Welcome to another Mathologer video. Today I'd like to invite you to chase down a particularly crazy mathematical rabbit
hole. It starts out with some curious properties of the golden ratio Phi and the
Fibonacci sequence and it will culminate in some crazy facts about some mutant
relatives the tribonacci constant and the tribonacci sequence and some
really, really really strange rabbits. By the way, this is Marty giggling in the
background, you remember him from last time I hope. Anyway just a reminder, the Fibonacci sequence starts with two 1s and then every
other Fibonacci number is the sum of the previous two Fibonacci numbers. So 1 plus
1 is 2, 1 plus 2 is 3, 2 plus 3 is 5, and so on. The tribonacci sequence starts
just like the Fibonacci sequence with 1, 1, 2 but then every tribonacci number
is the sum of the previous three. So the next number is 1 plus 1 plus 2 is 4, and then
1 plus 2 plus 4 is 7, and so on. As far as I know, the first connection between the
Fibonacci sequence and the golden ratio was discovered by the famous
mathematician and astronomer Johannes Kepler. He observed that the ratios of
consecutive Fibonacci numbers converge to phi. So this sequence of ratios here
converges to the golden ratio. Similarly the ratios of consecutive tribonacci numbers converge to the tribonacci constant. Wonderful isn't it? Now just imagine being the first to discover these facts :) Now what if your life
depends on figuring out the 1000s tribonacci number. Well, we have the first 9
tribonacci numbers listed and by adding the 7th, 8th and 9th we get
the 10th, by adding the 8th, 9th and 10th we get the 11th. So if we keep going we'll eventually get to the 1000 tribonacci number. But
going this way may be too slow to stop the sword of the executioner.
Luckily there's this absolutely mind-bending formula which skips all the
adding involved in building the tribonacci sequence and straightaway
gives us the 1000s tribonacci number. Whoa, and there's the tribonacci constant right at its core. (Marty) I'm not sure that this will be quick enough to stop the executioner. (Mathologer) Let's just hope it will. So the two square brackets encasing everything indicate that the 1000th
tribonacci number is the crazy number inside these brackets, rounded to the
nearest integer. If you're interested in a general formula just replace 1000 by n. For example, choosing n within easy
reach like 20, the number in the square brackets pans out to be this guy here.
Then rounding gives us 66012. Have you ever seen a weird
formula as this. I was definitely very impressed the first time I
saw it. You too right Marty? (Marty) Oh absolutely. (Mathologer) Yeah really crazy one. So, anyway my mission today is to explain
this formula and a corresponding very famous formula that plucks the nth
Fibonacci number out of thin air. I would look at this one here. Also pretty neat,
right? But not only that, what I'd like to do is to give you a taste of how someone
might actually discover a formula like this. So I'd like to take you on a fairly
complete journey of discovery that takes us from playing with some numbers and
noticing something remarkable to an educated guess for this Fibonacci
formula, then to its proof and finally to our monstrous tribonacci counterpart
of the Fibonacci formula. I'll finish off by talking about a few neat
tribonacci counterparts to some famous occurrences of the Fibonacci numbers and
the golden ratio in geometry and in nature. Okay, so here we go,
let's play a little bit with the golden ratio phi. In decimals this number starts
out like this. When we square this number something
interesting happens. Wait for it, cute isn't it. The only thing this has changed is that the 1 in front has turned into a 2.
So squaring phi gives something interesting and that suggests to someone
like me or Marty to also look at higher powers. Here are the first few. Hmm,
pretty messy at first glance but if we have a closer look we discover some
really nice Fibonacci action. Here, as can see, the two yellow numbers seem to
add up to the red one. The same is the case here and here and it sure looks
like this will be true forever and ever after. And so the sequence of numbers
seems to grow exactly like the Fibonacci sequence except that its two seed
numbers at the beginning are phi and phi squared instead of 1 and 1. But there
are more miracles hiding in the sequence of powers of phi. if you look further
down the list, scroll down and down and down and down there you actually notice
that these powers get closer and closer to integers. Pretty neat isn't it. So just
write down the integers closest to our powers, let's do it, and we get what? As
you can see, in the green these integer actually start out forming another
Fibonacci-like sequence but this time of integers. There in the green, right on top
3 plus 4 is 7, 4 plus 7 is 11, and so on. Just there at
the very top things don't work out, right? 2 plus 3 is 5 and not 4. To
make things work out even there we, well, cheat a little bit. We just replace the
2 by a 1. Okay 1 plus 3 is 4, fixed :) (Marty) Are you allowed to do that? (Mathologer) Who's gonna stop me :) Anyway the sequence
of numbers that starts out like this actually has a name it's the Lucas
sequence or the Lucas numbers named after the mathematician François Édouard Anatole Lucas (pronounced Luca) (Marty) So they are the LUCA numbers. (Mathologer) Yeah, actually they are the LUCA numbers. but everybody seems to call them the LUCAS numbers. Anyway I'll try to call him the Luca numbers. In
a way the sequence is the closest and most important relative of the Fibonacci
sequence and just like the Fibonacci sequence you often find this sequence in
nature. I already talked about this connection in another video. I'll leave a
link in the description, so check it out. Anyway all this number play suggest
that if you are interested in the Lucas numbers (Marty) Luka!!! (Mathologer) Okay, it's gonna be bad, you don't
have to go 1 plus 3 is 4, 3 plus 4 is 7, 18 times. What you do is simply compute phi to the power of 20 which is this guy here and round to the next integer which is 15,127 and this is actually correct
and turns out to be true in general. Really neat isn't it? We can write
this as a formula there we go. The nth Lucas number is phi to the
power of n rounded to the nearest integer. Well, remember, we cheated a
little bit and n has got to be greater than 1 but that's the only exception.
The similarly wonderful formula for the nth Fibonacci number that I
mentioned earlier is hiding just around the corner. To uncover it let's have a
look at the Fibonacci and Lucas sequence side by side. So we've got the Fiibonacci numbers on the left and the Lucas numbers on the right. Have a look at this: 1 plus
4 is 5, divided by 5 is 1, 3 plus 7 is 10, divided by 5 is 2, 4 plus 11 is 15
divided by 5 is 3, and it's actually quite easy to see that this will always
be true. (Marty) And it's easy to prove? (Mathologer) It's very easy to prove, even you can do
it :) In other words, the nth Fibonacci number can be calculated by adding the n minus
first Lucas number and n plus first Lucas number and then divide the sum by
5. But now we have this nifty power formula for the Lucas numbers and so
abracadabra ... yeah, now things in the brackets get closer and closer to
integers, right? Now what this means is that from some point on we can put both
the powers into one pair of brackets. Let's just do that and then with a bit
more arithmetic we can turn this formula into the one that I raved on about
earlier. That formula just included phi to the power of n, so let's isolate this
power. First here and then one more time here. Take out the nth power and using
that the exact value of phi is this guy here the green bit turns out to be equal
to root 5. Okay our goal is in sight so let's just pull the 5 inside the
brackets and now, without worrying too much about what can possibly
go wrong here, one final simplification and that's the formula we are after, very nice! (Marty) And have we cheated? (Mathologer) Not to worry, we will get to this. So let's see how well this works. So here are the powers of the golden ratio again. Let's divide
them by root 5 and calculate the nearest integers and spot on from the very
beginning! Works even better than for the Lucas
numbers. Alright, now our counterpart for this formula for the tribonacci
numbers looks like this where c is also a constant just like root 5.
One thing that follows very easily from these formulas is Kepler's discovery
about those Fibonacci fractions converging to the golden ratio and the
corresponding property of the tribonacci sequence. Maybe have a good putting
together a proof for the Fibonacci convergence. So that may be your first
homework. So some people here can definitely do this, so do this in the
comments. But of course all our nice results were just based on our
calculator experiments at the beginning. We still don't have any bulletproof
proof that the powers of phi really behave in this super nice way.
Turns out the reasons are even more amazing than the facts themselves. So let's
have a closer look and for that let's start at the beginning. What is phi? Well
there are a couple of different definitions but maybe the most popular
one is based on golden rectangles. A rectangle is a golden rectangle if when
you extendeding it by a square like this results in a larger similar rectangle. So
by just rotating the original rectangle 90 degrees and scaling up you can get
the large rectangle. Now phi is just the common aspect ratio of all golden
rectangles. To calculate this aspect ratio we start with a special rectangle
whose long side is phi units and whose short side is 1 unit. So the aspect ratio of
this rectangle is phi over 1 so that makes it into a golden rectangle. Now extend to
a large rectangle using a square. Its aspect ratio is the length
of its long side which is phi+1 divided by the length of its short side
which is phi. Since both rectangles are golden these
aspect ratios are equal. Multiplying the equation on both sides by phi gives a
quadratic equation. This equation instantly explains our strange squaring
property from the very beginning. Why? Because what this equation says is that
you get the square of phi by just adding 1 to phi. Nice isn't it and to figure out
what phi is we simply have to solve this quadratic equation here. Now you've all
been tortured to death with quadratic equations in school, right? (Marty) I like quadratic equations. (Mathologer) Okay I like them too. Anyway, quadratic formula, crank the handle and you get two
solutions, right? Which of these is the golden ratio? Well one solution is
positive the other is negative and since aspect ratios are positive phi has got to
be this guy there at the top, that one. And usually what people do at this
point is to discard the poor second solution as useless. Poor second
solution :) Which turns out to be a serious mistake as we'll see in a minute. For the
moment just remember that there is this second solution. Okay
now somehow the powers of phi seems to be a bit of a gold mine. So let's have a
closer look at them. To get phi cubed, for example, I'll just multiply this equation
here by phi. Alright, now you can spot these two phi squared's here and what this
means is that I can replace the second phi squared by phi plus 1. Let's do
that. Simplify, all right. Multiply the second equation by phi again and you get
the fourth power. Two phi squared's again. So we replace the second one by phi + 1
again, alright. Simplify, and you can keep on going like this
obviously. And so with those coefficients on the right we're definitely getting
into Fibonacci territory and we can easily prove a general formula. And just
to read it off let's have a look at the last one here the exponent here is 6
and 8 is the sixth Fibonacci number and five is the fifth Fibonacci number and so in general we get what? Well, the nth power of phi is equal
to the nth Fibonacci number times phi plus the n-1st Fibonacci number, very pretty too. Now, remember, we are after a formula for
the nth Fibonacci number so this looks very promising. But sadly it's not quite
good enough. Why? Because we cannot straightaway solve for the Fs, right? And
that's where the second solution that I asked you not to forget comes in. The
equation up there is a direct consequence of that very first identity
for phi, this one here. Now, for that second solutio,n let's call it phi red,
exactly the same identity holds, right? So what this means is that the second
equation is also true for phi red and instead of just phi. Now this conclusion
is really important so make sure you really really understand it. Okay so at
this point we've actually got these two equations to work with. Now we just
subtract the second equation from the first and that gets rid of one of the Fs
right? There gone and now we just solve for F and that gives us an exact
formula for the nth Fibonacci number. The denominator there again pans out to be
root 5 and here's the formula in all its glory. Whoa I still remember
that seeing this one twenty or thirty years ago and I've never been
able to forget it, neither the proof. So it's really really pretty. This formula
is called Binet's formula after a French mathematician Jacques Philippe Marie
Binet although it was already known one
century earlier to Abraham de Moivre which is one of those sad stories...
When you think about this formula it's really quite amazing that you see all
these irrational numbers that this formula is crawling with combining into
integers for those infinitely many possible values of n. But it gets better.
Remember that phi red is equal to -0.618... Since the absolute value of this number is less than 1 its powers converge to
zero. Let's have a look. So two, four, six. This shows that the
contribution of phi red to this formula diminishes quickly and actually explains
why the nth Fibonacci number is the closest integer to the nth power of phi
divided by a root five. You can just forget about this contribution. Wonderful
isn't it? (Marty) Wonderful. (Marty) Very good. In fact with what I've just shown you
it's also very easy to prove all those other facts that I mentioned at the
beginning. First that the nth Lucas number is the integer closest to the nth
power of phi. Then that two consecutive powers of phi add up to the next power
and so on. Now, if you know how to prove these facts or maybe if you are aware of
some of the other closely related facts that I did not get around to discussing
let us know in the comments okay, concerted effort and we will have phi covered completely by the end of this. Now just in case you're wondering, no I did not
forget about the tribonacci numbers they are up next and now it's all pretty easy.
Just like phi is one of the two zeros of this quadratic equation here the tribonacci
constant is one of the three solutions of this closely related cubic
equation. Now manipulating the quadratic equations first got us to this and then,
by dragging in the extra solution phi red we got Binet's formula. Now if we do
the same with the cubic equation and its three solutions we get this amazing
formula for the nth tribonacci number, an exact solution. So those extra
solutions of the cubic equation t green and t yellow are actually proper complex
numbers so there is i in there and irrational numbers, really a mess but anyway. And so
this precise formula with all those complex in irrational numbers arranging
themselves into the tribonacci numbers for all n is really a little miracle.
Well it's a bigger miracle than the one before. In any case, just like our red phi,
t green and t yellow have absolute values less than 1 which then also
means that their contributions to this equation die out quickly for larger n
and this gives the crazy formula that I promised you at the beginning that one
here. Well, here again in its irrational glory, really crazy. Okay I'll take a bow here, it's
just too good. As you all know the golden ratio and the Fibonacci numbers make
numerous appearances in geometry and nature. Some of these appearances have surprising counterparts for the
tribonacci numbers and the tribonacci sequence. For example, the icosahedron can be constructed by sticking three golden rectangles together at right angles and
therefore is full of golden ratios, have a look. A bit of an animation here, three
golden rectangles there, filled in with triangles that's an icosahedron, very
nice. Now it's only been discovered fairly recently that the so-called snub
cube, one of the Archimedean solids and a close relative of the icosahedron is
full of tribonacci constants. Now many of you will not be familiar with the
snub cube so here's an animation which shows how an ordinary cube can be
transformed into a snub cube. Basically the cube explodes and as its faces are
flying apart they are rotating a little bit until the gaps between them are of
just the right size and shape to be filled with bands of equilateral
triangles very pretty right. Now there are lots of tribonacci constants
present in all sorts of distance ratios and other statistics connected with the
snub cube. However, in many ways, the nicest tribonacci constant fact about
the snub cube is that exactly half of the permutations of these tribonacci
filled coordinates over there are the coordinates of the corners of a snub
cube. The other half of the coordinates well they describe the
corners of the mirror image of our snub cube. To me the way this video panned out
is very reminiscent of Alice's trip down the rabbit hole. Just like Alice notices
that strange rabbit in a waistcoat, we stumble across this curious squaring
property of phi and then, before we know it, we're tumbling head over heels into a
mathematical wonderland. Now speaking of rabbits, of course there
are those immortal rabbits that were first conjured up by Leonardo
Fibonacci in the 13th century whose population growth is captured by the
Fibonacci sequence. You probably all know this right but anyway: a pair of these
bunnies that gets born does not have any offspring while growing up fo a month
but then has one pair of baby bunny babies every month. Starting with one
pair the Fibonacci sequence tells you how many pairs there are month after
month. You are all familiar with this, right? So the final task for you is to
follow in Leonardo's footsteps and engineer an ideal mathematical tribonacci
rabbit population whose population growth is captured by the tribonacci
sequence. And that's it for today. As usual, let me know how my explanations
worked for you and discuss any problems or thoughts you might have in the comments.
Just watched it. Agree very, very pretty math.
It seems that if you keep growing the Fibonacci degree (tribonacci, quadrinacci, etc.) the ratio of successive numbers approaches 2.
https://en.wikipedia.org/wiki/Lucas_number
https://fr.wikipedia.org/wiki/LUC_(cryptographie)
https://en.wikipedia.org/wiki/Jacques_Philippe_Marie_Binet
another one to try maybe (-+-+-+)Bonacci ...
This topic kinda loses its magic once the profs made you solve third order linear difference equations by hand.
very nice presentation !