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, but are there any other flexible polytopes?

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I also know that there are polytopes having several realiztions 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.