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.