![\"Robert](\"http:\/\/blogs.ams.org\/mathgradblog\/files\/2010\/01\/Lang.jpg\" "\\"Robert")
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Robert Lang<\/a> speaks about mathematical origami<\/a> design<\/p>\n by Nicholas Neumann-Chun<\/a><\/p>\n Friday morning (the 15th) I was privileged to\u00a0hear several talks on the mathematics of origami<\/a>, including one by (my personal hero) Robert Lang<\/a>.\u00a0 Since early childhood I have been fascinated by origami, influenced primarily by my mother.\u00a0 Only recently have I discovered that origami is a serious mathematical discipline!<\/p>\n The talks were about different areas of mathematical origami. \u00a0Lang talked very directly about how the art of origami is related to, and feeds off of, mathematics. \u00a0Traditionally, the two have had little overlap. \u00a0But 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. \u00a0Akira Yoshizawa<\/a> took the ancient craft of origami and, among other things, introduced a standard folding notation. \u00a0This allowed artists to share and build upon their designs and ideas, and led to unprecedented advances in complexity and intricacy of origami models.<\/p>\n <\/p>\n Mathematicians are interested in paper-folding for many reasons. \u00a0If, instead of the standard compass and straightedge, we use folds, then we can construct many new things. \u00a0For example, the old problem of trisecting an angle (which is impossible with compass and straightedge), can be done using folds<\/a>. \u00a0Also 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.<\/p>\n 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. \u00a0To the uninitiated, this (like many problems in mathematics!) at first\u00a0seems daunting – almost impossible. \u00a0How can we possibly abstract an organic form into a foldable model? \u00a0Lang answered this question, breaking down the creation process into steps.<\/p>\n We start with the subject, perhaps a spider. \u00a0Then we make a very simple tree graph of the spider – basically a stick figure spider. \u00a0We make sure to include all the important limbs \u00a0(8 legs, etc.). \u00a0From 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. \u00a0From this base, it is rather easy to finish the model – polish it to make it look realistic<\/a>.<\/p>\n It is the step from the tree graph to the base model that is extremely complicated. \u00a0The first step in figuring out this process is to realize that flaps represent circles – or parts of circles – in the crease pattern. \u00a0Eventually, the problem reduces to circle packing in the standard origami square. \u00a0Lang, among others, has refined this process and has created a computer program, TreeMaker<\/a>, that will generate a guiding crease pattern that, if folded correctly, will produce the desired number of flaps. \u00a0This, and similar programs, has allowed origami design to achieve unprecedented complexity.<\/p>\n Some of my favorite of Lang’s models are: a\u00a0Klein Bottle<\/a> (I kid you not!), a rattlesnake<\/a>,\u00a0a\u00a0scorpion<\/a>, an assyrian bull<\/a>, an organist<\/a>, a fedora<\/a>, and a golden eagle<\/a>. \u00a0There are many, many more on his website<\/a>!<\/p>\n —<\/p>\n As you may have heard, there are numerous applications of origami to problems outside of fine art. \u00a0In 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. \u00a0For 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.<\/p>\n —<\/p>\n Again: any questions, comments, complaints? Diatribes, panegyrics, philippics? \u00a0Feel free to comment below!<\/p>\n