Toroflux paradox: making things (dis)appear with math
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- Опубліковано 21 вер 2024
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Today is all about geometric appearing and vanishing paradoxes and that math that powers them. This video was inspired by a new paradox of this type that Bill Russel from Bakersfield, California discovered while playing with a toroflux. Other highlights to look forward to: a nice new visual proof of Cassini's Fibonacci identity which forms a core of a very nice Fibonacci based paradox, the classic Get-off-the-the-Earth puzzle, and much more.
Here is the link to Daniel Walsh's blog post on the toroflux: danielwalsh.tum...
As usual thank you very much to Marty for all his nitpicking, Michael for his help with filming the video and Danil for looking after the Russian subtitles.
Today's t-shirt I got from here: www.teepublic....
The piece of music at the end is: English_Country_Garden from the free UA-cam music library.
There is a bit of a visual typo at 6:19: The pieces are fine but the grid in the background is not. Here is what it's supposed to look like www.qedcat.com/...
According to the the wiki page the inventor of the toroflux Jochen Valett who is German :) . en.wikipedia.o...
Enjoy :)
I can't pay any attention to the video because the t-shirt is TOO GOOD.
I agree, one of the best t-shirt I saw on youtube
Why I read it as T-series
I had to laugh so hard when I saw it xD
Alejandro Arévale: which two t-shirts did you hear last?
I'm just trying to figure out if he's a boxer or a praying mantis
I studied Math 50 years back, yet your explanations are still clear as ever they could be. Though I sometimes get lost in the theory, I still enjoy these presentations. Even now, I am getting to understand things that were as mystery to me all those years ago. Please continue the series.
Great, glad these videos work so well for you :)
"Why is he using such fat lines?" "...let's rearrange..." "Never mind, chocolate bar."
This video is as good as the last two combined ;-)
Nice
I wasn't fooled. It reminded me of the trick with the chocolate bar.
Well, it's exactly the same thing :) As I said this one has been around for over a 100 years in a number of different guises. Still, the way it comes together is really, really nice and people usually don't talk about the nice maths that makes it work. In fact, there is even some more nice mathematics hiding in this apart from that bits I talked about in the video. If you are interested maybe ponder the slopes of that extra square along the diagonal :)
@@Mathologer when I came across it, I hit pause and didn't hit play until I realized that the sections had different slopes.
First thing I thought of seeing this was the chocolate bar trick meme.
when you cut it and get one more extra bit
Thank you for the video. I can finally be the one to get the extra chocolate piece, hooray! :-)
Before I got to the explanation, I cut a 8X8 gold bar up and reassembled. Darn, I was going to be king.
Just wait until you get your infinite sheet of gold foil then! ^_^
Then double it by means of Banach-Tarski paradox!
Good call! I had not thought of that strategy.
I already ordered is from a guy in Nigeria who needed help getting it out of the country. Will be here any day now.
If I could cut a gold bar, rearrange it, and end up with 1 inch square more of gold, I would have a quick profit. I could simply repeat until I was the wealthiest man on earth. Then, becoming king would be the easy part.
Mathologer and 3Blue1Brown are the best math channels. I would love to see you guys doing a video about the Apollonian Gasket and their relationship to Pythagorean Triples. Both are related as well as they are to Quantum Mechanics. The Hofstadter Butterflie and population transfer respectively. Plus, whats cooler than fractals? circle fractals!!
Arturo Lozano 3b1b has a video about you can derive all possible Pythagorean triples by calculating the squares of complex numbers; IIRC, title is “All Possible Pythagorean Triples, Visualized”.
I also really loved infinite series ... a pity PBS canned it.
@Poyraz Pekcan how do i like your comment?
Infinite series wasn't that good, also Numberphile is popular and some speakers are very good, some are much worse and boring. 3b1b and Mathologer is a 1 person show, not a collection of smart people, therefore better by a scale of 10 or more :) Tadashi is awesome though :D
Blackpenredpen
For the reason that the two torus numbers have to be relatively prime, using the solar system analogy, every time the "planet" makes a full revolution around the "Sun", the knot will connect back on itself if and only if the "Moon" has made a whole number of revolutions. If the knot connects back on itself, the knot ends right there, so the first time at which both numbers are whole numbers is where the knot ends. Since both the "planet" and "moon" are orbiting at constant rates, the proportion between the number of orbits of each is constant. Since numbers that aren't relatively prime by definition always have a smaller set of whole numbers with the same ratio between them, the knot would have already ended before reaching those two numbers.
thanks!
This is basically the same conclusion I came to. Worded a bit differently, the two torus numbers must be relatively prime, otherwise you'd have more than one "loop" by the time you're done drawing the shape. For example, if you tried to make a 10/2, what you've actually made is two 5/1s.
It's still a contradiction.
@@MrLlama-gl2hk Same, for me. If the 2 numbers: a & b weren’t relatively prime, there would be numbers in the ratio: a/b, smaller, than our smallest whole numbers, in this ratio: a & b, which is impossible (same deal, as with the proof for: a^4 + b^4 = c²; which we established in the video of Fermat’s Last Theorem).
x = x + 1
programmer: meh
math teacher: "noise intensifies"
x+=1
@@flamingfearow7403 that's the nickname, x += 1;
imagine your parents call by yor full name x=x+1
Drogon Blue that would be so embarrassing...
@@flamingfearow7403 :)
@@aliph-null ... and all you want is to be called: x++
Find someone who looks at you like this guy looks at his twisted slinky toy @1:05
Justin Abramson pretty cool, isn‘t it?
Hmmmmmm, I thinks I just realized what I love about math ; the synergy between enlightened intuition & strict rationality.
This is a nice illustration of something that falls into the interplay between geometry and topology. The toroflux is a torus knot, as noted, but depending on the positioning (technically 'embedding') we give it geometrically, what we naturally think of as a 'coil' changes: in one case, it's one of the two numbers identifying the torus knot, but in the other embedding, the other number is the one that naturally corresponds to the geometric coils. The point is that our idea of a coil is a geometric one.
Ah Mathologer, nothing makes me happier than a new video from you guys.
Obi-Wan Reviews
May the FORCE be with your reviews!
A while back I was playing around with a rubber band and found something that's almost the reverse of the toroflux paradox. At the time I was more interested in the twisting of the band: for interesting topological reasons I won't go into here, you can take an untwisted closed rubber band with a flat cross-section, and loop it three times around your finger (or any ODD number of times, if it's long enough) without any extra twisting of the band. It ends up with sort of a braided appearance, but the band lies flat everywhere without twisting relative to the surface of your finger.
But the relevant thing here is how you can get there. If you take the untwisted band, lay it out so that it looks like a long, thin oval, and fold the two ends of the oval into the center on top of each other, the two sides of the oval will naturally pop out and then fold inward on each other beneath. That gets you a band that loops around multiple times with no extra twisting. It seems like the process ought to get you four loops--two for the ends of the oval and two for the sides; but if you then count them, you'll only find three.
I suppose it's to be expected that it's the reverse of the toroflux paradox, because here the bending forces make the band prefer to be in the state with three loops, once you start bending it at least.
You would have to be the happiest presenter on UA-cam very enjoyable. Thank you
Your permagrin is contagious.😁
Thanks Professor Polster! Great video, as always. 👍
Permagrin, I like that! =)
i really really dont know what im doing here. it's friday 4:07 am, i study communication in the university and i don't even speak in english
You're the cuddliest (smiling & joking) mathematician i've ever seen. Love your work, sir!
This channel is awesome. I discovered it 2 days ago and I can't stop watching videos!
Also, the explanations are super simple and interesting, which makes it easier for us to understand :)
Here’s one vanishing/appearing paradox: Take a cubical (100*100*100, since humans can differentiate about 100 tones of each primary colour (red, green, and blue)) colour space. It has 100³ = 1 million little cubies, each corresponding to a different hue. Now, take a cylindrical colour space (same 3 primary colours, 100 tones, each). As you travel along the perimeter, from your starting point (say, red: R:99, G:0, B:0), you’ll start adding the next colour (in this case, green); thus, incrementing the G-value (saturation of green), until it reaches the maximum value, 99 (now you have R:99, G:99, B:0, corresponding to yellow); then, you’ll start decrementing the R-value (saturation of red), until it reaches 0; so, you’re now at green (R:0, G:99, B:0). Repeat the process: Increment B till 99, decrement G till 0, Increment R till 99, decrement B till 0. You’re now back at red. That took 6 * 100 = 600 steps; thus, we conclude that the perimeter is 600 units around. Since the colour space is cylindrical, besides having a perimeter, corresponding to hue, it also has radius of the base, corresponding to saturation, and height, corresponding to brightness/lightness. Since humans can distinguish ~100 different tones/shades of each colour, the radius of the base and the height are both 100 units. But, this then means that the cylinder, all in all, has 600*100*100 = 6 000 000 pieces, corresponding to different colours/tones/shades/whatever; 6 times as many, as in our cubical colour space. Note that we didn’t add or subtract anything; but, somehow, we ended up with 6 times more options to choose from. 🤔
*EDIT:* I have cracked this paradox; but I’m gonna let you, the reader of this comment, work on it, yourself; rather, than just spoonfeed answers to you.
In other versions I've seen of the Fibonacci triangle (aka Curry) paradox, the very slightly kinky diagonal isn't hidden by a thick black line, but presented openly as the "hypotenuse" of just one right triangle. You still don't get to notice anything until you take a straight edge to it. I think the Fibonacci numbers are relevant only in that they allow the difference, the rhomb, to be expressed and seen as a whole number of unit squares. It's a great conjuring trick, in which the eye is deceived by spatial smallness instead of temporal brevity (quickness of the hand). And on the fact that you're not expecting such trickery anyway..
10:51
One beautiful way to do this is to diagonalize the matrix A = [1,1;1,0] to get the three matrixes PDP^-1, [-2/(1-sqrt(5)), -2/(1+sqrt(5)); 1, 1], [(1+sqrt(5))/2, 0; 0, (1-sqrt(5))/2], and [1/sqrt(5), 2/(sqrt(5)+5); -1/sqrt(5), -2/(sqrt(5)-5)] respectively (yes this part is messy and has lots of square roots but they drop out nicely in the next step). Since we care about the determinant of our matrix A^n we can simply find the det(PD^nP^-1) which is equal to det(P )*det(D^n )*det(P^-1 ). D^n is special because it only has diagonal entries which means that when raised to a power, n, the new matrix can be found by raising the entries to the power n. So the det(D ) is simply (-1)^2 and since P and P^-1 are inverses their determinants are also inverses therefore they equal 1 when multiplied together so the det(A^n) = (-1)^n and since A^n = fn+1*fn-1-fn^2 -A^n = fn^2-fn+1*fn-1 = (-1)^(n+1) Alright maybe not the cleanest but definitely fun!
Fun video! I think the reason why the two toroflux numbers must be relatively prime is because if they were not, it would consist of more than one connected part. This is the same as star polygons written as P/Q where P is the number of points (vertices) around the circumference, and Q is the number of points you skip when drawing edges from each P sub N to P sub N+Q. So the the 5-pointed Christmas star is 5/2 while the Jewish star of David is 6/2 consisting of two disconnected equilateral triangles.
That's exactly it :)
@@Mathologer On the topic of why torofluxes are always made using numbers differing by one, I imagine this means each loop is quite closely connected to its neighbours. If odd numbers differing by two were used, neighbouring loops would be very loosely connected via nearly half the overall wire. I imagine this would have an effect on how the toy would feel when used - which might seem similar to Melinda's suggestion of two disconnected parts.
I genuinely don't understand the concept of disliking these videos. I've never seen a UA-camr go into so much detail on anything.
I really love the infinite sums, products and fraction animation proofs and videos! I’d like to see more of those
Pretty sure there will be more. Did you already watch all the ones that are there ? :)
Mathologer I did indeed ( ^ω^ )
Always thought provoking 🙏👍🖖🤘
It's all about that initial 1 which we stop at. The Fibonacci geometry can continue inward but we start at integer one for convenience. This causes lopsidedness of one which gets watered down over time.
Fibonacci is the integer mapping of the golden curve.
A simple trick to make things truly disappear by using maths all of you guys can try:
Enter a random high school classroom and tell the students you’re going to give supplementary math to whoever wants to stay.
Won't always work. Depends on how many nerds are around.
I would stay
@@maxwellsequation4887 i would stay too
@2:42 You know you're talking to a German when the word fun sounds so incredibly unnatural.
Love the videos! Keep it up!
Thank you for the great videos! It would be amazing if you would start a series for elementary and middle school math. I think you would be a great teacher for kids (and adults too!).
The reason the slopes are similar is that the ratio of adjacent terms in the Fibonacci sequence is approximately phi (the golden ratio). The reason the areas are off by one is that the actual closed form expression for the nth term in the sequence is (phi^n - phihat^n)/✓5. This is the scalar version of the 1 1 0 1 matrix that was shown(using eigenvalues)
10:46 To get the Cassini's identity you just need to get the determinant of the matrices in the equation.
10:01 You can calculate the area multiplying 8*5 or also summing all the squares 1^2+1^2+2^2+3^2+5^2 so you can get the identity f(n)*f(n+1)=sum(i=0,n) f(i)^2
19:44 I think is a bit easy to understand why they need to be relative primes because otherwise while tracing the toroflux you will redraw the curve the number of times as the largest common factor.
As regards the rectangle challenge (the second, and easiest), I think it's also interesting that the blue area left over if you put the red rectangle inside the blue square is 8*3. How would you generalize this identity?
Just found this channel. LOVE IT! Love the animations, presentation, and t-shirt.
Snap! I haven't ascended past 3rd level of enlightenment yet and there is a new video already!
If you pay real close attention to the details, the extra square at the end of the video comes from the space between the pink square and the one inside it, you can see that before cutting and rearranging there's more space in between on the top side
SOLUTION OF CO-PRIME PROBLEM IN TORUS KNOT :
It's based on geometry and little modular algebra (which I'll not show here to prevent the mess equations).
Let R be the number of revolutions that a point on knot makes around the center of torus while making one complete cycle of knot & let T be the number of turns around the ring of the torus while making one complete cycle of knot. Now gcd(R,T) = k (say).
Let's think of a torus surface with T no.s of 2-D circular rings (around the 3-D ring of torus) which we will cut and join to make a knot.
Now number each ring as 1,2,...T.
Start with ring 1 and join it to (R+1)th ring and join this ring to the next Rth ring and repeat until you reach ring 1 again. You've made a knot with the number of revolutions (R') = R/k and the no. of turns (T') = T/k. Repeat the process for remaining circular rings if any.
If k=1 ( i.e. R &T are co-prime) then there will be one knot on torus. But if k>1, then there will be k number of knots on torus each with associated numbers R' and T' ( which are also co-prime).
This reminds my of musical intervals. Great video as always!
If you start at a cetain point, if the first number is whole, it means you're on the same inner-outer side of the torus as that point, and if the second number is whole, it means you are on the same angle around the circle. These 2 coordinates completely define any point on the torus, so combining the two, it means that the first time both numbers are whole, you've got to the very same point you started with, and if these numbers had a common factor it means it would have happened sooner..
And the determinant argument, of course..
Knots allow for points to overlap. The two number descriptor isn't necessarily unique
@@sergiosanchez5439 Can you give an example of such a knot? I went over all of the examples in the video and found no intersection.. also, suppose there are some finite number of intersections, just pick a point that isn't one of them, and i think the argument is valid..
By definition you must be able to follow the line in a knot from end to end without interruption since it's an embedding of a circle - there are no overlaps in a circle. If you take a circle and twist it into a figure of 8 it may seem as if it has an overlapping point in the middle, but in 3D space the two points in the middle are distinct.
Realy amazing how you explain you are crystal clear in subject
For those of you who won't be able to continue living without owning the t-shirt, here is where I got it from :) www.teepublic.com/t-shirt/2138490-funny-this-fibonacci-joke-is-as-bad-as-the-last-tw
This video is SO AWESOME! Full of SO many cool things!
10:50 apply the determinant on both sides and multiply by -1
Spot on. Supercool isn't it ? :)
@@Mathologer now i wanna see the graphical interpretation :P
@@cicciobombo7496 How about that:
Determinant of a 2x2 matrix measures the misalignement between two vectors, here (fn+1,fn) (fn,fn-1). You can draw these vector on the figure explained at 8:30.
Better yet, remember that this misalignement is the key to the magical trick?
Well, the most common graphical interpretatino of det(u,v) is that it represents the (oriented) area of the parallelogram defined by the vectors u and v*. And the missing slice from 5:56 is exactly one such parallelogram! So its area is det((13;5),(8;3)). It does not quite yield Cassini's equation, it yields f(n+2)f(n) - f(n+1)f(n-1) = +/- 1, but I suspect a different trick should exist to illustrate graphically the demonstration of Cassini's eq through det.
*no visual proof of that, but an elegant abstract proof that skipps all the computings (save for multiplying one by one)
For a second I thought this was my comment that I’d forgotten about because of the avatar and thought “wow I sure am smart”
@14:37 oh that's brilliant... no data is lost... only rearranged.
it' 11:30 pm, I go to work tomorrow at 6 am and still I am watching this drinking beer. My life is confusion.
We've all been there. Being there right now.
you sir, got yourself a new subscriber! beautiful explanation!
You should mention in this context the novel "Around the World in Eighty Days" by Jules Verne. (Edit: Spoiler Warning! Don't read the answers to this comment if you don't know the book yet. It's a great story with a mathematical surprise in it. You must have read it).
Nice :)
Me, not having read the book:
"Wait, how would you add time together as if it were length- oh. Oooooh!"
It's just the same as in the example with vertical the lines. The days are in average 18 minutes shorter during the travel in east direction. After 80 days this summes up to a whole day of 24 hours. Then he gained one day. From now on he also is one day older than his same age twin brother who did not travel. He just experienced one additional sun raise and sun set in his life during the same time.
Wouldn't the International Date Line come into play here? As soon as he crossed it, the extra day is subtracted. (but I don't know if there was such a thing defined when Verne wrote his story)
@ Of course, but he realised this when he was back from his journey. First he was very sad that he lost his bet by one day to make it around the world in 80 days. But then the story came to a happy end.
I serendipitously discovered your channel. Subscribed.
I know I am silly for saying this, but I figured out this "paradox" on the toroflux about 2 years ago, so seeing this video was pretty exciting! Someone gave me a toroflux a few years back, and then in my geometric topology class a couple years ago I was thinking about Lens spaces, and I had the toroflux lying around and saw it and was like "oh this is a torus knot, which one?" So I counted the two numbers and figured that out, but mine is a 13, 12 torus knot. And as a digression I thought "how does the extra longitude go into the meridians?" and so I held two adjacent meridians in the same spot and collapsed it and realized the same thing it as in this video. Though I didn't really think of it as a paradox since it just made sense after thinking about some basic knots all quarter. Great video as always!
Very cool. HI I'm the guy the that's from Bakersfield California. Bill Russell. You're observation of I'm assuming gravitational lensing is what you were referring to was a very interesting note. Thanks for the intelligent comment.
@@billrussell3955 Oh great idea, but sorry no, I was just talking about lens spaces en.m.wikipedia.org/wiki/Lens_space Which probably do have a connection to gravity though, since they are 3 manifolds and there is a correspondence with a certain class of lens spaces and (p,q) torus knots. But any connection to gravitational lensing was unintentional.
Thanks for the link. Just yesterday I was reading up on the Lagrangian. Lens space looks very similar to me in the mathematical explanation.
An interesting fact related to Quantum Mechanics and infinities: Only the Rational (countable) numbers are discrete, separate entities. The higher levels of infinity, such as the number of points on a line are Real (smoothly shading one to the next) numbers with no gaps in-between. The Cantor Diagonal Proof shows that there are an infinity of intermediate numbers between each Rational Number, so they cannot be discrete with sharp boundaries. Thus, for Quantum Mechanics to have discrete results of any test where all observers at any place and time in our universe agree that this specific result occurred, these observers all have to be in the same universe with no "fuzzy" edges. so that the number of possible results in any Quantum Mechanical test has to be a Rational Number result, which requires that Quantum Mechanics be made up of discrete STEPS, each one being a whole number or a fraction made up of the ratio of whole numbers so that all inputs are based on Rational Number inputs and give Rational Number outputs, as to the possible number of outputs, not the numerical value of any single output, of course, since those are based on universal constants such ass the speed of light. This limit on the number of possible outputs per test in Quantum Mechanics may indicate something fundamental on how the universe "calculates" the results of any test of a Physics problem. Is the "computer" that runs the universe running on "fixed point" math???
I watched this hungover and my head hurts more.
It’s OK. It’s just growth-pains.
Long ago I remember having a variation of that last puzzle made of wood as well. It had to go a certain way to open a box if I recall; or maybe to fully close it? Anyways, I think Mr. Puzzle's channel has several more variations of these in both 2D and 3D types which are fun to see. Really cool to hear why they work at long last. Never seen the toroid of steel rings like that before that's pretty cool. I liked the "inductor coil" you drew with 12 turns. My inner EE was joyous there.
Cool :)
Superb video. Its really amazing.
9:14 Obviously; a 3rd way to calculate the area is:
(21*34) - (8*13) = 714 - 104 = 610
When I saw the toroflux I was reminded of a magnetic flux around a moving charge. I wonder if this transformation (spatial transformation of the toroflux) is analogous to the lorentz transformation...
Your laugh really makes me smile 😃. Super video!
Jokes on you, I don't think I've ever heard a Fibonacci joke before!...
...wait, no... there's that one old pigeon meme, so that's one...
oh wait, and there's that one where they put the Fibonacci spiral on top of any picture they can find... so that's one.
And then there's the corny pun about counting to 2 in Fibonacci being as easy as 1, 1, 2... so there's 2. I guess ya got me.
Jokes on him, yup. I see the shirt.
Praepes i must've been living under a rock, but... what pigeon meme? :">
I see what you did there! Clever.
Well, once you've heard two, you've heard them all...
I'm waiting for them to make a series
This is related to turning the torus inside out. The two numbers that define the torus knot then switch places.
So I took a 2 inch diameter ring, like the kind you use to join together a bunch of index cards, and clipped it to my toroflux. In its 3D upright state, it was clipped through the inside of the torus. If you try to push it flat, it doesn’t go all the way, cause the ring has a smaller diameter than the toroflux, and you get this very pretty almost flat thing, that’s like what the toroflux would look like viewed from above if the hole in the middle was bigger, like in the example knots you showed.
Then I tried to turn it inside out. I gathered together the loops in one hand, trying to bring them all next to each other. And then something happened and the ring changed orientation and was on the outside of the torus. **Except** it had all but one of the loops through the ring!
Cool :)
If you travel arround the world one day (dis)appers - it depends, if you're travelling west to east or east to west ;)
Been reading Verne, have we, young man?
Not necessarily. You could cross the international date line as well.
If you travel _around_ the world you will cross the IDL... and that's precisely the point.
Spoiler alert: don't read this if you haven't already read Round the World in Eighty Days
Of you travel round in an instant, you gain or lose nothing. The IDL simply reverses all the 1 hour time shifts as you move from one time zone to another.
In the Jules Verne example, the travellers thought they had spent 1 day longer than they really had, because they had moved their watches on by an hour 24 times. Except for Passepartout's watch, which as is explained in the book, stayed on GMT and therefore would have counted the correct number of days (if they had calendar watched in those days).
In the book a lot of money turned on that mistake
A torus knot is an immersion p (or an embedding if you exclude singular knots) from the circle S1 to the torus T2 = S1xS1. The knot invariants described are the elements of the homotopy group of the torus, which is a free product of two integers Z * Z. By the functoriality of the fundamental group map pi1, the embedding p descends to a homomorphism pi1(p) from Z to Z * Z. Fix a generator a from Z, we see that the maps pi1(p): a = n+m -> (n,m) and pi1(p’): a = b(p+q)-> (bp,bq) are necessarily equivalent by the universal property of the free product (let g, h: Z -> Z^2 be homomorphisms that map a to its summands in the two different ways above, then they MUST descend to the same map on the free product Z * Z after quotienting due to the universal property), hence the torus knot invariants are coprime. Indeed, homotopy invariance means torus knots p, p’ with coprime torus knot invariants are honotopous.
(Area of the rearranged pieces increased): "What!?"
(Some time later both of them surrenders to the fact that the area do increase and decrease.)
(Reveals that they cheated): *Both of them, again* "What!?"
"You shocked?"
OMG, this is the ABSOLUTELY BEST T-shirt i've ever seen! I just burst into laughter.... ffs, couldnt stop for 2 mins, people in the cafeteria were staring at me as if i was weird...
I'm buying the shirt, i'm buying the hoodie, i'm buying the mug... Hell, i'd buy the underwear if they had! I'm completely sold!!!
Nice video! Do you plan make a video on hyperbolic functions anytime? I never see them get very much attention, if any, and when learning about them, both at university and A levels, we didn't really go over them very much, why they exist, or even what it means for a function to be hyperbolic.
Again, I'd like to say that's a beautiful marsterpiece of a video you made there.
One on hyperbolic functions would definitely be nice. It's sort of on my list. Already got a very nice hook to get it going :)
Oh brilliant, thanks for the reply!
While I would love to see a video on hyperbolic functions I think you grossly exaggerated how little attention they get.
I'll see myself out.
blackpenredpen has some very good videos on the meaning and relationships of hyperbolic and regular trig functions. ua-cam.com/users/blackpenredpenvideos
Oh yeah, I occasionally watch the pens, though clearly I gotta look at it more from the sound of things, he usually goes through weird questions.
If I worked at the toroflux factory, I'd pitch a plan to management to build the tori in the expanded configuration to save the material of one whole loop, and pocket a nice bonus for making the factory ~7% more efficient.
I'm pretty sure the 12-lines paradox is the reason paper money has the serial number TWICE. Try this with money....and the serial numbers on the two halves of your money won't match.
Mephistahpheles wouldn’t it be easier just noticing that the money was cut?
+Millton Manakeeper Yes, but still an interesting comment I think :)
Mathologer yes creative indeed
Sure, but I've USED damaged money. Newer bills are much harder to rip, but older bills could get damaged and taped back together. Not that this has happened often (to me, anyway).
There's lots of ways to tell a bill is counterfeit. The more security the better. Not that I would normally check the serial number, but if a bill was cut, THEN I would check it.
No doubt, laws vary by country regarding damaged money.
This video reminds me of a chocolate bar puzzle that was floating around several years ago. In the puzzle you ended up with an extra square of chocolate when you cut it a specific way. Just like this puzzle, the piece was created by sneakily taking a small portion from the inner squares based off the cut. They didn't go in depth about the math so this explains it a lot better.
Solution to the last puzzle:
*ahem* Spoiler alert, I guess.
The lines of the cross are a bit longer than the edges of the original square, so the entire area of the new square is slightly larger.
As the area of the four pieces stays the same, there has to appear a free area, just as big as that difference.
Well, but the difference is there, albeit extremely small at the angle shown in the video.
If you look closely, you can see it.
Neither of the frames in the examples shown is an exact match, so both configurations fit.
There is no "shrinking" between re-arrangements. There is enough thickness in the line (saw blade) to cover the area of the square in the middle and re-distribute it around the edges of the pieces.
That "line" (the missing wood removed by the saw blade) is called the "kerf", in case you're interested. This is why, when cutting wood, you always cut on one side (the "waste" side) of the measuring line: so the piece you've cut comes out the length that you measured.
Thank you, I work wood too (not very skilfully, but enough to know what a kerf is and how to cut a piece to measure). However the puzzle as originally proposed in the video has no saw blades, only a set of geometric figures including 2 very thick lines, just as there were very thick lines hiding the gaps/overlaps in the Fibonacci/Cassini rectangle/square rearrangements.
@@dlevi67 when the original square hase side length 1, then each cut has length 1/cos(phi) where phi is the tiliting angle of the cross; hence the new square has also side length 1/cos(phi) - which is typically larger than 1 (except for phi=0, then you cut the square into rectangles without any extra hole after reorientation, as the rectangeles are symmetric to 180° turns) and the four pieces can't fill it completely and a hole appears.
Oh thank to your video, I found an extension of Cassini's identity: F(n)^2 = F(n + k)*F(n - k) + (-1)^(n + k)*F(k)^2 , for every n and k positive integers, where F is the Fibonacci function :)
You made .999... = 1.0. After that, everything else is child's play. Thoroughly enjoyable.
Actually, that infinity trick I showed is a visual version of the argument that shows that 0.999... =1. To turn the recurring decimal 0.999... into a fraction, set M=0.999... . Multiply by 10 (that's like the pulling action in the video) which gives 10M=9.999... (with the "extra" 9 sticking out on one side like the square). Now subtract the first equation from the second (like cutting off the square) . 10M-M = 9.999... - 0.999... which is the same as 9M=9. Therefore M=9/9=1=0.999... . :)
That's a rather nice way of reconsidering the visualization ... to confirm, does this also show that there are, essentially, an infinite number of .xxx999... situations (0.24999...= 0.25; 1.73999... = 1.74, etc.)? It would appear so under any of the presentations, but especially the one above.
Yes, all terminating decimals have a second representation ending in an infinite sequence of nines.
Yeah but if I purchased something that came to $0.99 and I give them a dollar; I'm expecting my penny back unless if I'm feeling generous!
@@skilz8098
But if the price was $0.9999 then why would you be entitled to a penny of change? 99¢ ≠ 0.99999 repeating.
You can make a toroflux with numbers 1 and 100. Just wrap a Slinky into a torus, mesh the ends together, and it will roll over your arm etc. You can even squish it flat (for as flat as the material allows with all the overlaps)
19:45 Didn't think I'd see that little editing correction, eh? lol
I didn't understand a bit of this video, but kept watching because of your accent!!! I imagined it to be how Einstien would of sounded ( if he was bald).
I had to replay the first 10 seconds because I was reading your t-shirt joke first
:)
In your example, you would need a third cut perpendicular to the first 2 to create a ribbon you can pull from infinity - because otherwise you’d never cut all the way through. You’d be tearing like you do when you pull a tab from a supermarket bulletin board posting for a painting service.
*_Now let's have some fun_*
:D
*_Serious fun_*
D:
but wait...
*This is Matholger...*
:DDDDDDDDDDDDDDDDDDDD
Me at 8:25 : huh, that looks a lot like a golden ratio rectangle
Me at 8:45 when Mathologer reminds us that this is a representation of of the Fibonacci sequence: well of course it does, that's literally the definition as you approach infinity
Unfortunately, you made a mistake in the video. That's not how moon orbits work.
The important point with moon orbits is that we have conservation of angular momentum, meaning that (to first order) the rotational axis of the moon orbit does not change direction. That means you can't get a regular torus knot from the orbit. For that, the axis would need to rotate along with the planet.
Oh, and of course the lunar orbit tends to not fit an integer time into the planets orbit. See the Earth/Luna system, which you might be slightly familiar with.
Mathologer mentioned that the moon normally moves in the same plane as the earth snd the sun.
He was considering a hypothetical "what if the moon rotated perpendicularly", though he didn't make it that clear in the video.
"Unfortunately, you made a mistake in the video. That's not how moon orbits work." Sigh, REALLY ? :)
I, for one, was more than happy to imagine a real moon tracing out a torus knot, its axis rotating along. So I'm grateful to @Kai Henningsen for pointing out it's physically impossible. Damn conservation laws! But even with the axis direction constant, wouldn't the resulting orbit shape be a torus knot anyway? Topologically speaking.
+Fero Kuminiak I am very happy for someone to point out that this is physically impossible. However, as far as I am concerned, there is one aspect to this comment that makes it into the third dumbest overall in this comment section :)
"there is one aspect to this comment that makes it into"...
Which of the aspects do you have in mind?
"Start by choosing one of the Fibonacci numbers, let's say 8"
Nah, I'll go with 55. It's underappreciated because it's both a Fibonacci number and a triangle number, but overlooked because unlike 21 it isn't perf-
"Now make an 8x8 square"
You know what, 8's good. 8's good.
After seeing this video, I wanted to get a toroflux. I found one under the brand name Cosmic Coil, and it has 12/13 coils. While playing with it, I noticed that the flat state and the fully expanded state are both stable, but the fully expanded state is much more stable. Of course, this means that there is some partially expanded state that is an unstable equilibrium.
Folding a bandsaw blade converts one loop into three, but it's still just one ring. The mind gets confused when it sees 'loops' and thinks 'rings'.
Your t-shirts are always awesome
3:05 The slope of the green and purple shape is 3/5 and the slope of the red and orange triangles is 5/8. This means the combined triangles are not really triangles at all and they are missing 0.5 area for each one.
The T-shirt does it for me!
Wow, mathematics is so FUN - thank you for sharing this subject at one time I used to really hated it. Cheers.
I always enjoy your professional presentation. Good work my friend. I am 33 years old and I really want a toroflux now (to play with).. that is not a good sign.
Edit:
Just wanted to mention that back when I was in 11th grade, I stumbled upon a form of the fibnoacci sequence where each term is given as a function of the previous term and its index x(n+1)=f[x(n),n]. At the time, I thought it was something new, and I didn’t realize it’s trivial until a year or so later. So now, whenever someone mentions this sequence, I have to scream that embarrassing memory out of my head.
Trivial? If f[x(n),n] = x(n) + n, the sequence starts 1, 2, 4, 7, 11, 16, 22, 29, 37, ... which the OEIS identifies as the "maximal number of pieces formed when slicing a pancake with n cuts."
Hey Steve: by trivial, i meant it was done a long time ago... but my ego at the time convinced me i came up with something new.
I dunno about proving it mathematically, but I know that if you try to conjure a torus knot with those two values not being relatively prime, it is actually a smaller knot, where the largest common denominator is the number of times you travel around the valid knot to get your relatively not-prime copy of it.
Hello @Mathologer..... I have observed a peculiar fact regarding the square and rectangle that is shown in your video. Kindly observe the video at 4:50 timeline. Now observe the number of squares(complete squares), half triangles, less than half triangle and more than half triangles within the coloured region. Its different in both square and triangles. So, the total number of boxes that add up to give the area of the shaded region are different .So in my observation its this deviation that is producing the difference of 1 square unit in area.
Thank you for the video! All of you friends are super awesome! Oh, one moment of time that video was sad.
actually the difference in the counting is exactly how we count rotation of astronomical objects. If you count rotations of moon by looking at it from earth it’s 0. But if you count it by looking at the moon itself it’s 1. Corresponds to the 13 and 14 exactly. Similar for rotation of earth there is 365…. rotation counted by looking from sun, namely days, but there’s actually 366…. rotation done every revolution.
You blew my mind, I love this channel!
that infinity duplication trick is why we now know reality is discrete, if it wasn't discrete we could shave off infinitessimal chunks of energy and generate mass in this way
A big fan of your powerful lectures 😍
You simply take the determinant which gives you
f_(n-1)f_(n+1) - f_(n)^2 = (-1)^n then factor a negative one -> and give it to the other side which gives you the equation back
10:48
Determinant
Right side is -fn^2+fn+1 *fn-1
Left size is (-1)^n
fn^2+fn+1 *fn-1=(-1)^n-
(1+fn^2-fn+1 *fn-1=(-1)^(n
There is a mistake at 18:55 --> if the moon orbited at a right angle relative to the sun, its orbital path would not trace out a slinky knot. This is because the moon's angular momentum must be conserved over the course of the year. Thus, the plane in which the moon orbited would only intersect the sun every 6 months.
1) 5*8=40
2) 1²+1²+2²+3²+5²=40
--> A Fibonacci number times its predecessor is equal to the sum of the squares of all previous Fibonacci numbers
I must argue with your assertion at 19:55 ; Villarceau circles are very interesting. The fact that they are true circles for a 'geometric' torus is miraculous enough, but en masse they make up the very beautiful Hopf fibration.
The rearrangement of the pieces in the animation at the end is a neat way of proving Pythagoras' Theorem on the triangle formed by moving the cross so that one of the arms hits a vertex of the square.
You coul d have made me love math.
I think there's one other thing with the Fibonacci square puzzle. You cover why the difference is exactly 1, but not why the slopes are so close to each other (they're progressively closer approximations to 2-phi).
Also, for a square with side length F(n), with some trig and Fibonacci identities, the tangent of the angle between the two slopes can be shown to be (-1)^n/[2F(n-1)^2 - F(n-3)^2], which as expected decreases very quickly with n. With a little prodding of the numbers, you can get that the width of the extra parallelogram is no more than 0.75/F(n-1), or about 1/6 of a unit for the 8x8 square.
The one detail: The equation is over the matrix ring. The determinant of a matrix is an element of the base ring; specifically in the 2x2 case, it's ad - bc. But the determinant is multiplicative in the sense that det (AB) = (det A) (det B), and in particular (det A)^N = det (A^N). The determinant of the left side is F_(N + 1) F_(N - 1) - (F_N)^2; the determinant of the right side is (1*0 - 1*1)^N = (-1)^N.
You just exposed the shape of the Earth my friend.