[HN Gopher] An artist's perplexing tribute to the Pythagorean Th...
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An artist's perplexing tribute to the Pythagorean Theorem (2009)
Author : nyc111
Score : 78 points
Date : 2024-04-12 08:44 UTC (2 days ago)
(HTM) web link (mathtourist.blogspot.com)
(TXT) w3m dump (mathtourist.blogspot.com)
| caturopath wrote:
| The 2-3-4 right triangle. What's the problem?
| scoot wrote:
| 13 != 16
| dudeinjapan wrote:
| The triangle is not right, its wrong.
| lupire wrote:
| The triangle is right. It's the nuts are nuts.
| ploxiln wrote:
| (possible sarcasm detected ;)
|
| (A 2-3-4 triangle is _not_ a right triangle, no angle is 90o)
| Izkata wrote:
| 3-4-5 is a right triangle, not 2-3-4.
|
| The intent was apparently to use nuts to represent edges, but
| he put them on points instead.
|
| The artist's realization isn't even correct.
| baruz wrote:
| I believe you are responding to a joke.
| Izkata wrote:
| I figured they remembered it was three consecutive numbers,
| but misremembered which three.
| BlueTemplar wrote:
| 4-5-6 of course.
| jb1991 wrote:
| I still don't get it. The image is a 3-4-5 right triangle,
| which is mathematically fine. What do you mean by "nuts" and
| "points"?
| Izkata wrote:
| The image in the article is of hazelnuts (I originally
| wrote "stones" then quickly edited it), and it's not a
| 3-4-5 triangle.
|
| 3-4-5 describes the length of each side - if you count the
| lengths of the triangle drawn in the image (the lines of
| chalk visible between the nuts on each side), it's only
| 2-3-4. To get 3-4-5 you're counting the number of nuts on
| each side, but those aren't lengths - those are the number
| of points marking the start/end of each unit length.
| jb1991 wrote:
| I see, I think you are referring to the unequal spacing
| of the nuts on each side, i.e. the side with 5 nuts has
| them closer together than the other sides.
|
| I thought there was some point being made about the use
| of nuts vs. some other arbitrary item. Why does it matter
| they are hazelnuts and not something else?
| Izkata wrote:
| It doesn't. The entirety of my comment is that they're
| representing the wrong thing.
| partdavid wrote:
| No! X--X--X 0 1 2
|
| That diagram represents a length of 2, not a length of 3,
| see? Here's three: X--X--X--X 0
| 1 2 3
|
| It's not that the hazelnuts are somehow imperfectly laid
| out or are an imperfect representation. It's wrong in
| principle, not practice (I mean it's wrong in practice
| too but every representation is).
| jb1991 wrote:
| Thank you for literally explaining it to me like I was
| five, which apparently I am, I can't believe I missed
| that.
| lupire wrote:
| You didn't miss it. You were focusing on the lattice
| edges, and PP was focusing on the lattice points. You're
| both right (except for PP's "No!" which should be
| "Yes!").
| layer8 wrote:
| The artist meditated, he didn't realize.
| gerdesj wrote:
| The piccie has nuts at unit lengths and the first line of the
| article after the very short intro is:
|
| "The artwork references the idea of relating the lengths of the
| sides of a 3-4-5 right triangle ..."
|
| How on earth did you get 2-3-4 for a right angled triangle! I
| blame booze, drugs, a late night or perhaps a standard issue:
| "off by one" (this is HN after all) ...
| gerdesj wrote:
| Whoops: "we see a 2-3-4 triangle" in the article
| alephknoll wrote:
| You can have a 2-3-4 right triangle if you can find the right
| axioms for it.
| layer8 wrote:
| The triangle is right, but three nuts are left.
| topherclay wrote:
| Nice example of a fencepost error.
| lanna wrote:
| It would have worked if the nuts represented squares instead of
| points
| baerrie wrote:
| The rub is the thinking a length of 4 maps to four points when in
| reality, the points are 4,3,2,1,0, totaling 5. I feel like this
| could all be helped if in casual counting we started at zero,
| then our entire concept of where the measurements start would be
| more in line with math. I think often about these fundamental
| conflicts in how we casually think about numbers and how they are
| actually modeled in math
| aaplok wrote:
| To count the length of the segment, we count the number of unit
| segments that compose it. Start with 1 (the first segment) and
| count up to 4 (the fourth segment), yielding a length of 4.
| We're consistent with the units: we are adding up lengths to
| get a length (metres, yards, or whatever).
|
| Counting (ie, adding up) points gives a number of points, which
| isn't a unit for lengths. Starting the enumeration from zero is
| a hack to recover the previous process of adding up unit
| lengths.
|
| This hack only works in this specific context of conversion. If
| you want to count points (say, the number of corners in the
| triangle) you'd need to start from 1.
| sparky_z wrote:
| I'm sorry, but your suggestion doesn't make any sense to me.
| Saying "I have zero coconuts" would mean that you have a
| coconut? Would you have to say you had "negative 1" coconuts if
| you didn't have any? At a sandwich shop you would have to order
| a "no feet long" sub? And this is your plan to prevent
| confusion?
|
| All that would really do swap the meaning of a bunch of words
| around so that "zero" means "one", "one" means "two", etc. Then
| we'd have to call it a "Two three four" right triangle but
| remember to make it with "three" (four), "four" (five), and
| "five" (six) stones and we're right back where we started.
|
| The problem here is confusion about what quantity is actually
| being counted: the "fence posts" or the "fence lengths"? That's
| always going to depend on context - the speaker and listener
| have to be on the same page. There's no way to fix that by
| changing the number we count from.
| baerrie wrote:
| Where do I say anything about zero becoming 1? What I am
| talking about is that counting arbitrary periods in a given
| quantity always starts with zero whether you count it or not.
| If you ground all of those coconuts up and decided to count
| how many cups of coconut you have in that mass, well you
| would be starting with an empty cup measure, which is zero.
| It's not just fence posts
| lupire wrote:
| Lok up "ordinal" and "offset". They are two different
| concepts that cannot be merged.
|
| Always using offsets will just cause confusing elsewhere.
| YeGoblynQueenne wrote:
| It's "ordinal" and "cardinal" really. If you wanna be all
| mathsy and stuff.
| YeGoblynQueenne wrote:
| Sorry about the above comment. I'm being way too snarky
| in this thread. I hope the following makes up for the
| snark in my other comments.
|
| What I meant above is that, in set theory, there are
| ordinal numbers and cardinal numbers, both of which are
| natural numbers. The easiest way to understand them is in
| terms of arrays: the number of elements in an array is a
| cardinal whereas an index into the array is an ordinal.
|
| Ordinals start at 0, because that's the first natural
| number, and that's prooobably? why array indexing
| traditionally begins at 0 (I'm totally guessing).
| Cardinals also start at 0. So, 0 is the "cardinality" of
| the empty set, {}, which has no elements and so no
| ordinal. In the set {+}, 0 is the ordinal that
| corresponds to the position of the single element of the
| set, in the set {+,+} 1 is the ordinal for the last
| element in the set etc.
|
| How this is relevant to the article above is that the
| artist very clearly intended the number of nuts in the
| "squares" at the sides of the triangle to be understood
| as cardinals: the squares represent sets and the
| cardinality of each set corresponds to the surface of
| that square. Sets are not ordered so the artist is free
| to place their elements in any arrangement, including a
| square, and placing a set in the form of a square on the
| side of an edge of the triangle clearly signals that its
| cardinality corresponds to the square of that edge. So
| arranged, the three squares necessarily overlap, so the
| nut at each of the three points in the triangle must be
| counted twice, once for each square it participates in.
| Seen that way, the image is a visual representation of
| Pythagora's theorem for a triangle with sides 3, 4 and 5,
| with squares 9, 16 and 25, where 9 + 16 = 25 (so the
| bottom edge is the hypotenuse).
|
| On the other hand, the person who commented on the blog
| interpreted the number of nuts as ordinals, denoting the
| position of vertices in three lattices, represented by
| the "squares". The same person therefore interpreted e.g.
| the three nuts on the left edge of the triangle as
| standing for vertices indexed by ordinals 0, 1 and 2, and
| so representing a square lattice of side "2"; and so on
| for the other "squares". Seen that way, the image is a
| visual representation of a triangle with sides 2, 3 and
| 4, where the squares of the sides are 4, 9 and 16, where
| 4 + 9 [?] 16 and so the triangle is obtuse rather than
| right (so the lower side is no longer the hypotenuse,
| since its square is no longer the sum of the squares of
| the other two sides) and the image is not a correct
| visual representation of Pythagoras' theorem.
|
| I want to say that even having written down the latter
| interpretation it still sounds deliberately obtuse to me,
| but the real lesson I think is that there are always
| multiple interpretations of the same statement, or
| formula, etc, and it's useful to be able to see as many
| of them as possible. At the same time, there is usually
| one _intended_ interpretation and that 's the one that
| should be preferred. All that is formalised in First
| Order Logic, in the concept of an interpretation, that is
| an assignment of truth values (true or false) to all the
| atoms of a predicate. A FOL interpretation is uniquely
| identified by the set of atoms to which it assigns the
| value true, therefore the number of possible
| interpretations is equal to the cardinality of the
| powerset of the set of atoms of a predicate. That's a lot
| of a interpretations! That's why you need an intended
| interpretation (also a concept in FOL).
|
| On the other hand, I might be wrong. Maybe it _is_
| deliberately obtuse to count the surface of a square by
| ordinals, rather than cardinals. Well I don 't know.
|
| Bottom line is that in art, like in maths, one must
| always look for more than one way to see things and not
| assume that they know all the answers before they have
| asked all the questions.
| xanderlewis wrote:
| > that's prooobably? why array indexing traditionally
| begins at 0 (I'm totally guessing).
|
| More than just tradition, it's because (at least in C) if
| a is an array, it's effectively just a pointer and so
| a[i] = *(a + i) (which means the i'th element of a is
| just the contents of memory address a + i). In
| particular, we have a[0] = *a.
|
| The first element of the array lives at zero offset from
| the address pointed to by a, so is considered the
| 'zeroth' element.
| kergonath wrote:
| Yeah. It has nothing with intuition or mathematical
| sense, quite the contrary. It's just a quirk of a
| language that got transformed as "the natural way" after
| decades of reinforcement. There were other languages
| before C where array indices started at 1.
| YeGoblynQueenne wrote:
| That's right. Before, and after too. R for example has
| indices starting in 1. But I was thinking about the
| specific 0-based convention. I guess I was wrong about it
| though.
| YeGoblynQueenne wrote:
| Ah thanks. I got the wrong model for that. It's been
| years since I coded in C :)
| denton-scratch wrote:
| It's not a problem with confusing ordinals and cardinals;
| the artist's problem is, as others have said, a fencepost
| error.
|
| I suspect that what has annoyed people is that if all the
| hazelnuts are evenly-spaced, then the triangle the artist
| created is not a 3-4-5 triangle, it's a 2-3-4 triangle,
| which isn't a right triangle. To make it look like a
| right triangle, he's had to arrange the nuts with unequal
| spacing. He must have noticed that, and it should have
| annoyed him too.
|
| I suppose there's some geometry in which a 2-3-4 triangle
| is a right triangle; but I doubt the artist was exploring
| non-euclidian geometries.
|
| FWIW, I didn't notice the error immediately - but I did
| notice the uneven spacings.
| YeGoblynQueenne wrote:
| >> It's not a problem with confusing ordinals and
| cardinals; the artist's problem is, as others have said,
| a fencepost error.
|
| Yeah, my argument is that it's a fencepost error only if
| the numbers of nuts are interpreted as ordinals indexing
| the vertices of a lattice on the Cartesian plane, rather
| than cardinals counting the elements of a set, while the
| artist intended them to stand for cardinals. That is what
| the anonymous contributor to the 360 blog, mentioned in
| the article above, seems to have seen:
|
| _" To my eye," the commenter continued, "the hazelnut
| grids look exactly like pins on a Geoboard, or lattice
| points in the plane. And given that perspective on this
| image, we see a 2-3-4 triangle, an obtuse triangle, and
| squares of area 4, 9, and 16."_
|
| But this is clearly only one way to see things and
| there's nothing to make it more valid than the other,
| except of course that this one leads to an error which
| strongly implies it's not the right view.
|
| >> I suspect that what has annoyed people is that if all
| the hazelnuts are evenly-spaced, then the triangle the
| artist created is not a 3-4-5 triangle, it's a 2-3-4
| triangle, which isn't a right triangle. To make it look
| like a right triangle, he's had to arrange the nuts with
| unequal spacing. He must have noticed that, and it should
| have annoyed him too.
|
| Some other comments say something similar, but I don't
| understand it. What is the issue with spacing? As you
| point out it would be difficult to get a perfectly
| mathematically correct spacing with irregularly shaped
| solids, like hazelnuts. Which suggests that spacing was
| not part of the intended interpretation. Can you explain?
| dimask wrote:
| No, but if you wanted to count how many coconuts you had, you
| would start with 0 (pointing at nothing) and then you would
| continue by 1,2 etc pointing to each other coconut. This
| would allow you to use the same counting process to count
| something, even if there was nothing to count.
| YeGoblynQueenne wrote:
| OK, so I start counting on my fingers:
|
| thumb -> zero
|
| index -> one
|
| middle -> two
|
| ring -> three
|
| pinkie -> four
|
| See what you gone did. Now I have four fingers, like a
| Looney Tune.
| scoot wrote:
| _But when he created his pattern, he found that he had three
| stones left over. Finally, it dawned upon him that the surplus
| came from counting the corners of the triangle twice.
|
| [...]
|
| Bochner welcomed the rediscovery of this "discrepancy" so many
| years after he had created the artwork. Yet he also wondered
| "about the unwillingness to assume that I already knew what they
| had just discovered (do mathematicians still think all artists
| are dumb?)._
|
| Apparently so, because he failed to understand that what was
| being commented on was not the absence of three stones (or
| wallnuts), but rather of significantly more.
| ElevenLathe wrote:
| "counting the corners of the triangle twice" is just another
| way of saying he got the math wrong. It's just a fencepost
| error. Or am I missing something?
| dullcrisp wrote:
| The squares overlap on the corners so he only needed to use
| 47 stones to form the diagram. That's a separate issue from
| the stones not being evenly spaced and seeming to show that 4
| + 9 = 16.
| lupire wrote:
| You're not missing anything. The artist didn't understand the
| difference between fenceposts and fence, boundary and
| interior.
|
| When presented with beautiful evidence of his mistake, he
| failed to see what it was showing him.
|
| The art is good in that it's a puzzle to interpret the
| mistake and resolve the paradox, even if (especially if!) the
| artist doesn't understand what they created.
| brookst wrote:
| The artist claims it was intentional. Do you know
| otherwise?
| spacecadet wrote:
| As an artist who explores mathematics through multi-dimensional
| art and works with mathematicians, can confirm, they find us all
| dumb. But! I have genuinely intrigued a few too.
| gerdesj wrote:
| If the maths kiddies get too much, paint/draw/whatevs a
| spherical cow - do it in the style of Mr G Larson (don't forget
| the light frown). If they don't get it then find another
| mathematician - they are quite common and herds of them are
| reasonably easy to find.
|
| If you need to encourage one to eat from your hand, try a
| fourth order differential equation with e and i in it as a
| tempter and work on from there. Be very careful of straying
| into physics - if something useful comes up they are known to
| scatter. Very skittish, your wild mathematician.
|
| You are not dumb - no-one is dumb. You have a voice and are
| demonstrably not dumb and the dumb have hands and are hence not
| dumb. If the dumb don't have hands then it gets complicated but
| it is possible that they might not be dumb. Dumb as a synonym
| for stupid is dumb. Please don't be dumb and use the term dumb.
|
| Examples of your work please!
| spacecadet wrote:
| hahaha, thanks-
| edanm wrote:
| Can you give some examples of your work? This sounds very
| interesting.
|
| (Especially the multi-dimensional aspect, as I sometimes hang
| out with people who solve higher-dimensional Rubik's cubes and
| other similar puzzles!)
| spacecadet wrote:
| Funny you mention Rubiks cube, here is a project I worked
| with 2 fellow computational artists on. We generated the form
| using open cas.cade, then machined it all on a 5 axis VMC,
| then assembled with magnets and a steel ball barring that I
| pulled from a ship engine. Carrol and Ruza if yout out there,
| I miss you :(
|
| https://vimeo.com/322284709
|
| I dont put my work online anymore, there is no point, I do it
| for me, and it just attacts corporate thieves :)
| crubier wrote:
| There are only two hard problems in computer science: Cache
| invalidation, naming things, and off-by-one errors.
| karmakaze wrote:
| This is taking _artistic license_ too far. It 's different than
| an explosion making a sound in space in a movie. It's missing the
| core point of the thing, which could easily been illustrated. Bad
| math _and_ bad art.
| D13Fd wrote:
| I honestly don't see the problem. 5^2 = 3^2 + 4^2. Where is the
| error?
| dullcrisp wrote:
| The problem is that the marks seem to imply that a 2-3-4
| triangle is a right triangle, when in fact they're just
| unevenly spaced.
|
| To illustrate the theorem, each stone would need to account
| for the same amount of area, which the ones on the edges
| don't do.
| jb1991 wrote:
| The error is that, for example, the side with five nuts means
| that it is four segments long, not five, because of where the
| nuts are placed (nuts at the endpoint of the lines). The
| triangle is a 2-3-4 in this image, which is impossible
| because that is not a right triangle.
| YeGoblynQueenne wrote:
| The problem is that there is an obvious interpretation (count
| the nuts) and a less obvious interpretation (count the spaces
| between the nuts) and if you don't see the less obvious
| interpretation immediately the people who see it will claim
| it's the obvious interpretation and you're a mathematical
| ignoramous for not noticing it immediately, as they did.
|
| In truth that's probably coming from people who didn't
| immediately notice the obvious interpretation and had to
| squeeze their noggin hard to get it, and then got upset that
| they found it that hard and squeezed their nogging even more
| to find a less obvious interpretation to hold up and say
| "see, that's why I was confused, you're all wrong".
|
| I'm being very mean in this thread. It's because it's all a
| big example of the Ludic (sic) fallacy, that I just learned
| about today and can't stop laughing.
|
| https://en.wikipedia.org/wiki/Ludic_fallacy
| Asooka wrote:
| I would not claim it's more obvious, but this is an
| illustration on the cover of a mathematics journal. It
| should show mathematics, not an artistic misunderstanding.
| If people have to stop and think through the math to
| understand it, then it is doing its job properly. As it is,
| it just gives a bad name to the artistically minded.
| YeGoblynQueenne wrote:
| You can think through the maths but you don't have to.
| This is one of those cases were pattern recognition is
| enough and no more thought is needed. That is to say, a
| mathematician should (and has no excuse not to) have seen
| Euclid's illustration of Pythagora's theorem which the
| journal cover is a very obvious representation of, so
| there should be no confusion.
| chubot wrote:
| That's certainly a way to get a rise out of nerds lol
| lupire wrote:
| The most annoying kind of art is art that does something wrong
| and then mocks people explained the mistake.
|
| The artist clearly failed to understand the difference between
| boundary points and interior regions, and incorrectly puts the
| blame on "hazelnuts are not abstract 0 dimensional points"
| larodi wrote:
| Nailed it. He should've counted the centroids if the squares if
| he needs to do... counting in matter of something-squared,
| while he counts the sides which are not yet squared in his
| design. So the placement is off or rather he mixed metrics.
| smitty1e wrote:
| > Finally, it dawned upon him that the surplus came from counting
| the corners of the triangle twice.
|
| Stack the extra hazelnuts on the vertices. Problem solved.
| Applejinx wrote:
| Better yet, step them in to where they represent the center of
| the intended squares, and then there's room for the extra
| three.
| 082349872349872 wrote:
| mathematicians work up to isomorphism; artists work up to
| plausiblemorphism.
|
| (another example: logicians work in syllogisms; rhetoricians omit
| the middle term and work in enthymemes; artists provide nothing
| but a middle term)
| globalnode wrote:
| i dont see any problem with this 3-4-5 tri :)... i mean its
| obvious isnt it? ofc the vertices share a point, so what?
| gpvos wrote:
| The lengths of the sides do not correspond to 3 : 4 : 5
| proportions. 2 : 3 : 4 is not a Pythagorean triangle.
| dimask wrote:
| If the title was "meditation on math errors" it would have been a
| perfect art piece.
| Asooka wrote:
| It is still wrong and could have been trivially fixed by putting
| the stones in the centres of the 1x1 squares making the
| rectangles rather than at the corners. The fact that the artist
| had 3 stones left over should have clued him that what he did was
| incorrect, yet his fix was to make the piece even more wrong. A
| very powerful reminder that artists should never be allowed to
| make any important policy decisions, as they are totally bereft
| of logical thought.
| Asooka wrote:
| I see one artist has already downvoted me. If you feel the need
| to do so, you're part of the problem plaguing modern society.
| mondobe wrote:
| Google "Leonardo da Vinci".
| rhelz wrote:
| Its hilarious watching a group of some of the smartest people on
| the planet still not getting it, over and over. It just sails
| right under their heads :-)
|
| The nuts are discrete, point-like objects. But length is
| continuous, and so is area. The _whole point_ of the artwork is
| to point out that its possible to confuse discrete objects with
| continuous objects, and confuse 1 dimensional objects like
| lengths with 2-dimentional objects like area.....and have off-
| by-1 errors, and double-counting errors...basically, you can make
| every mistake that it is possible to make.....
|
| ....AND STILL not realize you are wrong, because in this
| particular test case, the numbers came out to be what you
| expected them to be. Its saying don 't do that.
| nvy wrote:
| >Its hilarious watching a group of some of the smartest people
| on the planet still not getting it, over and over. It just
| sails right under their heads :-)
|
| The hubris displayed in thinking that HN is "a group of some of
| the smartest people on the planet" is revolting.
| YeGoblynQueenne wrote:
| To summarise the opinions expressed in this thread, all with
| equal conviction:
|
| a) It's a fencepost error (corner nuts are counted twice)
|
| b) The nuts are unevenly spaced
|
| c) The line segment counts are 2, 3 and 4
|
| d) The artist is an artist so obviously wrong
|
| An other commenter said something about "the smartest people on
| the planet". I hear this stuff about HN very often and, oh boy.
| Just look at this thread. Smartest people? Come on. We're just
| good with computers and then only with the stuff that has to do
| with computers that we've seen a hundred times before (e.g. off
| by one error). We see a novel representation and lose our shit.
|
| Sometimes I find myself defending human intelligence against the
| wildest claims of some of the faithful of AGI on this site but in
| cases like this I have to wonder: are humans _really_ that smart?
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