magic studio

how the snake works #1

2008-02-23, 01:50 pm
Dies Saturnus
Luna wn 92%v

“However, nets are not quite as simple as the diagram in Figure 3 suggests. Strictly speaking, a net requires instructions concerning which edges on the boundary of the polygon that forms the net are to be pasted together.”

– from Unfolding Polyhedra by Joseph Malkevitch (CUNY)

If all you are given to begin with is a net like the image on the right, where do you get the instructions, where do they come from? Are they QBL, “received”  or are they hunted and captured directly? (& ultimately is there really a difference?)

If I had possession of the shape the lines of the cuts of the particular edges make, like a sigil in 3-d, I might could figure out the instructions. It would be a like a little deformed pipe-cleaner animal, but it would be a key.

Sometimes one net can be folded together in varying ways to make differently shaped solids.

“Figure 4 (This diagram shows a collection of 8 equilateral triangles, which when folded together in one way yields the regular octahedron while the same collection when folded a different way yields a non-convex octahedron which, consists of equilateral triangle faces.) Without the gluing instructions, we do not have enough information to guarantee what polyhedron the net really defines! (One can find examples of polygons (nets) without the gluing instructi’ons that can be folded to form two inequivalent convex polyhedra.)

Thank goodness I’m only having to deal with a cube here.