Origami design secrets mathematical methods for an ancient art
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Fold and unfold in half vertically and horizontally. Right: the new flat-foldable crease pattern. It is impossible for me to identify everyone who has contributed to my work, but some of the larger pieces come from the following, who I thank: Neal Elias, for his encouragement and for introducing me to the magic of box pleating and the realization that anything was possible in origami. To make the box longer, stretch the paper apart. Universal molecule for the polygon shown in Figure 11. But there is a more important difference: In the second form of the box, the raw edges of the paper are exposed on the top side of the box. Hull, The Mathematical Intelligencer, March 2005 This magisterial work, splendidly produced, covers all aspects of the art and science.

I'm also now able to better visualize how a crease pattern will collapse and what could come out of it. By this measure, valley, mountain, and crease are all part of a continuum of fold angle. Computerized solution offers an additional benefit: precision. The raw edge can actually take on any three-dimensional path whatsoever, as long as the mating part takes on the same path. Fold a rabbit ear from the lower layer. Correspondence between the parts of the folded model and the crease pattern.

The crease pattern for the small box. The analytical models are verified by the close agreement between the travelling waves predicted by the model and those measured in the experiments. On the other hand, as we increase the inset distance, there comes a point beyond which one or more of the reduced path constraints is violated. I'm not sure that it would interest everyone who picked it up. Subtrees and Subbases It can be shown that active paths cross each other only at leaf vertices.

Swing one flap over to the left. Reverse-fold the top four points out to the sides; mountain-fold the bottom pair out to the sides. Any given technique may contribute to some criteria and perhaps degrade others. By learning a variety of design techniques, the origami artist can pick and choose to apply those techniques that best contribute to the desired effect. Are there any other features on the square that we can identify on the base? The chapter presents the most common molecules, which are sufficient to construct full crease patterns for any uniaxial origami base. Fold some layers to the right.

The origamiadapted design process is illustrated and tested using three examples of preliminary design: an origami bellows to protect the drill shafts of a Mars Rover, an expandable habitat for the International Space Station, and a deployable parabolic antenna for space and earth communication systems. When he folds paper, he doesn't just make objects, he creates art. In the string-of-beads method, the tree is converted into a large polygon in which each corner is one of the leaf nodes of the tree, and each side is as long as the path between adjacent leaf nodes. C C C D H Left: tree with all nodes lettered. Reverse-fold the corner in and mountain-fold the edge on the existing crease.

So the box can be made longer by adding more paper to the starting rectangle. Fold a group of edges over to the right. Move that vertex the tiniest amount away from the center, changing nothing else, and the crease pattern becomes unfoldable or rather, un-flat-foldable; it can no longer be pressed flat without creating wrinkles. Reverse-fold one of the upper horns down and to the right. Engineers have managed to design several origami-adapted products that are innovative in their respective fields. Open out the two flaps to form small cups.

If you construct a box according to the prescription in Figure 12. Measure and mark off four points along the edges and three in the interior. Valley-fold downward a point in front and behind. Traditional Bases 53 Folding Instructions Stealth Fighter Snail Valentine Ruby-Throated Hummingbird Baby 5. These are illustrated in Figure 11. A base with narrow flaps will require many folds, no matter how you design it. A brief discussion on origami axioms will introduce origami geometry.

But as a starting point for origami, boxes are somewhat limited: you can only use them to make things that are, well, box-like. The results then are compared to common methods in the literature for assessment of standing to traveling wave ratio. Dotted lines are lengths that exceed their minimum value; solid green lines have lengths equal to their minimum value. Fold the corners over and over on existing creases. With corrections and improved illustrations, this new expanded edition also covers uniaxial box pleating, introduces the new design technique of hex pleating, and describes methods of generalizing polygon packing to arbitrary angles. However, it can be solved directly, algebraically.