Origami: Complexity in Creases (Again)

Bern and Hayes showed that some origami crease patterns can be used to encode NP -complete logic problems; solving the crease pattern would be equivalent …

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Origami: Complexity in Creases (Again)
In this origami composition, “Hummingbird and, Trumpet Vine,” Lang folded, the bird, each blossom, and, each leaflet from a single,, uncut square of paper. Tsune Abe’s trisection, method works for any, angle less than 90, . There, are other methods that, work for larger angles. This brainteaser uses, genders instead of “true”, and “false.” Four couples, attend a party: Pat and, Kim, Renay and Leslie, Lynn, and Lee, and Sydney and, Chris. Each couple consists, of a husband and wife,, though not necessarily in, that order. Each of the, following trios includes, two members of one sex, and one of the other, 1. Leslie, Lynn, and Sydney;, 2. Pat, Leslie, and Chris;, 3. Kim, Renay, and Lynn;, 4. Kim, Leslie, and Lee. If all the husbands get, together in one room and, all the wives in another,, who is in the room with, Renay? For the answer, see, page 44. Peter Messer’s construction, of the cube root of 2. Left: The traditional Bird, Base and Frog Base are, uniaxial bases. Right: John Montroll’s, famous Dog Base is not,, having two distinct axes. A folded uniaxial base, (right) casts a tree-graph, shadow. We can take, many paths between leaf, vertices P and Q, the, shortest of which (path A), is the same length as the, shadow. When the paper, is unfolded (far right), this, path becomes the crease, connecting P and Q. Origami subjects with, relatively large, rounded, bodies are not so well, suited to tree theory. This, figure, “Night Hunter,” uses, a mixture of tree theory, and intuition in its design. Top: The tree graph for a, six-flap base. Bottom: The skeleton of, the crease pattern that, will fold into the, corresponding base. Now, all we have to do is, construct the secondary, folds that will, for example,, bring together all the, points labeled B. Top left: The crease pattern for the triangle molecule. Top right: The molecule is formed by, folding along the angle bisectors and bringing the three points marked D together as you, flip the paper over. (Dashed lines are valley folds, dot-dashed lines are mountain folds.), Bottom left: The folded molecule. If you now folded tips B and C forward to meet tip A, all, of the original triangle’s edges would now lie on a common line. Bottom right: The, molecule’s tree graph. Below: The two four-node, graphs, and some, molecules that fold into, them. Heavy black lines, are valley folds, heavy, green lines are mountain, folds, and the light black, lines are “hinge creases,”, which may be mountains,, valleys, or unfolded,, depending on the flap’s, role in the model. A one-cut puzzle. Cut out, this rectangle and fold it, only on the solid lines, mountain folds are green, and valley folds are black. Then cut through all layers, along a line that runs from, point A to point B (from, dot to dot) and carefully, unfold all the pieces. Origami tree theory works, best for subjects that can, be approximated by a stick, figure. It is especially, suited to insects like this, cicada, in which each of, the legs, wings, antennae,, and body can be represented by an edge of the, tree graph. Calculating the crease, pattern for this scorpion, base required solving a, complex optimization, problem using CFSQP. Lang in front of the fivemeter Eyeglass prototype, at Lawrence Livermore, National Laboratory.
tree graph, valley folds, uniaxial base, mountain folds, origami, crease pattern, tree theory, folded uniaxial base, fivemeter eyeglass prototype, Origami tree theory, Heavy black lines, Renay, paper, larger angles, origami composition, Tsune Abe, folded molecule, uncut square, six-flap base, Trumpet Vine, triangle molecule, distinct axes, Peter Messer, method works, Dog Base, Frog Base, Top right, cube root, dot-dashed lines, page 44, Night Hunter, angle bisectors, hinge creases, one-cut puzzle, Lawrence Livermore, solid lines, original triangle, complex optimization, traditional bird, National Laboratory, green lines, common line, Sydney, room, trisection, brainteaser, genders, leaflet, blossom, Hummingbird

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