Circle packing, Origami, and RepRap

Eric Joisel's "Woman in Dress 2008"

Eric Joisel's "Woman in Dress 2008"

One of my interests outside MakerBot/RepRap is origami, the Japanese art of paper folding.  At it’s inception it was really only a hobby for the rich – the only ones who had access to such a luxury as paper.  Modern technical origami restricts one to only a single sheet of square paper transformed only through folding – no cuts, glue, tape, etc.

A lot of very interesting origami models have been developed over the years by some incredibly talented artists through a combination of experience and trial and error.  These kinds of models have so much personality they almost appear to be real – rather than mere squares of paper.  One of my favorite origami artists of this “genre” is Eric Joisel.  I still find it hard to believe his models, such as the “Woman in Dress 2008,” can really be made of just a single square of paper.

There’s been a movement in the last two decades to bring a more methodical and mathematical approach to design, sometimes with the assistance of a computer.  One of my favorite technical origami artists is Robert Lang).  His models tend to the more realistic, rather than representational (as with Joisel).  His insects, such as the stag beetle, are a prime example.

Lang has used his background in mathematics and origami to develop a giant collapsing/expandable lens – so that it could be transported to outer space and then deployed.  This allows a normal space craft to deploy a much larger lens than would otherwise be possible.

Robert Lang's "Stag Beetle BP, opus 477"

Robert Lang's "Stag Beetle BP, opus 477"

Unlike Josiel’s models which are typically totally unique and irreproducible by anyone (including Josiel!) Lang’s mathematically assisted models are usually carefully documented by diagrams or, increasingly frequently, crease patterns.  A crease pattern is what you would get if you totally unfolded a completed origami model.  Typically only the “major” structural folds are depicted in a crease pattern.

Brian Chan's "Attack of the Kracken"

Brian Chan's "Attack of the Kracken"

An interesting intersection between mathematics and origami is the problem of determining the most efficient manner of placing the most number of equally sized circle within the smallest possible area, called, “circle packing.”  The reason this is important to origami is that the center of each circle can be turned into an appendage.  Lang has developed a computer program that allows the creation of truly arbitrary proportions – any number of points with any kind of ratio of one appendage to another.

Brian Chan's crease pattern for "Attack of the Kraken"

Brian Chan's crease pattern for "Attack of the Kraken"

Even to someone who doesn’t have any experience with origami or the mathematics involved, the appearance of circles in crease patterns can start to make intuitive sense.  One of my all time favorite origami models is the “Attack of the Kraken” by Brian Chan.  (Check out the larger pictures of that model.  The entire thing is just one sheet of paper.  If you look closely you’ll see some of the tentacles are grabbing sailors!)  In addition to a picture of his final model he has also posted a picture of his crease pattern for that model, even labeling which parts of the model are derived from which regions of the sheet of paper.

I was reminded by all of this after reading Nophead’s recent post on circle packing as it relates to the optimal placement of resistors on a heated RepRap bed and reading Cory Doctorow’s Makers in which one of the characters is an avid origami folder.  As with RepRap, origami allows the production of complex models through an additive manipulation (layers with RepRap, folding with origami) of a simple material (plastic versus paper) in a methodical and repeatable process.

If you’re the least bit interested in origami, I’d highly recommend (in this order) Peter Engel’s “Origami from Angelfish to Zen,” a documentary on origami called “Between the Folds,” and Lang’s book “Origami Design Secrets.”  The Angelfish to Zen book is one of my favorite books for background on origami and a very accessible introduction to the hidden geometry  underlying even the most simple models (and real life too!).

One Response to “Circle packing, Origami, and RepRap”

  1. [...] Origami is another of my hobbies and it is all about design constraints.  The rules are simple1 – one square sheet of paper only manipulated by folding. 2  Yet, within these rules it is theorized that a sufficiently skilled artisan can design and fold any arbitrary figure.  I find folding origami to be at once cathartic and contemplative. 3 [...]

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