Reconstructing David Huffman's Origami Tessellations
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Reconstructing David Huffman’s Origami Tessellations Eli Davis Erik D. Demaine Martin L. Demaine Jennifer Ramseyer MIT CSAIL MIT CSAIL MIT CSAIL MIT CSAIL Cambridge, MA 02139 Cambridge, MA 02139 Cambridge, MA 02139 Cambridge, MA 02139 [email protected] [email protected] [email protected] [email protected] David A. Huffman (1925–1999) is best known in com- ing to the symmetries of one of the 17 periodic “wallpaper puter science for his work in information theory, particu- groups”. Origami historian David Lister [4, 5] attributes the larly Huffman codes, and best known in origami as a pio- origin of origami tessellations to Shuzo Fujimoto, a Japanese neer of curved-crease folding. But during his early paper schoolteacher of mathematics who self-published the first folding in the 1970s, he also designed and folded over a origami tessellation book Twist Origami in 1976. In 1966, hundred different straight-crease origami tessellations. Un- however, artist Ronald Resch patented several origami tes- like most origami tessellations designed in the past twenty sellation designs [6]. Huffman met Resch in 1968, and they years, Huffman’s straight-crease tessellations are mostly talked more extensively at University of Utah in 1973, while three-dimensional, rigidly foldable, and have no locking Huffman was on sabbatical and Resch was a professor there. mechanism. In collaboration with Huffman’s family, our Both Huffman and Resch may have been influenced by these goal is to document all of his designs by reverse-engineering discussions, though we have no clear documentation to this his models into the corresponding crease patterns, or in some effect. Huffman’s tessellation designs spanned the 1970s cases, matching his models with his sketches of crease pat- while he was a professor at University of California, Santa terns. Here we describe several of Huffman’s origami tessel- Cruz. lations that are most interesting historically, mathematically, and artistically. An explosion in artistic origami tessellations over the past two decades includes Chris Palmer’s work in fabric, Joel 1 Introduction Cooper’s representational sculpture, and Eric Gjerde’s book David A. Huffman was a brilliant mathematician, com- [7]. On the mathematics side, Lang and Bateman [8] recently puter scientist, and artist. Sadly, he wrote only one paper de- characterized a general family of flat-folded origami tessel- voted to mathematical paper folding, in 1976 [1]. His paper lations, while another recent result [9] can be interpreted as describes fundamentals of both straight and curved creases, algorithmic design of a family of 3D origami tessellations. in particular using a dual diagram to analyze both local be- havior and interactions among creases. Beyond his paper, Huffman’s tessellations are remarkably different in style Huffman designed and folded hundreds of models and sculp- from most modern tessellations as well as Fujimoto’s histor- tures, and took copious notes on his ideas and designs, but ical tessellations. The latter tessellations typically fold flat never published and only twice exhibited his work (at U. C. and are locked (not rigidly foldable): the finished piece is Santa Cruz in 1977 and at Xerox PARC in 1998). More re- two-dimensional and, from the folded form, it is impossi- cently, a 2012 exhibit [2] revealed several of his pieces to the ble to flatten back out into its original square without bend- public for the first time. ing along non-crease lines. Most of Huffman’s origami tes- In this paper, we focus on Huffman’s straight-crease de- sellations, on the other hand, are three-dimensional and not signs, particularly his work in the area now known as origami locked: they can be rigidly folded into their final form by tessellations. (Huffman’s pioneering curved-crease designs bending the material just at the crease lines. are the study of other ongoing work [3].) An origami tessellation is a folding design where both Our ongoing work, in collaboration with the Huffman the crease pattern and the folded state use repeated ele- family, aims to document, reverse engineer, and analyze ments to form a two-dimensional pattern, typically accord- David Huffman’s origami tessellation designs, of which there are over one hundred. In this paper, we detail the reverse-engineering process for one representative example, The authors are listed in alphabetical order. A preliminary version and detail (our reconstructions of) one family of closely re- of this paper appears in the ASME 2013 International Design Engineer- lated models. ing Technical Conferences & Computers and Information in Engineering Conference (IDETC/CIE 2013). Figure 1: David Huffman’s drawing of an origami tessella- tion crease pattern (date unknown) in which the triangle is nearly equilateral after being rounded to have corners on a square grid. Used with permission of the Huffman family. Figure 2: Our photograph of David Huffman’s design “Three Axis Woven Design” (date unknown), which was the basis of our reconstruction. 2 Reconstruction Process Our starting point is Huffman’s physical models, sculp- tion into tileable units, which are repeated to form the over- tures, and handwritten notes, which are being photographed, all model. From there, we reconstructed a crease pattern for scanned, and organized into an archive as part of the curved- each unit using a mixture of techniques, and then tiled a page crease analysis [3]. In some cases, we have both Huffman’s with those units. hand drawing of a crease pattern as well as his folded model, but often we have only one or the other. Our reconstruction process differs depending on whether we work mainly from a (scanned) hand-drawn crease pattern or (a photograph of) a 3 Reconstructing “Three Axis Woven Design” folded model. In either case, our goal is to produce an accu- For sake of example, we detail the reconstruction pro- rate vector drawing of the crease pattern, print it out (using a cess for one of Huffman’s most stunning origami tessella- Graphtec cutting plotter to score the crease lines), and fold it tions. Figure 2 shows our starting point, and Figure 6 shows to ensure that we correctly reconstructed the design. our finished reconstruction. This was one of our first re- When starting from a crease pattern, our process is fairly constructions from a photographed model. In fact, after fur- simple. We estimate the exact positions of the crease lines ther searching through the Huffman archives, we discovered by determining the underlying grid system—most patterns Huffman’s crease pattern for the model, shown in Figure 5, follow a regular square or triangular/hexagonal lattice— which turns out to be identical to our reconstruction, aside and measuring lengths of creases as accurately as possible. from Huffman’s rounding of equilateral triangles to a square Sometimes the resulting folding is not accurate enough, so grid. Nonetheless, it serves as an instructive example of our we adjust the creases and repeat. reconstruction process. One interesting detail is that, while Huffman’s designs Refer to Figures 3 and 4 for notation and step-by-step often appeared to have triangular or hexagonal symmetry, he partial reconstructions. drew his crease patterns on (square) graph paper, so he often rounded the triangular grid onto the square grid. Figure 1 Step 1. The inset hexagon is formed by a hexagon of shows a simple example. Mathematically, this rounding re- creases: A1;A2;A3;A4;A5;A6. moves some of the symmetry in the design, but in practice the difference is imperceptible. Our reconstructions treat this Step 2. The little orthogonal valley fold is crease B1. From rounding as an artifact of Huffman’s process and not part looking at the interior wall of the hexagon, we see that said of the mathematical design, and thus use an exact triangular interior wall is made out of right triangles (angles unknown) grid. so we know that segment B1 is perpendicular to segment A1. Reconstructing a design from a photograph is much more complicated. Huffman often explored many variations Step 3. In the intersection of A1 and B1 is segment C1, of a design, and we exploit this fact by modifying previous which runs behind the triangle that makes up the outer face reconstructions to determine many reconstructions, often by of the interior hexagon wall. We cannot actually see C1, but guessing and trying small variations until producing the de- because of the symmetry of the hexagon, we can infer that sired one. For the first design in each family, however, we it exists. The angle between B1 and A1 is 90 degrees. The needed a more precise strategy. First we broke each tessella- angle between B1 and C1 is also 90 degrees. Thus, the angle F P E P U D B B C G A Q C ’ E’ A A Q C A’ A S Q U’ P’ T R C ’ Figure 3: Points and lines of interest in Figure 2 for reconstructing Huffman’s “Three Axis Woven Design”. B C C B B D A A A A A (a) Step 1 (b) Step 2 (c) Step 3 (d) Step 4 P F F E P E C P E C C D Q D P P A A B B C S Q Q (e) Step 5 (f) Step 6 (g) Step 7 (h) Step 8 F F F C C U C' C' C' E’ G G U E' A' A' C R C R H C' C' U S T S T U' P' (i) Step 9 (j) Step 10 (k) Step 11 (l) Step 12 Figure 4: Step-by-step partial reconstructions of Huffman’s “Three Axis Woven Design”. between A1 and C1 is 180 degrees, meaning that A1 and C1 Step 5.