Is it possible to fold a piece of paper such that it has a smooth, wavy edge?

Question

Can you show mathematically why it is or isn't possible? By a "smooth, wavy edge", I mean a non-straight, undulating edge with no creases running into it, i.e., the wavy edge is the only crease in the paper.

Alexander, you've given an excellent reason why a paper folded onto ITSELF cannot have a wavy crease. However, what about partially folded paper, where the final surfaces have some angle between them?

I didn't say that the fold was complete and flat on the table.

I'm extending this question because I'd like to see if anyone can offer a mathematical opinion in the case of a partial fold, instead of a flat one.

Answer 1

Real World answer: Yes, I just did it. I made a nice little sine wave shape. I causes the paper to stand up like a low sloping roof with a curved edged top and undulations on the surface corresponding to the curves on the top.

Mathematical World: I don't know.

* It is fairly easy to imprint a sine wave curved on to a piece a paper. Find a nice sharp edged curved object. I used a CD holder. Press lightly and crease the paper for a 1/4 radius then repeat the process for a curve going in the other direction.

>>>Edit

Alexander may be right. He usually is. (And I confess that I learn a great deal looking at his work). But two premises of his proof here seem shaky.

First, he assumes that for every A, there is a unique A' on the other side of the fold. But if E is a wavy, curved edge, that is not necessarily true. The paper on either side of the curve is no longer flat, but curved. There also may be more than one point on E which is perpendicular to a line from A to E.

Second, ACB and A'CB may be right angles, but only if the paper is curved in a specific way. And the line CB does not necessarily lie on E except at points C and B.

>>>>>Edit 2

Yes, Mathmatically, you can.

You can only bend paper in one direction at a time (not like a sphere where you have to bend the surface in two directions at a time), but that direction can vary point to point. So, you can have a smooth wavy edge. (The edge, however, has to be 'flat')

You are also going to have to make sure that you don't turn the direction of the edge more than or equal to 180 degrees.

Good Intro on Guassian surfaces: http://www.geom.uiuc.edu/docs/doyle/mpls/handouts/node21.html

More math about Riemann curvature tensor and Ricci curvature and Gaussian surfaces (basically, how to mathmatically): http://www.slimy.com/~steuard/teaching/conversations/paperbend.html

Practical way to bend (engineering the bend): http://www.math.nmsu.edu/~breakingaway/Lessons/creasingpaper1/creasingpaper.html

Answer 2

Let me show why the edge is always straight line, (regardless of the 5th postulate).

Lets denote set of ponts on the edge as set E, and choose arbitrary point A not on the edge. The point on the other side of the fold corresponding to A is A'. Now unfold the paper, making it euclidean plane.

http://mathworld.wolfram.com/Elements.html

According to postulate 1 it is possible to draw straight line AA'.

Since points A and A' are on the other sides of continuous set E, there is point C from set E which belongs to line AA'.
Choose any other point B from set E.

Because the paper was originally folded, we have conditons
AC = A'C and
AB = A'B.

Triangles ABC and A'BC are congruent because all three of their sides are equal.

Angles ACB = A'CB, and are therefore right angles.
The straight line CB is therefore perpendicular to line AA', and because such straight line is unique all points from set E belong to this straight line CB :
{E} = CB


Playing a little bit with postulate 3 as shown here
http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI22.html
one can deduce, that sum of lines AB and AB' is greater than AA'

Answer 3

Yes. Look at the work of David Huffman, UC Santa Cruz faculty who died in 1999. The following link is to a copy of a 2004 New York Times article, and includes the surprising illustrations.

http://www.skypape.com/huffman.htm

Answer 4

Curling iron?