An artist's perplexing tribute to the Pythagorean Theorem (2009)
mathtourist.blogspot.com
mathtourist.blogspot.com
(A 2-3-4 triangle is not a right triangle, no angle is 90º)
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.
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 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?
X--X--X
0 1 2
X--X--X--X
0 1 2 3
"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) ...
[...]
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.
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.
Stack the extra hazelnuts on the vertices. Problem solved.
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.
The hubris displayed in thinking that HN is "a group of some of the smartest people on the planet" is revolting.
If I respond to this insult, though, that's not "curious conversation".
The moderation on this site is total horse shit.
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.
> "What I had stumbled upon was that physical entities (stones) are not equatable with conceptual entities (points)," Bochner said
Nope, that's not it.
(another example: logicians work in syllogisms; rhetoricians omit the middle term and work in enthymemes; artists provide nothing but a middle term)
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!
(Especially the multi-dimensional aspect, as I sometimes hang out with people who solve higher-dimensional Rubik's cubes and other similar puzzles!)
I dont put my work online anymore, there is no point, I do it for me, and it just attacts corporate thieves :)
And the resulting cube looks pretty awesome :)
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?
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.
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.
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 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"
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.
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.
Always using offsets will just cause confusing elsewhere.
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.
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.
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?
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.
thumb -> zero
index -> one
middle -> two
ring -> three
pinkie -> four
See what you gone did. Now I have four fingers, like a Looney Tune.
you would start with 0 (pointing at nothing)nothing -> zero
thumb -> one
index -> two
middle -> three
ring -> four
pinkie -> five
Hey, this seems like it might work! How many numbers are those? Let me count them.
On my fingers, duh.
zero -> thumb -> one
one -> index -> two
two -> middle -> three
three -> ring -> four
four -> pinkie -> five
five ->nothing -> six
So now I got six fingers! What am I, the illegitimate child of a Hapsburg royal?
Fucking internets...
Because while you might think you were being funny, you came off as fucking annoying. HTH!
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.
(As I recall, there was a lot of switching patrons after having pissed off the previous one. Anything else he did better on?)