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Comment by drc500free | Hacker News Reader

What makes sense to me is to think about something that DOESN'T roll.

Suppose I start in Greenwich, walk - without rolling - down the prime meridian to the south pole, up the international date line to the north pole, and back down the prime meridian to Greenwich.

How many rotations do I go through? One. I get a full rotation because I've followed the earth's curvature all the way around the globe once, even though I'm walking straight without rolling.

So the answer is "how many rotations due to rolling" plus "one bonus rotation for passing around the curvature of the circle."

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The way I thought about it was to ask what happens if you just have a coin going around a point, which would be the natural end point of reducing the size of the second coin. That of course would give you 1, so a good first hypothesis is that the total is just however many spins the two coins go, plus 1.

The video demonstrates the 1:1 coin size case and speaks about the curvature of the line it rolls on, but your thought experiment makes it a lot clearer. As the size of the other coin goes to 0, the amount "gnawed off" of the rolling coin will also go to 0 while obviously still rotating a full revolution.

I like this; it gives me a good way to think about the "from the circle's view" - Earth center looking out at you sees your feet close and your head far away, always; you don't rotate at all.

> I get a full rotation because I've followed the earth's curvature all the way around the globe once, even though I'm walking straight without rolling.

I don't follow. You aren't walking straight without rolling. You're constantly turning to keep the angle from the center of the earth to your head, as measured through your feet, fixed. This is not a necessary part of moving around the earth; you could maintain a constant orientation to e.g. the earth's axis of rotation instead. The one rotation that you're imagining isn't due to your travel around the earth; it's due to the rolling that you're pretending you aren't doing along the way.

I tried defining a parametric equation for the position of a point on the rim of the small outer circle; if it has radius r, and it starts at coordinates (3r, 0) lying tangent to a circle of radius 3r centered at the origin, and it takes 2pi units of time to roll around the larger circle, then its position at time t is

    (4r cos t, 4r sin t) - (r cos 3t, r sin 3t)

(unless I've made a mistake...?)

We then need to define what a "revolution" is. If we define it in what appears to me to be the obvious way, as having been completed whenever the vector from the center of the outer circle to the point that we're tracking on its rim is parallel to its initial value of (-r, 0), then this parameterization makes it clear that the zeroth revolution of the outer circle ends at time t = 0, the first ends at t = 2pi / 3, the second ends at t = 4pi / 3, and the third ends at t = 6pi / 3. Since the maximum value of t is 6pi / 3, it appears that there cannot be more than three revolutions.

However, I find the argument compelling that we should be able to get the number of revolutions by dividing the distance traveled by the center of the circle - 8pi - by the circumference of the circle, 2pi. This clearly tells us that there must have been four revolutions.

What was wrong with the parameterization approach?

Yes, but that intuition doesn't seem to work for the case where small disk is rolling around the inside of large one, not outside. What's the intuition for why we then subtract one rotation rather than adding one ?!

Very basically, to go the same direction, a small disk on the inside and the outside of a large disk need to spin opposite directions. When you're on the outside, your spinning direction agrees with the direction you go around the circle (i.e. the rotations add). When you're on the inside your spinning direction is the opposite of the direction you go around the circle (the rotation subtracts).

It’s concave rather than convex? It’s bending into itself instead of away from itself?

You're intuitively rolling the other way. Going on the surface means that you're falling forwards while going inside means that you're falling backwards.

I think you can always arrive at the correct answer by tracking the distance traveled by the center of the rolling object.

In the original SAT question, the center of the rolling circle is 4r from the center. That means it travels an arc of 8pi. Then assuming no-slip, that works out to 4 rotation.

In your case, rolling on the inside, the center of the rolling circle will only be 2r from the center. Using the same logic, this yields 2 rotations.

Yes - the relationship between distance traveled by center and number of rotations (= distance/diameter) is a nice generalization, although the guy in the video went so fast that I didn't understand his derivation of this relationship (related to speed, etc).

This is definitely the easier way to understand it. Imagine a point in the centre of the circle that is traveling. Draw the line that point takes, and then calculate using the new circle.

That intuition makes a lot of sense to me, especially if I picture the person's frame of reference.

Kind of reminds me (a layman) of winding numbers. I suppose there are topologically inspired variations of this problem that might be even more "paradoxical" (or perhaps just silly). If you moonwalk the second half, you undo your rotation? Or if you follow specially designed subterranean tunnels, you can end up doing 0 or negative rotations!

>How many rotations do I go through?

0. Walking in a circle is different from spinning. They are 2 separate things that should be counted separately.

Adding them together should be a type error.

No, GP does mean spinning there. As they walk around the earth, their head points away from the north pole, then away from the equator, then away from the south pole, then away from the equator, then away from the north pole again.

It happens that they also went around the earth, but the "rotations" in the part you quoted refers to the spinning.

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