Recently Mark McClure constructed and displayed the 261 unfoldings of the hypercube (tesseract) in response to the question, "3D models of the unfoldings of the hypercube?":

^{The first 9 unfoldings in Mark McClure's display}

Each of the 11 unfoldings of the cube form monohedral tilings of the plane, as so well illustrated in the "Etudes" video to which Igor Pak pointed:

A polyhedron that is the prototile of a monohedral tiling is called an

*isometric space-filler*: $\mathbb{R}^3$ can be tiled by congruent copies of that one shape (rotated and translated but not reflected).

Now that we have the unfoldings of the hypercube, analogy with the cube raises the question:

. Which (if any) of the 261 unfoldings of the hypercube are isometric space-fillers?Q1

Asking this question raises another:

. How can one determine if a given shape, in this case a polycube / 3D polyomino, is an isometric space-filler?Q2

**(**

*Update**7Dec2015*). Aside from the two hypercube unfoldings that Steven Stadnicki showed tile space (below), with a student I found two more that tile $\mathbb{R}^3$, including the Dali hypercube cross unfolding, confirming Steven's intuition ("I don't know if the 'Dali unfolding' tiles space, though I'd be surprised if it didn't.") We posted an arXiv note on the topic.

nottile space?'... – Steven Stadnicki Dec 8 '15 at 6:10