Not sure why anyone would be amazed by this. It is simple physics. The higher the grade, the faster the ball would go.

You missed the point of the demonstration of isochronous curves. Balls travelling faster on a steep curve v straight line isn't what amazed me. The vid is about describing how isochronous curves got its name: because of its "occurring at the same time" characteristic -- which is the definition of the word isochronous. That is: no matter where you start two balls on different sides of the curve, they will reach the middle at the same time. But what really amazed me was its relation to a circle. Having a curve with isochronous properties is one thing (seems reasonable such a curve would exist), but how it can be created out of a circle is very cool -- discoverable and usuable with the simplest of tools. I'm sure early geometry mathematicians and Egyptian architects were similarly amazed. btw: And your "simple physics" / "higher grade" point is about straight slopes. Half of that curve is a "lower grade" than the straight line path. What if the curve wasn't symmetrical (steep to start, but nearly flat for majority of the course), or was bowed the other way -- travelling above the straight line at first then falling? Here's a test of your "simple physics" knowledge: From point A to point B: Does a ball on *every* curve travel faster than the straight line slope (bowed up or down, with vertical midpoint anywhere on the horizontal plan)? If this is so simple, that should be a simple answer, right? And if the answer is No, then isochronous curves have that property while other curves don't, which is cool. (@B-Bob for the answer.)

I mean there was a physical demonstration of it in the video, over and over again. You really think you're going to get it across now, breaking it down in paragraph form? Spoiler

sorry, I was mainly wanting to add why the curve amazed me, which was the purpose of 2nd paragraph. The rest is a question...given the same X:Y starting point of a isochronous curve: does every curve beat the straight slope?

The slope on the curve you may find fascinating, but I just see simple physics. Starting the ball at any point in the curve is going to start it at a different angle, and therefore a different rate of speed. Recall "The Edge" at Waterworld vs. other slides? Be fascinated all you want. The idea of the rotating wheel creating the curve is interesting, but the balls hitting at the same time isn't; at least not to me.

Yeah, that's what I said. No big deal that such a curve exists, but what I find interesting (like you) is its required relation to a circle, and that Pendulums don't have this property (nor any other curve, ellipses/parabolas). woohoo, we agree! fwiw: the Greeks never identified this special curve existed. So, I think a lot of mathematicians were amazed to discover it. But @Roscoe Arbuckle be all like -- "meh" Spoiler: Other cool properties of Isochronous Curves ---------------------- btw: per my question to @B-Bob, I've found it is *also* the fastest curve. And any curve beats the straight line slope, even if it's leaning like a ship taking on water. Here's another cool property of the Isochronous Curve: The time it takes the ball to reach the midpoint is identical to the time it takes the ball to free-fall from the top of the circle that made it.

Don't know how to embed a facebook vid, but this is awesome https://deadspin.com/table-tennis-guy-pulls-off-the-last-ditch-block-of-a-li-1830784334

Not very intuitive how you embed, but basically just get the FB url of the video and naked paste it into a post here; bbs will handle the rest