Teenagers find an "impossible" proof of Pythagoras' Theorem
This might be old news to some but others may not have seen it:
https://www.theguardian.com/us-news/2023/mar/24/new-orleans-pythagoras-theorem-trigonometry-prove
The thread title is clickbait like a lot of the media articles since actually this proof isn't as special as the articles say. But ignoring that, it's still very impressive that these students managed to come up with this and I also think it's a great exercise for A Level students who are studying geometric series (I have given it to my own students).
Most importantly, this illustrates that there is an infinite amount of new maths waiting to be discovered and it's not only professional mathematicians who can find it. If you are at school and have an interest in maths, don't just stick to textbook questions, try getting a pen and paper out and see what you can discover.
Below is the proof in full. If you have studied geometric series, once the triangles have been constructed see if you can pause the video and complete the proof:
https://www.theguardian.com/us-news/2023/mar/24/new-orleans-pythagoras-theorem-trigonometry-prove
The thread title is clickbait like a lot of the media articles since actually this proof isn't as special as the articles say. But ignoring that, it's still very impressive that these students managed to come up with this and I also think it's a great exercise for A Level students who are studying geometric series (I have given it to my own students).
Most importantly, this illustrates that there is an infinite amount of new maths waiting to be discovered and it's not only professional mathematicians who can find it. If you are at school and have an interest in maths, don't just stick to textbook questions, try getting a pen and paper out and see what you can discover.
Below is the proof in full. If you have studied geometric series, once the triangles have been constructed see if you can pause the video and complete the proof:
Original post by Notnek
This might be old news to some but others may not have seen it:
https://www.theguardian.com/us-news/2023/mar/24/new-orleans-pythagoras-theorem-trigonometry-prove
The thread title is clickbait like a lot of the media articles since actually this proof isn't as special as the articles say. But ignoring that, it's still very impressive that these students managed to come up with this and I also think it's a great exercise for A Level students who are studying geometric series (I have given it to my own students).
Most importantly, this illustrates that there is an infinite amount of new maths waiting to be discovered and it's not only professional mathematicians who can find it. If you are at school and have an interest in maths, don't just stick to textbook questions, try getting a pen and paper out and see what you can discover.
Below is the proof in full. If you have studied geometric series, once the triangles have been constructed see if you can pause the video and complete the proof:
https://www.theguardian.com/us-news/2023/mar/24/new-orleans-pythagoras-theorem-trigonometry-prove
The thread title is clickbait like a lot of the media articles since actually this proof isn't as special as the articles say. But ignoring that, it's still very impressive that these students managed to come up with this and I also think it's a great exercise for A Level students who are studying geometric series (I have given it to my own students).
Most importantly, this illustrates that there is an infinite amount of new maths waiting to be discovered and it's not only professional mathematicians who can find it. If you are at school and have an interest in maths, don't just stick to textbook questions, try getting a pen and paper out and see what you can discover.
Below is the proof in full. If you have studied geometric series, once the triangles have been constructed see if you can pause the video and complete the proof:
haven't got time to check the video right now - seems to be about 30 minutes long which seems a bit "excessive" for a Pythagoras proof! - but a bit of googling suggests they haven't actually published their proof in detail.
Are they actually using trig, or is the "trig" just an aliasing of some ratios of lengths? Is the "geometric series" element of this just a rehash of this proof:
https://www.cut-the-knot.org/pythagoras/Proof100.shtml
Original post by davros
haven't got time to check the video right now - seems to be about 30 minutes long which seems a bit "excessive" for a Pythagoras proof! - but a bit of googling suggests they haven't actually published their proof in detail.
Other videos are closer to 10 minutes but the one I posted explains it in full and is accessible to A Level students (or high level GCSE students).
Are they actually using trig, or is the "trig" just an aliasing of some ratios of lengths? Is the "geometric series" element of this just a rehash of this proof:
https://www.cut-the-knot.org/pythagoras/Proof100.shtml
https://www.cut-the-knot.org/pythagoras/Proof100.shtml
I wouldn't call it a rehash since it's a different proof but there are similar (excuse the pun) ideas in it. Many news articles say that this is a brand new type of proof but that's not true.
My post was more targeted at school students because I know this kind of thing would have appealed to me when I was at school and I would have liked to try the proof myself.
Original post by Notnek
Other videos are closer to 10 minutes but the one I posted explains it in full and is accessible to A Level students (or high level GCSE students).
I wouldn't call it a rehash since it's a different proof but there are similar (excuse the pun) ideas in it. Many news articles say that this is a brand new type of proof but that's not true.
My post was more targeted at school students because I know this kind of thing would have appealed to me when I was at school and I would have liked to try the proof myself.
I wouldn't call it a rehash since it's a different proof but there are similar (excuse the pun) ideas in it. Many news articles say that this is a brand new type of proof but that's not true.
My post was more targeted at school students because I know this kind of thing would have appealed to me when I was at school and I would have liked to try the proof myself.
Agreed, it did look interesting. Reflecting the triangle at the start to get a double angle is similar to the sin(2x) identity proof using a circle, in fact if you did put the circle on there, radius c, the "lower side" would be the tangent. Then arranging the similar triangles into a decreasing geometric sequence has an archimedes parabola-triangle feel to it. Its perhaps a bit ironic that things like the archimedes precalculus area under a parabola isnt really taught, but then stuff like this makes headlines.
Thinking about it a bit more it has a fair bit of
https://en.wikipedia.org/wiki/Tangent_lines_to_circles#Tangent_lines_to_two_circles
in the proof which has a ukmt flavour about it and things are related by a geometric sequence
(edited 11 months ago)
Original post by Notnek
Other videos are closer to 10 minutes but the one I posted explains it in full and is accessible to A Level students (or high level GCSE students).
I wouldn't call it a rehash since it's a different proof but there are similar (excuse the pun) ideas in it. Many news articles say that this is a brand new type of proof but that's not true.
My post was more targeted at school students because I know this kind of thing would have appealed to me when I was at school and I would have liked to try the proof myself.
I wouldn't call it a rehash since it's a different proof but there are similar (excuse the pun) ideas in it. Many news articles say that this is a brand new type of proof but that's not true.
My post was more targeted at school students because I know this kind of thing would have appealed to me when I was at school and I would have liked to try the proof myself.
Using the same construction as the kids did but approaching it from a different way. So a circle of radius c centered on the top vertex so the sides of the large, grey triangle are c, x and y+c (hypotenuse) and its obviously a right triangle. Then the tangent - secant thm gives
x^2 = y(y+2c)
x^2 = y^2 + 2cy
x^2 = (y+c)^2 - c^2
so
x^2 + c^2 = (y+c)^2
which is pythagoras for the large, tgrey triangle. Also x is tan of the double angle (unit c/circle) etc ...
Edit - looking at davros link, it does have the same flavour when that argument is applied to the reflected double triangle as the dissection is parallel to the base etc. So I guess the novelty is the combination of the reflection (circle) and the geometric, rather than either being particularly novel in themselves. Having said that, the us president famous pythagoras proof was simply cutting a previous argument in half - this one is more interesting.
(edited 11 months ago)
Original post by Notnek
Other videos are closer to 10 minutes but the one I posted explains it in full and is accessible to A Level students (or high level GCSE students).
I wouldn't call it a rehash since it's a different proof but there are similar (excuse the pun) ideas in it. Many news articles say that this is a brand new type of proof but that's not true.
My post was more targeted at school students because I know this kind of thing would have appealed to me when I was at school and I would have liked to try the proof myself.
I wouldn't call it a rehash since it's a different proof but there are similar (excuse the pun) ideas in it. Many news articles say that this is a brand new type of proof but that's not true.
My post was more targeted at school students because I know this kind of thing would have appealed to me when I was at school and I would have liked to try the proof myself.
Finally had a look at this a couple of days ago to make sure I could understand / reproduce what's going on. Have to say that original video was hard work - I had to download it and watch over 4 or 5 periods. Really struggled with the guy's accent and couldn't actually understand some of the words he was using!
Leaving that aside, from a purely maths point of view it looked OK, but then I thought: hang on, he's assumed "wlog" that b > a at the start so he could write a series in powers of a/b and force convergence. BUT - what happens when angle alpha = beta = 45 degrees, so a = b? Now when we duplicate the triangles we get an infinite set of isosceles right-angled triangles with sides 2a, 2a and 2c, so each of the and are equal to 2c, and our convergence disappears!
Can this be patched up in this case?
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