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Let $P\subset\Bbb R^d$ be a convex polytope (a convex hull of finitely many points). Lets call it flexible, if it can be continuously deformed while

  • keeping its combinatorial type, and
  • keeping its edge-lengths.

I know that the $d$-cube is flexible in this sense. More generally, most (all?) zonotopes are flexible (see the comments). Also all polygons are flexible. But are there any others?

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I also know that there are polytopes having several realizations with matching edge-lengths (e.g. see the image here), but these realizations cannot be continuously deformed into each other while preserving all edge-lengths.

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  • $\begingroup$ Possibly: All zonohedra? $\endgroup$
    – Joseph O'Rourke
    Commented Oct 10, 2020 at 12:15
  • $\begingroup$ @JosephO'Rourke Oh I see. I suspect you mean to generate a polytope $$Z=\sum\mathrm{conv}\{-r_i,r_i\}$$ for some (generic) unit vectors $r_i$, and then you change these unit vectors continuously. So the question might be whether there are any that are not zonohedra/zonotopes (or polygons)? Is it still okay for me to edit the question? $\endgroup$
    – M. Winter
    Commented Oct 10, 2020 at 12:18
  • $\begingroup$ Sure, please edit as you see fit. $\endgroup$
    – Joseph O'Rourke
    Commented Oct 10, 2020 at 13:55
  • $\begingroup$ What about a pentagonal cylinder, i.e. $\{(\cos 2\pi n/5, \sin 2\pi n/5, \pm 1)\}$? Or a cylinder of any other polygon? $\endgroup$
    – user44143
    Commented Oct 10, 2020 at 16:53
  • $\begingroup$ @MattF. Right. A prism over any flexible polytope is flexible. Probably the same holds for the cartesian product of a flexible polytope and an arbitrary polytope. What about Minkowsi sums? It might be useful to find the "elementary" flexible polytopes. $\endgroup$
    – M. Winter
    Commented Oct 10, 2020 at 17:29

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