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Dissection problem

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331: 36: 219:. Most polygons cannot be equidissected, and those that can often have restrictions on the possible numbers of triangles. For example, 184:
may be dissected into any other polygon of the same area, using interior-disjoint polygonal pieces. It is not true, however, that any
149:(of one polytope into another). It is usually required that the dissection use only a finite number of pieces. Additionally, to avoid 145:) into smaller pieces that may be rearranged into a new figure of equal content. In this context, the partitioning is called simply a 372: 104: 46: 76: 17: 83: 158: 90: 421: 72: 61: 401: 396: 365: 391: 241: 177: 162: 154: 426: 97: 358: 346: 57: 197: 188:
has a dissection into any other polyhedron of the same volume using polyhedral pieces (see
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Stein, Sherman K. (March 2004), "Cutting a Polygon into Triangles of Equal Areas",
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Geometric problems involving the partition of a figure
165:. For instance, they may be restricted to being the 383: 223:states that there is no odd equidissection of a 366: 62:introducing citations to additional sources 133:is the problem of partitioning a geometric 373: 359: 161:, the pieces are typically required to be 52:Relevant discussion may be found on the 14: 384: 265: 325: 208:of equal volume (in any dimension). 29: 24: 25: 438: 305: 204:) in three dimension and any two 329: 159:Tarski's circle-squaring problem 45:relies largely or entirely on a 34: 196:possible, however, for any two 268:The Mathematical Intelligencer 259: 13: 1: 252: 345:. You can help Knowledge by 7: 230: 215:of equal area is called an 10: 443: 324: 242:Hilbert's third problem 178:Bolyai–Gerwien theorem 153:issues related to the 422:Mathematical problems 155:Banach–Tarski paradox 18:Dissection (geometry) 402:Geometric dissection 73:"Dissection problem" 58:improve this article 131:dissection problem 397:Euclidean geometry 280:10.1007/BF02985395 392:Discrete geometry 354: 353: 316:Dissection Tiling 247:Hinged dissection 237:Dissection puzzle 211:A partition into 123: 122: 108: 16:(Redirected from 434: 375: 368: 361: 339:geometry-related 333: 326: 299: 298: 263: 221:Monsky's theorem 192:). This process 180:states that any 118: 115: 109: 107: 66: 38: 30: 21: 442: 441: 437: 436: 435: 433: 432: 431: 382: 381: 380: 379: 322: 308: 303: 302: 264: 260: 255: 233: 119: 113: 110: 67: 65: 51: 39: 28: 23: 22: 15: 12: 11: 5: 440: 430: 429: 427:Geometry stubs 424: 419: 414: 409: 404: 399: 394: 378: 377: 370: 363: 355: 352: 351: 334: 320: 319: 312:David Eppstein 307: 306:External links 304: 301: 300: 257: 256: 254: 251: 250: 249: 244: 239: 232: 229: 217:equidissection 190:Dehn invariant 121: 120: 56:. Please help 42: 40: 33: 26: 9: 6: 4: 3: 2: 439: 428: 425: 423: 420: 418: 415: 413: 410: 408: 405: 403: 400: 398: 395: 393: 390: 389: 387: 376: 371: 369: 364: 362: 357: 356: 350: 348: 344: 341:article is a 340: 335: 332: 328: 327: 323: 317: 313: 310: 309: 297: 293: 289: 285: 281: 277: 273: 269: 262: 258: 248: 245: 243: 240: 238: 235: 234: 228: 226: 222: 218: 214: 209: 207: 203: 199: 195: 191: 187: 183: 179: 174: 172: 168: 164: 160: 156: 152: 151:set-theoretic 148: 144: 140: 136: 132: 128: 117: 106: 103: 99: 96: 92: 89: 85: 82: 78: 75: â€“  74: 70: 69:Find sources: 63: 59: 55: 49: 48: 47:single source 43:This article 41: 37: 32: 31: 19: 347:expanding it 336: 321: 274:(1): 17–21, 271: 267: 261: 210: 193: 175: 169:of disjoint 163:well-behaved 146: 130: 124: 111: 101: 94: 87: 80: 68: 44: 137:(such as a 386:Categories 296:1186.52015 253:References 198:honeycombs 186:polyhedron 147:dissection 84:newspapers 417:Polytopes 412:Polyhedra 288:117930135 213:triangles 206:zonohedra 200:(such as 171:open sets 54:talk page 407:Polygons 231:See also 167:closures 139:polytope 127:geometry 114:May 2024 182:polygon 98:scholar 294:  286:  225:square 135:figure 100:  93:  86:  79:  71:  337:This 284:S2CID 105:JSTOR 91:books 343:stub 202:cube 176:The 157:and 143:ball 129:, a 77:news 292:Zbl 276:doi 141:or 125:In 60:by 388:: 314:, 290:, 282:, 272:26 270:, 227:. 194:is 173:. 374:e 367:t 360:v 349:. 318:. 278:: 116:) 112:( 102:· 95:· 88:· 81:· 64:. 50:. 20:)

Index

Dissection (geometry)

single source
talk page
improve this article
introducing citations to additional sources
"Dissection problem"
news
newspapers
books
scholar
JSTOR
geometry
figure
polytope
ball
set-theoretic
Banach–Tarski paradox
Tarski's circle-squaring problem
well-behaved
closures
open sets
Bolyai–Gerwien theorem
polygon
polyhedron
Dehn invariant
honeycombs
cube
zonohedra
triangles

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