19 ms·
An artist's perplexing tribute to the Pythagorean Theorem (2009)
- caturopath 3y agoThe 2-3-4 right triangle. What's the problem?
- scoot 3y ago13 != 16
- dudeinjapan 3y agoThe triangle is not right, its wrong.
- lupire 3y agoThe triangle is right. It's the nuts are nuts.
- ploxiln 3y ago(possible sarcasm detected ;) (A 2-3-4 triangle is not a right triangle, no angle is 90º)
- Izkata 3y ago3-4-5 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 3y agoI believe you are responding to a joke.
- Izkata 3y agoI figured they remembered it was three consecutive numbers, but misremembered which three.
- BlueTemplar 3y ago4-5-6 of course.
- jb1991 3y agoI 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 3y agoThe 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 3y agoI 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 3y agoIt doesn't. The entirety of my comment is that they're representing the wrong thing.
- partdavid 3y agoNo! 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 3y agoThank you for literally explaining it to me like I was five, which apparently I am, I can’t believe I missed that.
- layer8 3y agoThe artist meditated, he didn't realize.
- gerdesj 3y agoThe 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 3y agoWhoops: "we see a 2-3-4 triangle" in the article
- alephknoll 3y agoYou can have a 2-3-4 right triangle if you can find the right axioms for it.
- layer8 3y agoThe triangle is right, but three nuts are left.
- deleted 3y ago[deleted]
- topherclay 3y agoNice example of a fencepost error.
- lanna 3y agoIt would have worked if the nuts represented squares instead of points
- baerrie 3y agoThe 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
- CRConrad 3y ago> I feel like this could all be helped if in casual counting we started at zero So when you've counted to one you've actually counted to two? I feel like some people have been damaged by C-like languages.
- aaplok 3y agoTo 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 3y agoI'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.
- scoot 3y agoBut 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 3y ago"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 3y agoThe 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 3y agoYou'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.
- spacecadet 3y agoAs 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 3y agoIf 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 3y agohahaha, thanks-
- edanm 3y agoCan 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 3y agoFunny 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 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 3y agoThere are only two hard problems in computer science: Cache invalidation, naming things, and off-by-one errors.
- deleted 3y ago[deleted]
- karmakaze 3y agoThis 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 3y agoI honestly don't see the problem. 5^2 = 3^2 + 4^2. Where is the error?
- dullcrisp 3y agoThe 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 3y agoThe 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 3y agoThe 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 https://en.wikipedia.org/wiki/Ludic_fallacy
- chubot 3y agoThat’s certainly a way to get a rise out of nerds lol
- lupire 3y agoThe 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 3y agoNailed 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.
- deleted 3y ago[deleted]
- smitty1e 3y ago> 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 3y agoBetter yet, step them in to where they represent the center of the intended squares, and then there's room for the extra three.
- 082349872349872 3y agomathematicians 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)
- nh23423fefe 3y agoits great because "artists" make gibberish and lie about the meaning which is another cognitive test AI passes now. the singularity is near.
- globalnode 3y agoi 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 3y agoThe lengths of the sides do not correspond to 3 : 4 : 5 proportions. 2 : 3 : 4 is not a Pythagorean triangle.
- dimask 3y agoIf the title was "meditation on math errors" it would have been a perfect art piece.
- Asooka 3y agoIt 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 3y agoI 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 3y agoGoogle "Leonardo da Vinci".
- CRConrad 3y agoWhat important political decisions did he ever make -- and did he get them right? (As I recall, there was a lot of switching patrons after having pissed off the previous one. Anything else he did better on?)
- rhelz 3y agoIts 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.
- CRConrad 3y agoNo, that's not the point of the artwork. The artist still doesn't even get that this is what's wrong with the artwork: > "What I had stumbled upon was that physical entities (stones) are not equatable with conceptual entities (points)," Bochner said Nope, that's not it.
- nvy 3y ago>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.
- rhelz 3y agoWell, on the basis of this reply, I'll have to agree that the readers are not uniformly very smart :-)
- nvy 3y agoWhat's awful about this site is because you couched your insult in superficially polite language, dang won't do anything. If I respond to this insult, though, that's not "curious conversation". The moderation on this site is total horse shit.
- YeGoblynQueenne 3y agoTo 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?
- deleted 3y ago[deleted]
- knappa 3y agoLooks like he figured out the problem by at least 2007: http://www.melbochner.net/exhibitions/gabbiano-rome-2007/ http://www.melbochner.net/exhibitions/gabbiano-rome-2007/