The golden rectangle and the familiar DIN A4 sheet of paper, mentioned last week, share the property of easily self-replicating. If we remove a square from the golden rectangle with a side equal to its smaller side, the remaining rectangle is similar to the first (and, therefore, is also golden).
If we take the shorter side of the initial rectangle as a unit and call its longer side x, for both rectangles to be similar, their sides must be proportional, so:
x:1 = 1:(x-1)
x2 – x – 1 = 0
x = 1.618… = Φ (the golden ratio)
In the case of the DIN A4 sheet of paper, self-replication is even simpler: by folding it in half, we obtain two DIN A5 sheets similar to the entire sheet. If we again take the shorter side of the sheet as a unit and call the longer side x, we will now have:
x:1 = 1:(x/2)
x²/2 = 1
x = √2 = 1.414…
The name DIN A4 comes from the German Institute for Standardization: DIN is the acronym for the Deutsches Institut für Normung, and the 4 indicates the number of times the original sheet, A0, measuring 841×1189 mm, must be folded to obtain the familiar 210×297 mm format, which roughly corresponds to the old folio , in the same way that A5 approaches the old page and A6 approaches the leaflet.
If in the attached figure we appropriately changed the placement of A3, A4 and the following, we would have the framework of a peaked quasi-spiral similar to the well-known golden spiral: the DIN pseudospiral. What can you say about her?
And, when it comes to folding, how many times do you think you can fold and redouble a sheet of paper? If you try, you will see that you will not be able to fold it more than seven consecutive times. The thickness of a normal sheet of paper is approximately one tenth of a millimeter, so after seven folds you will have in your hand a small and impractical billet about 13 mm thick. But if you could keep folding and refolding indefinitely, how many times would you have to fold a sheet of paper for the resulting billet to reach the Moon? What size would the sheet have to be for the operation to be possible (in the ideal case that the paper offered no resistance to bending)?
As for the pentacle, if you have not yet found the proportion between its area and that of the inverted pentacle inside, you have a second chance: keep in mind that in a pentacle there are 10 isosceles triangles, 5 acute angles (the points of the star) and 5 obtuse angles, and in all of them the ratio between the largest and smallest sides is the golden ratio (1.618…). And in a regular pentagon, the diagonal and the side are also in the golden ratio. What, then, will be the ratio of the area of the pentacle and that of its inner antipentacle? A warning: in a famous short story by Fredric Brown, a would-be witch is trapped by the devil for not knowing enough pentacular geometry, so…
From DIN folding to map folding
Returning to the art of folding a sheet of paper, a question no less interesting than how many times it is physically possible to do it is how many different ways we can fold it. If it is a blank sheet to be folded in half in the usual way (i.e., joining the shorter edges), the question is irrelevant; but if it is a printed sheet with images and/or texts on both sides, we can fold it in two ways, depending on which side we leave out, and when we fold it again we have two possibilities again, and so on. That is, we have 2ⁿ possibilities, with n being the number of doublings (which in the case of the DIN format always gives rectangles similar to the first).
But when in the real world we find a paper folded several times, it is not usually in successive folding of DIN-type mediators, as it would not be practical when, for example, displaying a map. And then things get considerably complicated.
In how many different ways can we fold a sheet of paper twice in a row if “elongated” folds are also valid, that is, joining the longest sides? And three times, four…?
And if we also contemplate folding that is not in half, like the ones we actually find on maps, we enter a complex and elusive field of combinatorics that mathematicians themselves describe as “irritating”: map folding. But that's another article.
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