The golden ratio spiral: visual infinite descent

The golden ratio spiral: visual infinite descent


Welcome to another Mathologer 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, the spiral here. Pretty good fit, hmm? The golden spiral picture captures some of
the amazing properties of one of mathematics’ superstars the golden ratio
Phi. However the one feature that this picture is most famous for is, sadly, just
a mathematical urban myth pushed and propagated by lots and lots,… and lots of
wishful thinkers. These people, I call them Phi-natics 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 spirals we 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 are arrived at 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 in flower heads do have a connection to the golden ratio,
However, in general, not even the spirals in 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. Phi-natics, 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 the spiral of root 2. Hands up, who has seen a
green golden spiral before? Anyway, these square 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 s square
spiral in just the right way it magically morphs 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 spiral of a number is really the geometric face 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. I’ll 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 is 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 3 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 this size. Next, also only one square of this size. Next, two of
those. Okay, then one, then two again. In fact, from this point on things repeat so
121212, forever. Neat! One way to convince ourselves that things
really repeat is to show that this blue rectangle here is also a root 3
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 starting
green rectangle, and so the pattern repeats. Okay let’s show that this blue
rectangle really is a root 3 rectangle. Remember that we started with
a root 3 rectangle and so the ratio of the sides is root 3. Put the first
square and so the dimensions of the remaining green area are … what? Well short side on top has lengths A minus B and the long side obviously B. Put the next square in and calculate its 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 3. 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.
Remember this 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 are 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 lengths
B, A-B and 2B-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 spiral 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 squares in
our spiral shrink to a point and therefore they must eventually have
side lengths smaller than the smallest possible positive integer 1. The only
way to resolve this contradiction is to conclude that the assumption we started
with, namely that root 3 is a ratio of positive integers is wrong. And so we
conclude that root 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 the 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 remove 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? What do you
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 numbers. Well,
obviously, it cannot have an infinite spiral, that is, its spiral must end
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 7s
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? Okay
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 lengths 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. Neat hmm? Okay, so that means that the rational numbers are exactly the numbers with a
finite spiral which then also implies that the irrational numbers are
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
infinite. The way we were able to show this for root 3 was by recognising
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 1s 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 2 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
fold 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 root 2 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 Phi, root 2, root 3, so all square rooty 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 that
are roots of quadratic equations with integer coefficients. These numbers are
usually referred to as quadratic irrationals. Now, the fact that a
periodic spiral implies that we’re dealing with one of these rooty
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 are ratios of
expressions like this. For example, we 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. 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?
Coincidence? I don’t think so. 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 3 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 1.7320… 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, 1.7320… and so on. Ok how many squares of side lengths 1 can we 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 0.7320… . Up on top we can also do something, we can rewrite things
like this. Now Mathologer 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 .7320 is 1.3360…, and so on Now, again, from the start. How many
squares of side lengths 1 fit into the green? Well, obviously one, the integer
part of 1.3360 … Focus on the remaining green rectangle and rescale everything such that it’s
short side becomes 1. There we go. Rewrite the top as before 1 over 1.3360 is 2.7320… How many squares can we cut off the green. Two 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 Mathologer 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?

100 Comments

  1. 12:17 I don’t understand the proof. You just showed one particular example and may not be general. Can someone please explain why is the relationship between rational numbers and finite spiral mutual?

  2. Burkard, why did you talk about infinite spirals this time instead of making the side lengths of your big rectangle the smallest possible integers commensurate with root three then using the smaller one to get a proof by contradiction like last time with the square triangles? Great video as always! :-*

  3. The two solutions of the quadratic equation shown are the golden ratio and the silver ratio and the integers are fibonacci numbers… Not exactly surprising, but still, that's pretty neat.

  4. It might be trivial. But I find it interesting that we draw 'irrational sized' shapes. There is no way to draw a rectangle precisely with irrational side lengths. I guess it's no different to drawing circles where the ratio of the diameter to the circumference is pi. It's just weird in a way that these things exist really as thought experiments.

  5. 11:40

    That's 48 by 27.

    When tried to help the neighbor kid with his math homework, I forgot to explain that a lot of stuff in math is the same as a lot of other stuff. 🙁

    O.K., the short side of the rectangle is 9 final squares and the long side is 16 final squares. I don't know what to do with that, I just had to figure it out.

    Darn you Mathologer!!!!!!!!! 😉

    Hey, you couldn't do that with a transendental number, could you?

  6. Great video as always ! Really interesting visualization !
    Random comment : As showing that any periodic spiral is related to a quadratic formula is simple but the reverse is hard, is it possible to create the cryptographic function from it?

  7. I finally finished the video, and found that it got more puzzles xD

    Puzzle 3 (equation):
    1) I got -243 310×3 + 117 493x² – 65 062x + 103 952 = 0
    2) And…. I was about to calculate Delta' — 4(b² – 3ac) — then I decided to put my equation in a graph calc (graphsketch.com), I thought I was going to see the golden ratio as a solution, but no… I lost faith here x)

    Puzzle 4:
    Fibonacci sequence !

    Puzzle 5: no idea… I checked the ratios and the squares, they seems roughly good

  8. What's wrong with the satellite picture at the very end? Maybe it's that clouds follow the curvature of the earth, so we are seeing a projection of a "spherical" spiral. The golden spiral is strictly planar, right?

  9. Hi I am big fan, mathologer . I love maths and discovering new stuff in math. I ask a lot of questions in the class but my whole class lauphs at me and my teacher scolds me for asking useless questions . what should I do?

  10. Hey Mathologer fans, i've got an assumption and now want to know if it is true that the harmonic series ist the slowestly diverging to Infinity series with real partial sums of all diverging to infinity series?

  11. 12:04 first puzzle solution the general proof of gcd

    I won't say long and you guys will all understand

    Euclidean Algorithm

  12. i think, that there is unfortunately a slight room for misunderstanding (solely due to notation) regarding what you state at 4:20 and what you prove at 9:19. maybe simply writing '√3 is rational – false' or '√3 in Q – false' or '√3 =/= p/q' in the future would be more unmistakeable.
    that being said, outstanding math lesson sir.
    i give it a thought, i narrow my eyes, i tighten my face, i take off my hat, i put on a smile, and i nod in admiration.

  13. 11:50 So the connection between the GCD of A and B in the original rectangle, and the smallest side length in the square sequence, is based on the Euclidean algorithm for calcuting a GCD, and the division theorem.
    The divison theorem is simple. For all integers a,b: a = qb + r, where 0<=r<b
    ==> https://en.wikipedia.org/wiki/Euclidean_algorithm#Procedure
    The tl;dr is that when we start with the rectangle of sides A and B, with A>B, then we compute the side length of the next square by using the Euclidean algorithm in disguise. First square has length B. The next square's length can be thought of as the remainder of A/B, because you can make some number of squares with B, say q of them, but once you can't make another square with length B, and the next square must have length r1=A-qB, where q is how many whole B lengths we can fit inside A… so it looks like the division theorem above. If we continue this process again, the next square will have length r2=B-wr1, with quotient w for the division of B by r1..
    The Euclidean algorithm part arises because it operates on a theorem that states that GCD(A,B)=GCD(B,r1)=GCD(r1,r2)=…=.GCD(r_n-1, rn)=rn, where rn is the last nonzero remainder.
    The sequence will terminate with some smallest side length rn, for the smallest square side length, but the Euclidean algorithm let us trace the equalities back up and realize that GCD(A,B)=rn

    For 1920/1080:
    1920 = 1*1080 – 840
    1080 = 1*840 + 240
    840 = 3*240 + 120
    240 = 2*120 + 0
    Then 120 = rn = GCD(1920,1080)

    Thx Euclid

  14. ugh feet.. top 10 oversexualized body parts which are actually repulsing: 1) feet. and there are no other entries yet.

  15. At 21:35, not sure if this is "wrong", but the cyclone is spinning clockwise. It must be in the southern hemisphere.

  16. Rooty numbers! lol … I'm not hearing that like a bogan … I'm hearing it ryhme with fruity numbers.

  17. Hello. I would like you to explain how to "construct" Spot It game. If you don't know it, it is Group of round cards, each card has 8 images, any two cards has one and only one repeated image. It is marvelous, What a great math in there, but I tried a lot to create a math model to do it and I failed.

  18. Please correct me if I'm found to be incorrect but, the answer I've got for the two aspect ratios is that x=25.33445188… or x=-24.33445188… and the similarity is something I would like you to exsplain, if you want.

  19. Whoops! Your "root 2 spiral" at 3:10 needs another large square stuck onto the left side of the given rectangle.

  20. Are these spirals connected with infinite fractions? I remember you said in video about Pi that if a number can be represented as such an infinite fraction with integer denominators and numerators = 1, it is irrational. Numberphile says that Phi is infinite fraction 1 + 1 / (1 + 1 / ( 1 + …) ). It seems that everything matches.

  21. Love your videos, but at some point, given the resemblance of the opening chords, and that every one of your videos therefore instantly makes me think of this classic song, you just have to make a video with links to "Babooshka", or perhaps something more esoteric. I was thinking something like a video on nested radicals à la nested Russian Babyshka dolls, or something.
    Keep on making the videos.
    Schöne Grüße einer Elsässer aus Großbritanien.
    https://www.youtube.com/watch?v=6xckBwPdo1c

  22. I just discovered the continued nested radical sqrt(3+sqrt(3+sqrt(3+… converges on the ratio constant of the sequence a(n) = a(n-1) + 3*a(n-2)

  23. Question about the irrationality proof of sqrt(3): What tells us that the square filling goes on infinitely, i.e., that there is no point at which filling in a square fills the area completely? (as it would be the case if the ratio A/B was deliberately chosen to be rational)

  24. One answer to final question: The spiral should not be not tangential to the lines where corners of squares meet. They are in your picture because you have drawn circular arcs rather than a logarithmic spiral.

  25. I'm a little confused… how can you have A/B = root 3 ? Obviously A and B can't be integers? How do you get/calculate the squares?

  26. Even by the standards of your channel, this was an absolutely exceptional video. It's a masterful example of clear explanations. Awesome.

  27. This is an invitation to see a theory on the nature of time! In this theory we have an emergent uncertain future continuously coming into existence relative to the spontaneous absorption and emission of photon energy. In this process we even have an objective reason for the start of the Fibonacci numbers 0, 1, 1,… with the t = 0 and the positive +1 and negative –1 representing the positive and negative of electromagnetic waves with everything being based on one geometrical process. In this theory the future is not random it is based on a process of spherical symmetry forming and breaking. Spherical symmetry forms the low entropy that we see if we look back in time at the ‘big bang’ and also forms the potential for ever greater symmetry formation that we have in cell life with the Fibonacci spiral being visible almost everywhere in nature! This is because if the quantum wave particle function Ψ or probability function is reformulated as a linear vector then all the information I have found says that each new vector is formed by adding the two previous vectors together this forms the Fibonacci Sequence 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ∞ infinity!

  28. 17:03 I don't understand where these numbers come from, how can these be all side lengths of the resulting squares?
    If x is a fixed (just unknown) number, then there are only two different sizes of squares in the spiral, which surely isn't true. The numbers at 17:32 confuse me even more. If the numbers before really were ALL side lengths of the resulting squares, then what are the new numbers?
    Is x not fixed, but a stand-in for the general 1*x rectangle? At 16:55 he says "in this particular x-rectangle…" Did he mean the exact rectangle we see on the screen or the general 1*x rectangle?
    I understand the point he is getting at afterwards (x has to be a quadratic irrational, because it is the solution to a quadratic equasion), but I miss that one step.

  29. 15:27 So if the original piece of paper was A1 size, what A size would you give the new rectangle with the two squares removed?

  30. 0:42 Actually, there are no squares in the true golden spiral, but rectangles are. When you say "good fit" indeed the spiral goes through blue points, but it's clearly visible that true golden spiral extends beyond square's side. The sides of the squares nor tangents neither normals to the spiral at blue points as you draw. You repeat usual pictures of "good fitting of golden spiral by squares", but those pictures are misleading.

  31. What? 😲😞😟… Are you really saying it is an urban legend that cats know the Fibonacci sequence as evidence by them seeking out positions of the golden spiral? …. How can this be …. 😒🙁😔😢😢😭,,

  32. Pythagoras (Πυθαγόρας) is the first mathematician we know, that is "responsible" for some of these maths.
    Awesome vid, as usual.

  33. You asked at 10:49 what happens when the footprints reach the bottom? Clearly they stop when the smallest box is smaller than one foot.

  34. This orientation of squares in rectangle reminded me, parallel and serial connection of resistance in a circuit. I checked and found out continued fractions are utilised in solving electrical circuits, http://www.eeeguide.com/continued-fractions-method/

  35. You keep skipping steps that are important for understanding both the math and the logic you're trying to explain. You also went from trying to prove something then showing that it can't be proven true without the lead up that the entire point was to disprove it. Too many jumps in math and logic in this video.

  36. I don’t see what this guy is talking about. All the examples he showed us would produce an awkward, forced , unefficient form of a spiral. Unnatural in my opinion. Instead of continuous growth he had blocks that would be wasted simply because it would only serve to connect Two arcing blocks. Simply trying to prove that you are able to stuff numbers into a particular formula will not create the same results that took so many factors to find its place and settle on an elegant and efficient solution. You really need to to see the big picture. One that surpasses mathematics, although it is a language that comes closest to explaining it all it still remains to be just a language.

  37. The fact that the side length of the smallest square is the greatest common divisor for the two numbers is related to the Euclidean algorithm. When we find the squares, we are dividing A by B until we're left over with a remainder, A – xB, where x is the quotient of A/B. Then we repeat the same process, dividing B by the previous remainder and generating a new remainder. After enough iterations of the process, we wind up with a division problem where the remainder is 0. Since every side length for every square prior to that was the quotient of the two previous side lengths, we know that the final side length fits evenly into every other side length. Since the process can't be extended any further, we know that that must be the smallest possible side length that fits evenly into all the other side lengths. Hence, the greatest common divisor.

  38. If you just cut off one square from a A4 paper, you'll get a 'silver' rectangle. If you cut off 2 squares from a silver rectangle, you'll get another silver rectangle.

  39. Great explanation on continued fractions! It's interesting how you use geometric modelling instead of the more common algebraic proofs 🙂

  40. I'm glad that I'm not the only one who rolls their eyes when someone forces a golden spiral onto an image to "prove" it's well-designed or "natural".

  41. Irrational numbers are number which cannot be represented in a/b form. How did you write root(3) as a ratio of two integers?

  42. Euclid! … and the proof of his algorithm proves that the smallest square is the GCD. Done … by reference!

  43. How can you determine if a continued fraction will go on forever without becoming periodic? And/or how can you tell how long the period might be if it's a very long period?

  44. 1:57 I believe that the pentagon and hexagon are fundamental to the understanding of the universe physically as well as spiritually. The pentagon is symbolic of the golden ratio, harmony, and God, while the hexagon with all of its equilateral triangles is symbolic of the 6 pointed jo0 star and godlessness (human=God man=woman truth=lies). Notice the pattern of the flower; at the center/singularity, this is the God creation/start point for the golden spirals, but as you get farther away from the center, the more hexagons form.
    1,1,2,3,5,… the fifth fibonacci term is 5 which is symbolic of the pentagon and (1+sqrt(5))/2 and the two previous terms (which also happen to be primes) 2×3=6 is symbolic of the hexagon.

  45. I want to play around with these ideas. Is there a computer program that will generate these rectangles for me with varying ratios? I would like to study and create a similar proof that PI and e are Irrational. If I can find a tool, it would be better than graph paper, and allow faster study and more insights.

  46. Would this work for 3 dimensions as well? i.e. for cube roots? My first thought when I saw the infinite spiral was if pi could be drawn like that. Then I remembered it can't because pi is transcendental.

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