The name rhymes with "binocular", and it's a channel intended to teach math with the help of visuals and animation wherever possible.
Henri Poincaré once said about mathematicians, "Matter does not engage their attention, they are interested in form alone." And in one sense, math can be defined as the study of pure form ("morphḗ"). This channel is meant to be a morphocular, a lens on math, that clarifies and illuminates its often confusing and daunting concepts to reveal the beautiful thing it really is. I hope this channel lets a little of that beauty shine through to you.
Mechanical Engineer here: Your negative space to determine the shape has a problem in that the "peaks" and "troughs" of gears don't touch. Making them touch causes problems in their function. If I recall correctly the problem is the gears will bind. Gear contact is a more complex interaction then you seem to realize. Gears can drive in both direction, your clover gear can't drive the square gear without perfect friction. Gears work better with lower friction not higher friction, those are not gears. Without friction the clover will rotate the square so the close side is perpendicular to the line between the axels and then spin freely wit the square staying stationary. The contact point on gears jump, they don't trace out the shape of the gears in one continuous motion. In fact if your gears don't rotate both ways you only need about half of each tooth! Less the half of the perimeter would be a contact surface! I think you designed noncircular friction rollers. Which are not actually gears. Some people list them as a type of gear but they just are not; although they are a direct competitor for the same applications. Edit: you comment on the similarity to the wheel for a square is your "gear" is allowing some slippage to adjust for constant angular speed by allowing specific amount of slippage without actually supplying a mechanism to control the amount of slippage. In a gear the slippage is controlled by the geometry of the teeth, that is why teat have their peculiar shape.
i would think that we could observe that if the wheel moved to the center then was moved a bit over it would there on the line and that would be a simple observation
I'm sure this wouldn't actually be practical, but if all you need is constant rotation at the input and the output and not necessarily at every intermediate step, you could use wheels as gears, right? It seems like it should be possible to cancel out all the jerkiness so that the input and output rotate together smoothly, with only internal gears moving at non-constant rates.
great video but I will say it would have been nice to see the gear partners spinning like gears once the problem was solved, instead of rolling around each other. still great stuff though keep it up
Isn't the rotation property of complex numbers more a result of multiplying any number pairs by coordinates on the unit circle, rather than the fact of one of the coordinates being imaginary?
The slippage is more perpendicular to the axel-line for the gears than expected considering the hook the video claims in the beginning, but also the gears in the example seem to break contact before they would drag against eachother by dezign Also, misinformation should be determined by the listener and not governed by a central body or even someone else, that’s how fascizm begins, hence The First Amendment
cannot be asked to watch half an hour of yipyap someone tell me where he answers the question that is the title, which is the only reason i clicked on this video, to find out what gear shape meshes with a square, i click and i'm greeted with a half hour long video, it doesn't require 30 whole minutes to tell me what gear shape meshes with a square.
Imaginary numbers looks like x,y coordinates. Why its written as a+bi when its just vector (a,b)? Its main feature is rotation by multiplying on i, which can be done with usual vector with similar rotation rule.
I would argue that the curve with the line along the y-axis is still connected with the path definition (Disclaimer: I am not a mathematician, so feel free to correct me if I'm misusing terminology; more notes at bottom). There are only two possible scenarios for the "endpoints" of the separated curves without the vertical line (these "endpoints" reflect the y-value of the function immediately to the left and immediately to the right of x=0). The endpoints are either located at effectively the same point to create what would normally be a removable discontinuity or located at different y-values to create what would normally be a jump discontinuity. These types of discontinuity usually refer to functions where the limits at the x-value are known. In this case, the limits from the left and right are undefined due to the oscillating pattern. However, the function is bounded between two y-values, so we know that the "endpoints" of both sides will each have a y-value between -1 and 1. This would mean that adding a line at x=0 that ranges from y=-1 to y=1 should attach to both disconnected "endpoints" regardless of where they are positioned relative to the y-axis. Although we cannot determine the precise values of the path function as it meets the y-axis, similar to the way that we cannot determine a finite limit at x=0, an indefinite path function must exist that traces the vertical line to connect the two segments. Note: This is the result of a bunch of concepts and theorems back from Calculus 1 that have been scrambled together from memory to try to make something that resembles a decent explanation. The main inspiring concepts aside from discontinuity were the Intermediate Value Theorem and the Squeeze Theorem. It's also 2:30am as I'm finishing this (I got hyperfixated on number stuff again), so please let me know if there's something that doesn't make sense, and I'll try my best to clarify later
18:50 just an aside: would that graph be described as a45 degree rotated parabola? My mind went in the direction of a graph for an equation similar to y=1/x where x is greater than 0. I don’t remember the specifics or how to test in this instance, but would that actually satisfy the requirements for a parabola?
Have you ever heard of a Wankel engine? It's a convex triangle spinning inside of a shape I can't describe. You should do a video on it it is very interesting how they solved that clipping issue you talked about.
I was playing around with one of my game projects written from scratch in Java last year, and was making a 2D GUI system (bear in mind I don't have a strong mathematical background at all - thanks lackluster school system ...). In it, I was playing around and wanted to make functional gears, but I couldn't find any useful information on generating gear meshes, so I ended up just coding something that could generate an N-toothed gear with a given radius that was visibly passable as a gear instead. Life ended up happening, and I haven't really revisited that project for a while. After watching this video just now, however I had an idea to help me generate better looking (and hopefully better working) gears: take an arbitrary shape and make it orbit around a circle (using [Math.atan2(y, x) * -2], removing pixels from the circle where the shape overlaps (this part should prove challenging for me lol). I think that ought to result in a shape that was 'traced out' with the outline of the original starting shape, no? I'll admit I'm just throwing stuff together to see if it works lol, I know there's probably a much simpler and better way to programmatically tackle this problem.
Didn't the last video end with the statement that constant angular velocity is beyond the scope of the video? Now I'd like to see the development to constant torque. I was also hoping for the mirrored wheel/gear example leading to the triangular cutter that you showed last time but for this new condition.
