David Lister on why a crease is a straight line

John Cunliffe asks why it is that when a crease is made in a sheet of paper the result is a straight line, rather than a curve (British Origami No.121, page 13). It is obvious what he means, but it only applies when we are considering plane sheets of paper. For a start, if you fold a sheet of paper in half the crease is only a straight line as long as the folded sheet is kept flat. Once you bend the folded sheet of paper, (so that it is no longer in a single plane) the crease becomes a curve!

Then again, you can fold a curved crease in a sheet of paper: it is a thing paper sculptors do as a matter of course. The simplest way is to score the paper with a semi sharp point to mark the line of the intended crease before you fold it: then it becomes easy. The paper can then be folded along the crease. BUT, the paper will not lie flat – in other words it will no longer lie in a single plane. Paul Jackson has demonstrated creases of this kind at his teach-ins at recent Conventions of the Society.

Three dimensional origami often produces curved fold, especially of the first sort (the “Susan” and “Super Susan” are simple examples). The second sort appear when you try to sharpen up creases in folds of things like dishes in the style of Philip Shen and they can be found on close examination in three-dimensional animal folds, usually formed towards the end of the folding process and often not pressed sharp. They are one of the tricks which Yoshizawa exploits to make his animals so lively: this is often why his models are so difficult to copy.

But this is all begging the question: why does the simple fold in a plane sheet of paper make a straight line? The answer is given in Martin Gardner’s “More Mathematical Puzzles and Diversions” (1961) page 128 where he writes:

“The very act of folding raises an interesting mathematical question. Why is it that when we fold a sheet of paper, the crease is a straight line? High-school geometry texts sometimes cite this as an illustration of the fact that two planes intersect in a straight line, but this is clearly not correct because the parts of a folded sheet are parallel planes. Here is the proper explanation, as given by L. R. Chase in the American Mathematical Monthly for June-July 1940:

“Let p and p’ be the two points of the paper that are brought in coincidence by the process of folding; then any point on the crease is equidistant from p and p’, since the lines ap and ap’ are pressed into coincidence. Hence the crease, being the the locus of such points a, is the perpendicular bisector of pp'”.

So now you know! Yet… in some ways this begs the question. It all turns on the Euclidian proposition that the locus of all points equidistant from two other points is a straight line, and also the fact that Euclidian geometry confines itself (at this level at any rate) to the geometry of the simple plane. Go a little deeper and you may start asking why the locus of all points equidistant from two other points is a straight line. If you do, you may get into some very deep water or perhaps ascend to some very rarified planes.