The golden ratio spiral: visual infinite descent

by birtanpublished on August 17, 2020

welcome to another mythology video the Golden Spiral over there is one of the most iconic pictures of mathematics the background of the picture is the special spiral of squares and the golden spiral itself is made up of quarter circles

Inscribed into these squares overall this quarter circle spiral is a very close approximation of the true golden spiral which is a logarithmic spiral that passes through these blue points here this spiral here pretty good

Fitting our golden spiral picture captures some of the amazing properties of one of a mathematic superstar the golden ratio Phi however the one feature that this picture is most famous for is sadly just the mathematical urban myth

Pushed and propagated by lots and lots and lots of wishful thinkers these people I call them fanatics will assure you that the spiral that you see in Nautilus shells are golden spirals which is simply not true same thing for

Spirals in spiral galaxies cyclones and most other spirals found in the wild what is true is that just like our quarter circle spiral a lot of disparity observe in nature are approximately logarithmic spirals however there are

Infinitely many different logarithmic spirals and most of the logarithmic spirals found in nature are not even remotely golden in fact most of the pictures that are supposed to prove the golden nature of naturally occurring

Spirals arrived it by roughly fitting a really thick golden spiral to some suitably chosen and doctored picture having said that sometimes spiral patterns that we observe in nature like for example those and flower heads do

Have a connection to the golden ratio however in general not even the spirals and flower heads are golden and the connection is established in different non spiral ways if you're interested I've linked to some articles that debunk

A vast portion of the golden spiral in nature story fanatics sorry to disappoint what I'd like to do in the following is to focus on some true and truly amazing features of this picture which even a

Lot of mathematicians are not aware of what will be important for us about this picture is the curious spiral of squares at its core in fact as far as today's story is concerned the sole function of the golden spiral spiral is to highlight

This square spiral it turns out that not only the golden ratio but in fact every positive real number has an Associated square spiral for example here's a spiral of root 2 ends up who has seen a green golden spiral before anyway these

Squares spirals which can be finite or infinite are very easy to construct and provide a wealth of insight into the nature of numbers for example I'll show you that if you look at root 2 square spiral in just the right way it

Magically moves into a so called infinite descent proof of the irrationality of root 2 in fact I show you a simple characterization of the irrational numbers in terms of their square spirals

And use this characterization to pin down and visualize the irrational nature of many famous numbers to finish off I'll show you how the squares power of a number is really the geometric phase of the so-called simple continued fraction

Of that number those guys here anyway ready for some really amazing and beautiful mathematics let's go okay to start with let me show you how the square spiral of a number is constructed and why if the resulting

Spiral is infinite the number has to be irrational our first focus on the number root 3 to construct the spiral we start with a root 3 rectangle like this one here a root 3 rectangle as a rectangle with sides a and B whose aspect ratio a

Over B is equal to root 3 trivial but important observation if you scale a root 3 rectangle you get another root 3 rectangle now here's the first square of the root three square spiral here's the second

Square of the spiral the third the fourth the rule is that the next square is the largest square that fits into the remaining green area fitted in such a way that it continues the right turning spiral so next is this then this and

This and so on pretty straightforward right let's quickly go back to the beginning and count the number of squares of each size that we come across in this spiral okay first square again there's only one square of the size next

Also only one square of this size next and two of those okay then one then two again in fact from this point on things repeat so one two one two one two forever neat mm one way to convince ourselves that things really repeat is

To show that this blue rectangle here is also a root three rectangle just like the green one we started with this means that new squares fit into the blue rectangle in exactly the same way as they do in the start in green rectangle

And so the pattern repeats okay let's show that this blue rectangle really is a root three rectangle remember that we started with a root three rectangle and so the ratio of the sides is root three put the first screen and so the

Dimensions of the remaining green area are what well short side on top has length a minus B and alongside obviously B put the next square in uncalculated side lengths in exactly the same way now the third square and now let's check

That the aspect ratio of the blue rectangle is really root three this aspect ratio is what well this now some straightforward algebra divide both the numerator and denominator by B that does not change the ratio but remember a over

B is equal to root 3 the standard trick to get rid of the root in the denominator is to multiply the bottom and the top by a 2 plus root 3 like that just in case you have not seen this trick in action let's highlight the

Denominator the highlighted product is of the form u minus V times u plus V which of course is equal to u squared minus V squared which in this case is 2 squared minus root 3 squared and so you can see the square root in the

Denominator vanish remembered is clearing the denominator of roots trick it really comes in handy very often in maths anyway now just go on algebra autopilot and you'll see that the whole expression

Simplifies to root 3 wonderful at this point we're ready to draw a couple of pretty amazing conclusions let's start by using our square spiral to prove that root 3 is irrational this also ties in nicely with what I did in the last video

Okay if root 3 was rational that is if root 3 could be written as a ratio of positive integers a and B then the rectangle with sides a and B would be a root 3 rectangle now we just calculated the lengths of the sides of

The first couple of squares right now since a and B are supposed to be integers these three side lines B a minus B and two B minus a would have to be integers as well in fact it's very easy to see that this continues the side

Lengths of all the infinitely many squares in our Sparrow must be some integer multiple of a or B minus some other integer multiple of B or a like down there integer times a minus integer times B this implies that all the side

Lengths of all the squares all the way down are positive integers but and regulars have heard me say this a lot this is impossible why well the infinitely many our spiral shrink to a point and

Therefore they must eventually have sidelines smaller than the smallest possible positive integer one the only way to resolve this contradiction is to conclude that the assumption we started with namely that root three is a ratio

Of positive integers is wrong and so we conclude that 3 is irrational that is a really really pretty proof don't you agree but it is much more than that why because all sorts of things we've just

Said stay true beyond a special case of root 3 for example it's really easy to see that if we start with any rectangle with integer sides and if we move squares according to our recipe then all those squares in the spiral must also

Have integer sides this means the same proof by contradiction shows that any number with an infinite square spiral must be irrational so for example the golden ratio Phi is irrational because it's spiral is also infinite now here's

A really pretty way to picture what we've accomplished the essence of our proof by contradiction is called an infinite descent because our assumption that a rational number has an infinite spiral implies the existence of an

Impossible infinitely descending or decreasing sequence of positive integers very nice but also notice that you can actually see the impossible infinite descent in the spiral by interpreting the squares as steps of an ever

Descending spiral staircase there's our infinite spiral staircase and the footsteps of someone going for the infinite descent what's going to happen when they reach the bottom one I think anyway to round off this part of the

Video just remember that if we can show that a number has an infinite square spiral then we've also shown that this number is irrational so what about the spiral of a rational number well obviously it cannot have an infinite

Spiral it is its spiral must and after a finite number of steps but how does it end well let's have a look at an example the aspect ratio of the rectangular frame of this video is 1920 over 1080 that means

That this rectangle is a rectangle that corresponds to the rational number 1920 over 1080 and so as you can see the square spiral of this number consists of only 7 squares so the spiral ends because when we place the 7 square the

Rectangle we started with is completely covered there is no space left for an eighth square here's an interesting fact the side lengths of the smallest square in this finite square spiral is the greatest common divisor of the numbers

1920 and 1080 puzzle for you show that this is true in general second puzzle for those of you in the know which super famous Greek mathematician is responsible for some closely related mathematics ok so we can be sure that

The square spiral of a rational number is finite how about going the other way is it also true that every finite spiral comes from a rational number well let's see say I give you a finite spiral like this one there here's how you can

Determine its aspect ratio first we scale things so that the smallest square has side lengths 1 then it's clear that the next larger square has side length 1 plus 1 plus 1 is 3 then we can see that the largest square has side lengths 3

Plus 3 plus 1 is equal to 7 and finally that the top side of our rectangle is of length 3 plus 7 is equal to 10 and so our rectangle has aspect ratio 10 over 7 and of course we can do exactly the same for any finite spiral to show that it

Corresponds to a rational number neither ok so that means that the rational numbers are exactly the numbers with a finite spiral which then also implies that a rational number exactly those numbers with an infinite

Spiral that's a pretty amazing characterization of rational and irrational numbers don't you think definitely made my day the first time I read about this now to actually use this characterization of irrational numbers

To prove that a particular number such as Phi is irrational we somehow have to show that it's associated spiral is an infinite the way we were able to show this for route 3 was by recognizing that the square spiral repeats in turn this

Was possible because we were able to show that while building the spiral we come across rectangles with the same aspect ratio now it's very easy to see that this also happens for the golden ratio Phi in fact this repeating

Property is part of the definition of the golden ratio that is a rectangle is golden if when you cut off a square like this you end up with a scaled-down version of the original so since things repeat after cutting off

One square this also means that Phi has the simplest possible square spiral with every square size occurring just once and the Associated sequence of integers being all ones like that anyway just remember things repeat for Phi so next

Time someone asks you why the golden ratio is irrational just point at the closest golden spiral and say infinite descent in an ominous voice okay as a final repeating example here is root two and here's a nice little root 2 factoid

That I actually did not know myself until recently all these pink rectangles are root 2 rectangles right of course an a4 piece of paper is basically a root 2 rectangle what this means is that if you fall the paper in half you get a scaled

Down version of the original that is another root 2 rectangle but did you know that you also get another root 2 rectangle when you cut off two squares like this there another route to rectangle very

Cool maybe not earth-shatteringly cool but I enjoy little mats moments like this almost as much as the really deep stuff okay at this point it's natural to ask for which numbers this works so which numbers have a repeating spiral

Well the examples so far were 5 root 2 root 3 so all square root numbers in fact it turns out that the numbers with repeating spiral are exactly the numbers of this type and when I say of this type

I mean all positive irrational numbers there are roots of quadratic equations with integer coefficients these numbers are usually referred to as quadratic Irrational's now the fact that a periodic spiral

Implies that we're dealing with one of these routine numbers is pretty easy and was first shown by one of the usual suspects Leonhard Euler on the other hand showing that every quadratic irrational has a repeating square spiral

Is not super hard but it's definitely a little bit fiddly so let me just show you a sketch of the easy direction periodic spiral implies quadratic irrational so let's say X is a number with a repeating spiral then in this

Particular X rectangle all side lengths of the resulting squares look like this so integer times X minus another integer or integer minus another integer times X this means that the aspect ratios of the rectangles that we come across during

Spiral building our ratios of expressions like this for example could have something like that now we said the spiral repeats what this means is that two of these aspect ratios have to be the same hmm but obviously after

Multiplying through with the denominators any such equation simplifies to a quadratic equation and so X as a solution of this quadratic equation is a quadratic irrational easy-peasy

Lemon squeezy puzzle for you what's the solution to the equation over there and what do all the coefficients in this equation have in common a coincidence okay now at the start of this video I claimed that the square spiral of a

Number is really the same thing as the simple continued fraction of the number to explain this correspondence let's have another look at the root three spiral okay here comes the magic to get the continued fraction you just take the

Sequence of numbers of squares of each size at the top and do this so root 3 is 1 plus 1 divided by 1 plus 1 divided by 2 plus and so on very cool right but how does this work well let me finish off this video by explaining what I do is to

Run the standard algorithm for generating the infinite fraction and our algorithm for building the spiral side by side this will make it clear why we are getting the same sequence of green numbers okay root 3 is equal to 1 point

7 3 2 0 and so on let's rescale the short side of our root 3 rectangle to make it length 1 then the long side is equal to root 3 that is 1 point 7 3 2 and so on okay how many squares of side lengths 1 can be fit

Well obviously just one the integer part of root 3 next let's have a look at the rectangle that remains let's rescale everything so that the short side of the green rectangle becomes 1 the scale factor that does the

Trick is 1 over 0.732 0 up on top we can also do something we can rewrite things like this now mythology regulars will be familiar with this maneuver everybody else just think about it for a moment ok all under

Control great so good anyway this gets the continued fraction going on top now 1 over 0.732 0 is 1.33 6 0 and so on now again from the start how many squares of side links one fit into the green well obviously one the integer

Part of one point three three six zero and so on focus on the remaining green rectangle in rescale everything such that it's short side becomes one there we go rewrite the top as before one over point three three six zero is two point

Seven three two zero etc how many squares can be cut off the green – of course and so on as you can see the sequence of numbers that corresponds to the spiral is exactly the sequence of numbers in the denominators of the

Infinite fraction and with this transition to simple continued fractions understood you're ready for the mythology video dedicated to continued fractions and some of the other amazing insights they offer into the nature of

Numbers for example the amazing pattern in the continued fraction of the number e the continued fraction of pi and the curious observation that the golden ratio the number was the simplest spiral and continued fraction is the most

Irrational number etc and that's it for today except here is one more puzzle apart from the golden spiral what else is wrong with this picture here you

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