Mathematics and Origami

Origami is the art of paper folding, which is often associated with Japanese culture. The goal is to transform a flat square sheet of paper into a finished sculpture through folding and sculpting techniques. Modern origami practitioners generally discourage the use of cuts, glue, or markings on the paper.

More than just an art, it has received a considerable amount of study in mathematics. Anybody familiar to paper folding would appreciate the wide ranging implications in Euclidean geometry. Just like Euclid’s axioms, paper folding has its own axioms – The Huzita-Hatori Axioms:

1. Given two distinct points p1 and p2, there is a unique fold that passes through both of them.
2. Given two distinct points p1 and p2, there is a unique fold that places p1 onto p2.(The fold gives the perpendicular bisector of the line segment p1 p2).
3. Given two lines l1 and l2, there is a fold that places l1 onto l2.
4. Given a point p1 and a line l1, there is a unique fold perpendicular to l1 that passes through point p1.
5. Given two points p1 and p2 and a line l1, there is a fold that places p1 onto l1 and passes through p2.
6. Given two points p1 and p2 and two lines l1 and l2, there is a fold that places p1 onto l1 and p2 onto l2.
7. Given one point p and two lines l1 and l2, there is a fold that places p onto l1 and is perpendicular to l2.

One can easily visualize or try these out to verify these axioms. Let’s begin with Haga’s theorem: Suppose you have a square sheet of paper and you want to divide its side in any arbitrary rational ratio. To a casual eye, it might seem impossible to do so without measuring the side with a ruler. Let’s say you want to divide a side (say BC) of a unit square ABCD into 3 equal parts.Take an adjacent side AB and fold A toB forming a crease P. Now, AB=AP. Now bring D to P and make a valley fold. It should look like the diagram shown below:

Now we have AP=PB=½Let’s call AR=x , so DR=RP=(1-x) Triangle ARP is a right angled triangle and we can use the pythagorean theorem (I forgot to mention that paper has zero Gaussian curvature). We have

$$ AR2 + AP2 = RP2X2 + (½)2 = (1-X)2 $$

Which gives X=⅜. We can see that triangles ARP and BPQ are similar, which gives:

$$ AR/AP=PB/BQ(⅜) / (½) = (½) / (BQ)BQ=⅔ $$

Eureka! We finally managed to fold a paper into thirds. Similarly if we divide AB into k and (1-k) and do the same thing, we get BQ=(2k/(1+k)).

Origami and paper folding has wider applications in real life than you might have imagined. Firstly, folding minimizes space and it can reduce the space taken by solar panels in satellites. Nature uses folding to pack leaves and flowers in a bud.

Certain folds such as the Miura fold give a lot of strength to the paper and can be used in architecture (plus it looks cool).

If you were not convinced that mathematics was art before, you should be now. Even if you don't have the faintest idea what Tensor contraction is, I'm sure that you will be able to appreciate the beauty in naturally occurring patterns. Now tell me, if you've ever folded a paper in your life, are you a math person?