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It's often claimed that it's possible to wrap gifts in normal 6-sided boxes "perfectly", meaning that the seam on the back side matches the pattern on the paper it overlaps. I'm convinced that it's possible to prove, mathematically, that this is not possible, but my math skills are too deteriorated to actually do so.

It seems to me that, in order to accomplish a "perfect" wrapping job, the length of the diameter of the box to be matched must be a multiple of the length of the pattern and that no amount of folding can overcome this requirement. Can anyone prove that this is true?

This may be a little more trivial than this site usually caters to, but it seems seasonally appropriate...


As a last minute note, it occurred to me this morning that, each time you go around the box, the diameter of the outermost later increases slightly, which means that, given an infinite amount of paper, you could increase the diameter sufficiently to create a match. I seriously doubt that anyone actually wraps gifts in inches of paper though...

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You might enjoy John Conway et al, "The Symmetries of Things". A great book that addresses much of this. –  Ross Millikan Dec 25 '12 at 15:57
    
For discussions related to this circle of ideas look at Joseph O'Rourke's wonderful book: How to Fold It, The Mathematics of Linkages, Origami, and Polyhedra, Cambridge U. Press, 2011. There is also the more technical book by Demaine and O'Rourke, Geometric Folding Algorithms. –  Joseph Malkevitch Dec 25 '12 at 16:18
    
In light of the holiday season you might enjoy looking at this video about gift wrapping a cube: youtube.com/watch?v=TNqc2yWZztE –  Joseph Malkevitch Dec 25 '12 at 16:27

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You are correct that you need the perimeter of the end rectangle (which becomes the circumference of the cylinder of paper-it looks like you don't need to match the pattern on the ends). It depends upon what you mean by no amount of folding can fix this. I can certainly construct a cylinder of paper that is larger than required and properly matched. A mountain fold and valley fold spaced by the difference between my cylinder circumference and the box circumference will make a tight fit. You may complain that I have just moved the pattern mismatch to the fold instead of the seam.

Another approach that will sometimes work is to make the long axis of the paper spiral around the box. You can then lengthen the circumference as required to make a match along the seam. It may then not match along the short axis, however. If the circumference of the box is $c$ and the offset is $h$, the length from edge to edge is $\sqrt {c^2+h^2}$. If this is a multiple of the long axis repeat and $h$ is a multiple of the short axis repeat you are there.

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why can the box not be perfectly wrapped?

instead of thinking in mathmatical terms to prove this, think of some logic..

Since the box contains six faces, all with the same dimensions creating a cube, we can look at this as being one face to the multiple of six creating a square.

The roll of gift wrap may only come in a larger area of a square or rectangle, but the object of this is to create a perfectly wrapped box, each overlap will be consistant.

So by theory we can cut out one face, perfectly with equal dimensions and multiply that to an identical number of six..

If you were to cut the gift wrapping paper into the layout of an unfolded box, like a large cross like formation for a better image, then each overlap will consist of two meeting faces. This will not create the apperence of a perfect wrapped box, because in my eyes,there should be one face that is void of any tape, or creases.

But back to theory that can evenly wrap a box by creating a sketch outline of gift paper would create a perfectly congruent wrapped box with equal dimensions.

The best possibility for not receiving the perfect box is human error. Always human error.

If you are asking to disprove the perfect but by only cutting the gift paper once to create a rectangle that can be folded around the surface area of a cube with equal dimensions of overlap and apperence, then the perfect box is unconcievable. since the gift wrap would be more abundent at the origin of contact and wrapping.

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