Phi and the TRIBONACCI monster

by birtanpublished on August 21, 2020

welcome to another mythology 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 a Fibonacci sequence and it will culminate

In some crazy facts about some mutant relatives that trip on a chi constant and a trip on a cheese sequence and so I'm really really really strange from oh by the way is Marty giggling in the background you remember him from last

Time I hope anyway just a reminder the Fibonacci sequence starts with two ones and then every other kubinashi 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 Tribuna Chi sequence starts just like the Fibonacci sequence with 1 1 2 but then every triple-action numbers to some of the previous 3 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 trip onaji numbers converge to the true Bonacci 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 triple notch in number well we have the first 9 triple action numbers listed and by adding the sevens eights nines we get tense

By adding the 8's 9s and 10s we get 11s so if we keep going we'll eventually get to the 1000 strip Renacci number but going this way may be too slow to stop the sword of the execution all right luckily there's this absolutely

Mind-bending formula which skips all the adding involved in building the true production sequence and straightaway gives us the 1000s try Bonacci number whoa and there's the two energy constant

Right at its core let's just hope it will so the two square brackets encasing everything indicate that the 1000 to production number is the crazy number inside these brackets rounded to the nearest integer if you're interested in

A general formula just replace one thousand by n okay for example choosing n with an easy reach like 20 the number in the square brackets pans out to be this guy here then rounding gives us sixty six

Thousand twelve I've ever seen a weird formula as this I definitely had I was very impressed by this the first time I saw you two right 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 true Bonacci counterpart of the Fibonacci formula and then I'll finish off by talking about a few neat chubachi 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 okay when we square this number something interesting happens wait for it cute

Isn't it the only thing this has changed is that the one in front has turned into a 2mm so squaring Phi gives something interesting and that suggests to someone like me or Marty to also look at higher

Powers so here are the first few mmm pretty messy at first glance but if we have a closer look we discover some really nice Fibonacci action here can see the two yellow numbers seem to add up to the red one the same as 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 it's two seed numbers at the beginning are Phi and Phi

Squared instead of one and one 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 bit 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 they 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 for you know to make things work out even there we well

Cheat a little bit we just replaced the 2 by the one ok one plus three is four fixed who's gonna stop me anyway this 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 Francois a doís and at all Luca yeah yeah actually that a Luke are numbers but everybody seems to call him 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 the sequence in nature I already talked about this connection in another video I'll leave a link in the description so check it out

Okay anyway all this number play suggests that if you are interested in the Lucas number 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 5 to the power of 20

Which is well this guy he and router next integer which is 15,000 127 and this is actually correct and turns out to be true in general really neat isn't it so we can write this as a formula there we go

So the enth luka number is 5 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 exceptionally there's 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 lucus sequence side-by-side so we've got a few Bonacci numbers on the left and the luca 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 proof it's very easy

To prove even you can do it in other words the NZ Bonacci number can be calculated by adding the n minus first leuco number and n plus first local number and then divide the sum by 5 but now we have this nifty power formula for

Aloka numbers and so abracadabra yeah now things in the brackets get closer and close to integers right now what this means is that from some point on we can put both the powers into one pair of brackets so 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 ok our goal is inside so let's just pull the 5 inside the brackets and now without worrying too much about what can

Possibly wrong here one final simplification and that's the formula we after very nice nobody 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 toluca numbers all right now our counterpart for this formula for the two

Bonacci 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 converge into the golden ratio and the

Corresponding property of the tribology sequence maybe have a good putting together a proof for the Fibonacci convergence so that's maybe the first homework ok so some people can definitely do this or do it 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 bullet proof proof led the powers of 5 really behaved in this super nice way turns out the

Reasons are even more amazing than 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 extended by a square like this result 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 a common aspect ratio of all golden rectangles to calculate this aspect ratio we start with a special rectangle whose long satisfy units and whose short side is one unit so the aspect ratio of this

Rectangle is five over one so that makes it into a golden rectangle now we extend to the large rectangle using a square its aspect ratio is the line of its long side which is five plus one divided by the length of its short side

Which is Phi since both triangles 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 Phi by just adding 1 to Phi nice in it and to figure out what Phi is we simply have to solve this quadratic

Equation here now we've all been tortured to death with quadratic equations in school right ok I like them too of course 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 fires got to be this guy there at the top that one yeah and usually what people do at this point is to discard the poor second

Solution as useless for a second solution which turns out to be a serious mistake as real 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 fire cube for example I'll just multiply this equation here by Phi all right now you can spot these two v Squared's here and what this means is that I can

Replace the second Phi squared by five plus one let's do that simplify all right multiply the second equation by Phi again and you get the fourth power to five Squared's again so we replace the second one by five plus one again

All right simplify and you can keep on going like this obviously and so with those coefficients on the right we're definitely getting into a 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 six and eight is the sixth Fibonacci number and five is the fifth gotcha number and so in general we get what well the ends power of Phi is equal

To the nth Fibonacci number times Phi plus the N minus first Fibonacci number very pretty – now remember we're 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 s 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 fire this one here now for

That second solution let's call it Phi red exactly the same identity holds right what this means is that the second equation is also true for fire red instead of just five 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 all right they're gone and now they 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 whoa they're still remembered seeing this one like 20 or 30 years ago

And I've never been able to forget it neither the proof so it's really really pretty this formula is called Benes formula after a French mathematician Jacques Philippe Marie Binet although it was already known one century earlier to

A brand moivre which is one of those set stories of no okay 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 five red is equal to minus zero point six one eight and so on since the absolute value of this number is less than one its powers converge to zero

Look so two four six this shows that the contribution of firered 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 wonderful wonderful 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 enthalpy is the

Integer closest to the nth power Phi then that two consecutive powers of fire 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 will have fire covered completely by the end of this now just in case you're wondering no I did not forget about the Fibonacci numbers they up next and now

It's all pretty easy just like Phi is one of the two zeros of this quadratic equation here the trip Renacci constant is one of the three solutions of the closely related cubic equation now manipulating quadratic equations first

Got us to this and then by dragging in the extra solution Phi red we got Benes formula now if we do the same with the cubic equation and it's three solutions we get this amazing formula for the ends true Bonacci number an exact solution so

Those extra solutions of the cubic equation T green and T yellow are actually proper complex numbers it is I in there and imagine numbers really mess but anyway and so this precise formula with all those complex in irrational

Numbers arranging themselves into the two Bonacci numbers for all N is really a little miracle well it's a bigger miracle than anyone before in any case just like our red Phi T green and T yellow have absolute values less than

One which then also means that their contributions to this equation diode quickly for larger n and this gives the crazy formula that I promised you at the beginning that one here well he again and it's irrational

Glory really crazy okay don't 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 in nature some of these appearances have surprising counterparts

For the true Bonacci numbers and a trinity 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 bit

Of an animation here three golden rectangles they're filled in with triangles it's like was a he drum 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 treble nachi constants now many of you will not be familiar with the snap 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 chip renacchi constants present in all sorts of distance ratios and other statistics connected with the snub cube however in many ways the nicest true Bonacci constant fact about the snub cube is that exactly half of

The permutations of these children are filled coordinates over there are the coordinates of the corners of a snub cube and the other half of the coordinates well they describe the corners of the mirror image of our snob

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 a Liana human achi 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 for months 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 a month after months you are all familiar with this right so the final task for you is to follow in Leonardo's footsteps and

Engineer ideal mathematical tripping archie rabbit population whose population growth is captured by the terminology sequence and that's it for today as usual let me know how my explanations work for you and discuss

Any problems or thoughts you might have in the comments you

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