# How do the 3d models of the hypercube nets relate to the pairings?

In "Unfolding the Tesseract" Journal of Recreational Mathematics, Vol. 17(1), 1984-85. (Journal link.), Peter Turney presents the pairings of an unfolded tesseract.

In "3D Models of the Unfoldings of the Hypercube," Mark McClure provides a wonderful Mathematica notebook that, among other things, produces 3d models of the $261$ nets of the hypercube.

How do McClure's 3d models relate to Turney's and McClure's own pairings?

• What do you mean by the term "pairings"? – Joseph O'Rourke May 16 at 23:57
• Tumey, in his JRM article presents 23 trees with 8 nodes each that represent the possible unfoldings of the hypercube into 3 space. Based on these trees he shows and explains the possible pairings that can be modelled by 3d models. – user124366 May 18 at 17:08
• The pairings are what Turney presents in his paper. I have a more detailed explanation but it will not fit in this comment box. – user124366 May 18 at 17:33
• I have determined the answer. Apparently, McClure performs all of his evaluations in the same order so his 1st tree corresponds to his 1st 3d model and so forth. He also provides facial graphics (in the same order) that show which edges are cut for each net. Turney gives a procedure that maps a given net to its (his) corresponding tree figure. – user124366 May 22 at 21:03
• If you have found a satisfactory answer, you should post it as an answer below and accept it by clicking the checkmark so that the question does not remain unanswered. – j.c. May 23 at 14:53