© Cambridge University Press Cambridge

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

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

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. Buckminster, 53 isotopy, 184

Gardner, Martin, 232, 254 joint, 9 Gauss, Carl Friedrich, 303 free, 29 Gauss—Bonnet formula, 304 interior, 9 Gauss—Bonnet theorem, 304, 395, 411, 429 pinned, 9, 29 Gaussian elimination, 52 Justin, Jacques, 169 geodesic, 358, 359, 372 closed, 305, 372–375 k-connected, 47, 49, 308, 339 disk, 303, 383 Kawahata, Fumiaki, 240 loop, 372 Kawasaki, Toshikazu, 169 periodic, 373 Kawasaki’s theorem, 199, 203, 236, 296 polygon, 366, 372 generalized, 200, 202 quasigeodesic, 373, 375 in higher dimensions, 438 triangulation, 108 Kelvin, Lord, 30

Index 467

Kempe, Alfred Bray, 2, 24, 31, 35 Maclaurin’s trisectrix, 285 Kempe chains, 31 Maekawa, Jun, 169, 240 kernel Maekawa’s theorem, 203, 227 null-space, 51 Malkevitch, Joseph, 418 of star unfolding, 367 Manhattan towers, 332 Klein bottle, 355, 365 manifold knitting needles, 88, 89, 114, 135, 151, 153, 154 4D, 440 doubled, 96 of metrics, 352 and interlocking, 123 simplicial, 440 orthogonal, 436, 437 map folding, see folding, map, 4 knot Margulis napkin problem, 239 tame, 92 Maxwell, James Clerk, 43 trefoil, 89, 90 Maxwell—Cremona theorem, 58, 100, 101, 109 trivial, 89 mechanism, 12, 18, 20 knot theory, 80, 88–90 1-dof, 109, 111 definition, 9 Laman’s theorem, 45, 46 degrees of freedom, 17, 155, 157 Lang, Robert, 2, 169 free space of, 17 Latin cross, 386, 403 kinematics, 12 23 polyhedra, 408 planar, 29 in 4D, 439 pseudotriangulation, 109, 110 5 edge-to-edge foldings, 390, 391, 403 medial axis, 266, 370, 371 85 general foldings, 403 median link, 62 no rolling belts, 411 Meguro, Toshiyuki, 240 lattice metric embedding, 158–160 intrinsic, 348, 352 triangular, tetrahedral, 160 polyhedral, 348, 349, 373 lemniscate, 30 supremum, 174, 185 length ratio, 89, 91, 154 Mira, 289 Lenhart, William, 87 Mitchell, Joseph, 87 line tracking, 66, 71, 73, 80, 93, 94 Miura Map, 293 4-bar, 67 Mobius¨ strip, 184 simple, 66 moduli space, 11, 37 linear bounded automaton, 24 molecule, see origami, molecule, 267 linear programming, 54, 56, 118 monotypy, 352 link, 9 Montroll, John, 238 fat, 91, 92 motion planning, 17–19, 155 of polygon, 345 obstacles, 17 topological, 128, 139 sampling-based, 155 linkage, 1, 9, 10, 131 motion, trivial, 18, 43, 54 4D, 437 mountain fold, 57, 58, 103, 169, 203, 225 extrusion, 437, 438 mountain—valley assignment, 222 graph, 44 combinatorics, 208 orthogonal, 137 definition, 169 planar, 10 and flat foldings, 193, 201, 205, 207 refolding, 113 global, 214 rigid, 9, 10 in Kawasaki’s theorem, 203 simple, 10, 23 for map, 224 Lipschitz necessary properties, 203 -continuous, 106 overlap order, 222 constant, 106 random, 209 local dimension, 44 unique, 214 locked mountain—valley pattern infinitesimally, 113, 114 mingling, 195–197, 210 strongly, 115–117, 119 mouth flip, 81 within ε, 114 multiplicator (Kempe), 32–35, 39 locked chain, see chain, locked, 9 lower bounds, 23 de Sz. Nagy, Bela, 74, 76, 77 Lubiw, Anna, 399 net, 2, 299, 300, 306, 312, 317, 321, 436 Lundstrom¨ Design, 306, 307 convex, 327, 328 Lyusternik—Schnirelmann theorem, 374 for cuboctahedron, 299, 300

468 Index

net (cont.) 22.1: Fewest Nets, 308 definition, 2, 299 22.2: Overlap Penetration, 309 for few vertices, 321 22.3: General Nonoverlapping Unfolding of fewest, 308 Polyhedra, 321 Hamiltonian, 328, 329 22.4: Edge-Unfolding Polyhedra Built from orthogonal creased, 432–436 Cubes, 331 shared, 423 22.5: Edge-Unfolding for Nonacute Faces, 331 for snub cube, 3 22.6: Edge-Unfolding for Nonobtuse for truncated icosahedron, 301 Triangulations, 332 vertex-unfolding, 333 22.7: Vertex Grid Refinement for Orthogonal nonlinear optimization, 243, 246 Polyhedra, 332 Not-All-Equal clause, 218, 219 22.8: Refinement for Convex Polyhedra, 332 NP,NP-complete, NP-hard, 22 22.9: Vertex Unfolding, 338 NP-complete, weakly, 25 23.1: D-Forms and Pita-Forms, 354 null-space, 51 23.2: Practical Algorithm for Cauchy Rigidity, 357 objective function, 104, 316, 318 24.1: Star Unfolding of Smooth Surfaces, 371 in linear programming, 54 24.2: Closed Quasigeodesics, 374 obstruction diagram, 92, 93 24.3: Closed Quasigeodesic Edge-Unfolding, Open Problem 375 3.1: Continuous Kempe Motion, 39 25.1: Folding Polygons to (Nonconvex) 3.2: Noncrossing Linkage to Sign Your Name, 40 Polyhedra, 384 4.1: Faster Generic Rigidity in 2D, 46 25.2: Finite Number of Foldings, 396 4.2: Generic Rigidity in 3D, 47 25.3: Polynomial-Time Folding Decision 4.3: Realizing Generically Globally Rigid Algorithm, 402 Graphs, 49 25.4: Volume Maximizing Convex Shape, 418 5.1: Pocket Flip Bounds, 76 25.5: Flat Foldings, 423 5.2: Shortest Pocket-Flip Sequence, 76 25.6: Fold/Refold Dissections, 424 5.3: Pops, 81 26.1: Higher-Dimensional Fold-and-Cut, 6.1: Equilateral Fat 5-Chain, 91 438 6.2: Unlocking Nested Chains, 97 26.2: Flattening Complexes, 438 6.3: Polynomial Number of Moves, 113 26.3: Ridge Unfolding, 441 6.4: Characterize Locked Linkages, 113 open problems, xii 6.5: Self-Touching Chains, 119 Oppenheimer, Lillian, 168 7.1: Unlocking Chains by Cutting, 123 orbit, of joint, 28, 31 7.2: Interlocking a Flexible 2-Chain, 126 orientable manifold, 184, 355, 357, 365 8.1: Extreme Span in 3D, 135 orientation determinant, 124, 130 8.2: Flat-State Connectivity of Open Chains, origami, 167 137 base, 2, 237, 240, 241, 243, 244 8.3: Flat-State Connectivity of Orthogonal uniaxial, 2, 242–244 Trees, 141 bird, 191 9.1: Locked Length Ratio, 154 checkerboard, 238, 239 9.1: Locked Unit Chains in 3D?, 153 color reversal, 234, 236, 238, 239 9.2: Locked Fixed-Angle Chains, 154 computational, 94, 169 9.4: Locked Unit Trees in 3D, 154 crane, 242 9.5: Complexity of Protein Folding in Other curved, 292, 296 Lattices, 160 deer, 3 9.6: PTAS Approximation Scheme for Protein design, 2, 168, 169, 171, 232, 240 Folding, 161 final folded state, 2 9.7: Protein Design, 164 flat 12.1: 3D Single-Vertex Fold, 212 4D, 437 14.1: Map Folding, 224 foldability, 170 14.2: Orthogonal Creases, 231 folding motion, 169 14.3: Pseudopolynomial-Time Map Folding, fractal, 175 231 history, 167, 168 15.1: Seam Patterns, 236 horse, 232 15.2: Efficient Silhouettes and Wrapping, 237 isometry condition, 172, 173, 184, 185 18.1: Continuous Flattening, 279 lizard, 242 18.2: Flattening Higher Genus, 281 uniaxial base, 242, 252 18.3: Flattening via Straight Skeleton, 283 mathematical, 168, 169, 172 21.1: Edge-Unfolding Convex Polyhedra, 300 molecule, 264, 265, 267

Index 469

paper, 169, 172, 184 polygon folding, see folding, polygon, 5 1D, 173 polygraph, 13 bicolored, 232 polyhedral complex, 438 circular, 182 polyhedral graph, 57 disconnected, 182 polyhedral lifting, 57, 58, 101–103, 109, 110 rigid, 195, 212, 279, 292, 293, 436 polyhedron scorpion, 241 circumscribed, 340 silhouette, 171, 189, 232, 237, 254 continuum, 383, 412, 415, 416, 420 stacking order, 176, 201, 205, 218, 222 dome, 322–327 tesselation, 293 doubly covered polygon, 349, 352, 381, 423 tree method, see tree method, 240 extrusion to, 329 triangular twist, 218 inscribed, 340 universal molecule, 250, 252, 253, 255, 267 nonconvex, 283, 306, 309, 311, 341, 355, 358, waterbomb 384 uniaxial base, 243 nonorthogonal, 433, 434, 440 zebra, 233 orthogonal, 329, 332, 431, 432 OrigamiUSA, 168 orthostack, 329, 330 ortho-, see polyhedron, ortho-, 329 orthotree, 330 osculating plane, 296 orthotube, 330, 331 overbracing, 46, 57 pita, 412, 413 overlap penetration, 308, 309 random, 315 Overmars, Mark, 76, 157 simple, 308 simplicial, 308, 334, 406 P, 2 2 star-shaped, 318 pantograph, 12, 13, 32, 39 tetramonohedron, 424 paper, see origami, paper, 173 truncated icosahedron, 299, 301, 312 parallel offset, see polygon, offsetting, 266 ununfoldable, see unfoldable, polyhedron, partition, 25, 133, 136, 228, 229, 433 318 definition, 25 volume, 279, 348, 355, 418, 419 path planning, 11, 27 volume polynomial, 355 path-connected, 178, 184 polymer, dentritic/star, 131 Peaucellier linkage, 30, 31, 35–37, 39 polynomial-time approximation scheme (PTAS), pebble game, 46 26, 161 peptide bond, 14, 148 pop, vertex, 81 permiter halving, see folding, perimeter-halving, prismatoid, 283, 321–324, 380 382 smooth, 323, 324 petal of conic, 287 prismoid, 322, 323 Petersen’s theorem, 308 probabilistic method, 12 piano mover’s problem, 17 probabilistic roadmap, 15, 154, 155, 157 pita form, 353, 354, 414, 418 projection, simple, from 3D, 84, 85, 119, 120 pivot proper crossing, 173 edge, 141 protein backbone, 14, 15, 131, 148, 153, 157 pivoting, 80 protein design, 161, 164 planarization, of tensegrity, 100 protein folding problem, xi, 15, 16, 131, 154, Platonic solid, xi, 423, 424 159 pleating, 293, 294, 419 pseudotriangulation, 46, 105, 108, 109, 111, 112 pocket flipping, 74, 76, 78, 80 expansive motion of, 109, 110 polygon, 10 flip, 110 invertible, 64 method, 107, 113, 213 noninvertible, 64 minimally infinitesimally rigid, 109 nonsimple, 310 minimum, 108 not-foldable, 5, 381 pointed, 46, 108, 110 offsetting, 251, 266 and rigidity, 110 orthogonally convex, 231 PSPACE, PSPACE-complete, PSPACE-hard, 22 regular, 289 self-crossing, 80 quadratic equation, 287, 290 shrinking, 250–252 quadrilateral, 79, 80, 322, 336, 407, 418 simple, 10, 299, 310 arch, 83 spherical, 145, 212, 213, 342, 343 flat, 314, 390, 408, 409 star-shaped, 81, 88, 360, 367, 372 flexible, 83, 126, 127 wrapping, 423 spherical, 213

470 Index

rabbit-ear fold, 267 sloppy, 116 Ramachandran plot, 157, 158 testing, 44, 49 rank-nullity theorem, 52 theory, 43, 47, 53 reachability, 11, 20, 23, 59 roadmap algorithm, 16, 18, 19, 21, 22, 27, 114, 124 in circle, 68 robot arm, 10 for n-link arm, 61, 63 rolling belt, 393, 395 region, 60 characterize, 396 realization, 11 definition, 394 of graph, 44, 45 double, 395 of metric, 348, 351 lemma, 409 of polyhedron, 339, 350 not in Latin cross, 408, 411 stick, 90 and perimeter-halving, 384, 402 unique, 49, 350 ring, 421 reconfiguration, 11, 20, 23, 59 for square, 415 in confined region, 67 Ross, Betsy, 254 of convex polygon, 70 Rote, Gunter,¨ 88, 316 path, 20 round-robin, 77 reconstruction ruler folding, carpenter’s, 4, 25, 26, 133, 433 of hexahedra, 406 NP-complete, 25 of linkage, 48, 403 spherical, 213 of octahedra, 407, 408 in triangle, 69 of polyhedra, 339, 340, 354, 357, 403, 406, 418 of tetrahedra, 406 Sabitov’s algorithm, 354–357, 407 refinement Sallee, G. Thomas, 328 of offset graph, 267 scale optimization, 246, 247, 255 of surface, 330, 332, 333 Schlickenrieder, Wolfram, 315 vertex grid, 332 Schwartz, Jack T., 17 Resch, Ron, 293, 296 self-crossing, 10, 70, 71, 115 reversor (Kempe), 34 closed geodesic, 374 ribosome, 148, 149, 151 shortest path, 358 Riemannian manifold, 184, 359 self-intersection, 10 rigid semialgebraic set, 19, 44 chain, 124–126, 130 sensor networks, 48 edge, 132 separation puzzle, 21 linkage, 43, 44 set-sum, 60 motion, 132 Sharir, Micha, 17 net, 434–436 Shephard, Geoffrey, 300, 327 origami, see origami, rigid, 279 Shimamoto, Don, 40, 415 partially, 138, 140, 148 shortest path polyhedra, 43, 145 algorithm, 362 self-touching configuration, 116 Chen and Han’s algorithm, 364 sphere, 352 on convex polyhedron, 351, 358, 359, 361 surface, 352 geodesic, 358, 359, 372 rigidity metric, 173, 182, 184, 185, 348 first-order, see rigidity, infinitesimal, 52 on nonconvex polyhedron, 364–366 generic, 44–48 source unfolding, 360 generically global, 48, 49 star unfolding, 316, 366, 371, 440 global, 48, 342, 350 sign alternation, 72, 118, 342–344 hierarchy, 53 silhouette curve, 19 infinitesimal, 37, 48, 49, 51, 52, 56, 57, 116, 342 simple polygon, see polygon, simple, 10 algorithm, 52 simply connected, 310, 333 definition, 52 span, 133, 134, 146, 147 of polyhedron, 347 extreme, 133, 135 self-touching, 116, 118, 119 flat, 133, 134, 433 tensegrity, 54, 56, 116 Spriggs, Michael, 308 matrix, 50, 51, 56 spring monotone chain, 120, 122 rank, 52 square minimally generic, 46–48, 52 chain confined to, 67, 69 minimally infinitesimal, 109, 110 flexible, 51 redundant generic, 49 folding to polyhedra, 411, 412, 414, 415, 419, second-order, 52 421

Index 471

gift-wrapping cube, 237, 238 locked, 24, 94, 95, 114, 115 map folding to, 224 self-touching, 117 nested, 294 unit, 154 origami paper, 169 manifold, 337 perimeter increase, 239 method, 171, 240, 242, 254, 255 teabag, 418 metric, 2, 241, 243, 244 Steffen, Klaus, 347 molecule, 267, 269 Steinitz, Ernst, 143, 342 monotone, 96 Steinitz’s lemma, see Cauchy’s arm lemma, 143 orthogonal, 138, 139, 141 Steinitz’s theorem, 308, 339 petal, 94–96, 114, 117 stereoisomer, 342 radially monotone, 96 straight edge and compass, 34, 168, 285–288, 290 ridge, 359 straight skeleton, 255–260, 266, 281 shadow, 242–244, 247, 249, 251, 261, 262 3D, 282 spanning, 308, 311, 315, 361, 366 gluing, 282, 283 disk packing, 269 method, 256, 259, 263 minimum length, 316 perpendiculars, see crease, perpendicular, 256 number of, 315, 431 skeletal collapse, 283, 284 unit, 96, 154 subdivision, 283 TreeMaker, 2, 3, 240, 246, 247, 254 stress, 54–56, 101 trisection, see angle trisection, 285 duality, 56, 99, 110 Turing machine, 23 equilibrium, 54–58, 100, 116, 118 turn angle, 60, 102, 145–147, 149, 203, 373 everywhere-zero, 55, 58, 99, 101, 103, 110 two-kinks theorem, 62, 63 strictly expansive, 96, 97 strongly polynomial time, 54 unfoldable strut, 53, 54, 99, 140 polyhedron, 318 sliding, 116 unfolding subspace topology, 184 band, 322, 379 surface convex Hamiltonian, 329, 430 curved, 184, 293 dome, 328 developable, 191, 296, 352 edge, 306, 311, 313, 331–333, 375 torsal ruled, 191 of band, 379, 380 Sylvester, J. J., 30 bound, 431 symmetric axis, see medial axis, 370 convex, 327 deﬁnition, 299, 306, 333, 338 teabag problem, 296, 418 of deltahedron, 331 tensegrity, 53–58, 98–100, 109, 110 evidence against, 313 tesseract, 438, 439 evidence for, 312 tetrahedron, spiked, 318–320 nonconvex polyhedron, 309, 310 Thurston, William, 31, 36, 81, 376 open problem, 300 topological proof method, 124, 127 of orthostack, 330 TouchCAD, 306 of prismatoid, 380 Towers of Hanoi, 21 prismoid, 322 translator (Kempe), 32, 34, 35, 39 reﬁned, 332 trapezoid special classes, 321 double-sided, 314 of tetrahedron, 314 twisted, 83, 84 general, 306, 307, 320, 321, 362, 366, 369, 440 tree grid, 330 cut overlapping, 308, 314–317 combinatorial, 429 random, 315 geometric, 429 ridge, 441 diameter, 96 source, 307, 359–361, 367 gluing, 392–395 in higher dimensions, 440 combinatorial type, 426, 427 star, 307, 358, 366–370 of convex polygon, 411, 421 computation, 370 exponential number of, 396 in higher dimensions, 440 four fold-point, 427 nonoverlap, 316, 366, 368 geometric, 429 of smooth surface, 371 leaves, 426, 428 vertex, 330, 333–335, 337, 338, 440 partial, 399 volcano, 321–324, 326 path, 412 uniaxial base, see origami, base, uniaxial, 2

472 Index

universality theorem, 27, 29, 31, 39, 48, 290 vertex unfolding, see unfolding, vertex, 333 ununfoldable Voronoi diagram, 370, 371 manifold, 321, 334 and cut locus, 369 polyhedron, 312, 318, 319, 334 Watt, James, 29, 30 valley fold, 57, 58, 103, 169, 203, 225 Watt linkage, 30 van der Waals force, 158 Whitesides, Sue, 87, 88 vector, equilibrated, 340 winding number, 188, 200 vertex workspace, 10, 12, 14, 155, 156 chain, 9 nonﬂat, 78, 358 Yoshizawa, Akira, 168