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Comment by pmayrgundter | Hacker News Reader

None of the explanations gave me an intuition for it except the circle rolling down a straight line with the same length as its circumference.

A roll down the line will rotate the circle once. Coming back the same. But rolling around one corner will add half a circumference, and another half for rolling around the other. So you get 2 * 0.5 extra circumferences, and so + 1 C. Somehow that helps with the other polynomials for me too. Super cool.

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Triangle is probably easier to see (120 deg rotation when you turn the corner). Then you can see that the key is the sum of exterior angles, and this holds as you increase num sides. In the limit, you get a circle.

Oh wow! Hadn't heard it like this before, but now I like this way the best. I can finally see clearly where that "extra" rotation comes from.

Having watched the video, yours is the only intuitive explaination that talks to me.

The best intuition for me was the YouTube comment that said to imagine the rolling coin/circle is picked up on its edge, in 3D, rolling like a unicycle wheel. It turns forwards to roll and measure distance, and it turns sideways to travel around the circular path. i.e. "N" rotations for circumference distance on one axis (the straight line distance) plus one full rotation on a different axis around the circle. When the coin is lying flat, both have to happen on the same axis.

To me the intuitive explanation is that if they were gears, with shafts fixed in a block, the small wheel would turn 3 times per turn of the big wheel, but in the given scenario the small gear also has to go around the big one, hence the R/r+1 solution.

Yeah. I think that’s the intuition I have about this too. It rotates once by virtue of going around, and three times by virtue of the size of the big circle.

If you imagine a very tiny circle it’d still rotate a full time.

Yes, but that line of thought doesn't really help when the small disk is rolling around the inside of the large disk, not the outside.

For my inner eye, I can see that the circle will rotate half a circumsphere to get around the corner.

But I can’t really explain why. It’s the same distance.

I get that when a road turns left, it’s longer on the right side, but these are lines that have no width and so there’s no stretch and by definition we’ve kept the length the same.

This is the right way to think about it. You get half at the beginning and half at the end so overall you get an extra rotation.

Do you mean polygon, or did I miss something relating to a polynomial solution?

What corner? What shape only has 2 corners?

A line (which is really a rectangle in this case).

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