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Joseph O'Rourke
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Example. $k=5$, $d=3$, $100$ (substantive) iterations; center of mass pink, time growing downward:
       InchWorm100
It appears the center of mass performs a scaled random walk, scaled as a function of $k$ and $d$. Perhaps scaled by about $(d/2)/k$...?

Joseph O'Rourke
  • 150.8k
  • 36
  • 358
  • 958