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?

$\quad\quad$<img src="https://i.sstatic.net/FP0Vh.png" width="270" />

I also know that there are polytopes having several realiztions with matching edge-lengths (e.g. see the image [here](https://mathoverflow.net/a/372979/108884)), but these realizations cannot be continuously deformed into each other while preserving all edge-lengths.