Each of the white circles are really just moving back and forth along a straight line. The second animation shows the same exact motion, but with straight lines added to make the linear motion obvious. Inspired by the Tusi-couple.

Mathematica code:

Manipulate[
Graphics[
{{Black,
Disk[{0, 0}, 1.05]},
Table[
{White, Opacity[o],
Line[
Table[1.05 s {Cos[n*2 Pi/m], Sin[n*2 Pi/m]}, {s, -1, 1, 2}]]},
{n, 1, m, 1}],
Table[
{White,
Disk[
Cos[2 Pi (t + n/m)] {Cos[n*2 Pi/m], Sin[n*2 Pi/m]}, .05]},
{n, 1, m, 1}]},
PlotRange -> 1.1, ImageSize -> 500],
{{m, 16}, 1, 20, 1},
{{o, .5}, 1, 0},
{t, 0, 1}]
 
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