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:

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

Asking this question raises another:

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

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.

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    $\begingroup$ I wondered about your motivation! The second tiling that is illustrated in that video shows that there is a natural approach to generating the tiling, when the tree associated with the unfolding is the path on 6 vertices. There are four such unfoldings of the cube into the plane. Of the 261 unfoldings of the tesseract into space, there are 24 whose associated tree is the path graph on 8 vertices. I wonder if a similar approach would yield tilings of space by these? Presumably, a visualization of the unfolding process would help determine this - something I was considering working on next week. $\endgroup$ – Mark McClure Mar 5 '15 at 13:06
  • $\begingroup$ The note looks fantastic - I really like your Dali tiling. With several of the unfoldings confirmed to tile, I suppose the next natural question is 'have any been confirmed to not tile space?'... $\endgroup$ – Steven Stadnicki Dec 8 '15 at 6:10
  • $\begingroup$ @StevenStadnicki: Thanks! Indeed, that is the next natural question, and it is posed as the 3rd open problem on p.19. $\endgroup$ – Joseph O'Rourke Dec 8 '15 at 11:16

Here is an illustration of Steven Stadnicki's first tiling of a (portion of a) "2-cell thick infinite plane":


  • $\begingroup$ Gorgeous! May I ask how this was made? $\endgroup$ – Steven Stadnicki Mar 8 '15 at 4:55
  • $\begingroup$ I used Mathematica, with opacity less than 100% to permit transparency. $\endgroup$ – Joseph O'Rourke Mar 8 '15 at 13:21

For your first question, I haven't gone through all of the unfoldings, but the first one there clearly tiles space: it even tiles a 2-cell thick infinite plane (which can then be stacked to tile space), something like this: $$\begin{array}{cccc} a & b & c & d & e & f \\ a & g & c & d & e & h \\ a & g & i & d & e & h \\ a & g & i & j & e & h \\ k & g & i & j & l & h \\ k & m & i & j & l & n \\ \\ \\ a & a & a & a & e & e \\ o & g & g & g & g & h \\ p & p & i & i & i & i \\ q & q & q & j & j & j \\ k & k & k & k & l & l \\ r & m & m & m & m & n \\ \end{array}$$ As for your second question, the traditional answer is 'find a region that the polyomino tiles, which in turn tiles the space by translation'. A good example of this is the second unfolding shown here. We can couple the given shape with a copy that's been rotated 180 degrees about two axes to get a more 'solid' shape with a rotational symmetry:

$$\begin{array} . & . & a & a & . & . \\ a & a & a & b & b & b \\ \\ \\ b & b & b & a & a & a \\ . & . & b & b & . & . \\ \end{array}$$

Now, a copy of this shape can be paired with another, rotated copy to form a shape that can tile a 2x2xinfinity 'rod' by translation: $$\begin{array} . & . & c & c & d & d & d & d & d & d \\ c & c & c & c & c & c & d & d & . & . \\ \\ \\ c & c & c & c & c & c & d & d & . & . \\ . & . & c & c & d & d & d & d & d & d \\ \end{array}$$

Something similar (essentially, building shapes that fit an extended Conway criterion) can certainly be done with some of the other 'biplanar' unfoldings, but I haven't gone through them to try and figure out which ones are amenable to it. I don't know if the 'Dali unfolding' tiles space, though I'd be surprised if it didn't.

Meanwhile, it's at least conceivable that one of the unfoldings tiles space only aperiodically, but that would be a shock; AFAIK, no single polyomino has been shown to tile only aperiodically (though sets of polyominoes can model Wang tilings, and so tile only aperiodically) in either two or three dimensions.


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