Відео

How tf did they find identical gears if they couldn't use a first one

I think it's worth noting that the gear ratio has to be rational. This becomes very clear when using the 'carving out' visualization, since the source gear needs to make exactly a whole number of rotations for each orbit (rotations of the partner gear). The gear "ratio" is then a rational number: GR = (N_rot_source / 1). Also, as was touched upon, there are many restrictions on the initial parameters for this to work; such as the rotational speed of the source gear. Imagine a shape, size and distance such as what was portrayed in the video, if the source gear spins super-fast, then the envelope will approach a circle. Furthermore, if the shape of the source gear is weird, then the envelope may no longer represent the best partner gear, even if there exists a perfectly viable option. (I'm pretty sure)

morpph:i can use this same trick infenatly so i can always identify a number you missed me:ill just order them like 0.00...01 to 1 0.00...02 to 2 and so on down the line

Some very niche conspiracy theorist out there is watching just the first part of this video and shouting "Yes! I knew gears aren't real! They're mathematically impossible, even the engineers say so! All advanced machines work with ropes and pulleys!"

5 mins in and i thought Taylor series made sense this guy just convinced me it shouldn’t work at all 😅

I love how some of them ends up as geneva mechanisms. Also great video. Gonna experiment myself with a custom slicer based on the knowledge you provided me and attempt to 3D print them :)

Nice videos. Thanks.

One of the things that drives me crazy about higher math education goes something like this: I've encountered all of these thought experiments and challenges in classes before but they've never told me why I'm learning them or what use they have to people above me. It's not until I read about something I'm not familiar with or listen to a random UA-cam video that I recognize why they were asking me these questions in the first place. Like, I get that I should be trying to figure them out but I can't go 10 weeks with 3 classes 4 days/wk all exploring results experimentally. Same with novel proofs. I only have so much time.

Is this why so many mathematically significant numbers, like Pi, Euler's number, Feigenbaum's constants, etc, are all transcendental, simply because the probability of a random number being part of an uncountable set is basically 1 compared to the probability of being part of a countable set? Just luck?

I watched the whole ad to support you

Quantized depth of roads. Neat.

awesome video bossman!

The only thing you're forgetting to account for is backlash

Bold of you to call yourself a "mathematician".

Weak and pitiful and very discombobulated.

The perfect sequel does not exi…

This video was 22 minutes of pure joy

Not that the naming matters to mathematical topic, but if you asked me, I'd say you are switching the definitions of wheel and gear!

the egg gear at 27:00 reminds me of the sofa problem

Your intuition about some kind of "self-intersection" of the envelope is on the point for the artifacts. Just like how zero derivative is necessary but not sufficient for a maxima or minima, the envelope condition used is necessary but not sufficient for the type of envelope wanted here. If the curve traces out some kind of interior envelope, that will be caught too, and mess up the result. Additionally, the full failure is probably since not all positions of the gear necessarily have to correspond to being part of the envelope. That is, the gear at the positions for which the formula fails is entirely inside of the envelope, not touching it. I'm also not sure it would handle correctly the cases where multiple points or sections of the gear shape at a position are part of the envelope. In any of those cases however, the real-world implications is that the parameters set up are impossible to construct a normal gear for. Either the force transfer is not in the correct direction to couple the motions, and/or the gears would physically separate and not transfer motion. It may still be useful for things like cam systems, where the motion wanted is to pause (while the gears are not in contact), like in watch escapements or film projector reels, or if the intent for the gearing is to synchronize motion rather than transfer forces.

Very nice!!! Can we make a pair gears which are of "similar" shape? Btw, I think the interpretation of omega is slightly incorrect ua-cam.com/video/FUHkTs-Ipfg/v-deo.htmlsi=p9snlIDc3Es4S8FW

11:00 Bethesda

I played with this almost forty years ago. I came super close. I figured out the Cauchy method on my own, but only just heard of it in this video. I stopped short of the gamma function before I was distracted by other things and never looked back.

Your explanation of the envelope is fascinatingly similar to the math behind (I think) splines (or was it bezier curves?). Very interesting!

bravo#

i noticed the variable angle velocity in ep 1, and now i feel proud of myself

I absolutely love the little notations on the bottom right to direct further study, but I wish they stuck around longer. I keep having to scan back and pause very carefully to read them. I would like them to last long enough for me to notice there is a notation, tab into youtube, click on the video, and hit the pause buttom. They currently seem to be onscreen for less than two seconds. Assuming each of the actions four actions were instant, but required me to react to it at about average human reaction time of ~273 milliseconds, I recon I'd need a little over a second (1092 milliseconds) of the notation being onscreen. Given I am not The Flash, I think I'd prefer a 2 to 3 second window. I'm not sure how to make this smooth along with the rest of your transitions, but given your obvious skill at video editing in the rest of the video, I hope this is enough information to go on.

Do you have to use peicewise construction to create a non-analytic function? I guess the problem is that since any algebraic formula compsed of entirely analytic components is also analytic, you need to find some sort of "fundamental non-analyitic" function or operator. In other words, an answer to the question: "what is the simplest non-analytic function?"

I am an mechanical engineer and you are not. The axiom of gear making is that gears ROLL and not SLIDE! Of course they slide but the reason is not what you are babbling. The reason is that is extremely hard to get all the machining tolerances right. Therefor the machining error is the cause not the math/design. We design gears in the way that price/performance is acceptable but always with a thought that the design itself is perfect. Again, manufacturing errors make gears slide. Do study a little more. I don't know what your background is, but stay within your field of expertise, please.

0:00 never

0:00 never

0:00 never

Side note: Most calculations of these complex mathematical functions are done using CORDIC or a similar derived method in hardware.

Did anyone else catch the “I hate this channel” at 1:56

is there a way too have the shape of the source as a variable and equal to the partner?

Great video!

Thank you very much Very amazing and philosophical topic 🎉🎉❤❤

Camus' theorem would give good insight. 27:00 The error can be interpreted as being caused by the gear gets inside-out in some point. It is interesting problem that how much the gear's projection can gouge out its pair-gear without causing errors or slipping through.

Damn I saved this in a playlist and put off watching it until now, I'm glad I got around to it. This is so amazing, as a math minor some of these concepts eluded me during university, particularly everything after 13:00 , but within 20 minutes you literally finally made me understand and truly appreciate analytic, holomorphic functions and their 'equivalence' in complex analysis

the algorithm breaks down at parameters that would result in an actually physically impossible geartrain.

Could you have also found the curve by using the same method as the one used to find locals, but instead of setting the requirement to 0, set it to a continuous function? Or maybe that's what you did just with a different approach?

An important note about the common misconception that gears 'do not slip' - A pair of linked constant-speed gears can be *modeled* as a pair of circles that do not slip when rolling against each other, when modeling the gear system for its ratios of speed of rotation etc. This is likely the origin of the 'no slip' mistake when talking about gears themselves, as many discussions of gears only focus on the abstraction of the relative radii of non-slipping circles rather than the physical shapes of the 'teeth' of the gears. On the flipside, if we look at a 'conventional gear' (a circular wheel with 'teeth' projections), and then we were to take a sort of 'limit' as the size and number of the teeth grow large but the teeth themselves get smaller, the wheels instead start to look like 'rough circular surfaces' rather than 'toothy wheels'. And two rough surfaces rolling against each other would be expected to have a high amount of friction between them - that is to say, they would 'not slip' across each other. Thus, as conventional gears 'approach' circles physically, the necessary slippage of the macroscopic surfaces against each other of the toothy wheels transitions into the 'non-slip' microscopic surfaces having good friction against each other as two 'circles rolling without slipping'. So you sort of end up with the 'no slip' misconception showing up on 'both ends' of the abstraction. :)

OH MY GOD I JUST WATCHED THIS AFTER TAKING PARAMETEIC AND POLAR EQUATIONS IN MATH CLASS I UNDERSTAND IT

Wow Ureca!

Now determine f(s)!

There is no merit to the background noise.

Can anyone help me find the song starting around 18:52 in the video? 😅

Thoughts from only having watched the first 3 minutes: - You can create this result with a simpler sequence. 1-1+1-1+1… = 0, because each pair sums to 0. But if you “rearrange” the sequence to do two pluses followed by one minus, 1+1-1+1+1-1… = infinity, because with each triplet you are adding 1. - When you say that the rearrangement doesn’t affect the sum of finite terms, but can affect the sum of infinite terms, I think that’s incomplete. The reason is that the type of rearrangements you are doing actually cannot be done with finite terms. Take your first example of two positive terms followed by a negative term. You cannot apply this rearrangement to a finite set, because you will run out of positive terms and still have half the negative terms over. You get to do this with infinite sets because every subset of the set is also infitely large, so you can pick from them unequally without running out.

I wish you pronounced "envelope" like an English-speaker and not like a French-speaker.

I very much like the fact that this video combats the suggestion that it is normal for a function to be determined by its Taylor series, or even that its Taylor series says anything at all about a smooth function away from the point where the Taylor series is taken; bravo for exposing a dirty secret that mathematicians don't like to talk about. (At the same time they take care to not say anything that directly contradicts it; in fact this is the reason that theorems like Taylor's always say something more complicated than that what one might naively expect them to say.) However you do fall into the trap, contradicting your own main point, by saying that the reason functions like exp and the trigonometric functions are in fact everywhere equal to their Taylor series, lies in the rate of growth of their Taylor coefficients (8:20 : "how quickly the higher order derivatives grow as their order increases"). That is just plain wrong: while that rate does determine the radius of convergence of the Taylor series, and therefore whether the series converges to any function at all away from the point x of the series, it does not say whether (supposing it does converge) the function determined by the Taylor series equals the original function. Indeed there are infinitely many smooth functions with _identical_ Taylor series in a given point, and at most one of them is determined by that Taylor series (there is one if and only if the series has nonzero radius of convergence; then this one is called the analytic function in the bunch). What one needs to control in order to say things about values at other points than x is not (only) the rate of growth of the Taylor coefficients at x, but the behaviour of the function and its derivatives _as the point of evaluation varies._ Indeed, no information about those values at x give you _any_ control of those values at some other point x', however close to x. (To make this precise: for a given Taylor series at x, once can chose any Taylor series at all at another point x', and there will always be a smooth function that has those two Taylor series at x and at x'.) This is hard to get one's head around, since for instance the second order derivative _describes_ (something about) how the first order derivative varies in the _neighbourhood_ of x: nonetheless it (nor any higher order derivatives at x) _control_ the variation of that first order derivative in that neighbourhood. But this is just a paradox (of sorts) about limits: while the limit of a family of values tells something deep about that family as a whole, it does not say anything about (give any control over) any individual member of that family. It is for this reason that Taylor's theorem (and I don't really think it was his) must necessarily talk about values of derivatives in a whole interval, not just at x (it does so by speaking of just one value, but at an unspecified point of the interval). So the reason that exp and sin and friends are equal to there Taylor series cannot possibly be found in that Taylor series itself (the higher order derivatives), as you assert. Instead it is due, depending on what one takes as definition of those functions, either to the fact that they are _defined_ by that (everywhere converging) power series (making it a tautology), or if you define them to be the solution of a particular differential equation, by the fact that they are assumed to satisfy this same differential equation _everywhere._ (A smooth function f with the same Taylor series as exp at x=0 would still satisfy f'(0)=f(0), obviously, but it would no satisfy f'=f on any interval containing 0.) A fun fact is that one can interpret the differential equation directly as a relation among the coefficients of the Taylor series (so without ever considering a function that satisfies the differential equation) that entirely determines those coefficients and therefore the Taylor series (in fact one is working with _formal_ power series here), so it is true that the Taylor series S of exp at x=0 is the unique formal series that (formally) satisfies S'=S and has constant term 1. While pointing out this error in the video, let me add another much minor one. When illustrating "pathological" smooth functions that are not analytic, you show what appears to be the graph of the Weierstrass function (7:48), but that is a "pathological" function of a different kind: it is everywhere continuous but nowhere differentiable. It is therefore not smooth (lacking even the first derivative; smooth functions must have derivatives of all orders, everywhere), and therefore has no Taylor series to begin with. (It is true that among the everywhere continuous functions, such "pathological" functions are the rule, and smooth functions are the extreme exception; the analytic functions are the even more extreme exception among the smooth ones.)