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 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. 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 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 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 2. PLUS 1 times 3 is 3. 2+3, that’s 5. 3,4,5 ,
3 squared + 4 squared = 5 squared. Impressed. 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 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 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 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 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 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. 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 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. 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. 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. 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. 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 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. 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. 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. 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. 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? 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 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? Here is one more crazy fact that will become
important later when we get to proving things. 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? 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, 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. 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. 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 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. 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 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, 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. 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 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. 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 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. 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 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. 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. Here we go How long is this vertical side?
Well, let’s count. 1, 2, 3, 4, 5, 6. Just 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. 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. 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 angled triangles with vertical and horizontal
short sides. Blow up by a factor of four. There, 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. 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. 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 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. 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 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. 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. 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 solution to this puzzle at the very end of this
video :) See whether you can read my mind :) 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. 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 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? Well let’s see. If there are such integers A and
B, then we can rewrite this equation like this: 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 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. 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. Let’s first climb up on the right. 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, 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. 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. 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. 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 :) 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, 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 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 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 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, 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? 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 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) 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 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 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 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 radius of a parent is always smaller than the
incircle radius of any of its three children. 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. One of the things to prove is that every
primitive Pythagorean triple is in the tree. 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. 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 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 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, 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. 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 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.
