Puzzle time - Beetles edition

[jon-nyc]

infinite

[Klaus]

I think so, too. But how can they turn around each other infinitely often while only traveling a finite distance? (pinging @Zeno!)

[Axtremus]

:::

“Orthogonality” (90°) in the n-polygon case ==> inside angle of the polygon (n-2)*180°/n

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[jon-nyc]

:::

Along the lines of Ax’s post: You can calculate how much distance is traveled for every unit reduction of R, with R being the distance from a bug to the center of the rotating polygon.

In the case of a square:

Think of the initial square rotated and superimposed on an x,y coordinate system such that each bug is sitting on one of the axes. Infinitesimally, the bug on the positive x axis moves in a direction 45° off the vertical. You can draw the infinitesimal right triangle formed by the x axis, the distance he traveled, and the line from his new position back to the x axis. As he moved along the hypotenuse of that infinitesimal triangle, he covers sqrt(2) distance for every unit of radius reduction.

When you frame the orthogonality that way, you can see how it generalizes - that 45° angle changes as does the ratio of distance traveled to radius reduction.

:::

[Larry]

If you walk them backwards it says Paul is dead..