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Cambridge University Press 978-0-521-85757-4 - Geometric Folding Algorithms: Linkages, Origami, Polyhedra Erik D. Demaine and Joseph O’Rourke Index More information Index 1-skeleton, 311, 339 bar, 9 3-Satisfiability, 217, 221 base, see origami, base, 2 α-cone canonical configuration, 151, 152 Bauhaus, 294 α-producible chain, 150, 151 Bellows theorem, 279, 348 δ-perturbation, 115 bending λ order function, 176, 177, 186 machine, xi, 13, 306 antisymmetry condition, 177, 186 pipe, 13, 14 consistency condition, 178, 186 sheet metal, 306 noncrossing condition, 179, 186 beta sheet, 158 time continuity, 174, 183, 187 Bezdek, Daniel, 331 transitivity condition, 178, 186 blooming, continuous, 333, 435 bond angle, 14, 131, 148, 151 Abe’s angle trisection, 286, 287 bond length, 148 accordion, 85, 193, 200, 261 active path, 244, 245, 247–249 cable, 53–55 acyclicity, 108 CAD, see cylindrical algebraic decomposition, 19 additor (Kempe), 32, 34, 35 cage, 21, 92, 93 Alexandrov, Aleksandr D., 348 canonical form, 74, 86, 87, 141, 151 Alexandrov’s theorem, 339, 348, 349, 352, 354, Cauchy’s arm lemma, 72, 133, 143, 145, 342, 343, 368, 381, 393, 419 377 existence, 351 Cauchy’s rigidity theorem, 43, 143, 213, 279, 339, uniqueness, 350 341, 342, 345, 348–350, 354, 403 algebraic motion, 107, 111 Cauchy—Steinitz lemma, 72, 342 algebraic set, 39, 44 chain algebraic variety, 27 4D, 92, 93, 437 alpha helix, 151, 157, 158 abstract, 65, 149, 153, 158 Amato, Nancy, 157 convex, 143, 145 amino acid, 158 cutting, xi, 91, 123 amino acid residue, 14, 148, 151 equilateral, see chain, unit, 91 analytic isomorphism, 39 fixed-angle, 9, 132, 145, 147, 149, 154 angle flat state, 135, 137, 147 deficit, 303 interlocked, 124 solid, 303 protein, 15, 131, 148 space, 112, 113, 154 span, 133, 135 trisection, 33, 34, 285–289 unit, 153 annulus, 59–61, 437 flattenable, 135, 136, 150, 151 anticore, 371, 372 flexible, see flexible, chain, 124 approximation algorithm, c-, 160–162 folded state, 69 arc, see chain10 locked, 9, 20, 86, 88, 153, 154 arch algorithm, 81–83 not in 2D, 96, 105, 109 Archimedean solids, 312, 313 in 4D, 92 arm, 10 definition, 11, 86 robot, 9–12, 14, 16, 20, 59, 63, 131, 155, history, 88 156 from knot, 90 Aronov, Boris, 350 orthogonal, 436 assembly planning, 20 of planar shapes, 119 463 © Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-85757-4 - Geometric Folding Algorithms: Linkages, Origami, Polyhedra Erik D. Demaine and Joseph O’Rourke Index More information 464 Index chain (cont.) for deer, 3 locked hexagon, 90, 91 faces, 242 monotone, 85, 120–122, 138 flap, 242, 243 orthogonal, 138, 141, 142, 151 flat-foldable, 193, 199, 203, 205, 207–209, 214, polygonal, 4, 9, 10, 14 217, 222, 258 random, 151, 153 for lizard, 252 self-touching, 119 local, 169 simple, 91 piecewise-Ck, 185 unit, 69, 91, 96, 138, 153 simple, 194 unlocked, 86 single-vertex, 193, 198–201, 210, 212, 215 chains tesselation, 293 interlocked, 9, 24, 114, 123, 124, 130 for turtle, 258 separated, 123 unfoldable, 4, 212 circle, osculating, 301, 302 crease points, 175, 185 circular to linear motion, 31, 40 crimpable pair, 195, 196 Collins, George E., 19 crossing collision detection, 155, 157 geometric, 179 composition order, 179 of reflections, 437 cube of rotation matrices, 211 built from, 331 configuration, 10 doubling, 289 of chain, 149, 150 snub, 3 cis, 148 wrapping of, 237, 238 free, 11, 173 cuboctahedron, 299, 300, 333 lockable, 151 curvature, 147, 301–303, 352 self-touching, 115–117, 119 elliptic, 303 semifree, 11, 173 Gaussian, 191, 303, 304 trans, 148 geodesic, 304, 352 configuration space, 9, 11, 17, 20, 59, 94 hyperbolic, 303 for origami, 192 negative, 303 torus, 61 in origami, 199 conformational map, 157 and shortest paths, 358 connected sum, 90 and star unfolding, 368, 370 Connelly, Robert, 88, 347 unfoldable, 318, 319, 321 constructibility, geometric, 285 principal, 302 constructible coefficient, 287, 289 curve constructible number, 288, 289 monotone, 96 contractive motion, 109 slice, 376–378 contraparallelogram, 32–35, 37–41, 346 smooth, 145, 147, 301, 352 convex decomposition, 237, 247, 250, 253 cut locus, 359, 360, 367, 369, 371, 440 convex optimization, 104, 105, 340 cylindrical algebraic decomposition, 18, 19 convex programming, 113 convexification Dali, Salvador, 439 in 4D, 93 de Bruijn, Nicolaas, 125 of active-path decomposition, 247, 248 decision problem, 11 of chain, 87, 88, 97, 107, 111–113, 213 deflation, 78–80 by flipping, 74, 76 degrees of freedom, 17, 29 of polygon, 80–82, 94, 97 Dehn, Max, 342 core, 367, 371 deltahedron, 331 crease Descartes, Rene,´ 303, 305 from Alexandrov gluing, 350 development curved, 292, 296 of curve, 358, 371, 372, 375–378 hinge, 242 manifold of, 352 perpendicular, 252, 258–260, 263, 267, 282, 283 of polyhedra, 145, 299, 380 crease line, 227, 285 D-form, 296, 352, 353, 354, 419 crease pattern, 2, 169, 170 differential equation, of motion, 104, 105, 107 in 1D map, 226 dihedral angle, 132 in 2D map, 225, 226, 228 in blooming, 333 in 3D paper, 437 of crease, 212 corridor, 259–261, 263 in lifting, 57 for crane, 170 of polyhedron, 329, 340, 341, 407, 408, 431, 434 © Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-85757-4 - Geometric Folding Algorithms: Linkages, Origami, Polyhedra Erik D. Demaine and Joseph O’Rourke Index More information Index 465 random, 153 fewest nets problem, 308 sign changes, 342 flat folding, see folding, flat, 4 dihedral motion, 14, 131, 132, 138, 148 flat state, 133, 136 local, 132 flat-state connected, 136, 137, 141–143 Dijkstra’s algorithm, 158, 363 flat-state disconnected, 136, 138, 141 continuous, 362, 363 flattening directed angle, 77 4D, 280 disk packing, 240, 246, 255, 263, 264, 266, 267, 281 continuous, 279 dual, 267 polyhedra, xi, 279, 281, 283 equal-radius, 246 polyhedral complexes, 438 dissection, 372, 423, 424 trees, 154 hinged piano, 423, 437 flexibility double banana, 47 generic, 44–47 doubling infinitesimal, 52, 99, 103, 109, 116 of chain, 90 sloppy, 116 of tour, 251 flexible of tree, 97 Bricard’s octahedron, 346, 347 duality chain, 124, 126, 127, 129 stress, see stress, duality, 56 Connelly’s polyhedron, 347 Durer,¨ Albrecht, 2, 3, 299, 312 linkage, 43–45, 47, 116, 125 The Painter’s Manual, 2, 299 octahedron, 345 polyhedra, 345 ear polyhedron, 345–348, 357 of polygon, 189 Steffen’s polyhedron, 347, 348 of triangulation, 109 surface, 293 edge flipturn, 76, 80, 81 exterior, 90 fold flat, 57, 58 book, 225, 289 frozen, see rigid, edge, 132 complex, 225, 231 sequence, 363 crimp edge unfolding, see unfolding, edge, 5 1D, 194, 195, 197, 226 Edmonds, Jack, 224 for error correction, 247 Elephant Hide paper, 295 single vertex, 204, 205, 207, 215 elliptic distance, 112 end, 194, 197, 226 energy point, 382, 383, 392 in HP model, 158 simple, 170, 224–231, 290 minimization, 15, 113, 154, 158, 159 1D, 226 for unlocking, 111–113 all-layers, 225–228 equation one-layer, 225–227 cubic, 287, 289, 290 some-layers, 225, 226, 228 quartic, 289 fold-and-cut, 254, 255, 280 equilateral triangle 3D, 280 face, 331 multidimensional, 278, 280, 281, 438 folding, 394, 419 theorem, 254, 278 as obstacle, 67, 69 foldability Erdos,˝ Paul, 74 flat, 217 Erickson, Jeff, 441 1D, 197, 226 Euclid, 341 fold-and-cut, 278 Euler NP-hard, 217, 221 characteristic, 305 single-vertex, 200, 207, 208 formula, 342, 344, 345, 406 global, 170, 217, 222 method, 106 local flat, 214, 216 Euler, Leonhard, 43, 347 simple, 230 Eulerian tour, 239, 251 folded state, 1, 172 Eulerian walk, 337 of 1D paper, 175 EXP,22 of 2D paper, 183 expansive motion, 73, 88, 96–99, 107, 112, 121 final, 169 exponential map, 186 flat, 4, 169, 193, 224 rolling between, 189, 190 face path, 334, 336, 337 silhouette, 189 Farkas’s lemma, 56 free, 173, 184, 185, 212 © Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-85757-4 - Geometric Folding Algorithms: Linkages, Origami, Polyhedra Erik D. Demaine and Joseph O’Rourke Index More information 466 Index folded state (cont.) gift wrapping, 232, 237 of higher-dimensional paper, 437 Gluck, Herman, 347 origami, 169, 210, 212 gluing of protein, 15 Alexandrov, 349, 350, 381, 393, 396, 397, 402 semifree, 177 algorithm of uniaxial base, 243 decision, 402 folding edge-to-edge, 387 of carton, 14 general, 399, 402 flat, 169, 170, 193, 221, 222, 236, 280, 423 edge-to-edge, 386, 389, 396 1D, 193, 198, 204, 205 gradient descent, 111 cell complex, 437 graph of convex paper, 236 induced subgraph, 45–47 definition, 4 orthogonal, 138, 140, 141 of D-form, 354 outerplanar, 325 and dissection, 423 Grunbaum,¨ Branko, 299 of equilateral triangle, 419 for fold-and-cut, 254, 263 Hamiltonian cycle, 159, 160 of map, 224 Hamiltonian path, 160, 329 of n-gon, 413 Hamiltonian triangulation, 233, 235 of oriented tree, 262 Hamming distance, 159 perimeter, 239 Hart, George, 331 of polyhedron, 279, 281 Hart’s inversor, 33, 39, 40 single-vertex, 199, 204, 206 Hatori’s axiom, 285, 288 of square, 412 Hayes, Barry, 281 free motion, 174, 185 Henneberg construction, 47, 48 of Latin cross, 402 Henneberg, Ernst Lebrecht, 47 of map, 4, 224, 227, 228, 231 Henneberg’s theorem, 48 1D, 228 Heron’s formula, 348 motion, 182, 187, 189–191 Hirata, Koichi, 396, 399, 408, 423, 424 perimeter-halving, 382, 383, 394, 418, 420 homeomorphic, 27 nonconvex polyhedron, 384 Houdini, Harry, 254 perimeter-halving, 412 HP model, 15, 158–162, 164 of polygon, 5, 381, 431 Huffman, David, 295, 296 convex, 411 Huzita, Humiaki, 285 random, 382, 384, 385 Huzita’s axioms, 169, 285, 286, 288 regular, 289, 412 hydrophobic—hydrophilic, see HP model, 158 seam of, 236 hypercube, 439 of shopping bag, 292, 293 HyperGami/JavaGami, 312 of square, 411, 414 of strip, 233 infinitesimal motion, 49–54, 56, 99, 109, 110 unique, 162–164 inheritance property, 153 unique optimal, 161, 164 inside-out, 63, 64, 67, 86, 437 forbidden disk, 146 instability, degree of, 63 four-color theorem, 31 intractable problem, 22 Francesca’s formula, 348 inverse kinematics, 12, 156 Fukuda, Komei, 316 isometric function, 172, 174, 185 Fuller, R.
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  • On Rigid Origami I: Piecewise-Planar Paper with Straight-Line Creases

    On Rigid Origami I: Piecewise-Planar Paper with Straight-Line Creases

    On Rigid Origami I: Piecewise-planar Paper with Straight-line Creases Zeyuan He, Simon D. Guest∗ January 5, 2021 Abstract We develop a theoretical framework for rigid origami, and show how this framework can be used to connect rigid origami and results from cognate areas, such as the rigidity theory, graph theory, linkage folding and computer science. First, we give definitions on important concepts in rigid origami, then focus on how to describe the configuration space of a creased paper. The shape and 0-connectedness of the configuration space are analysed using algebraic, geometric and numeric methods, where the key results from each method are gathered and reviewed. Keywords: rigid-foldability, folding, configuration 1 Introduction This article develops a general theoretical framework for rigid origami, and uses this to gather and review the progress that researchers have made on the theory of rigid origami, including other related areas, such as rigidity theory, graph theory, linkage folding, and computer science. Origami has been used for many different physical models, as a recent review [1] shows. Sometimes a "rigid" origami model is required where all the deforma- tion is concentrated on the creases. A rigid origami model is usually considered to be a system of rigid panels that are able to rotate around their common boundaries and has been applied to many areas across different length scales [2]. These successful applications have inspired us to focus on the fundamental theory of rigid origami. Ultimately, we are considering two problems: first, the positive problem, which is to find useful sufficient and necessary conditions for a creased paper to be rigid-foldable; second, the inverse problem, which is to approximate a target surface by rigid origami.