Welcome to another Mathologer video.
After the last insane Mathologer marathon in which Marty and I proved that e and pi are transcendental numbers I needed a bit of light relief. And maybe
you do, too, right? So how to relax? Well we'll have fun with some of the most
spectacular mathematical vanishing and materializing paradoxes, tricks and
illusions. This video was inspired by physics student Bill Russell from
Bakersfield, California who contacted me about a new type of vanishing paradox. He
chanced upon this paradox while considering magnetic fields, spring
constants and playing with a mysterious object that features on the Mathologer
channel page, that curious blurry, bubbly thing over there. What is it? Well let's
see. (music) pretty cool isn't it? It's called a toroflux or Torofluxus or flowing torus. It's an amazing toy, a long flat ribbon
or spring metal coiled into a very special closed helix. When draped around
an arm or rope its overall shape can be seen to be essentially that of a bagel
or, in maths lingo, a torus. The bending energy of the spring metal makes it cling to
the rope and, oriented like this, it will slowly start rolling down the rope. As it
does so, it will rotate both around the rope and around the circle at its core.
You can see what's happening in this animation taken from a great toroflux
blog post by Los Angelos physicist Daniel Walsh. The toroflux is also very
interesting for mathematical reasons. I'll say more about that at the end of this
video. For now let me just show you the amazing property that Bill stumbled
across. Let's count the number of coils. Alright, 13. Great. Now collapse the
toroflux and let's count a number of coils again. Okay count and what have we got? 14, one more than previously. What's going on here? Where did the additional coil come from?
I'll explain this paradox later on, but first let me tell you about some of my
other favourite appearing and disappearing tricks and the mathematical magic that
powers them. Time for some fun, serious fun :) Okay, let's start with a classic
powered by the Fibonacci sequence: 1, 1, 1+1 = 2, 1 + 2 = 3, 2 + 3 = 5,
etc. you all know this sequence and you may be sick of it :), right? But, anyway, begin by
choosing one of the Fibonacci numbers, say 8. Next make an eight times eight 8 x 8
square now 3 and 5 are the two Fibonacci numbers that come before 8
and so they add up to 8. We can use 3 and 5 to cut up the square
like this here is one of the segments of length 3 in this dissection down
there and here are the other ones. So it's really crawling with these things.
And here all the segments of length 5. Ok now rearrange those four pieces into a
rectangle like this. Neat, hmm. Notice how the three unit segments pair up nicely in
the middle. Okay so all fits together nicely. For what comes next
I better show you this rectangle and the square we started with in one picture.
Ok, time to calculate some areas. How about the square? That's a hard one? Well 8
times 8 that's 64. How about the rectangle? The short side is 5 and how
about the long side? 5 plus 8 that's 13, the next Fibonacci number. That means that
the area is 13 times 5 and that's 65. Amazing, huh :) Just a little cutting and
rearranging and we get a larger area. Where did the extra area come from? It
gets even more unbelievable. Notice that all our cutting and pasting was based on
the Fibonacci number 8 and the three neighboring Fibonacci numbers 3, 5 and 13. It turns out that similar magic can be created by any other Fibonacci number.
For example, let's begin with 13 and it's Fibonacci neighbors 5, 8 and 21.
Then our cutting and rearranging creates this square and rectangle here. Okay
and the areas are, well, 13 squared at the top that's 169. Alright and 21 times 8 at the bottom that's 20 times 8, 160 plus 8 is 168. What,
instead of gaining a unit square this time we've lost one. Are you amazed? (Marty and Michael) I am amazed. What's going on? To recap starting with an 8 by 8 square we gained 1 unit square. Then starting
with 13 by 13 we lost one. What's next? Well maybe you can guess. Starting with
the next Fibonacci number 21 it turns out that again we gain one. And after
that we lose one, and so on. Okay, so what's happening here? Well first
of all it's time to fess up: we cheated! :) (Marty and Michael) What? (Burkard) You are shocked? My god! Of course, without
cheating the areas before and after must be the same. Here's the rectangle drawn
without the thick black edges and you can can probably guess now what's happened.
As you can see, the rearranged pieces don't quite fit together which was
previously hidden by the thick edges. In fact, the pieces leave a thin sliver
along the diagonal uncovered. That sliver has exactly area 1, the extra
bit which makes a total area apparently jump from 64 to 65. Cheeky, hmm. In the next instance, where we lose a square, sliver by sliver, the non-cheating picture looks
like this. A bit harder to see but this time, instead of a gap there's a very
thin overlap along the diagonal. Let me highlight it for you by pulling
things apart a bit. So, pull apart just wiggle a bit. Okay so our Fibonacci
magic boils down to cheating but it is really magical, really ingenious cheating.
But of course this is Mathologer so even in a relaxing holiday video I can't just
show you a mathematical trick and then wave my hands in the air. I've actually
got to show you why it is true. But no real mathematical seatbelts needed for
this one. Just hold your coffee steady. Okay, so here we go. As we've seen already,
the fact that our four puzzle pieces almost fit nicely together comes from,
you guessed it, the way Fibonacci numbers work, the fundamental Fibonacci fact
that two consecutive Fibonacci numbers sum to the next one. Algebraically the
plus or minus one difference in areas is the difference between a Fibonacci
numbers squared and the product of its two neighboring Fibonacci numbers. This
fact has its own name it's called Cassini's identity and is named after its
discoverer, the mathematician, astronomer and engineer Giovanni Domenico Cassini.
You may be familiar with the Cassini space probe. Well it's the same Cassini.
Okay, anyway, we can write Cassini's identity like this or, to be more precise,
we can rewrite the right side to capture the alternating plus and minus like this.
Cassini's identity shows why the Fibonacci puzzle works and while
preparing this video I stumbled across a really, really nice pictorial proof of
Cassini's identity which I just have to share with you. Here we go. Start with a
unit square and place another unit square next to it. Now place a square on
top. What's its side length? Well, obviously, 1+1=2. Now attach
a square on the right. Side length 1+2=3. Square on top Side
length 2+3=5, and so on, alternating between attaching the
squares to the right and on the top. There's 8 ... 13 ... 21 ... 34.
Well, obviously, the sequence of squares is a very pretty geometric counterpart
of the Fibonacci sequence. Let's now look at two consecutive squares. And
let's calculate the combined area of these squares in two different ways. The first
is the obvious one. Just calculate the area of each square and sum. That gives
13 squared plus 21 squared. Okay, for the second way, we
consider the total area as the sum of the areas of two rectangles. Okay, there
they are. First the large rectangle. So that's one with area 34 times 13. And
what about the smaller one? Well let's have a look. It's area is 21 times 8. Almost done. We just have to rearrange a
little. Move the 21 squared to the right side and move to 21 times 8 to the left.
Okay, here we go. To finish, we make things look a little more symmetrical by
pulling out a - 1 on the right side. Okay, what this says is that the two
consecutive Cassini differences only differ in a minus and the same is true
for any two consecutive Cassini differences. For example, repeating our
calculation for the two green squares gives this, and so on. And so we know that
all Cassini differences have the same size, just alternating in sign. And so,
simply verifying that one of the difference is 1, which is completely
obvious, proves Cassini's identity at all levels.
How super nice pretty was that? Now just in case you know some matrix algebra
there's another really cute way of proving Cassini's identity based on the
matrix equation over there. This identity allows you to calculate the Fibonacci
numbers just using powers of the 1 1 1 0 matrix on the right. Puzzle for you: this
matrix identity is just one step away from Cassini's identity. What's that one
step? Well if you know tell us in the comments. And here's one more challenge
for you: calculate the area of the big red rectangle in two different ways to
come up with another interesting Fibonacci identity. Okay that was a great
appearing/disappearing trick but, of course, it involved cheating. On the other
hand, if you're a master of infinity, then you can make things appear and disappear
without cheating. Let me just show you one nice simple
example of this sort of mathematical magic. Over there is an infinite half-plane. So what you see there is supposed to go on forever in this direction. Okay,
let's now make two parallel cuts like this. Take the resulting infinite strip
and move it to the right. Slice off the square there and make the cuts invisible
again. Alright, so we end up with our original half-plane plus an extra square.
And we created this extra square just by making three cuts and rearranging the
resulting pieces. Very simple and pretty amazing when you think about it. Was this
cheating? No, that's simply the way infinity works. As some of you will know
there are many more such infinity tricks much more impressive than our slicing
trick. If you're interested in some highlights definitely check out the
Vsauce video on the Banach-Tarski paradox and the Mathologer video
dedicated to these sorts of infinity paradoxes. Ok, back to a Planet finite and
time to start zeroing in on the toroflux and there's still more tricks to
see along the way. Over there I've drawn 13 line segments. Now watch this.
Whoa, all of a sudden there's only 12 lines left. Where did that 13th line go?
Hmm well there it is again. Now many of you will have guessed the trick but the same
basic trick can be much better hidden. Here it is using people instead of lines.
There are eight baseball players to start with. Now watch this. Let's count the
players again. 9, one more player! Pretty cool isn't it? Here's another
really famous and moderately racist example. The Get-off-the-earth paradox.
What do we see here? Well 13 Chinese swordsmen arranged around the globe. Okay there's 13. Now let's turn the globe and everything on it. Alright. Let's count the swordsman again and we get 12, one's gone missing. Okay let's make him appear again. There 13 swordsmen again. In fact, if we keep rotating in the
clockwise direction, we can get more swordsmen: 14, 15 well those guys there are getting a bit iffy but anyway. The Get-off-the-Earth puzzle was
published in 1896 by Sam Lloyd the Erno Rubik of the 19th century. Sam Lloyd
was incredibly ingenious and prolific, the creator of many puzzles that are
still puzzling millions of people today His Get-off-the-Earth puzzle sold over 10
million copies and Lloyd is also responsible for popularizing the
Fibonacci cheat that I showed you earlier and a super famous 14-15 puzzle
that I already talked about in another video. Many of Lloyd's puzzles were based
on simple principles, just really well disguised. His Get-off-the-Earth puzzle,
for example, is really no different than our lines paradox and here's the simple
explanation. No individual line vanishes or gets created. What happens is that
this red cut here creates 12 line segments above the red line and another
12 below. Then the 12 below are recombined with the 12 above to form 12
new lines, each a little longer than the ones we started with. The increase in
lengths of the individual lines is barely noticeable but of course these
little increments together sum to exactly the lengths of each of the
original lines. More generally, the vast majority of geometric vanishing and
appearing paradoxes are based on the cunningly disguised redistribution of
length, area or volume. That's true for all the paradoxes that I discussed so far
and it's also true for the toroflux paradox. Remember, expanded like this we count 13 coils. Collapsed we count 14. It's
actually not surprising that there's a difference. Remember that if you just
give the collapsed toroflux a little bit of a nudge, it expands all by itself.
In particula,r all the coils expand. However, the steel wire definitely does
not expand. So the total lengths of all the coils must remain the same and so if
our coils have grown larger, we should not be surprised that we've also ended
up with fewer coils. However, unlike all the other real-world paradoxes we've
discussed, the toroflux transition between the collapsed and the expanded
state is continuous. Doesn't this strike you as strange? How can you possibly move continuously between 13 and 14 coils. Surely, either jump or you
don't, right? There's definitely something extra paradoxical about our toroflux
paradox. I already mentioned that when you drape the toroflux around
something like a rope or your hand, then overall the toroflux will look like a
torus (that's of course where it gets its name from, right).
In fact, the toroflux is what mathematicians refer to as a torus knot, a
closed loop that lies on the surface of a torus. There are infinitely many
different torus knots. Here are just three examples. This one is not very knotty,
it's basically just the slinky with its ends joined together, okay. Alright then
the next one is definitely knotty but not so windy. It's called a trefoil knot (for pretty obvious reasons). Finally, here's an example that's very
windy, very knotty and very rainbowy. Okay, any torus knot can be pinned down
by two numbers. The first number is the total number of times that you wind
around the ring of the torus as you travel around the loop. For example, for a
slinky this number is 12. How can we see this. Well moving from here to here we've
looped once around the torus and so the number of times the slinky winds around
the torus just equals the number of points on the outer equator and that's
12 for the slinky. In the case of a trefoil knot this number is 3 and for
our windy, knotty, rainbowy example the number turns out to be 15. The
second number associated with a torus knot is the number of times you loop around
the point in the very middle as you travel once along the knot. In the case
of the trefoil knot this number of revolutions is 2. Let's convince
ourselves of this. Okay once around, here we go, and a second time and we're back to where we started from. Great. What about a slinky? Well, obviously,
here the number is just 1. For our complicated knot this number is 7.
Okay, now here's a nice idea. A couple of videos ago I talked about the possible
orbits of moons of planets revolving around a sun. I had the planet and the
moon always moving in the same plane (as they do). On the other hand, if the
circular orbit of the moon is at right angles to the orbit of the planet, the
overall orbit of the moon around the sun will be a torus knot. If we do this with
the moon over there, then the moon's orbit is our slinky knot. Nice, huh? So in
this sun-planet-moon model of torus knots our two numbers just count the
number of times the moon orbits the planet and the number of times the planet orbits around the sun as the moon
completes one full journey. Now it's not too difficult to see that the two
numbers of a torus knot are always relatively prime, that is, they don't have
a common factor different from 1. In our examples this amounts to 2 and
3 for the trefoil being relatively prime, the same for 1 and
12 for the slinky, and also for 7 and 15 for the complicated knot.
There's another challenge for you: prove this relatively simple relatively
prime fact in the comments. Also, given any ordered pair of relatively prime
numbers, there's a torus knot with these numbers as the knot's numbers. Of course,
unless the two numbers are both 1, corresponding to the World's most boring
torus "knot", being relatively prime means in particular that the two numbers of a
torus knot must be different. Okay, what then are the numbers of the toroflux?
Well perhaps you've already guessed. The first number, the number of times the toroflux coils around the torus is, well, let's count it again, there we go, 13. Alright then collapsing the toroflux doesn't change the number of times the
wire winds around the middle, and so the second number is what? Well, let's remind ourselves: 14. What this means is that when we count the coils of
the toroflux before and after the collapse we are really counting two very
different quantities. This also becomes apparent if we perform the collapse
really, really slowly. Okay, let's do it. The red markers on the toroflux
indicate the 13 loops, giving its first torus knot number. I've labelled the
markers in the order that we would come across them if we travel along the
toroflux. Currently the labels are touching the sheet of plexiglas that I've placed on
top. Okay, in fact, as I collapse the toroflux by pressing down on the plexiglas
these markers will always touch the plexiglas and will always form a circle,
even when the toroflux is fully collapsed, like now. Okay, let me just show
you where the markers are now. Okay zoom in. There they are. To count the coils in this
collapsed state, we can count along this radius. Right we would basically just do
this and then count along there. And what we also see is that all the coils that we
count this way are as long as the circumference of the circle in front of
us- this long, right. However, the distance from marker 1 to marker 2 along the
wire is longer than that. Let's just see what it does. To be precise, it's longer by 1/13th of the circumference. And since all 13 distances between consecutive markers
feature this 1/13th excess, together these excesses add up to the extra 14s coil
that we count in the collapsed state. And that's how the length redistribution
from expanded to collapsed state works for the toro flux. And this also explains
why there actually is no discontinuous jump from 13 to 14 coils
during our continuous transition. We're really just counting different things.
Paradox solved :) I made the very nice torus knot pictures
with the amazing software package KnotPlot. If you're not familiar with it
definitely check it out. And here's what the toroflux looks like in KnotPlot. I
own about 10 toroflux toys, all made by different companies. The torus knot
numbers differ slightly from toy to toy but for each toy it's two numbers differ
by 1, as in the example I showed you. So a natural question to ask is whether it
is possible to build a functioning toroflux with a different difference, for
example, a toroflux based on this torus knot here whose numbers are 13 and 15. What do you think? I could go on for hours talking about other amazing mathematical
appearing/disappearing acts involving infinity, higher dimensions, and so on. But
let's call it a day with one last particularly nice trick, fully animated
and it's got funky music again. And that's it for today. Time for me to
vanish.
Mister Mathologer I don't feel so good. . .
Check out the t-shirt !!!!
Tokamaks are magnetic toruses, with one defining feature being the "safety factor", q, profile which is essentially defined as the ratio of toroidal (n) to poloidal (m) winding numbers on each nested torus surface.
q can take any value, including integers, and rational or irrational numbers. I don't think I've ever heard of m and n being required to be relatively prime.
Is q related to torus knots? They seem incredibly similar but also distinct, so I think I must be missing something.
Watched it earlier it’s great
Amazing! Thanks for sharing!
I actually just walked past some of these at the store and thought, “interesting” but didn’t buy one. Going back tomorrow to get a couple.
I'm puzzled by that last thing he shows with the squares...any hints?
i got one of those !!!