compared
| to the sphere. If the fence is small, the answer is the same as
| the plane version. If the fence is as large as possible (an
| equator) then the rope needs to be precisely one quadrant in
| length, equalling the "radius" of the fence. If the fence
| encloses more than half the sphere... well, if it encloses
| _all_ of the sphere (that is, it is a small circle with the
| "inside" declared to be the outside) then the rope is again one
| quadrant, so half the "radius" of the fence.
|
| More interesting is a space-goat tethered to the interior of a
| hollow sphere, hypersphere etc.; no closed-form solutions for
| higher-dimensional cases, but the answer tends to sqrt(2) as
| the number of dimensions approaches infinity.
| alecbz wrote:
| I feel like "closed form" becomes a lot less special when you
| realize that things as simple as sin, cos, log, or even just sqrt
| don't have "closed forms" in the sense of "able to be expressed
| in terms of 'simpler' functions".
| analog31 wrote:
| Indeed, there's always a convention as to what the building
| blocks are. Like chemists don't look for how the quarks are
| configured -- they are satisfied to know how the atoms are
| arranged. In some cases, finding out what the building blocks
| are is an interesting problem in itself, for instance what
| things are computable with geometric construction.
| svat wrote:
| The paper this article is about:
|
| * _A Closed-Form Solution to the Geometric Goat Problem_ , by
| Ingo Ullisch in _The Mathematical Intelligencer_.
| https://doi.org/10.1007/s00283-020-09966-0
|
| The illustration in the article (I mean the top half of
| https://d2r55xnwy6nx47.cloudfront.net/uploads/2020/12/Grazin...)
| was really helpful to understand what the question is. I know
| Quanta Magazine articles sometimes get some negative comments
| here on HN from readers who expect a different kind of exposition
| than what is suitable for a magazine of that sort, but for my
| part I'm really grateful to Quanta Magazine for bringing
| attention to papers like this, and writing nice articles about
| them with history, good illustrations, quotes from other
| mathematicians, etc.
|
| For the impatient, Wikipedia has a short article on the problem
| and earlier progress:
| https://en.wikipedia.org/w/index.php?title=Goat_problem&oldi...
| From this it's more clear in what sense this is progress: whereas
| earlier the answer r=1.1587284730181215178...
| (https://oeis.org/A133731) was known via more than one formula of
| the form r = (some function of r), only now do we have a closed-
| form expression of the form r = (some expression not involving
| r).
| cochne wrote:
| Not really an exact or closed form solution since it contains an
| integral. It is an explicit solution compared to the implicit one
| we had before.
| adwn wrote:
| > _Not really an exact [...] solution_
|
| But it _is_ exact.
|
| > _Not really [a] closed form solution_
|
| It's my understanding that what is and isn't "closed form" is
| rather arbitrary. Functions which are used frequently - like
| exp() - are elevated to closed form status, and yet you can't
| evaluate exp() in a finite number of steps. So how is the
| explicit solution to the goat problem objectively different?
| cochne wrote:
| Yes I also feel that way to some degree, but I just never
| considered an integral to be closed form. There is some
| argument to be made for exp, as it is considered an
| 'elementary function'. I was going off this statement from
| wikipedia on closed form expressions
|
| "It may contain constants, variables, certain "well-known"
| operations (e.g., + - x /), and functions (e.g., nth root,
| exponent, logarithm, trigonometric functions, and inverse
| hyperbolic functions), but usually no limit, differentiation,
| or integration."
|
| Edit: I think if you say an integral is closed form, you must
| also admit that a limit is closed form, since an integral is
| defined in terms of limits (though technically more
| restrictive). In that case, you should also admit that we
| already had a closed form expression for this number, as it
| could be expressed as a limit of an iterative process.
| adwn wrote:
| > _There is some argument to be made for exp, as it is
| considered an 'elementary function'._
|
| But exp() is defined as the limit over an infinite sum, so
| why does _it_ get to be an elementary function?
|
| My point it that the distinction between closed form and
| non-closed form is arbitrary, and that there is no
| qualitative difference. In fact, limit and integral are
| just (higher-order) functions as well - and rather
| ubiquitous ones, so why aren't they considered elemental?
| [deleted]
| zaptheimpaler wrote:
| Finally! This was getting to be a real problem in my grazing goat
| as a service business.
| skinkestek wrote:
| I think that is an actual service provided in Norway.
| codetrotter wrote:
| You are correct. I remember seeing some goats at Oscarsborg
| Fortress and I was told that they were there on loan or on
| hire. Looked it up now and it checks out.
|
| The articles in the two links below affectionately refer to
| the featured goats as the "coast goat commando", which I
| think is just lovely :)
|
| https://www.oblad.no/badebyen/kystgeitkommandoen-til-
| oscarsb...
|
| https://www.oblad.no/badebyen/umb-geitene-pa-
| sommerjobb/s/2-...
|
| Both of these articles are in Norwegian but basically they
| talk about the importance of keeping the vegetation in check,
| and that the goats are great at this, as well as the social
| value that goats provide to visitors. The goats featured here
| are owned by a University and were rented out to the
| Oscarsborg Fortress.
|
| There are pictures of the goats also in the articles.
|
| PS: For anyone not familiar with Oscarsborg check out the
| following links for some info and pictures:
|
| https://en.wikipedia.org/wiki/Oscarsborg_Fortress
|
| https://www.forsvarsbygg.no/no/festningene/finn-din-
| festning...
|
| https://en.wikipedia.org/wiki/Battle_of_Drobak_Sound
|
| Using the defence batteries at Oscarsborg Fortress, the
| Norwegian coastal defence successfully sank the German heavy
| cruiser Blucher on 9 April 1940, forcing the German fleet to
| fall back.
|
| https://en.wikipedia.org/wiki/German_cruiser_Blucher
| hinkley wrote:
| Goats as a Service has been around longer than The Cloud. I
| think you already missed that gold rush.
| _jal wrote:
| You laugh, but growing up, my family used goats to keep brush
| down on our property, and would occasionally loan them to
| neighbors for the same.
| mhh__ wrote:
| https://www.telegraph.co.uk/technology/google/5297097/Google.
| ..
| z3t4 wrote:
| Intuitively the formula should have the length of the rope,
| radius of the fence, and PI. And intuitively a rope length of the
| radius would cover half the circle, but a quick test of putting
| two goats in there shows they can't eat all the area. So the
| formula probably have a minus in it. Next I would try to figure
| out how large the goat circle outside the fence is because then i
| would also know the area inside...
| b0rsuk wrote:
| I wonder if someone comes up with a neat formula for the
| perimeter of an ellipse. Keppler gave up...
| pas wrote:
| There are neat formulas, but those are just approximations, as
| an exact (closed form) is provably not possible.
|
| https://www.reddit.com/r/math/comments/8gw9on/is_it_possible...
|
| https://www.youtube.com/watch?v=5nW3nJhBHL0
| nullc wrote:
| Why do I have to go to Wikipedia to see the solution?
|
| Would it have been that hard to stick
| https://wikimedia.org/api/rest_v1/media/math/render/svg/5b74...
| in the article?
| ABeeSea wrote:
| From a journalistic standpoint, including the formula would
| require a longer digression into contour integrals than the one
| clause the article currently contains.
|
| It's also not clear what license that image is available under.
| blu_ wrote:
| > Content is available under CC BY-SA 3.0 unless otherwise
| noted
|
| Just including the actual solution in an article about this
| problem would be IMO a much better choice.
| segfaultbuserr wrote:
| > _It's also not clear what license that image is available
| under._
|
| A plain image of mathematical equation is not copyrightable,
| it's literally the MathJax output of a LaTeX equation
| ({\displaystyle r=2\cos \left({\frac {1}{2}}{\frac {\oint
| _{|z-3\pi /8|=\pi /4}z/(\sin z-z\cos z-\pi /2)\,dz}{\oint
| _{|z-3\pi /8|=\pi /4}1/(\sin z-z\cos z-\pi
| /2)\,dz}}\right)}). It does not have any copyrightable
| artistic design. And to the nitpickers - the pixmap output of
| a font is also not copyrightable under U.S. copyright laws.
| Even if it is, Computer Modern is available under a free
| license. And even if it's not, pure facts - such as math
| formulas - are not copyrightable, it would be trivial to
| write down the identical equation using another program and
| font.
|
| Copyright is not an inevitable, divine, or natural right, it
| is only an utilitarian tool adopted by the Constitution and
| lawmakers "to promote the Progress of Science and useful
| Arts." Thus, there always exist things that cannot be
| copyrighted. It's also why fair use of copyrighted works is
| conditionally allowed (not relevant to this case). The
| tendency of people to assume that every single piece of data
| is automatically controlled exclusively under copyright is
| frustrating.
| melling wrote:
| Probably because including equations dissuades the average
| person from reading a book or article
| blueflame7 wrote:
| Articles about math in general dissuade the average person
| from reading
| tuatoru wrote:
| From TFA: * Of course, it won't upend textbooks or revolutionize
| math research, Ullisch concedes, because this problem is an
| isolated one. "It's not connected to other problems or embedded
| within a mathematical theory."*
|
| A island peak hinting at a submerged continent of mathematics.
|
| Unfortunately since our brains evolved (under a regime of calorie
| cost vs survival benefit) and are therefore limited, we might
| never discover the continent.
| Invictus0 wrote:
| Plaudits to Steve Nadis for pulling every goat joke known to man
| for this article.
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(page generated 2020-12-10 23:00 UTC)