Toroflux paradox: making things (dis)appear with math
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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.tumblr.com/post/20...
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.com/t-shirt/213...
The piece of music at the end is: English_Country_Garden from the free KZhead 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/misc/8grid.png
According to the the wiki page the inventor of the toroflux Jochen Valett who is German :) . en.wikipedia.org/wiki/Toroflux
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
This video is as good as the last two combined ;-)
Nice
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.
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 :)
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! :-)
"Why is he using such fat lines?" "...let's rearrange..." "Never mind, chocolate bar."
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++
Hmmmmmm, I thinks I just realized what I love about math ; the synergy between enlightened intuition & strict rationality.
Find someone who looks at you like this guy looks at his twisted slinky toy @1:05
Justin Abramson pretty cool, isn‘t it?
Just found this channel. LOVE IT! Love the animations, presentation, and t-shirt.
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
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 :)
You blew my mind, I love this channel!
Ah Mathologer, nothing makes me happier than a new video from you guys.
Obi-Wan Reviews May the FORCE be with your reviews!
Your permagrin is contagious.😁 Thanks Professor Polster! Great video, as always. 👍
Permagrin, I like that! =)
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.
the best channel hands down!
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..
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^ )
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!).
Thanks for all you do!
you sir, got yourself a new subscriber! beautiful explanation!
Snap! I haven't ascended past 3rd level of enlightenment yet and there is a new video already!
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
Realy amazing how you explain you are crystal clear in subject
You would have to be the happiest presenter on KZhead very enjoyable. Thank you
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 watched this hungover and my head hurts more.
Your best video to date, thank you.
This video is SO AWESOME! Full of SO many cool things!
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).
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?
As always, an excellent, entertaining, and enlightening video.
A big fan of your powerful lectures 😍
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”
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.
You're the cuddliest (smiling & joking) mathematician i've ever seen. Love your work, sir!
Superb video. Its really amazing.
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
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.
This reminds my of musical intervals. Great video as always!
super satisfying to watch
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 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.
@14:37 oh that's brilliant... no data is lost... only rearranged.
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 colour/tone. 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; almost 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. 🤔
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. kzhead.infovideos
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.
19:45 Didn't think I'd see that little editing correction, eh? lol
I serendipitously discovered your channel. Subscribed.
Your laugh really makes me smile 😃. Super video!
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.
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
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)
Superb content....as usual
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
(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?"
Your t-shirts are always awesome
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.
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.
I had to replay the first 10 seconds because I was reading your t-shirt joke first
:)
Thank you for this great video. It puts a smile in my day. Its just right.
:)
Thank you for the video! All of you friends are super awesome! Oh, one moment of time that video was sad.
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.
*_Now let's have some fun_* :D *_Serious fun_* D: but wait... *This is Matholger...* :DDDDDDDDDDDDDDDDDDDD
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
Excellent video! Just beautiful, beautiful!
The one step - take determinants.
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?
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.
@2:42 You know you're talking to a German when the word fun sounds so incredibly unnatural. Love the videos! Keep it up!
What a fabulous video.
Your channel ist just great :)
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 :)
Holy hell I was in Bill's class when he must of contacted you, but we never knew!
I like the image at 9:35 it made me think about what the practical implications are of using these sets for mip mapping in computer graphics. Instead of the regular old halfing.
Very beautiful, right up to the last second :)
Just WOW! I'm amazed; Very well done Mathologer 🙂👍 Greetings from Germany :D
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.
Excellent video. Thanks!
Another great video, Thanks. This took me to KnotPlot which is fun and from there I got to find an implementation of a 4D Rubik's Cube. The rubik's cube is right up your street so when are you going to take us to this other dimension. I can't wait!
Just amazing!
The T-shirt does it for me!
Wonderful - as usual.
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.
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 :)
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.
Wow, mathematics is so FUN - thank you for sharing this subject at one time I used to really hated it. Cheers.
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.
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 toroflux reminds me of symmetrical components in power systems engineering (Fortescue). You get harmonic pairs with positive and negative sequence components - e.g. you find the fifth harmonic with a positive sequence and the seventh harmonic with a negative sequence always turn up together. This has the same basic logic behind it.
The final square in the last demonstration meddled with the dimensions of the sum square. After the rotation, the outer edges are shorter (or longer, not sure which) due to the extra area now being the gap square.
one heck of an introduction
I genuinely don't understand the concept of disliking these videos. I've never seen a KZheadr go into so much detail on anything.
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.
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 :)
9:14 Obviously; a 3rd way to calculate the area is: (21*34) - (8*13) = 714 - 104 = 610
Your t-shirts are so good
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.