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# Toroflux paradox: making things (dis)appear with math

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.