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0:00
Welcome to another Mathologer video. In 2007 a simple beautiful connection between two seemingly unrelated mathematical gems was discovered. However, it appears that this discovery has largely gone unnoticed and is actually in danger of being forgotten. So, I thought let’s do something about this sorry state of affairs and Mathologerise this discovery. Show it to the world and save it from oblivion :) Ready for something new? Alright, what are those two mathematical
0:34
gems? Well, first there is the iconic identity. 3 squared plus 4 squared equals 5 five squared, the smallest instance of the double-miracle that two integer squares can add up to another integer square and that therefore by the converse of Pythagoras theorem there are right-angled triangles all of whose sides are integers. Like the famous 3,4,5 right-angled triangle.
0:57
Of course Pythagorean triples like 3, 4, 5 are regulars on Mathologer and something that people have been obsessing about for thousands of years. Okay, 3,4,5 and company that’s the first gem. Our second mathematical gem is the super famous Fibonacci sequence. There, 1, 1, 2, 3, 5, 8, 13 and so on. You all know the drill… 1 plus 1 is 2. 1 plus 2 is 3. 2 plus 3 is 5, and so on. Starting with two 1s, in the Fibonacci sequence two consecutive terms add up to the next
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term. Easy peasy. Again, just like 3 squared plus 4 squared = 5 squared, people have been pondering this sequence that miraculously shows up all over maths and nature for ages. But what can those two mathematical gems possible have in common? Not obvious at all, right? Well, let me explain. Okay, let’s do something unusual. Let’s bend the Fibonacci sequence. Wait what? Bend? Yes, like this. For the moment, let’s focus on those first four consecutive Fibonacci numbers and move
2:05
the rest out of the way. There. We’ll come back to the rest of the sequence in a moment. Okay, bending the Fibonacci sequence that’s new. Now, for the connection with 3-4-5 we’ll pick pairs of these numbers and look at their products. Why pick pairs of these numbers and look at their products? Don’t worry and just run with it for the moment :) For example, pick that 1 and the 3, 1 times 3 is 3. Another pair. 1 times 2 is 2. Double that and you get 4. Next 1 times 2 is
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2. PLUS 1 times 3 is 3. 2+3, that’s 5. 3,4,5 , 3 squared + 4 squared = 5 squared. Impressed.
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Maybe not? But wait, wait, we are not finished yet. There are a few more ways to pick pairs of numbers in the red block and all of them tell us something important about the 3, 4, 5 triangle. First let’s look at the incircle of the triangle. What’s the incircle? That’s the circle that touches all three sides of the triangle. Sort of the heart of a triangle. There that’s it. In our example, the radius of the incircle is 1. And picking the pair of numbers on
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top we get 1 times 1 is 1. Then there are the three excircles, the outer circles, sort of the airbags of the triangle :) Each of these excircles touches one side of the triangle, and extensions of the other two sides. For example, here is the excircle that touches the hypotenuse and touches these extensions of the other two sides of the triangle. The radius of this excircle is 6 and 3 times 2 is 6. Two more excircles. There, Radius 2 and
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1 times 2 is 2. Finally, this excircle has radius 3 And 1 times 3 is 3. Pretty unexpected that the first four Fibonacci numbers have so much to tell us about 3 squared plus 4 squared = 5 squared. Let me summarise all this. So there are six ways to pick two numbers. And all of them tell us something important about the 3,4,5 triangle. Top two, radius of the incircle, bottom two, radius of the excircle of the hypotenuse, and the two diagonal products, this one and that one, give the
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other two excircle radiuses. Then this vertical product gives the 3, one of the short sides of the triangle, that one here doubled is the 4 and the sum of the diagonal products is the 5. Super neat, right? And one more thing. The product of all four numbers is the area of the triangle. There, 1 times 1 times 2 times 3 equals 6. And the area of the triangle is 1/2 base times height, and so in
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this case 1/2 3 times 4 which is also 6. NICE :) But believe it or not we are just getting started. Let’s bring in the next Fibonacci number 5. These four new numbers translate into the second most famous Pythagorean triple in exactly the same way. 5, 12, 13. 5 squared plus 12 squared = 13 squared. A lot of you will also be very familiar with this one from school. Let’s just double-check that all
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our products really work out again. There 1 times 5 is 5. That’s the 5 in 5 squared. 2 times 3 is 6, doubling that gives 12, 12 squared. And 1 times 3 is 3, 2 times 5 is 10, 3 +10 is 13, 13 squared. Fantastic :) And what about all those circles. Well let’s see The radius of the incircle is 2, and 1 times 2 at the top is 2. Tick! That radius of the hypotenuse excircle really is 5 times 3 equals 15. Nice. 1 times 3, the radius is 3. And finally. 2 times 5, that last radius is 10.
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All works perfectly :) Area also works out. Wheel in the next Fibonacci number 8. And again this block of Fibonacci numbers also produces one of our nice Pythagorean triples. 16, 30, and 34. And all the radius business works out too. In fact, this will be true forever and ever after, as we push the Fibonacci sequence through our red 2x2 block. All pretty amazing. But there is a lot more. A LOT :) Things to look forward to in the rest of the video are
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Pythagorean triples growing on some bizarre Fibonacci infused tree, a Pythagorean fractal, the mysterious Feuerbach circle in action (ever heard of the Feuerbach circle ? :), some fun real-life applications of Pythagorean triples and some more extra-special Fibonacci action.
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Before I get going just quickly. The discovery of this beautiful connection between Fibonacci and Pythagoras and most of what I’ll be talking about in the following was published by H Lee Price and and Frank R. Bernhart in the half-forgotten preprints that I mentioned earlier. I’ll put links to these preprints in the description of this video. At this point, I’d also like to thank Colin Pountney for drawing my attention to these preprints.
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Now let me show you a bizarre tree of Pythagorean triples which was originally discovered in 1963 by the Dutch mathematician F.J.M. Barning (don’t know what the F.J.M. stands for). The discovery that this Pythagorean triple tree has natural Fibonacci origins was only pointed out in those half-forgotten preprints that I am popularising today.
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So, let me grow this Pythagorean triple tree for you, using some serious Fibonacci juice :) The root of this tree is what’s up there in the corner: that Fibonacci box and its associated Pythagorean triple 3, 4, 5. We grow the tree by first making three copies of the box.
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Let’s highlight the three pairs of numbers that correspond to the three ex-circles. Those ones there. Get rid of the other numbers and push everything to the top Now we complete the three boxes al la Fibonacci. There, 3+1 is 4 and 1 plus 4 is 5. Now just as our original Fibonacci box translates into a Pythagorean triple plus all that nice radius magic, so does this new Fibonacci box. Let’s just check this for the numbers 15, 8 and 17. 3
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times 5 is 15. 1 times 4 is 4 double that is 8. 1 times 5 is 5. 3 times 4 is 12, 12 +5 is 17 works. Again complete this box a la Fibonacci. 3+2 is 5. 2 plus 5 is 7. And we get another Pythagorean triple. Again, 1 plus 2 is 3. 2 plus 3 is 5. We already encountered this Fibonacci box earlier, that one just gives the second most famous Pythagorean triple 5, 12, 13 again.
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Let’s keep growing the tree. The three children of 3, 4, 5 have themselves three children each. There, make three copies on the left box. Highlight the three pairs that correspond to the ex-circles. Get rid of the other numbers. Push everything to the top. Complete all the boxes a la Fibonacci. More Pythagorean triples. The miracle keeps happening. Repeat. And keep on going forever by spawning more and more children to create an infinite Pythagorean triple tree.
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Now, of course, it’s amazing that all this works out forever and ever after. But the real killer is that this tree produces ALL Pythagorean triples and that every Pythagorean triple appears exactly once. Really? :) Well, actually, I am lying a bit here, but just a bit. The tree does not produce all the Pythagorean triples, it produces all the essentially different Pythagorean triples, the so-called primitive Pythagorean triples. Quick explanation. There, 3, 4, 5 again.
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To start with, since swapping the 3 squared and the 4 squared doesn’t really changed anything substantial, I’ll consider the Pythagorean triples 3, 4, 5 and 4, 3, 5 to be the same. In general, of the two ways of writing a Pythagorean triple, only one can appear in the tree. Next, if we scale the triangle by a factor 2, we automatically get another Pythagorean triple.
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If we scale by a factor of 3 we get another Pythagorean triple. And so on. Now 3, 4, 5 is the smallest Pythagorean triple that corresponds to these particular shape right-angled triangles and all the other Pythagorean triples like that are multiples of 3, 4, 5. We call Pythagorean triples that are minimal in this respect primitive Pythagorean triples. And so once you know all the primitive Pythagorean triples you know all Pythagorean triples. Makes sense, right?
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Anyway, as I said, our miracle tree contains all the primitive Pythagorean triples exactly once. And since absolutely every Pythagorean triple is a multiple of a primitive one, the tree really captures all Pythagorean triples. Wonderful isn’t it :) I’ll sketch a proof why the tree has this property at the end of the video. Here is a little challenge for you. 153, 104, 185 is a primitive Pythagorean triple. What are the four numbers in the Fibonacci box that goes
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with this Pythagorean triple? Leave your answer in the comments of this video. Bonus question: how to navigate from 3, 4, 5 to the triple over there in the tree in terms of left, middle, right children. So what I am after are instructions like: starting from 3,4,5 go left child, then middle child, then middle child again, something like that. Can you do it?
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Here is one more crazy fact that will become important later when we get to proving things.
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Turn all the columns of our boxes into fractions like this. There, now we have a tree of fractions or, more precisely, a tree of pairs of fractions. And what’s special about this tree? Well, to start, all the fractions are less than one. That’s pretty obvious. What’s not obvious at all is the fact that this tree contains every positive fraction less than 1 exactly once. Really?
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Again, I am lying a little bit. The tree does not contain every positive fraction less than 1 exactly once, it contains every REDUCED fraction less than 1 exactly once. So the tree contains the reduced fraction 1/2. There. But the tree does not contain 2/4 or 3/6 or any other fraction that reduces to 1/2. Again, the tree only contains reduced fractions less than one and every one of these exactly once. Noice. What this also means is that, in terms numbers,
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this tree contains every rational number in the interval from 0 to 1 exactly once. Very surprising. Just think about how densely packed the rational numbers are in this interval, but somehow, miraculously, here we find these fractions nicely separated out on this tree.
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If this reminds you of the Farey tree, another mathematical superstar, you are on the right track to another beautiful insight. You can read about this connection in another article that I link to in the description of this video. Before we put our Pythagorean triple tree under the microscope here is a bit of an interlude. Let me show you a totally different type of Pythagorean tree. It’s all sort of self-explanatory and so I’ll just run an animation and play some music.
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I’ve known these pretty Pythagorean trees for ages but never really looked closely. Anyway these are nice fractal constructs that are very much worth knowing about. And, of course, if you are looking for a special Christmas tree this Christmas season, how about the one over there :) I should mention that I produced this spectacular Pythagorean Christmas tree using some Mathematica code in a post by chyanog on stack exchange. There is a link to this
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Mathematica code in the description of this video. At the end of this video I’ll also show a very pretty animation of a spectacular spiral that combines the Fibonacci sequence and Pythagoras in a closely related ingenious way that I myself also just found out about recently. Something else to look forward to. Also, here is a very nice puzzle for you to ponder.
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If the area of the bottom square of the tree over there is 1, what’s the total area of this little tree? Leave your solutions in the comments. Back to our Pythagorean triple tree. As I said, it contains every primitive Pythagorean triple exactly once. Amazing, right? But there is another aspect to this tree that in my eyes is just as amazing. It’s a stunning way to visually capture the simple parent-child growth mechanism. In effect this allows to
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grow this tree purely in terms of triangles and their properties without any numbers involved. Definitely one of the prettiest pieces of geometry I’ve encountered in recent years. Really something for connoisseurs. Let me show you. Okay, let’s have a closer look at the three children of this Pythagorean triple combo. Quick reminder, in the red box different pairs of numbers correspond to…: top two numbers the incircle, bottom two numbers the excircle of the hypotenuse,
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and the diagonals, this one and that one, they correspond to the other two excircles. Incircle, hypotenuse excircle, other two excircles. All under control? Great! So, 3,4,5 at the bottom that’s the parent and the three guys on top, those are the children. How are the children related to the parent geometrically? Let’s first look at the blue child.
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3, 2 on top, so the incircle radius of this child triangle is 3 times 2 is 6. In the parent triangle 3, 2 is at the bottom and therefore corresponds to the excircle of the hypotenuse. There. Now, in theory the child triangle should have sides 21, 20 and 29
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like that one here. Let’s see whether our theory really works out in practise. NICE, that circle really fits perfectly :) Okay, what about the incircle of the green child? 3 times 1 is 3. In the parent 3,1 corresponds to the excircle with radius 3, that one there. And the right-angled child triangle that goes with it has sides 15, 8 and 17.
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And for the pink child things pan out like this. Now, in the tree, children have children. So let’s also quickly animate the pink child having children Well that’s quite visual and pretty the way the tree grows in terms those circles. However, what’s not clear in the pictures on the right is how you can also see the triangles themselves and not just their incircles without referring to the Pythagorean triples up there on the left, right? Now the amazing thing is that
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you can actually describe the complete growth mechanism including the children triangles purely in terms of the pictures over there. No need to look at any numbers. A real geometrical miracle. Let me show you. Okay, let’s pencil in the centers of all the circles.
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As you can see the four circles don’t touch. At this point the Feuerbach circle, another mathematical superstar makes its appearance. In 1822 the German mathematician Karl Wilhelm Feuerbach discovered something that the ancient Greek triangle enthusiasts had overlooked, that if you take any triangle and mark the midpoints of its sides. Then the circle through these three points touches the incircle and all three excircles. Feuerbach’s circle has
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all sorts of other magical properties. For example, in the special case of a right triangle the circle also always passes through the right angle corner. Anyway, eventually I’ll dedicate a whole Mathologer video to Feuerbach’s circle. Main thing for us at the moment is that there always is this special circle that touches all the four circles that we are interested in.
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For the moment let’s remove the incircle. Okay, connect the center of Feuerbach’s circle to the centers of the three ex-circles. Now extend these three connections to three right angled triangles with horizontal and vertical short sides like this Here comes the very much not obvious killer insight. Scale these triangles by a factor of four and you get the right-angled children triangles. Seeing is believing. Let’s check this for the pink child.
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Here we go How long is this vertical side? Well, let’s count. 1, 2, 3, 4, 5, 6. Just
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now I said we have to scale these triangles by a factor of 4 and so 4 times 6 is 24. Yep, there 24 is the length of one of the sides of the pink child triangle. What about the other side. Count again. 1 and a bit. Actually, this bit is three quarters and so this side is 1.75 long and 4 times 1.75 is 7. Works again. So, really, if you scale this triangle by a factor of four you get a right-angled triangle with sides 7 24 and 25 which will have the pink circle as incircle.
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Brilliant. How about the other two triangles. Well, real quick. 1, 2, 3, 4, 5, 6, 7. 4 times 7 is 28. Aaaaand … 28 works. This side is 11.25 long and 4 times 11.25 is 45. Works again. How long is this one? Well 12. And 4 times 12 is 48. And finally this side 13.75 and 4 times 13.75 is 55.
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Okay, now for some visual candy. Let’s animate scaling the triangles by a factor of 4 and fitting the incircles. Super pretty don’t you think? The whole growth mechanism purely in terms of geometry. Of course, that was all a bit fast and so again, here is how we find the three children of a parent. The parent triangle is the one in the middle. Draw in the Feuerbach circle of this triangle. Connect the center of Feuerbach’s circle to the centers of the excircles. Extend to right
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angled triangles with vertical and horizontal short sides. Blow up by a factor of four. There,
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the 5, 12 13 parent triangle in the middle surrounded by its three children. Very nice isn’t it? But there is one more AHA moment hiding in all this. Remember that I got rid of the incircle earlier. Well let’s put it back in at the point where we are connecting the excircle centers to the center of the Feuerbach circle. There. Now let’s also connect the Feuerbach circle center to the incircle center. And extend to a right-angled triangle. Have a close look at this new triangle.
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Ring any bells? Doesn’t it look like a scaled down 3, 4, 5 triangle? What’s going on here? Well, in the tree the parent is connected to its three children. But it’s ALSO connected to ITS parent. And now it turns out that what we see in front of us is the connection to the parent of the parent which in the case of our 5, 12, 13 triangle is the 3,4,5 triangle.
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And it turns out this always works. For any triangle in our Pythagorean triple tree We can reconstruct ITS parent triangle by connecting the centers of the incircle and the Feuerbach circle
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extending the connection to a right-angled triangle with horizontal and vertical short sides. And scaling this triangle by a factor of 4. Pure magic :) Nice question to ponder: All of the entries in the tree have a parent except for 3,4,5. Right? What happens if you apply the parent construction to 3,4,5? How do things fail there. As usual, leave your answers in the comments.
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Whenever I do one of these pure maths video there are always people who don’t seem to able to appreciate anything unless they are also given some real-world application. Okay, so here are two fun and important real world applications of Pythagorean triples. Do you remember this person from another video? That’s my father. When I was 7 or 8 years old I was doing some home improvement project together with him. I think the task of the day consisted in tiling the floor of
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a bathroom. To tile a room you usually tile row by row starting along one of the walls. However, this only works out nicely if the walls of the room meet at pretty much perfect right angles. On that day, my father introduced me to a simple way of checking that this is really the case.
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If that’s our room, mark 3 meters from one of the corners along one of the walls. Then mark 4 meters from the corner on the other wall. Now measure the distance between the marks. If you get 5 meters you can be pretty sure that you are dealing with a right angle.
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A very simple, but at the same time very powerful trick everybody should know about. Actually, who among you already did know this trick? Anyway thank you very much papa for this important life lesson :) Okay here is a little puzzle for you which is meant teach another life lesson :) Have a look at this. Can you think of a mathematical reason for me to give this pearl necklace to my wife as a present? :) I’ll include a pictorial
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solution to this puzzle at the very end of this video :) See whether you can read my mind :)
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Okay let’s climb up the tree along a few distinguished routes and see what we find. First let’s climb up right in the middle of the tree and note down the Pythagorean triples that we encounter along the way. What’s special about these Pythagorean triples? Well, pretty obvious. The first two numbers always differ by 1.
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This will continue forever as we ascend the tree. And, in fact, we’ll encounter all the Pythagorean triples with this special property along the way. What does this mean for the triangles? Well, for example, for the blue one the two short sides are pretty much the same length. This means that the triangle will be very close to being an isosceles right triangle, half of a square. And
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the higher we climb, the triangles will get closer and closer to being exactly isosceles. Of course, since the short sides always differ by one we’ll never get a triangle that’s perfectly isosceles, we just get arbitrarily close. Interesting, isn’t it. So what’s a good question to ask at this point? Well, for example: Are there any isosceles triangles with integer sides? In other words, are there Pythagorean triples the first two numbers of which are the same? What do you think?
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Well let’s see. If there are such integers A and B, then we can rewrite this equation like this:
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This means that an isosceles triangle with integer sides exists if and only if the square root of 2 can be written as a fraction, if and only if root 2 is a rational number. However, you probably all know that this is not the case. The square root of 2 is an irrational number. And so we conclude that right-angled isosceles triangles with integer sides don’t exist. Bummer ! :) However, all this is not a complete write off. Since our special triangles get pretty
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close to being isosceles we can use the numbers in the Pythagorean triples to approximate square root 2 by fractions. For example 169 divided by either 119 or 120 should be a number fairly close to root 2. Let’s check. There… pretty close. And of course just as the triangles get closer and closer to being isosceles, the corresponding approximations to root 2 as fractions get arbitrarily good.
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The family of triples that we come across on our way up the middle of the tree is called Fermat’s family named after the great 17th century mathematician Pierre de Fermat who once challenged some of his colleagues to find some members of this family of triples. Cool. Now, ascending on the far left and right of the tree produces two more famous families of triples.
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Let’s first climb up on the right.
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In this family the largest number and the next largest number differ by 1. This family is actually named after Pythagoras who is credited with the discovery of this special family. Of course, just as with Pythagoras’s theorem which was definitely not discovered by Pythagoras we cannot be sure who originally discovered this family of triples. The Babylonians definitely did know a lot about Pythagorean triple 2000 years before Pythagoras and so who knows :) Anyway,
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to see what’s happening in terms of the associated triangles in the case of this family, let’s draw these special triangles such that the long short side is always the same length. Have a look. There the 3,4,5 triangle. 5,12,13 scaled looks like this. So what happens is, that rescaled like this, the triangles become thinner and thinner, as we get higher and higher up in the tree. In fact, very soon the triangles will appear like a straight line.
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Interesting. Finally, Plato’ family named after the ancient Greek philosopher Plato. That’s the family of triples that you get when you climb up on the far left.
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In this family the largest number is two larger than one of the smaller numbers. So, the outer families are very similar. Rescaled, the triangles will also devolve into line segments.
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There are a couple of interesting questions one can ask at this point. For example: What other interesting families of Pythagorean triples are hiding in the tree? Do interesting families correspond to climbing the tree in certain regular patterns of taking left, middle or right branches? And so on. Lots of room for exploration and discovery if you are interested :)
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Well this is Mathologer and so I definitely still owe you some proofs for at least a couple of the things I talked about today. So for the keen among you, let’s run through some pretty proofs. Okay, start with any two positive integers at the top of our box and fill in the rest as usual a la Fibonacci. So, A plus B that’s what? Well, A+B :) And B+ (A+B) that’s A+2B. At the very beginning of this video we chose consecutive Fibonacci numbers for A and B,
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but later on in the tree most of the choices for A and B were not Fibonacci numbers. In fact, at this point in time most of us would probably suspect that no matter what positive integers we choose for A and B, our pair picking will always produce a Pythagorean triple plus all the magic radius stuff, as long as we complete the box a la Fibonacci. That this is true is actually not hard to check. But before we do this, let’s recast things in a slightly different way which
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will make things look a lot nicer in the long run and which will link what we’ve been talking about so far perfectly with a famous theorem that goes back more than two millennia. There, what we’ll do is to is to choose the numbers on the right instead of those at the top to kick things off in the box. Again pretty much anything goes in terms of choosing U and V, except that V always has to be chosen greater than U. Obvious, on the right A+B is always greater than B. Okay
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let’s start by choosing U and V instead of A and B. It will become clear in a minute why this is a good idea. Okay, U+V goes there. What about the upper left corner. Think about it for a moment. You got it? Fibonacciwise we are moving in the reverse direction, so instead of adding U and V, we have to subtract. That number up there should be V-U. Some of you may have some doubts so let’s just quickly double check the Fibonacci action in the box. There V-U plus U, that’s really V
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and U + V equals U+V :) So the box is really filled in a la Fibonacci. Now, let’s calculate the three numbers in the corresponding would-be Pythagorean triple. The first number in the triple is the product of the circled numbers, so V minus U times V+U. And you probably remember from high school maths torture time that V-U times V+U is V squared minus U squared. Okay, next number. Well,
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that’s just U times V. That’s it? No remember we have to double that. And so we get 2 times U times V. Third number. Remember for that one we have to add the diagonal products. U times (V+U) plus V-U times V. Simplify on algebra autopilot. And so in theory this should be a Pythagorean triple, right?
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And that this equation really always holds no matter what numbers we choose for U and V you can easily check by expanding everything on the left and everything on the right and then comparing the results. Again, if you choose 1 and 2 for U and V you get our prototype Pythagorean triple 3, 4, 5 and if you chose U and V to be 2 and 3 you get the other famous triple 5, 12 ,13. Now if you’ve been around for a while maths wise you may recognise this equation as something
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really ancient, something that already popped up around 300 BC, more than 2000 years ago, in Euclid’s Elements, the most successful and influential series of maths textbooks off all time. There in book 10 of the Elements, Euclid proves something very interesting. He proves that we get all primitive Pythagorean triples exactly once by choosing U and V in all possible ways such that 1) U is less than V 2) that one of U and V is odd and the other even and 3)
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that the two numbers have no common factors other than 1. Again, U less than V, one odd the other even, no common factors. As long as we make such a choice we get a primitive Pythagorean triple and we get every such triple exactly once this way. So they really already knew how to make all primitive Pythagorean triples ages ago. Pretty amazing. And with all the talk about primitive and exactly
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once, doesn’t this ancient result seem like the perfect starting point to prove that our tree really gives all the primitive Pythagorean triples exactly once? Well let’s have another close look at our tree. Let’s focus on the right sides of the boxes, that’s where the Us and the Vs live. As you can check, all the conditions in Euclid’s theorem are always satisfied, at least in this part of the
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tree. The number on the top is always smaller than the one at the bottom, one of the numbers is odd and the other one is even and the two numbers don’t have any common factors other than 1. And so to prove that the tree really features all the primitive Pythagorean triples exactly once, we only have to show that any possible Euclid choice of U and V pops up exactly once on the right side
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of one of the boxes in the tree. That’s a really nice way of making the proof manageable, isn’t it? Actually completing the proof is a pretty routine exercise and not worth working through in detail in this video. Let me just point out two delicious insights that you’ll come across in the proof if you look it up or make it up yourself. The first delicious insight is the fact that the incircle
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radius of a parent is always smaller than the incircle radius of any of its three children.
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Pretty obvious both in terms of the algebra and the geometry. The incircle radius is smaller than excircle radiuses. The second insight is that the 3,4,5 triangle has incircle radius 1 and that the incircle radiuses of all other right-angled triangles with integer sides are greater than 1. Anyway, just to give you a taste of the proof here is how these insights are used in the proof.
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One of the things to prove is that every primitive Pythagorean triple is in the tree.
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For this you first show that every primitive Pythagorean triple has a unique primitive parent. In turn that parent then has a primitive parent, and so on with the corresponding triangles having smaller and smaller radiuses. Also, it turns out that every primitive triangle with incircle radius greater than one has a primitive parent. And this means that our string of primitive triples necessarily ends in 3,4,5 whose triangle has incircle radius 1.
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And this means that what we see over there is a path in the tree from the 3,4,5 root to the primitive Pythagorean triple we started with, showing that all such triples are contained exactly once in the tree. For more details about this proof and all the other proofs that I’ve omitted or only glossed over today check out the links in the description of this video. Okay, so I started by showing you how the Fibonacci sequence translates into 3,4,5 and infinitely
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many other Pythagorean triples. However, it soon became clear that what’s important about this translation was the growth rule of the Fibonacci sequence and not the numbers in the Fibonacci sequence. In fact, as we just proved, starting with any two positive integers at the top of the box and filling in the rest a la Fibonacci does the trick. And so the Fibonacci numbers are not that special in this respect. However, it turns out that there is something extra special
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about the Pythagorean triples produced by the Fibonacci numbers after all. Have a close look at these first four Pythagorean triples produced by the Fibonacci sequence. Can you see what’s special? No? I give you an hint: Have a look at the numbers on the right sides of the equations. Can you see it now? Yes, all those numbers are also Fibonacci numbers. In fact, if we keep on going we get every second Fibonacci number starting with 5. There, 5, 13, 34, 89, and so on,
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every second Fibonacci number. This is very surprising considering how the third numbers in the triples are produced from the four Fibonacci numbers in the box. Right, that third number is this product plus that product. Why should that be another Fibonacci number? Not clear at all. Well it’s late in the game and so let me just show you. There, going through the usual motions, the four consecutive Fibonacci numbers in the box translate into this Pythagorean triple.
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Now that third complicated number there on the right. That complicated expression turns out to be always equal to another Fibonacci number, to be precise the expression on the right is equal to this Fibonacci number here. Very surprising isn’t it? Also think about it like this. In the Fibonacci sequence two consecutive terms always add up to the next term. But who would have thought that any four consecutive terms in the Fibonacci sequence
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always combine into another Fibonacci number in this simple way. Even when you think you’ve seen everything Fibonacciwise, the Fibonacci sequence always seems to have one more trick up its sleeve. Anyway, that’s it for today. Except, I also wanted to show you a really nice new spiral that combines the Fibonacci sequence and Pythagoras. This one is due to the maths teacher and artist Eugen Jost.
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