Welcome to another Mathologer video A
squared plus B squared equals C squared. Forget about Euler's formula and company,
Pythagoras's theorem beats them all in just about every conceivable way, at
least in my books. Ok so finally a Mathologer video about THE theorem
of theorems. My main mission today is to chase down the all-time greatest,
simplest, no-brainer proof of Pythagoras's theorem, just for you. At the same time,
I'll introduce to some truly gorgeous crazy and divine Pythagorean facts off
the beaten track. And to finish off the video I'll tell you a little bit about a
new book by me and Marty that's just appeared. Before we get started with the
maths let's do a little debunking. Pythagoras's theorem is a misnomer. The
basic fact about right-angled triangles named after Pythagoras was not
discovered by Pythagoras at all and was actually known to the ancient
Babylonians at least a thousand years before Pythagoras was born. Also, although it's often claimed that Pythagoras was the first to have produced a rigorous
proof of this theorem there actually does not seem to exist any
evidence to support this claim, and plenty of maths historians doubt it. Now
Pythagoras was the leader of a very influential mathematical cult, the
Pythagoreans, who were in the habit of attributing all results originating with
them to Pythagoras himself and it's likely that because of this practice
Pythagoras's name came to be attached to the theorem. Anyway it's too late now,
it's all gone, and so I'll also call THE theorem Pythagoras's theorem or simply
Pythagoras. Okay let's get started by looking at a couple of gorgeous proofs
of Pythagoras's theorem. Here's my take and one of the simplest, most beautiful and
most popular. Let's make four copies of a right-angled triangle with short sides A
B and hypotenuse C. Now we arrange these the four triangles in a square which at the
same time creates a smaller tilted square inside.
So the small square is what you get when you partially cover this big blue square
by the triangles. The small square has sides of length, what? Well, the length is
just the length of the hypotenuse of our triangle which is C and so the area of
the small square is equal to C squared. Let's shift one of the triangles. Now how
large is the modified blue area that you see here. Well, the triangle covers just as much blue as before and so the new blue area
is just as large as before, C squared. Let's do some more shifting. How large is
the new blue area? Well, obviously, still C squared, right? Now comes the punchline.
Shift again. Obviously the blue area is still C squared. However, it's also A
squared plus B squared and so A squared plus B squared equals C squared.
Super neat isn't it :) Now it all seems pretty convincing, but it's definitely
important to stress that for this argument to count as a bulletproof
proof we'd have to check some more details. For example, in this diagram here
we need to check that the two square looking shapes are really squares no
matter what right triangle we start with. Not hard but not a step we can afford to
skip. After all, looks can be deceiving, that's as true in maths as it is in life.
Just to give you an idea what a more complete proof looks like, let me show
you a proof of Pythagoras's theorem in the most famous maths textbook ever written,
the 2,000+ year old Elements by the ancient Greek mathematician Euclid.
Here's a translation. Whoa what happened here. It just looked very neat, didn't it,
but no now it's really pretty off-putting isn't it? And actually also
pretty hard to understand, just based on this diagram and the text. On the other hand,
we can also capture the gist of this classic proof (2,000 years and still
going strong) with a pretty animation. Now if Euclid had an animation, it's been
lost so here's mine :) So Euclid proves Pythagoras
by showing that the two orange figures have the same area, and the same for the
two blue figures. Then we've got orange plus blue on top which is A squared plus
B squared is equal to orange plus blue at the bottom which is C squared. Here's
my take on Euclid's argument. The little orange square is what you get when you
cover this larger orange region by this white copy of our right-angled triangle.
Then shifting the triangle as before does not change the area, right? The same
can be done with the blue square. Let's do it. Now the orange parallelogram is
just this larger orange figure here after being covered by the little red
triangle and shifting the triangle the arrow stays unchanged again and the same
with the blue parallelogram. And with pretty much the same diagram as before
materialising we are done. There's the C squared again and it is now also clear
that the orange and blue squares we started with are equal in size to these
orange and blue rectangles. Over the millennia, hundreds of different proofs
of Pythagoras's theorem have been found and there's even a book that lists 371
of them. And that's definitely 371 more than most people know:) That's the book.
I've got about 20 of the most beautiful proofs memorised. (Marty) Really? (Burkard) Really 20. at
least :) Let me now show you another of my my favourites. Here's our A-B-C triangle again.
Let's scale it up by a factor C. The new triangle has sides AC, BC and CC. We now
scale another copy of the original triangle by B. Okay let's do it. Right, and
one more copy of the triangle, we scale that one by A, there we go. Now let's have
a close look and so what we see is that these two sides are the same. And so are
these other two. And this means that the three triangles fit together into a
rectangle like this and so A squared plus B squared equals C squared.
Absolutely gorgeous, isn't it and there are lots and lots of other proofs that are
just as beautiful as the ones I've shown you. For example, you should check out the
proofs discovered by Leonardo da Vinci and the one discovered by American
President Garfield. If you're keen to find out more there's no need to dig up
the book I mentioned earlier. Instead have a look at the nicely Illustrated
list of about 120 proofs on the absolutely amazing cut-the-knot website.
So there I am just scrolling really quick. Oh, just in case the current American
president Donald Trump ever tweets his own proof please someone let me know
immediately :) Alright, all those proofs that I've shown you so far are
great but none of them is the simplest. I know that this will sound very strange
but the simplest proof for Pythagoras's theorem is a proof of a very very
general super theorem that features Pythagoras theorem as just one of
infinitely many special cases. So what I'm saying is that proving the general
theorem which comprises infinitely many special cases is easier than proving
Pythagoras's theorem which is just one of these special cases. Super weird,
well we'll see. Now what's this super theorem? Well Pythagoras says that if we
attach three suitably scaled copies of a square in the same way to the sides of a
right-angled triangle, then the two smaller squares together equal the larger square.
It turns out, and that's the super theorem, that the same is true if we
replace the squares by any other shape. For example, the areas of the little semicircles add exactly to the area of the large one, the areas of the little
pentagon's add exactly two the area of the large one, and so on.
That's Pythagoras Pythagoras :) That the super theorem is true
actually follows from Pythagoras's theorem. To see this, let's just focus on the semicircles and the squares. It's clear that every one of the semicircles occupies
the same fraction of the corresponding square, in fact a quick pi calculation,
and that's an exercise for you, shows that every one of the semicircles has
about 39 percent of the area of the corresponding square. But since the two
smaller squares add to the larger square 0.39 times the smaller squares adds to
0.39 times the larger square, like that, and that's the same as saying that the
two smaller semicircles add to the large semicircle. So Pythagoras for
squares implies Pythagoras for semicircles. But, using exactly the same
argument, it also follows that Pythagoras for squares implies Pythagoras for all
other shapes. In fact, even stronger and again using exactly the same argument,
Pythagoras for any shape whatsoever implies Pythagoras for all other shapes.
Now just to make sure that there's no misunderstanding what I mean by this, let
me rephrase what I just said: if we can prove from scratch that Pythagoras is
true for some special shape, then this proof implies immediately that
Pythagoras is true for all other shapes. How amazing is that.
But this insight then prompts a natural question: Is there one best, super-terrific perfect shape for which it's easiest to see that Pythagoras is true,
It turns out there is. Have a look at this. Wait for it... DONE, that's the easiest
proof of Pythagoras. Now just in case you think I've gone crazy note that you're
actually looking at three triangles and not just two, okay,
and that the two smaller ones obviously add to the larger one, AND that
the three triangles are similar because they have the same angles, AND that they
attach a la Pythagoras to the original red triangle like this. Really done.
There, the simplest proof of Pythagoras using the simplest imaginable shape, the
right-angled triangle itself. Pretty bloody amazing, as we say Down Under :)
Definitely something for real maths connoisseurs to savor. Okay before we get
into Pythagorean super facts here's a little puzzle for you. You go to a pizza
shop and as usual they have small, medium and large pizzas. It turns out that one
small plus one medium together cost exactly the same as one large pizza. Now,
using only a pizza knife decide which is the better deal,
the smaller plus medium combo or the large pizza by itself? Leave your answer
in the comments and while you are at it also let me know which of the proofs of
Pythagoras you like best. Now Pythagoras is great but it only
works for right-angled triangles. What about other triangles. Well there's also
a really nice generalisation of Pythagoras hidden in school mathematics
that applies to all triangles. For this let's have another look at Euclid's
proof. It turns out that equality of areas of the same colour in this diagram
generalises to other triangles like this. So in this diagram any two rectangles of
the same colour are equal. Now with a little trigonometry the proof is quite
simple. Let me just show you how you can see that the two green rectangles really
have the same area. So we just label the sides A,B, C and the angle opposite C gamma.
Then you can read straight off this diagram here that the two green
rectangles both have area A times B times cos gamma, yet another exercise
for you, you'll be very busy today :) And so the green rectangles have equal
area and with that insight we get the sum of the top two squares A squared plus B
squared minus the two green bits, so minus two times A B cos gamma equals C
squared. Of course most of you will have encountered
this Pythagoras under the name Cosine rule.
So, in the case that the angle is 90 degrees cos is zero and the cosine rule
turns into standard Pythagoras. When it comes to the cosine rule the next best
triangle angles beside 90 degrees are 60 and 120 degrees.
Why? Because cos 60 is 1/2 and cos 120 is -1/2. For example, for 60
degrees the cosine rule takes on this particularly nice form and for 120 degrees
we get a plus sign. There, almost as pretty as the 90 degree Pythagoras, and
who knows maybe in a world of giant bees in which the hexagon's 120 degree angles
dominate this is THE Pythagoras :) Anyway here's another little puzzle for you. One
remarkable thing about Pythagoras is the existence of Pythagorean triples.
These are triples of positive integers A, B, C such that A squared plus B squared
equals C squared, like, for example, 3 squared plus 4 squared equals 5 squared.
How about the 60 and 120 degree Pythagoras. Can you find non-trivial
positive integer triples ABC that satisfy these other two Pythagorean
equations? People always think that since Pythagoras has been around for thousands
of years they know everything worth knowing and there's nothing new to be
discovered about it. WRONG! New proofs and fascinating Pythagorean facts are
still being discovered and there's so much beautiful stuff that hardly anybody
has heard of. For example Marty and I just recently discovered a proof of
Pythagoras's theorem that seems to be new. (Marty) Really? (Burkard) Yeah, actually that was new to me too
but actually if you look at the cut-the-knot site in the list there's
a proof by us and I had a look: yeah, actually, what we did there amounts to a
new proof :) Anyway it's there. Anyway, to finish off here's just a little
gallery of really interesting Pythagorean flavored theorems that will
culminate in my all-time favourite in this respect de Gua's theorem. So A
squared plus B squared equals C squared but if this height here is D then 1 over
A squared plus 1 over B squared equals 1 over D squared. Hands up, who knew that? (Marty) My hand was down. (Burkard)
Well it's really easy to prove, give it a try. Now theorem 2. This one works
for arbitrary triangles. Drawing these connections here X, Y, Z, then X
squared plus Y squared plus Z squared is equal to A squared plus B squared plus C
squared ... times 3 :) and every single one of these new green triangles here has the
same area as the red one we started with. Not easy to prove but feel free to give
it a try anyway. Theorem 3. Suppose we had started with an A-B-C right-angled
triangle and then throw squares on the sides of the green triangles like this.
Then the yellow square is equal to C squared and 5 times the yellow square
is equal to the sum of the 2 orange squares. Pretty amazing :) Why 5? Why, anything nice?
And we can keep on going. Adding the pink squares here the sum of the two small
pink squares is equal to the large pink square. How unexpected and how pretty is
that and there seem to be many similar relationships to be discovered if we
keep extending this way. Ok now to really finish off let me tell you about higher-dimensional counterparts of Pythagoras's theorem. The first counterpart many of
you will be familiar with. To begin let's restate Pythagoras in terms of
rectangles. In a rectangle with sides A and B and diagonal C, well A squared plus B squared equals C squared. This version of
Pythagoras has one straightforward counterpart in 3d where we replace the
rectangle with a rectangular box like this. Now if its sides are A, B, C and its
diagonal is D then A squared plus B squared plus C squared equals D squared.
And this works in all dimensions. So for a 4d box with sides A, B, C, D and diagonal E
we've got a squared plus B squared plus C squared plus D squared equals E squared.
Okay most of you would know this but did you know that there is a second totally
different and totally crazy way in which Pythagoras generalizes to all higher
dimensions. It is called de Gua's theorem and the 3d version was only discovered
in the 18th century by the mathematician Jean-Paul de Gua de Malves. To motivate it we interpret this right angle triangle as what we get when we cut a corner of a
rectangle like that. In 3d cutting off the corner of our box we get this.
That's a triangular pyramid with the red cut forming the base of this pyramid and
we've also got three faces that are all right-angled triangles. Now if A, B and C
are the areas, so no longer distances, so the areas of the right angled triangles
and D is the area of the red base, then de Gua's theorem says that A squared
plus B squared plus C squared equals D squared. I absolutely love this one. I
find it so surprising that this should work for squared areas instead of
squared distances and not only in 3d but also in all higher dimensions where
what's squared are volumes and hyper volumes. I often assign the easy proof of
the 3d facts in assignments at uni so maybe you'd like to try your hand at
this one too. As I said, lots and lots and lots of homework for you today. Again
these are just some examples of tons of stunning Pythagorean facts. If you
enjoyed this video make sure to also check out some of the links in the
description and in particular the cut-the-knot list of proofs which also contains Marty my new proof of Pythagoras. And
that's it for today except I also wanted to tell you about a
new book by me and Marty which has just been published by the American
Mathematical Society. For about eight years Marty I wrote a weekly maths
column for one of the big newspapers here in Australia and this book is a
collection of 64 of these articles all with an Australian theme. The book has
come out really really nice, it's in full color, looks great and so if you like the
videos here's an opportunity to bridge the time between videos and to find out
about the prehistory of the Mathologer. In general have a look at www.qedcat.com that's Marty's and my website for all things mathematical, especially our mathematical
Movie Database our mathematical movie clip collection, our other books, articles
etc. And that's really finally it for today.
If someone cannot understand how math can be beautiful show him this video.
The suspense is intense! What are the pythagorean triples for the 60 and 120 triangles!?? I'll have to actually do the exersise.
I wish his book was available in ebook.
This made me think of the part in My Brain is Open: The Mathematical Journeys of Paul Erdos, where 17 y/o Erdos meets 14 y/o budding mathematician Andrew Vazsonyi and asks him how many proofs of the Pythagorean Theorem he knows. Vazsonyi knew one, Erdos announces that he knows 37. Vazsonyi says Erdos was not boasting; "The concept was not applicable to him".
I think he knew that many proofs because he loved math and wanted to know everything about it.
I lost my shit at the 6 minute part. That was a thing of beauty. Just finished teaching both scale factors and Pythagorean Theorem to my students. Now I have to find a way to show this part.
Can someone show me the proof for X2 + Y2 + Z2 = 3(A2 + B2 + C2)? Because Iām only getting 2(A2 + B2 + C2)