A close friend showed me an intuitive pattern that given a k-dimensional convex polytope P, if one doubles every sidelength to generate a $P'$ it is possible to fit $2^k$ copies of $P$ within the region of this $P'$ by translation and rotation.
How to prove this though?
I was thinking an inductive argument would make a lot of sense, for example we can try to show it is true for all $k$ dimensional simplices and then somehow make the (getting hand-wavy here) argument that we have a region $2^k$ times larger than the original shape and then from there try to describe a packing scheme.
But even for something as simple as a pentagon what is the packing strategy that results in 4 pentagons fitting into a pentagon of twice the sidelengths?