I came across an image, that show really simple unfold of 4-dim cube. https://arxiv.org/pdf/1512.02086.pdf here at #2.1, and 120 here https://page.mi.fu-berlin.de/moritz/mo/198722/unfoldings.html. Does anyone know how to fold it? Thanks for the help!


1 Answer 1


Since your first link is to my own paper, I will try to respond to your slightly unclear question:

Does anyone know how to fold it?

Our paper showed various hypercube unfoldings that tile $\mathbb{R}^3$, including the Dali cross, and one unfolding that further unfolds to tile the plane. I gather you are seeking additional unfoldings of those unfoldings into $\mathbb{R}^2$. In which case, this correspondence from Andrew Winslow, triggered by our paper, may help:

"Stefan Langerman and I have found some unfoldings of the Dali cross that tile the plane. Attached are pictures of three examples. They all tile the plane isohedrally using $180^\circ$ rotations, i.e., they satisfy Conway's criterion."

Below I show one:


Added. In case what you seek is how to fold the $L$-shape from 2D to 3D, Fig.16 is explicit:


Or perhaps you are asking how to fold the 3D $L$-shape to the 4D hypercube, i.e., which faces glue to which?

  • $\begingroup$ Mark McClure's Mathematica calculation and display of the $261$ hypercube unfoldings: 3D models of the unfoldings of the hypercube?. $\endgroup$ Oct 12, 2020 at 23:16
  • $\begingroup$ Excuse me for misunderstanding. At wikipedia en.wikipedia.org/wiki/Tesseract, in image gallery block you can see gif with unfolding projection of 4-dim cube into Dali cross. And I asked about something similar. I can't understand which edges we need to glue, if we want get tesseract. $\endgroup$ Oct 13, 2020 at 7:08
  • $\begingroup$ @MichaelKhoroshikh: Of course you cannot fold to get a tesseract except in 4D. I'll add another image, but I still may be misunderstanding your question. $\endgroup$ Oct 13, 2020 at 12:39
  • $\begingroup$ @MichaelKhoroshikh: I apologize for misunderstanding your question. I now see that what you must be asking is: How can that $L$-shape in 3D be refolded to form the 4D hypercube? Which faces glue to which? You want instructions on how to reverse the unfolding from 4D to 3D. That I don't know. You could use McClure's Mma program to see how the $L$-shape is achieved, and reverse-engineer from there. $\endgroup$ Oct 13, 2020 at 13:49

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