I've got a problem with that video that starts at 4:15. He seems to jump to the conclusion that for every midpoint there is only 1 distance. But that midpoint is formed by picking 2 points on the edge, and one could easily pick two other points on the edge that have the same midpoint (but have a different distance). He did not address that point at that point in the video, and for the next 2 minutes I kept raising that point in my mind. After he continued down that path not addressing that point, I felt that I must have missed something, or that more intelligent math viewers would have solved that open question in the mind in seconds and I am not mathematically inclined enough to be the target audience. And I stopped watching that video.
I think good educational videos are the result of a process where a trial audience raises such points and the video gets constantly refined, so that the end video is even good for people who question every point.
He addresses this at 9:00 in the video. You're thinking of a function graph, but he never made a function. He just sets up a visualization of a set of 3D points.
this is not a conclusion that he jumps to! all that is stated is that there is a mapping from every pair of points on a curve to a set of 3D coordinates specified by their midpoints and distances. there is no requirement for uniqueness here. in fact, the whole point of this is to turn the search for an inscribed rectangle into the search for two pairs of points on the curve that have the same midpoint and distance --- this is stated just 1 min 15 seconds after the timestamp that you point out.
The function defined in the video is "Given a pair of points A and B on the curve, output (x, y, z), where (x, y) is the midpoint and z is the length of the segment connecting A and B", and the pictures are of its image, not its graph. But if you define it visually, then it's very natural to misunderstand it the way you did, since the picture looks a lot like a function graph of a function which takes midpoints (instead of pairs of points) and returns the distance corresponding to that midpoint (which is not well-defined, as you pointed out). If this happens, the viewer is then completely lost, since the rest of the video is dedicated to explaining that the domain of this function is a Möbius strip when you consider it to consist of unordered pairs of points {A, B} (as one should).
Ultimately, if you don't have a 100% formal version of a given statement, some people will find a interpretation different from the intended one (and this is independent of how clever the audience is!). I think 3Blue1Brown knows this and is experimenting with alternate formats; the video is also available as an interactive blog post (https://www.3blue1brown.com/lessons/inscribed-rect-v2) which explicitly defines the function as "f(A, B) = (x, y, z)" and explains what the variables are.
The fact that "given a large enough audience (even of very smart people), there will be different interpretations of any given informal explanation" is a key challenge in teaching mathematics, since it is very unpredictable. In interactive contexts it is possible to interrupt a lecture and ask questions, but it still provides an incentive to focus on formalism, which can leave less time for explaining visualizations and intuition.
> I think good educational videos are the result of a process where a trial audience raises such points and the video gets constantly refined, so that the end video is even good for people who question every point.
It would be at least as long as a one-semester course in typical math major then.
To address your specific question: he doesn't assume each midpoint has only one distance at all. He doesn't say it and the visualization doesn't show it as so.
he maps two points (by using their midpoint) and a distance to the (x,y,foo) if it was two different points with the same midpoint but different distance it would map to (x,y,bar)
I think good educational videos are the result of a process where a trial audience raises such points and the video gets constantly refined, so that the end video is even good for people who question every point.