Coin rotation paradox #3

I again edited the Wikipedia article about the coin rotation paradox (link) to add two references. One was to the book Mathematical Fallacies and Paradoxes and the other to a YouTube video about when the two circles are different sizes. I also removed the template message about the article lacking sources. 

Tusi couple

Obviously the center of a circle rolling along a straight line travels a straight line. But I had not imagined that a point on the perimeter of a rolling circle could travel a straight line. It can and does if a circle rolls within another circle twice as big. This was noted by a 13th century Persian astronomer and mathematician. Tusi couple.

The same attribute with eight points on the perimeter of a rolling circle traveling straight lines is shown here, here (YouTube), and here (YouTube). The last gives some mathematical analysis. Guessing, the "illusion" is that the moving circle is imaginary in the two YouTube videos, since the moving dots are not connected to one another. Part of the first YouTube video shows the dots being added one at a time. The illusion is a rolling square while there are only four dots. 

Coin rotation paradox #2

Four months ago I edited the Wikipedia article about the coin rotation paradox (link), adding the new section Analysis and Solution. I explained why the moving coin does what it does. I wrote about it on this blog here May 29.

More recently somebody else (user name Cmglee) added another section about one coin revolving around another, but the two coins having unequal radii. While the person gives the correct formula, why it is correct is not explained. The person also gave a reference, but the reference neither gives the formula nor explains why it is true for any two radii. Therefore, I added a new section to the Wikipedia Talk page as follows.

The formula R/r + 1 is correct. However, the reference – author Y. Nishiyami – doesn't give the formula, much less prove it is true for any R and r. Also, Nishiyama says, “Separating revolution from rotation is helpful for understanding, but doing so does not provide a fundamental solution.” This latter part is not true. For a coin with radius r to make one revolution around a stationary coin with radius R, the center of the moving coin travels a circular path with radius R + r. (R + r)/r = R/r + 1. Likewise, the circumference of the path is 2*pi*(R + r)/(2*pi*r) = R/r + 1 times the circumference of the revolving coin. Rotation is irrelevant. [End]

The key to this solution is to focus on the center of the moving coin. The center travels a circular path with radius R + r and hence circumference 2*pi*(R+r) in order to make one full revolution around the stationary coin. How many rotations it makes, even zero, while doing so is irrelevant. (Rotation versus revolution.) If it "rolls without slipping or sliding," then the number of rotations will also be R/r + 1. 

Coin rotation paradox

The coin rotation paradox described here has existed for a long time. Before February 21, 2020 the Wikipedia page gave no explanation of why the moving coin rotates twice as it makes one revolution around the fixed coin. I did not find a good explanation anywhere else on the Internet either. I constructed my own solution. Then I edited the page to explain why, adding the following new section. 

Analysis and Solution

From start to end the center of the moving coin travels a circular path. The edge of the stationary coin and said path form two concentric circles. The radius of the path is twice either coin's radius. Hence, the circumference of the path is twice either coin's circumference. To go all the way around the stationary coin, the center of the moving coin must travel twice the coin's circumference. How much the moving coin rotates around its own center en route, if any, or in what direction – clockwise, counterclockwise, or some of both – has no effect on the length of the path. That the coin rotates twice as described above and focusing on the edge of the moving coin as it touches the stationary coin are distractions.

At the time of writing this, there is a warning that says the page lacks citations and has unsourced material. Like I said above, I did not find anything to cite or use as a source. I solved it on my own. 

Wikipedia can be edited at any time by almost anybody. I hope nobody removes or ruins the above section. Anyway, I hereby document it in my blog, which is far less subject to being altered by others.