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.
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?
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).
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!").
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) ...
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.
"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?
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.
> "What I had stumbled upon was that physical entities (stones) are not equatable with conceptual entities (points)," Bochner said
That's because the relevant conceptual and physical entities when contemplating lengths of sides of triangles, which is what Pythagoras' theorem is about, are lines, not points.
>The artist claims it was intentional. Do you know otherwise?
His reply reveals that he still doesn't get it, which to me seems to prove that it can't have been intentional.
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.
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.
I agree. And I’ll also say that whether the artist intended this as the meaning of the work is irrelevant. Art once created exists apart from its creator and can be analyzed and criticized and understood in ways the artist never expected or intended. So even if the artist just made a mistake here the art itself is able to evoke the much deeper and more interesting meaning which you read into it.
>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.
If anybody took it as an insult, I apologize. It was not meant that way at all.
It was meant to be ironic. The reason they didn't get the artwork wasn't because they were dumb, it was because they were smart. Kind of like how Goedel's technique works on the strengths of a formal system, not its weaknesses.
It's easy to outsmart yourself, and I've fallen into this trap more than anybody else. For example, it's harder to debug code than write code, so if you write the cleverest code you can, you won't be able to debug it.
I can't tell you how many times I've had to replace code I wrote, which so slickly used the latest C++ feature, or was so "elegantly" written, with the dumbest, dorkiest code possible which would solve the problem, just so I could actually deliver a robust and bug-free solution.
This particular artwork so elegantly captures the moral of this story, and in a very visually appealing way.
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.
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.
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 :(
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?
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.
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.
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.
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.
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.
> The problem is that there is an obvious interpretation (count the nuts) and a less obvious interpretation (count the spaces between the nuts)
Only if one goes with your "obvious" interpretation the artwork should obviously not be called anything to do with Pythagoras, since his theorem is about side lenghts, i.e. lines (spaces). Not points (nuts).
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"
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.
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
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.
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.
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
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.
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.
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.
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.
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.
>> 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?
> while the artist intended them to stand for cardinals.
Oh stop it with that BS already. He fucking obviously intended them to stand for line lenghts, or he wouldn't have arranged them in a right (but wrong) triangle.
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.
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.
