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Every group who ever managed to build a building with a rectangular foundation figured out the relation between the side lengths and the diagonal. The Egyptian

Every group who ever managed to build a building with a rectangular foundation figured out the relation between the side lengths and the diagonal. The Egyptians used Pythagorean triples to measure right angles way before the birth of Pythagoras. But that's not a theorem, just an observation. It becomes a theorem when you prove (i.e. explain why) this relationship always holds, based on more evident things. The Babylonian tablet mentioned in the article doesn't seem to do anything like that, whereas the Greeks definitely did (we don't know whether Pythagoras himself did it, as no writing of his survives, but later Greeks knew how to do it, and attributed it to Pythagoras).

WalterBright 3y ago
> Every group who ever managed to build a building with a rectangular foundation figured out the relation between the side lengths and the diagonal. I don't see why that was necessary. You can get pretty far with eyeballing it and custom cutting to fit.
- saithound 3y ago
  A "you" can, and many individual "you"s presumably did that, to general satisfaction. But the implicit assumption you made is that eyeballing and cutting is easy, while triangles are hard, even though it's the other way around, especially with stone age tools. But we're talking about groups of people, "they", who all build a lot. Such groups tend to have a few people who make these observations, and then the observations proliferate, because they are both way easier to use than the alternative hack-work, and yield much more aesthetically pleasing results. Especially in the stone age when you nornally don't have easy access to anything at all with a right angle (unlike in modern construction) and you build stuff out of clay bricks, where minor inaccuracies inevitably add up and make your life much harder down the road. Tying together three pieces of string with prescribed ratios, pull it tight was a very easy way to get a right angle compared to anything that came before. It's necessary in the sense that stone age construction is so much easier if you know about it, and so much harder if you don't. Those who didn't come up with it didn't do such construction, because doing difficult things is harder than doing easy things.
  - dylan604 3y ago
    >A "you" can, and many individual "you"s presumably did that, to general satisfaction I was chatting with the person that was in charge of marking the fields for the local youth soccer league. I volunteered to help one weekend, and one of the first things he asked was if I knew what a 3/4/5 triangle was since it was the only way to know you'll be squared. I never did figure out to what level he was dead panning his joke or if he was even meaning for it to be a joke. Either way, I laughed.
    - xxs 3y ago
      3/4/5 is considered a wood working trick - that many beginners would not know (according to youtube, at least)
      - dylan604 3y ago
        I learned it in geometry in like 10th grade. No wood involved.
  - WalterBright 3y ago
    You can easily accurately lay out a large square without knowledge of Pythagorean Theorem. 1. lay out a straight line 2x in length. 2. find midpoint (easy by drawing an arc with a string from each end point. Basic compass & ruler technique. 3. draw another arc from the midpoint, of radius 1x. Try different spots on the arc until it is equidistant from each of the endpoints of the line. 4. voila! an accurate right angle. Laying out the rest of the square is now trivial.
  - gregjor 3y ago
    I think you mean bronze age.
    - saithound 3y ago
      I don't think I do. I'm fairly confident that the people of the chalcolithic did enough large scale construction (and pottery and watching the night skies and so on) to have figured a whole lot of empirically accessible geometry out. The _article_ happens to be about written sources from the Bronze Age Babylonians, which lets us glimpse at their accumulated knowledge. But there's no reason to believe this knowledge was particularly new at their time, and this was a clay tablet equivalent of an arXiv preprint.
- fsckboy 3y ago
  > I don't see why that was necessary. You can get pretty far with eyeballing it and custom cutting to fit. i'm not saying at all that I know the answer, but eyeballing and cutting is good for a patio, but on a ziggurat or pyramid scale it seems you don't do so much eyeballing or cutting, and more planning how much material and how many slaves you're going to need for how long, and where to put the doors so the passageways will meet up, that sort of thing.
  - WalterBright 3y ago
    I've seen many Mayan structures in Mexico, and they all look like they were eyeballed.
    - posterboy 3y ago
      I'm currently in a brick build house which has a good ten percent slope, because the earth sank, but it hasn't collapsed. I'm pretty sure its not older than a hundred years and I'm amazed it still stands because it sure wasn't designed like this.
  - rebolek 3y ago
    Please, stop this slave nonsense. Pyramids were built by highly skilled workers, it was prestigious job.
    - PawgerZ 3y ago
      I remember reading about the worker homes and tombs, and I know that skeletal records also showed the workers had much muscular strain. So, it's impossible to deny that there were skilled laborers working on the pyramids, but just because some of the laborers were proven to be skilled and not slaves, doesn't mean slaves weren't used. The scale doesn't seem right to me. I still can't fathom how that much could be done without using slave labor. If I remember correctly, they layed a block like every 6 minutes for 20+ years. Do you know how many tombs of laborers were found, or where I could find more aobut that information? I'm very novice when it comes to Egypt
- danjc 3y ago
  I built a tree house for my kids. I didn't exactly eyeball it but my amateur technique and planning left me having to do a lot of compensation for things being out of square. Everything becomes a custom cut, often in multiple dimensions and all earlier errors cascade all the way to the end.
  - berkes 3y ago
    > Everything becomes a custom cut, often in multiple dimensions and all earlier errors cascade all the way to the end. Unrelated, but that sounds exactly like many software projects I've been unfortunately part of.
- ReptileMan 3y ago
  Ancient Mesopotamian laws sometimes flayed people alive for incompetent or corrupt work. Those types of mild punishments are good incentive to use the right angles. Pun intended.
  - WalterBright 3y ago
    Money works a lot better than flogging.
deleted 3y ago
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deleted 3y ago
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sn41 3y ago
Proofs are of course the gold standard. But even if others discovered the rule, their contribution should not be held to be trivial. A modern analogy: today, people are very happy to use LLMs and Transformers without anyone "proving" that they work. Right now, philosophically, they are at the level of empirical observations (perhaps not even that). Does that mean that today's AI researchers should get no credit when at a later date? I am not sure. Empirical discovery of a rule is also no trivial thing.
- Turing_Machine 3y ago
  Pointing out that it's not a theorem doesn't mean it's been trivialized. If there's no proof, it's not a theorem, by definition. That doesn't make empirical observations worthless -- far from it!
- saithound 3y ago
  Yes, when people in the 4000s write clickbait holonet articles about how the Master Theorem of Neural Network Scaling was not actually invented by the Muskovites of Mars, but was found written on an Ancient American tablet containing the works of Hoffmann et al., I hope somebody will offer substantial corrections. That's not in any sense a value judgment of the empirical work done at DeepMind. Nor does it stop anybody from writing a better article which explains that the Babylonians (and many others) used the empirical observations underlying the theorem, while explaining that this did not constitute mathematical proof.
  - outrun86 3y ago
    Found on an ancient tablet as the work of Schmidhuber et al. but ok
geraneum 3y ago
From the article: > Pythagoras is immortally linked to the discovery and proof of a theorem that bears his name – even though there is no evidence of his discovering and/or proving the theorem. Simply quoting. It seems like > Greek definitely did is not a universally held belief.
- saithound 3y ago
  Your quote from the article does not suggest what you seem to think it suggests. The Greeks definitely were able to prove the Pythagorean theorem, and the Greeks definitely though Pythagoras specifically knew how to prove it: e.g. Proclus II states this in writing explicitly, as does Euclid 300 years later, and numerous ancient sources inbetween. It would be hard for this not to be the consensus. We can't know if Pythagoras really did, simply because we have no surviving writings directly from Pythagoras - as I stated explicitly in my previous post.
  - geraneum 3y ago
    You are actually right. You didn’t claim that the Greek did it first and that is what I doubted in the first place. I misread your comment. That fact that the Greek could prove the theorem at some point is not unlikely even without evidence. It’s as likely as others doing it well before them.
fsckboy 3y ago
> Pythagorean triples to measure right angles ... But that's not a theorem, just an observation. It becomes a theorem when you prove (i.e. explain why) this relationship always holds, based on more evident things. The Babylonian tablet mentioned in the article doesn't seem to do anything like that according to the article, this Babylonian tablet does come closer to theorem than you are suggesting. They weren't using Pythagorean triples, rather they figured out that the diagonal of a unit square is the square root of 2, and knew how to calculate that: 1 + 24/60 + 51/602 + 10/603 = 1.414213
- hyperthesis 3y ago
  Agree it's closer but still an instance.
  - fsckboy 3y ago
    depends on the method they used to come up with the formula. It's not obvious that the sum of a "more complicated than you can do in your head" series of fractions would be either the square root of two or the the diagonal of a square, let alone both. This is a fragment of a single clay tablet. If they had a systematic method of coming up with this convergent series, there might have been a stack of tablets showing square root of three, five, etc. It seems pretty highly suggestive to me.
    - ssener2001 3y ago
      I agree.Obtaining this formula by observing is more difficult than obtaining it by proving it. Besides, it is an issue that is already needed in field work. If you have pen, paper and a good number system, it shouldn't be too difficult to find the formula with shapes.
      - loa_in_ 3y ago
        I imagine back then things like number systems and methods of proving were just made up as they go.
dclowd9901 3y ago
3,4,5
orra 3y ago
> It becomes a theorem when you prove (i.e. explain why) this relationship always holds, based on more evident things. [...] whereas the Greeks definitely did (we don't know whether Pythagoras himself did it Wait, really? I thought the proofs back then were geometric, so only proved specific instances. And so it wasn't until algebra, maybe trig, was discovered it was properly proven?
- n4r9 3y ago
  There are fully general and valid geometric proofs, eg http://mathandmultimedia.com/wp-content/uploads/2010/02/pythagrean41.png http://mathandmultimedia.com/wp-content/uploads/2010/02/pyth...
  - orra 3y ago
    Despite labels $a$, $b$, $c$, I don't see how it's fully general. At best you could say it works for similar triangles, if you have that concept. But I don't see how it proves anything if $a$ were say doubled, and $b$ kept the same.
    - n4r9 3y ago
      I'm not sure what you mean, sorry. It's definitely fully general. It demonstrates that: * For any real positive values $a$ and $b$ * let $T$ be the (unique) right-angled triangle with non-hypotenuse sides of length $a$ and $b$ * then $a$^2 = $b$^2 = $c$^2 * where $c$ is the (unique) length of the hypotenuse of $T$. You could give me the instance $a$ = 2, $b$ = 3 and it will prove it for this case. Then you could later double $a$ and give me the instance $a$ = 4, $b$ = 3 and it will prove it just as easily for this case.
usrusr 3y ago
Why would you need numbers for right angles? You need a straight line and a string. But I agree with the second paragraph: there's a huge difference between a procedure that is handed down as part of "this is how we estimate a building project" and a theorem that is declared universal truth and base for all kinds of other theorems. Even if they are exactly the same thing.
- detourdog 3y ago
  How useful are universal truths compared to getting shit done? I only see theorems as useful for complex societies. The son of a gem dealer would have the time to work out universal truths. The reputation of everyone doing it without numbers would reveal the pattern of the universal truth. Finally this looks like it was all done using cuneiform. Which brings up questions of notation and the language to describe a square root.
  - usrusr 3y ago
    Getting shit done is the antithesis of progress because it goes hand in hand with "if it was good enough for my father, it will be good enough for me, who are you to disrespect (the methods of) my father!"
    - detourdog 3y ago
      I agree but dying for ideals isn’t much progress. Progress is a luxurious goal.
  - empath-nirvana 3y ago
    What everyone is talking past here is that the Babylonians discovered what engineers call a "rule of thumb", and engineering is focused on getting things done using rules of thumb. The best possible rule of thumb is a mathematical proof or a scientific discovery, but it's by no means necessary. . Observing that something always holds and even having a formula for it is not the same thing as having a proof, and the proof is what makes it mathematics and geometry and not "just" engineering. The babylonians had a rule of thumb -- the greeks discovered the theorem -- and more than that, they seem to have invented the mathematical/geometrical proof as a concept, along with formal logic. Without that mental framework, it's hard to say that the babylonians proved anything or had any theorems at all, only collections of rules of thumb. It's quite likely that lots of babylonians sort of independently and intuitively understood _why_ it must be true, but they don't seem to have ever written it down. Not that there's anything wrong with having rules of thumb -- it's a huge achievement to even notice and collect and teach those things, all the stuff around you is built relying on them. I encourage everyone to watch this series of videos. https://www.youtube.com/watch?v=_ivqWN4L3zU https://www.youtube.com/watch?v=_ivqWN4L3zU
skypanther 3y ago
I took a course at the university of Buffalo in the early 80s about pre-roman examples of Pythagorean geometry in henge monuments and early structures. Basically the class was watching slide shows of the professors vacations across Europe and his photos of old ruins. Then he'd show us a line drawing of the structure and pick out what at times seemed random points that created 3-4-5 triangles. He'd exclaim 'see? They knew about the Pythagorean theorem before Pythagoras!' Our only graded assignment was to write a paper where we "discovered" a similar example. I picked a picture of some random ancient church and "found" the right triangle in it's foundation. I got an A. Definitely just an "observation" no theorum involved.

WalterBright 3y ago

> Every group who ever managed to build a building with a rectangular foundation figured out the relation between the side lengths and the diagonal. I don't see why that was necessary. You can get pretty far with eyeballing it and custom cutting to fit.

saithound 3y ago
A "you" can, and many individual "you"s presumably did that, to general satisfaction. But the implicit assumption you made is that eyeballing and cutting is easy, while triangles are hard, even though it's the other way around, especially with stone age tools. But we're talking about groups of people, "they", who all build a lot. Such groups tend to have a few people who make these observations, and then the observations proliferate, because they are both way easier to use than the alternative hack-work, and yield much more aesthetically pleasing results. Especially in the stone age when you nornally don't have easy access to anything at all with a right angle (unlike in modern construction) and you build stuff out of clay bricks, where minor inaccuracies inevitably add up and make your life much harder down the road. Tying together three pieces of string with prescribed ratios, pull it tight was a very easy way to get a right angle compared to anything that came before. It's necessary in the sense that stone age construction is so much easier if you know about it, and so much harder if you don't. Those who didn't come up with it didn't do such construction, because doing difficult things is harder than doing easy things.
- dylan604 3y ago
  >A "you" can, and many individual "you"s presumably did that, to general satisfaction I was chatting with the person that was in charge of marking the fields for the local youth soccer league. I volunteered to help one weekend, and one of the first things he asked was if I knew what a 3/4/5 triangle was since it was the only way to know you'll be squared. I never did figure out to what level he was dead panning his joke or if he was even meaning for it to be a joke. Either way, I laughed.
  - xxs 3y ago
    3/4/5 is considered a wood working trick - that many beginners would not know (according to youtube, at least)
    - dylan604 3y ago
      I learned it in geometry in like 10th grade. No wood involved.
- WalterBright 3y ago
  You can easily accurately lay out a large square without knowledge of Pythagorean Theorem. 1. lay out a straight line 2x in length. 2. find midpoint (easy by drawing an arc with a string from each end point. Basic compass & ruler technique. 3. draw another arc from the midpoint, of radius 1x. Try different spots on the arc until it is equidistant from each of the endpoints of the line. 4. voila! an accurate right angle. Laying out the rest of the square is now trivial.
- gregjor 3y ago
  I think you mean bronze age.
  - saithound 3y ago
    I don't think I do. I'm fairly confident that the people of the chalcolithic did enough large scale construction (and pottery and watching the night skies and so on) to have figured a whole lot of empirically accessible geometry out. The _article_ happens to be about written sources from the Bronze Age Babylonians, which lets us glimpse at their accumulated knowledge. But there's no reason to believe this knowledge was particularly new at their time, and this was a clay tablet equivalent of an arXiv preprint.
fsckboy 3y ago
> I don't see why that was necessary. You can get pretty far with eyeballing it and custom cutting to fit. i'm not saying at all that I know the answer, but eyeballing and cutting is good for a patio, but on a ziggurat or pyramid scale it seems you don't do so much eyeballing or cutting, and more planning how much material and how many slaves you're going to need for how long, and where to put the doors so the passageways will meet up, that sort of thing.
- WalterBright 3y ago
  I've seen many Mayan structures in Mexico, and they all look like they were eyeballed.
  - posterboy 3y ago
    I'm currently in a brick build house which has a good ten percent slope, because the earth sank, but it hasn't collapsed. I'm pretty sure its not older than a hundred years and I'm amazed it still stands because it sure wasn't designed like this.
- rebolek 3y ago
  Please, stop this slave nonsense. Pyramids were built by highly skilled workers, it was prestigious job.
  - PawgerZ 3y ago
    I remember reading about the worker homes and tombs, and I know that skeletal records also showed the workers had much muscular strain. So, it's impossible to deny that there were skilled laborers working on the pyramids, but just because some of the laborers were proven to be skilled and not slaves, doesn't mean slaves weren't used. The scale doesn't seem right to me. I still can't fathom how that much could be done without using slave labor. If I remember correctly, they layed a block like every 6 minutes for 20+ years. Do you know how many tombs of laborers were found, or where I could find more aobut that information? I'm very novice when it comes to Egypt
danjc 3y ago
I built a tree house for my kids. I didn't exactly eyeball it but my amateur technique and planning left me having to do a lot of compensation for things being out of square. Everything becomes a custom cut, often in multiple dimensions and all earlier errors cascade all the way to the end.
- berkes 3y ago
  > Everything becomes a custom cut, often in multiple dimensions and all earlier errors cascade all the way to the end. Unrelated, but that sounds exactly like many software projects I've been unfortunately part of.
ReptileMan 3y ago
Ancient Mesopotamian laws sometimes flayed people alive for incompetent or corrupt work. Those types of mild punishments are good incentive to use the right angles. Pun intended.
- WalterBright 3y ago
  Money works a lot better than flogging.

deleted 3y ago

[deleted]

deleted 3y ago

[deleted]

sn41 3y ago

Proofs are of course the gold standard. But even if others discovered the rule, their contribution should not be held to be trivial. A modern analogy: today, people are very happy to use LLMs and Transformers without anyone "proving" that they work. Right now, philosophically, they are at the level of empirical observations (perhaps not even that). Does that mean that today's AI researchers should get no credit when at a later date? I am not sure. Empirical discovery of a rule is also no trivial thing.

Turing_Machine 3y ago
Pointing out that it's not a theorem doesn't mean it's been trivialized. If there's no proof, it's not a theorem, by definition. That doesn't make empirical observations worthless -- far from it!
saithound 3y ago
Yes, when people in the 4000s write clickbait holonet articles about how the Master Theorem of Neural Network Scaling was not actually invented by the Muskovites of Mars, but was found written on an Ancient American tablet containing the works of Hoffmann et al., I hope somebody will offer substantial corrections. That's not in any sense a value judgment of the empirical work done at DeepMind. Nor does it stop anybody from writing a better article which explains that the Babylonians (and many others) used the empirical observations underlying the theorem, while explaining that this did not constitute mathematical proof.
- outrun86 3y ago
  Found on an ancient tablet as the work of Schmidhuber et al. but ok

geraneum 3y ago

From the article: > Pythagoras is immortally linked to the discovery and proof of a theorem that bears his name – even though there is no evidence of his discovering and/or proving the theorem. Simply quoting. It seems like > Greek definitely did is not a universally held belief.

saithound 3y ago
Your quote from the article does not suggest what you seem to think it suggests. The Greeks definitely were able to prove the Pythagorean theorem, and the Greeks definitely though Pythagoras specifically knew how to prove it: e.g. Proclus II states this in writing explicitly, as does Euclid 300 years later, and numerous ancient sources inbetween. It would be hard for this not to be the consensus. We can't know if Pythagoras really did, simply because we have no surviving writings directly from Pythagoras - as I stated explicitly in my previous post.
- geraneum 3y ago
  You are actually right. You didn’t claim that the Greek did it first and that is what I doubted in the first place. I misread your comment. That fact that the Greek could prove the theorem at some point is not unlikely even without evidence. It’s as likely as others doing it well before them.

fsckboy 3y ago

> Pythagorean triples to measure right angles ... But that's not a theorem, just an observation. It becomes a theorem when you prove (i.e. explain why) this relationship always holds, based on more evident things. The Babylonian tablet mentioned in the article doesn't seem to do anything like that according to the article, this Babylonian tablet does come closer to theorem than you are suggesting. They weren't using Pythagorean triples, rather they figured out that the diagonal of a unit square is the square root of 2, and knew how to calculate that: 1 + 24/60 + 51/602 + 10/603 = 1.414213

hyperthesis 3y ago
Agree it's closer but still an instance.
- fsckboy 3y ago
  depends on the method they used to come up with the formula. It's not obvious that the sum of a "more complicated than you can do in your head" series of fractions would be either the square root of two or the the diagonal of a square, let alone both. This is a fragment of a single clay tablet. If they had a systematic method of coming up with this convergent series, there might have been a stack of tablets showing square root of three, five, etc. It seems pretty highly suggestive to me.
  - ssener2001 3y ago
    I agree.Obtaining this formula by observing is more difficult than obtaining it by proving it. Besides, it is an issue that is already needed in field work. If you have pen, paper and a good number system, it shouldn't be too difficult to find the formula with shapes.
    - loa_in_ 3y ago
      I imagine back then things like number systems and methods of proving were just made up as they go.

dclowd9901 3y ago

3,4,5

orra 3y ago

> It becomes a theorem when you prove (i.e. explain why) this relationship always holds, based on more evident things. [...] whereas the Greeks definitely did (we don't know whether Pythagoras himself did it Wait, really? I thought the proofs back then were geometric, so only proved specific instances. And so it wasn't until algebra, maybe trig, was discovered it was properly proven?

n4r9 3y ago
There are fully general and valid geometric proofs, eg http://mathandmultimedia.com/wp-content/uploads/2010/02/pythagrean41.png http://mathandmultimedia.com/wp-content/uploads/2010/02/pyth...
- orra 3y ago
  Despite labels $a$, $b$, $c$, I don't see how it's fully general. At best you could say it works for similar triangles, if you have that concept. But I don't see how it proves anything if $a$ were say doubled, and $b$ kept the same.
  - n4r9 3y ago
    I'm not sure what you mean, sorry. It's definitely fully general. It demonstrates that: * For any real positive values $a$ and $b$ * let $T$ be the (unique) right-angled triangle with non-hypotenuse sides of length $a$ and $b$ * then $a$^2 = $b$^2 = $c$^2 * where $c$ is the (unique) length of the hypotenuse of $T$. You could give me the instance $a$ = 2, $b$ = 3 and it will prove it for this case. Then you could later double $a$ and give me the instance $a$ = 4, $b$ = 3 and it will prove it just as easily for this case.

usrusr 3y ago

Why would you need numbers for right angles? You need a straight line and a string. But I agree with the second paragraph: there's a huge difference between a procedure that is handed down as part of "this is how we estimate a building project" and a theorem that is declared universal truth and base for all kinds of other theorems. Even if they are exactly the same thing.

detourdog 3y ago
How useful are universal truths compared to getting shit done? I only see theorems as useful for complex societies. The son of a gem dealer would have the time to work out universal truths. The reputation of everyone doing it without numbers would reveal the pattern of the universal truth. Finally this looks like it was all done using cuneiform. Which brings up questions of notation and the language to describe a square root.
- usrusr 3y ago
  Getting shit done is the antithesis of progress because it goes hand in hand with "if it was good enough for my father, it will be good enough for me, who are you to disrespect (the methods of) my father!"
  - detourdog 3y ago
    I agree but dying for ideals isn’t much progress. Progress is a luxurious goal.
- empath-nirvana 3y ago
  What everyone is talking past here is that the Babylonians discovered what engineers call a "rule of thumb", and engineering is focused on getting things done using rules of thumb. The best possible rule of thumb is a mathematical proof or a scientific discovery, but it's by no means necessary. . Observing that something always holds and even having a formula for it is not the same thing as having a proof, and the proof is what makes it mathematics and geometry and not "just" engineering. The babylonians had a rule of thumb -- the greeks discovered the theorem -- and more than that, they seem to have invented the mathematical/geometrical proof as a concept, along with formal logic. Without that mental framework, it's hard to say that the babylonians proved anything or had any theorems at all, only collections of rules of thumb. It's quite likely that lots of babylonians sort of independently and intuitively understood _why_ it must be true, but they don't seem to have ever written it down. Not that there's anything wrong with having rules of thumb -- it's a huge achievement to even notice and collect and teach those things, all the stuff around you is built relying on them. I encourage everyone to watch this series of videos. https://www.youtube.com/watch?v=_ivqWN4L3zU https://www.youtube.com/watch?v=_ivqWN4L3zU

skypanther 3y ago

I took a course at the university of Buffalo in the early 80s about pre-roman examples of Pythagorean geometry in henge monuments and early structures. Basically the class was watching slide shows of the professors vacations across Europe and his photos of old ruins. Then he'd show us a line drawing of the structure and pick out what at times seemed random points that created 3-4-5 triangles. He'd exclaim 'see? They knew about the Pythagorean theorem before Pythagoras!' Our only graded assignment was to write a paper where we "discovered" a similar example. I picked a picture of some random ancient church and "found" the right triangle in it's foundation. I got an A. Definitely just an "observation" no theorum involved.