# Mathematical Origami Design

Robert Lang speaks about mathematical origami design

Friday morning (the 15th) I was privileged to hear several talks on the mathematics of origami, including one by (my personal hero) Robert Lang.  Since early childhood I have been fascinated by origami, influenced primarily by my mother.  Only recently have I discovered that origami is a serious mathematical discipline!

The talks were about different areas of mathematical origami.  Lang talked very directly about how the art of origami is related to, and feeds off of, mathematics.  Traditionally, the two have had little overlap.  But starting in the mid-20th century, said Lang, there was a renaissance in the art of origami that made it possible for mathematicians to become involved.  Akira Yoshizawa took the ancient craft of origami and, among other things, introduced a standard folding notation.  This allowed artists to share and build upon their designs and ideas, and led to unprecedented advances in complexity and intricacy of origami models.

Mathematicians are interested in paper-folding for many reasons.  If, instead of the standard compass and straightedge, we use folds, then we can construct many new things.  For example, the old problem of trisecting an angle (which is impossible with compass and straightedge), can be done using folds.  Also interesting is that the crease pattern of a model – that is, the pattern of lines that would be left if a model were completely unfolded – must be 2-colorable.

Most interesting, and what Lang spent most of his time talking about, is how mathematics can be used to help guide the creation of ever more complicated figures, especially representational models.  To the uninitiated, this (like many problems in mathematics!) at first seems daunting – almost impossible.  How can we possibly abstract an organic form into a foldable model?  Lang answered this question, breaking down the creation process into steps.

We start with the subject, perhaps a spider.  Then we make a very simple tree graph of the spider – basically a stick figure spider.  We make sure to include all the important limbs  (8 legs, etc.).  From this tree graph we can produce a crease pattern that guides us to folding a “base,” something with the correct number of free flaps that can then be folded into the limbs of the finished model.  From this base, it is rather easy to finish the model – polish it to make it look realistic.

It is the step from the tree graph to the base model that is extremely complicated.  The first step in figuring out this process is to realize that flaps represent circles – or parts of circles – in the crease pattern.  Eventually, the problem reduces to circle packing in the standard origami square.  Lang, among others, has refined this process and has created a computer program, TreeMaker, that will generate a guiding crease pattern that, if folded correctly, will produce the desired number of flaps.  This, and similar programs, has allowed origami design to achieve unprecedented complexity.

Some of my favorite of Lang’s models are: a Klein Bottle (I kid you not!), a rattlesnake, a scorpion, an assyrian bull, an organist, a fedora, and a golden eagle.  There are many, many more on his website!

As you may have heard, there are numerous applications of origami to problems outside of fine art.  In Lang’s words, anything that needs to be “small for the journey, large for the destination” has the potential to be designed using origami methods.  For example, space telescopes benefit from packing tightly for their launch into orbit, but then being able to unfold in order to have a large mirror for collecting light.

Again: any questions, comments, complaints? Diatribes, panegyrics, philippics?  Feel free to comment below!

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### 2 Responses to Mathematical Origami Design

1. Diana says:

Wow, great pictures! I can’t believe he could get eight legs out of a single sheet of paper without cutting it. I want to see the video of how it folds up!

A recent Williams thesis by Tomio Ueda ’06 dealt with origami folding of thick paper (rather than some sort of idealized, 0-thickness paper). This apparently has applications to the sort of outer space applications that you mention — folding up the solar panels for flight, then unfolding them at the destination.

2. Math Game Reviews says:

Absolutely fantastic! Really inspiring to see what can be accomplished and in what diverse areas of life, all through practical application of math!
Thanks for the link to his website – he’s even got origami dinosaur skeletons.