544:
31:
993:
1483:
841:
2038:
360:
2753:
1654:
1758:
The digits in the "odd place-intervals", i.e. between digits 2 and 2 − 1 are not restricted and may take any value. This fractal has upper box dimension 2/3 and lower box dimension 1/3, a fact which may be easily verified by calculating
1664:
The box-counting dimension is one of a number of definitions for dimension that can be applied to fractals. For many well behaved fractals all these dimensions are equal; in particular, these dimensions coincide whenever the fractal satisfies the
1197:
1737:
1341:
1746:. The upper box dimension may be bigger than the lower box dimension if the fractal has different behaviour in different scales. For example, examine the set of numbers in the interval satisfying the condition
834:
are not exactly identical, they are closely related to each other and give rise to identical definitions of the upper and lower box dimensions. This is easy to show once the following inequalities are proven:
988:{\displaystyle N_{\text{packing}}(\varepsilon )\leq N'_{\text{covering}}(\varepsilon )\leq N_{\text{covering}}(\varepsilon /2)\leq N'_{\text{covering}}(\varepsilon /2)\leq N_{\text{packing}}(\varepsilon /4).}
431:
660:
589:
1933:
700:
255:
805:
1808:
1870:
1903:
775:
832:
1561:
111:
227:
1925:
1837:
728:
617:
247:
1091:
1686:
1301:
1266:
1231:
146:
1507:
An interesting property of the upper box dimension not shared with either the lower box dimension or the
Hausdorff dimension is the connection to set addition. If
748:
475:
451:
383:
187:
79:
1478:{\displaystyle \dim _{\text{upper box}}(A_{1}\cup \dotsb \cup A_{n})=\max\{\dim _{\text{upper box}}(A_{1}),\dots ,\dim _{\text{upper box}}(A_{n})\}.}
2771:
2217:
1055:) may be easily calculated explicitly, and that for boxes the covering and packing numbers (defined in an equivalent way) are equal.
2140:
2105:
485:
2120:
2043:
These examples show that adding a countable set can change box dimension, demonstrating a kind of instability of this dimension.
623:
the fractal, or in other words, such that their union contains the fractal. We can also consider the intrinsic covering number
388:
2777:
193:
the set. The box-counting dimension is calculated by seeing how this number changes as we make the grid finer by applying a
2633:
2590:
2033:{\displaystyle \dim _{\text{box}}\left\{0,1,{\frac {1}{2}},{\frac {1}{3}},{\frac {1}{4}},\ldots \right\}={\frac {1}{2}}.}
2711:
1071:
626:
558:
2793:
2277:
669:
662:, which is defined the same way but with the additional requirement that the centers of the open balls lie in the set
2191:
ImageJ and FracLac box counting plugin; free user-friendly open source software for digital image analysis in biology
355:{\displaystyle \dim _{\text{box}}(S):=\lim _{\varepsilon \to 0}{\frac {\log N(\varepsilon )}{\log(1/\varepsilon )}}.}
1496:
sequence of sets. For example, the box dimension of a single point is 0, but the box dimension of the collection of
2836:
1037:
within a certain distance of a chosen center, and one counts such balls to get the dimension. (More precisely, the
2067:
1033:
is placed, and the ball definition can be formulated intrinsically. One defines an internal ball as all points of
2831:
2443:
2299:
1770:
1504:
by comparison, is countably stable. The lower box dimension, on the other hand, is not even finitely stable.
2736:
2184:
FrakOut!: an OSS application for calculating the fractal dimension of a shape using the box counting method
1842:
2785:
2344:
2210:
1875:
1649:{\displaystyle \dim _{\text{upper box}}(A+B)\leq \dim _{\text{upper box}}(A)+\dim _{\text{upper box}}(B).}
1021:, and defines boxes according to the external geometry of the containing space. However, the dimension of
780:
753:
2570:
2262:
2052:
1026:
1010:
810:
532:
87:
2731:
2726:
2516:
2448:
2062:
203:
1078:
and also measure how many bits or digits one would need to specify a point of the space to accuracy
2489:
2466:
2401:
2349:
2334:
2267:
1670:
1501:
999:
524:
481:
2162:
1908:
1820:
713:
602:
232:
2716:
2696:
2660:
2655:
2418:
2190:
1666:
1192:{\displaystyle \dim _{\text{box}}(S)=n-\lim _{r\to 0}{\frac {\log {\text{vol}}(S_{r})}{\log r}},}
1085:
Another equivalent (extrinsic) definition for the box-counting dimension is given by the formula
2132:
189:, imagine this fractal lying on an evenly spaced grid and count how many boxes are required to
2826:
2759:
2721:
2645:
2553:
2458:
2364:
2339:
2329:
2272:
2255:
2245:
2240:
2203:
2159:
2136:
2101:
2057:
1743:
1732:{\displaystyle \dim _{\text{Haus}}\leq \dim _{\text{lower box}}\leq \dim _{\text{upper box}}.}
163:
155:
59:
55:
2676:
2543:
2526:
2354:
2146:
1005:
The advantage of using balls rather than squares is that this definition generalizes to any
620:
190:
39:
1279:
1244:
1209:
119:
2691:
2628:
2289:
2150:
2097:
1497:
1018:
552:
527:. Only in very special applications is it important to distinguish between the three (see
82:
2386:
1074:, in that they measure the amount of "disorder" in the metric space or fractal at scale
543:
2706:
2650:
2638:
2609:
2565:
2548:
2531:
2484:
2428:
2413:
2381:
2319:
2125:
733:
460:
436:
368:
172:
64:
2820:
2560:
2536:
2406:
2376:
2359:
2324:
2309:
1489:
707:
152:
1754:, all the digits between the 2-th digit and the (2 − 1)-th digit are zero.
2805:
2800:
2701:
2681:
2438:
2371:
1006:
194:
114:
17:
2766:
2686:
2396:
2391:
30:
2619:
2604:
2599:
2580:
2314:
2072:
1674:
1680:
The box dimensions and the
Hausdorff dimension are related by the inequality
1677:
are all equal to log(2)/log(3). However, the definitions are not equivalent.
2575:
2521:
2433:
2284:
2167:
2094:
Mathematica in Action: Problem
Solving Through Visualization and Computation
1059:
596:
523:
The upper and lower box dimensions are strongly related to the more popular
998:
These, in turn, follow either by definition or with little effort from the
2476:
551:
It is possible to define the box dimensions using balls, with either the
454:
249:
required to cover the set. Then the box-counting dimension is defined as
2506:
2423:
2226:
1067:
730:
one can situate such that their centers would be in the fractal. While
2494:
2183:
159:
149:
34:
Estimating the box-counting dimension of the coast of Great
Britain
1062:
of the packing and covering numbers are sometimes referred to as
365:
Roughly speaking, this means that the dimension is the exponent
2199:
2195:
2127:
Fractal geometry: mathematical foundations and applications
1044:
definition is extrinsic, but the other two are intrinsic.)
433:, which is what one would expect in the trivial case where
547:
Examples of ball packing, ball covering, and box covering
528:
426:{\displaystyle N(\varepsilon )\approx C\varepsilon ^{-d}}
1673:, lower box dimension, and upper box dimension of the
1936:
1911:
1878:
1845:
1823:
1773:
1689:
1564:
1344:
1319:
The upper box dimension is finitely stable, i.e. if {
1282:
1247:
1212:
1094:
844:
813:
783:
756:
736:
716:
672:
629:
605:
561:
463:
439:
391:
371:
258:
235:
206:
175:
122:
90:
67:
1810:
and noting that their values behave differently for
2745:
2669:
2618:
2589:
2505:
2475:
2457:
2298:
2233:
1047:The advantage of using boxes is that in many cases
531:). Yet another measure of fractal dimension is the
516:, while the lower box dimension is also called the
2124:
2032:
1919:
1897:
1864:
1831:
1802:
1731:
1648:
1477:
1295:
1260:
1225:
1191:
987:
826:
799:
769:
742:
722:
694:
655:{\displaystyle N'_{\text{covering}}(\varepsilon )}
654:
611:
583:
496:. The upper box dimension is sometimes called the
469:
445:
425:
377:
354:
241:
221:
181:
140:
105:
73:
2186:(Does not automatically place the boxes for you).
584:{\displaystyle N_{\text{covering}}(\varepsilon )}
1492:stable, i.e. this equality does not hold for an
1396:
1127:
695:{\displaystyle N_{\text{packing}}(\varepsilon )}
285:
1066:and are somewhat analogous to the concepts of
2211:
1817:Another example: the set of rational numbers
1307:which have a center that is a member of
1303:is the union of all the open balls of radius
8:
1523:is formed by taking all the pairs of points
1469:
1399:
1029:, independent of the environment into which
1013: — one assumes the fractal space
1500:in the interval has dimension 1. The
555:or the packing number. The covering number
2218:
2204:
2196:
169:To calculate this dimension for a fractal
2017:
1993:
1980:
1967:
1941:
1935:
1913:
1912:
1910:
1883:
1877:
1850:
1844:
1825:
1824:
1822:
1792:
1784:
1772:
1720:
1707:
1694:
1688:
1625:
1600:
1569:
1563:
1460:
1444:
1422:
1406:
1384:
1365:
1349:
1343:
1287:
1281:
1252:
1246:
1217:
1211:
1163:
1151:
1142:
1130:
1099:
1093:
971:
959:
941:
926:
908:
896:
871:
849:
843:
818:
812:
788:
782:
761:
755:
735:
715:
677:
671:
634:
628:
604:
566:
560:
462:
438:
414:
390:
370:
335:
300:
288:
263:
257:
234:
205:
174:
121:
97:
93:
92:
89:
66:
2189:FracLac: online user guide and software
1803:{\displaystyle \varepsilon =10^{-2^{n}}}
1515:are two sets in a Euclidean space, then
1009:. In other words, the box definition is
542:
29:
2772:List of fractals by Hausdorff dimension
2084:
1335:} is a finite collection of sets, then
484:does not exist, one may still take the
27:Method of determining fractal dimension
229:is the number of boxes of side length
1865:{\displaystyle \dim _{\text{Haus}}=0}
1742:In general, both inequalities may be
7:
1898:{\displaystyle \dim _{\text{box}}=1}
1660:Relations to the Hausdorff dimension
800:{\displaystyle N'_{\text{covering}}}
2131:. Chichester: John Wiley. pp.
770:{\displaystyle N_{\text{covering}}}
827:{\displaystyle N_{\text{packing}}}
25:
2754:How Long Is the Coast of Britain?
486:limit superior and limit inferior
1241:, i.e. the set of all points in
488:, which respectively define the
106:{\displaystyle \mathbb {R} ^{n}}
2163:"Minkowski-Bouligand Dimension"
1268:that are at distance less than
222:{\displaystyle N(\varepsilon )}
2778:The Fractal Geometry of Nature
1640:
1634:
1615:
1609:
1590:
1578:
1466:
1453:
1428:
1415:
1390:
1358:
1169:
1156:
1134:
1114:
1108:
979:
965:
949:
935:
916:
902:
886:
880:
861:
855:
689:
683:
649:
643:
578:
572:
401:
395:
343:
329:
318:
312:
292:
278:
272:
216:
210:
135:
123:
54:, is a way of determining the
1:
1927:, has dimension 1. In fact,
1072:information-theoretic entropy
44:Minkowski–Bouligand dimension
1920:{\displaystyle \mathbb {R} }
1832:{\displaystyle \mathbb {Q} }
723:{\displaystyle \varepsilon }
612:{\displaystyle \varepsilon }
242:{\displaystyle \varepsilon }
2794:Chaos: Making a New Science
2068:Weyl–Berry conjecture
1206: > 0, the set
2853:
518:lower Minkowski dimension
514:upper Minkowski dimension
113:, or more generally in a
1669:(OSC). For example, the
148:. It is named after the
1839:, a countable set with
539:Alternative definitions
457:) of integer dimension
2786:The Beauty of Fractals
2034:
1921:
1899:
1866:
1833:
1804:
1733:
1650:
1479:
1297:
1262:
1227:
1193:
989:
828:
801:
771:
744:
724:
696:
656:
613:
585:
548:
471:
447:
427:
379:
356:
243:
223:
183:
142:
107:
75:
52:box-counting dimension
35:
2053:Correlation dimension
2035:
1922:
1905:because its closure,
1900:
1867:
1834:
1805:
1734:
1651:
1480:
1298:
1296:{\displaystyle S_{r}}
1263:
1261:{\displaystyle R^{n}}
1233:is defined to be the
1228:
1226:{\displaystyle S_{r}}
1194:
1068:thermodynamic entropy
990:
829:
802:
772:
745:
725:
710:open balls of radius
697:
666:. The packing number
657:
614:
586:
546:
533:correlation dimension
472:
453:is a smooth space (a
448:
428:
380:
357:
244:
224:
184:
143:
141:{\displaystyle (X,d)}
108:
76:
33:
2732:Lewis Fry Richardson
2727:Hamid Naderi Yeganeh
2517:Burning Ship fractal
2449:Weierstrass function
2092:Wagon, Stan (2010).
2063:Uncertainty exponent
1934:
1909:
1876:
1843:
1821:
1771:
1687:
1562:
1342:
1280:
1245:
1210:
1092:
842:
811:
781:
754:
734:
714:
670:
627:
603:
559:
502:Kolmogorov dimension
461:
437:
389:
369:
256:
233:
204:
173:
120:
88:
65:
2490:Space-filling curve
2467:Multifractal system
2350:Space-filling curve
2335:Sierpinski triangle
1671:Hausdorff dimension
1502:Hausdorff dimension
1488:However, it is not
1000:triangle inequality
934:
879:
796:
642:
525:Hausdorff dimension
506:Kolmogorov capacity
494:lower box dimension
490:upper box dimension
48:Minkowski dimension
18:Minkowski dimension
2717:Aleksandr Lyapunov
2697:Desmond Paul Henry
2661:Self-avoiding walk
2656:Percolation theory
2300:Iterated function
2241:Fractal dimensions
2160:Weisstein, Eric W.
2030:
1917:
1895:
1862:
1829:
1800:
1729:
1667:open set condition
1646:
1475:
1293:
1276:(or equivalently,
1258:
1223:
1189:
1141:
1017:is contained in a
985:
922:
867:
824:
797:
784:
767:
740:
720:
692:
652:
630:
609:
581:
549:
467:
443:
423:
375:
352:
299:
239:
219:
179:
138:
103:
71:
36:
2837:Hermann Minkowski
2814:
2813:
2760:Coastline paradox
2737:Wacław Sierpiński
2722:Benoit Mandelbrot
2646:Fractal landscape
2554:Misiurewicz point
2459:Strange attractor
2340:Apollonian gasket
2330:Sierpinski carpet
2121:Falconer, Kenneth
2058:Packing dimension
2025:
2001:
1988:
1975:
1944:
1886:
1853:
1723:
1710:
1697:
1628:
1603:
1572:
1447:
1409:
1352:
1237:-neighborhood of
1184:
1154:
1126:
1102:
962:
929:
899:
874:
852:
821:
791:
764:
743:{\displaystyle N}
680:
637:
569:
498:entropy dimension
470:{\displaystyle d}
446:{\displaystyle S}
378:{\displaystyle d}
347:
284:
266:
182:{\displaystyle S}
164:Georges Bouligand
156:Hermann Minkowski
74:{\displaystyle S}
56:fractal dimension
16:(Redirected from
2844:
2832:Dimension theory
2677:Michael Barnsley
2544:Lyapunov fractal
2402:Sierpiński curve
2355:Blancmange curve
2220:
2213:
2206:
2197:
2173:
2172:
2154:
2130:
2112:
2111:
2089:
2039:
2037:
2036:
2031:
2026:
2018:
2013:
2009:
2002:
1994:
1989:
1981:
1976:
1968:
1946:
1945:
1942:
1926:
1924:
1923:
1918:
1916:
1904:
1902:
1901:
1896:
1888:
1887:
1884:
1871:
1869:
1868:
1863:
1855:
1854:
1851:
1838:
1836:
1835:
1830:
1828:
1809:
1807:
1806:
1801:
1799:
1798:
1797:
1796:
1738:
1736:
1735:
1730:
1725:
1724:
1721:
1712:
1711:
1708:
1699:
1698:
1695:
1655:
1653:
1652:
1647:
1630:
1629:
1626:
1605:
1604:
1601:
1574:
1573:
1570:
1498:rational numbers
1484:
1482:
1481:
1476:
1465:
1464:
1449:
1448:
1445:
1427:
1426:
1411:
1410:
1407:
1389:
1388:
1370:
1369:
1354:
1353:
1350:
1302:
1300:
1299:
1294:
1292:
1291:
1267:
1265:
1264:
1259:
1257:
1256:
1232:
1230:
1229:
1224:
1222:
1221:
1198:
1196:
1195:
1190:
1185:
1183:
1172:
1168:
1167:
1155:
1152:
1143:
1140:
1104:
1103:
1100:
994:
992:
991:
986:
975:
964:
963:
960:
945:
930:
927:
912:
901:
900:
897:
875:
872:
854:
853:
850:
833:
831:
830:
825:
823:
822:
819:
806:
804:
803:
798:
792:
789:
776:
774:
773:
768:
766:
765:
762:
749:
747:
746:
741:
729:
727:
726:
721:
701:
699:
698:
693:
682:
681:
678:
661:
659:
658:
653:
638:
635:
618:
616:
615:
610:
590:
588:
587:
582:
571:
570:
567:
476:
474:
473:
468:
452:
450:
449:
444:
432:
430:
429:
424:
422:
421:
384:
382:
381:
376:
361:
359:
358:
353:
348:
346:
339:
321:
301:
298:
268:
267:
264:
248:
246:
245:
240:
228:
226:
225:
220:
188:
186:
185:
180:
147:
145:
144:
139:
112:
110:
109:
104:
102:
101:
96:
80:
78:
77:
72:
46:, also known as
40:fractal geometry
21:
2852:
2851:
2847:
2846:
2845:
2843:
2842:
2841:
2817:
2816:
2815:
2810:
2741:
2692:Felix Hausdorff
2665:
2629:Brownian motion
2614:
2585:
2508:
2501:
2471:
2453:
2444:Pythagoras tree
2301:
2294:
2290:Self-similarity
2234:Characteristics
2229:
2224:
2180:
2158:
2157:
2143:
2119:
2116:
2115:
2108:
2100:. p. 214.
2098:Springer-Verlag
2091:
2090:
2086:
2081:
2049:
1954:
1950:
1937:
1932:
1931:
1907:
1906:
1879:
1874:
1873:
1846:
1841:
1840:
1819:
1818:
1788:
1780:
1769:
1768:
1716:
1703:
1690:
1685:
1684:
1662:
1621:
1596:
1565:
1560:
1559:
1456:
1440:
1418:
1402:
1380:
1361:
1345:
1340:
1339:
1334:
1325:
1317:
1283:
1278:
1277:
1248:
1243:
1242:
1213:
1208:
1207:
1202:where for each
1173:
1159:
1144:
1095:
1090:
1089:
1064:entropy numbers
1043:
1019:Euclidean space
955:
892:
845:
840:
839:
814:
809:
808:
779:
778:
757:
752:
751:
732:
731:
712:
711:
673:
668:
667:
625:
624:
601:
600:
562:
557:
556:
553:covering number
541:
459:
458:
435:
434:
410:
387:
386:
367:
366:
322:
302:
259:
254:
253:
231:
230:
202:
201:
171:
170:
118:
117:
91:
86:
85:
83:Euclidean space
63:
62:
28:
23:
22:
15:
12:
11:
5:
2850:
2848:
2840:
2839:
2834:
2829:
2819:
2818:
2812:
2811:
2809:
2808:
2803:
2798:
2790:
2782:
2774:
2769:
2764:
2763:
2762:
2749:
2747:
2743:
2742:
2740:
2739:
2734:
2729:
2724:
2719:
2714:
2709:
2707:Helge von Koch
2704:
2699:
2694:
2689:
2684:
2679:
2673:
2671:
2667:
2666:
2664:
2663:
2658:
2653:
2648:
2643:
2642:
2641:
2639:Brownian motor
2636:
2625:
2623:
2616:
2615:
2613:
2612:
2610:Pickover stalk
2607:
2602:
2596:
2594:
2587:
2586:
2584:
2583:
2578:
2573:
2568:
2566:Newton fractal
2563:
2558:
2557:
2556:
2549:Mandelbrot set
2546:
2541:
2540:
2539:
2534:
2532:Newton fractal
2529:
2519:
2513:
2511:
2503:
2502:
2500:
2499:
2498:
2497:
2487:
2485:Fractal canopy
2481:
2479:
2473:
2472:
2470:
2469:
2463:
2461:
2455:
2454:
2452:
2451:
2446:
2441:
2436:
2431:
2429:Vicsek fractal
2426:
2421:
2416:
2411:
2410:
2409:
2404:
2399:
2394:
2389:
2384:
2379:
2374:
2369:
2368:
2367:
2357:
2347:
2345:Fibonacci word
2342:
2337:
2332:
2327:
2322:
2320:Koch snowflake
2317:
2312:
2306:
2304:
2296:
2295:
2293:
2292:
2287:
2282:
2281:
2280:
2275:
2270:
2265:
2260:
2259:
2258:
2248:
2237:
2235:
2231:
2230:
2225:
2223:
2222:
2215:
2208:
2200:
2194:
2193:
2187:
2179:
2178:External links
2176:
2175:
2174:
2155:
2141:
2114:
2113:
2106:
2083:
2082:
2080:
2077:
2076:
2075:
2070:
2065:
2060:
2055:
2048:
2045:
2041:
2040:
2029:
2024:
2021:
2016:
2012:
2008:
2005:
2000:
1997:
1992:
1987:
1984:
1979:
1974:
1971:
1966:
1963:
1960:
1957:
1953:
1949:
1940:
1915:
1894:
1891:
1882:
1861:
1858:
1849:
1827:
1814:even and odd.
1795:
1791:
1787:
1783:
1779:
1776:
1756:
1755:
1740:
1739:
1728:
1719:
1715:
1706:
1702:
1693:
1661:
1658:
1657:
1656:
1645:
1642:
1639:
1636:
1633:
1624:
1620:
1617:
1614:
1611:
1608:
1599:
1595:
1592:
1589:
1586:
1583:
1580:
1577:
1568:
1486:
1485:
1474:
1471:
1468:
1463:
1459:
1455:
1452:
1443:
1439:
1436:
1433:
1430:
1425:
1421:
1417:
1414:
1405:
1401:
1398:
1395:
1392:
1387:
1383:
1379:
1376:
1373:
1368:
1364:
1360:
1357:
1348:
1330:
1323:
1316:
1313:
1290:
1286:
1255:
1251:
1220:
1216:
1200:
1199:
1188:
1182:
1179:
1176:
1171:
1166:
1162:
1158:
1150:
1147:
1139:
1136:
1133:
1129:
1125:
1122:
1119:
1116:
1113:
1110:
1107:
1098:
1041:
996:
995:
984:
981:
978:
974:
970:
967:
958:
954:
951:
948:
944:
940:
937:
933:
925:
921:
918:
915:
911:
907:
904:
895:
891:
888:
885:
882:
878:
870:
866:
863:
860:
857:
848:
817:
795:
787:
760:
739:
719:
691:
688:
685:
676:
651:
648:
645:
641:
633:
608:
580:
577:
574:
565:
540:
537:
510:limit capacity
466:
442:
420:
417:
413:
409:
406:
403:
400:
397:
394:
374:
363:
362:
351:
345:
342:
338:
334:
331:
328:
325:
320:
317:
314:
311:
308:
305:
297:
294:
291:
287:
283:
280:
277:
274:
271:
262:
238:
218:
215:
212:
209:
178:
162:mathematician
137:
134:
131:
128:
125:
100:
95:
70:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2849:
2838:
2835:
2833:
2830:
2828:
2825:
2824:
2822:
2807:
2804:
2802:
2799:
2796:
2795:
2791:
2788:
2787:
2783:
2780:
2779:
2775:
2773:
2770:
2768:
2765:
2761:
2758:
2757:
2755:
2751:
2750:
2748:
2744:
2738:
2735:
2733:
2730:
2728:
2725:
2723:
2720:
2718:
2715:
2713:
2710:
2708:
2705:
2703:
2700:
2698:
2695:
2693:
2690:
2688:
2685:
2683:
2680:
2678:
2675:
2674:
2672:
2668:
2662:
2659:
2657:
2654:
2652:
2649:
2647:
2644:
2640:
2637:
2635:
2634:Brownian tree
2632:
2631:
2630:
2627:
2626:
2624:
2621:
2617:
2611:
2608:
2606:
2603:
2601:
2598:
2597:
2595:
2592:
2588:
2582:
2579:
2577:
2574:
2572:
2569:
2567:
2564:
2562:
2561:Multibrot set
2559:
2555:
2552:
2551:
2550:
2547:
2545:
2542:
2538:
2537:Douady rabbit
2535:
2533:
2530:
2528:
2525:
2524:
2523:
2520:
2518:
2515:
2514:
2512:
2510:
2504:
2496:
2493:
2492:
2491:
2488:
2486:
2483:
2482:
2480:
2478:
2474:
2468:
2465:
2464:
2462:
2460:
2456:
2450:
2447:
2445:
2442:
2440:
2437:
2435:
2432:
2430:
2427:
2425:
2422:
2420:
2417:
2415:
2412:
2408:
2407:Z-order curve
2405:
2403:
2400:
2398:
2395:
2393:
2390:
2388:
2385:
2383:
2380:
2378:
2377:Hilbert curve
2375:
2373:
2370:
2366:
2363:
2362:
2361:
2360:De Rham curve
2358:
2356:
2353:
2352:
2351:
2348:
2346:
2343:
2341:
2338:
2336:
2333:
2331:
2328:
2326:
2325:Menger sponge
2323:
2321:
2318:
2316:
2313:
2311:
2310:Barnsley fern
2308:
2307:
2305:
2303:
2297:
2291:
2288:
2286:
2283:
2279:
2276:
2274:
2271:
2269:
2266:
2264:
2261:
2257:
2254:
2253:
2252:
2249:
2247:
2244:
2243:
2242:
2239:
2238:
2236:
2232:
2228:
2221:
2216:
2214:
2209:
2207:
2202:
2201:
2198:
2192:
2188:
2185:
2182:
2181:
2177:
2170:
2169:
2164:
2161:
2156:
2152:
2148:
2144:
2142:0-471-92287-0
2138:
2134:
2129:
2128:
2122:
2118:
2117:
2109:
2107:0-387-75477-6
2103:
2099:
2095:
2088:
2085:
2078:
2074:
2071:
2069:
2066:
2064:
2061:
2059:
2056:
2054:
2051:
2050:
2046:
2044:
2027:
2022:
2019:
2014:
2010:
2006:
2003:
1998:
1995:
1990:
1985:
1982:
1977:
1972:
1969:
1964:
1961:
1958:
1955:
1951:
1947:
1938:
1930:
1929:
1928:
1892:
1889:
1880:
1859:
1856:
1847:
1815:
1813:
1793:
1789:
1785:
1781:
1777:
1774:
1766:
1762:
1753:
1749:
1748:
1747:
1745:
1726:
1717:
1713:
1704:
1700:
1691:
1683:
1682:
1681:
1678:
1676:
1672:
1668:
1659:
1643:
1637:
1631:
1622:
1618:
1612:
1606:
1597:
1593:
1587:
1584:
1581:
1575:
1566:
1558:
1557:
1556:
1554:
1551: +
1550:
1546:
1542:
1538:
1534:
1530:
1526:
1522:
1518:
1514:
1510:
1505:
1503:
1499:
1495:
1491:
1472:
1461:
1457:
1450:
1441:
1437:
1434:
1431:
1423:
1419:
1412:
1403:
1393:
1385:
1381:
1377:
1374:
1371:
1366:
1362:
1355:
1346:
1338:
1337:
1336:
1333:
1329:
1322:
1314:
1312:
1310:
1306:
1288:
1284:
1275:
1271:
1253:
1249:
1240:
1236:
1218:
1214:
1205:
1186:
1180:
1177:
1174:
1164:
1160:
1148:
1145:
1137:
1131:
1123:
1120:
1117:
1111:
1105:
1096:
1088:
1087:
1086:
1083:
1081:
1077:
1073:
1069:
1065:
1061:
1056:
1054:
1050:
1045:
1040:
1036:
1032:
1028:
1024:
1020:
1016:
1012:
1008:
1003:
1001:
982:
976:
972:
968:
956:
952:
946:
942:
938:
931:
923:
919:
913:
909:
905:
893:
889:
883:
876:
868:
864:
858:
846:
838:
837:
836:
815:
793:
785:
758:
737:
717:
709:
705:
686:
674:
665:
646:
639:
631:
622:
606:
598:
594:
575:
563:
554:
545:
538:
536:
534:
530:
526:
521:
519:
515:
511:
507:
503:
499:
495:
491:
487:
483:
480:If the above
478:
464:
456:
440:
418:
415:
411:
407:
404:
398:
392:
372:
349:
340:
336:
332:
326:
323:
315:
309:
306:
303:
295:
289:
281:
275:
269:
260:
252:
251:
250:
236:
213:
207:
200:Suppose that
198:
196:
192:
176:
167:
165:
161:
157:
154:
153:mathematician
151:
132:
129:
126:
116:
98:
84:
68:
61:
57:
53:
49:
45:
41:
32:
19:
2806:Chaos theory
2801:Kaleidoscope
2792:
2784:
2776:
2702:Gaston Julia
2682:Georg Cantor
2507:Escape-time
2439:Gosper curve
2387:Lévy C curve
2372:Dragon curve
2251:Box-counting
2250:
2166:
2126:
2093:
2087:
2042:
1816:
1811:
1764:
1760:
1757:
1751:
1741:
1679:
1663:
1552:
1548:
1544:
1540:
1536:
1532:
1528:
1524:
1520:
1516:
1512:
1508:
1506:
1493:
1487:
1331:
1327:
1320:
1318:
1308:
1304:
1273:
1269:
1238:
1234:
1203:
1201:
1084:
1079:
1075:
1063:
1057:
1052:
1048:
1046:
1038:
1034:
1030:
1022:
1014:
1007:metric space
1004:
997:
703:
663:
619:required to
592:
550:
522:
517:
513:
509:
505:
501:
497:
493:
489:
479:
364:
199:
195:box-counting
168:
115:metric space
51:
47:
43:
37:
2797:(1987 book)
2789:(1986 book)
2781:(1982 book)
2767:Fractal art
2687:Bill Gosper
2651:Lévy flight
2397:Peano curve
2392:Moore curve
2278:Topological
2263:Correlation
1547:and adding
197:algorithm.
2821:Categories
2605:Orbit trap
2600:Buddhabrot
2593:techniques
2581:Mandelbulb
2382:Koch curve
2315:Cantor set
2151:0689.28003
2079:References
2073:Lacunarity
1675:Cantor set
1555:. One has
1315:Properties
1025:should be
706:number of
599:of radius
597:open balls
595:number of
385:such that
2712:Paul Lévy
2591:Rendering
2576:Mandelbox
2522:Julia set
2434:Hexaflake
2365:Minkowski
2285:Recursion
2268:Hausdorff
2168:MathWorld
2007:…
1948:
1786:−
1775:ε
1722:upper box
1714:≤
1709:lower box
1701:≤
1632:
1627:upper box
1607:
1602:upper box
1594:≤
1576:
1571:upper box
1490:countably
1451:
1446:upper box
1435:…
1413:
1408:upper box
1378:∪
1375:⋯
1372:∪
1356:
1351:upper box
1178:
1149:
1135:→
1124:−
1106:
1060:logarithm
1027:intrinsic
1011:extrinsic
969:ε
953:≤
939:ε
920:≤
906:ε
890:≤
884:ε
865:≤
859:ε
718:ε
687:ε
647:ε
607:ε
576:ε
416:−
412:ε
405:≈
399:ε
341:ε
327:
316:ε
307:
293:→
290:ε
270:
237:ε
214:ε
2827:Fractals
2622:fractals
2509:fractals
2477:L-system
2419:T-square
2227:Fractals
2123:(1990).
2047:See also
1750:for any
1543:is from
1535:is from
1494:infinite
1042:covering
932:′
928:covering
898:covering
877:′
873:covering
794:′
790:covering
763:covering
708:disjoint
640:′
636:covering
568:covering
455:manifold
158:and the
2571:Tricorn
2424:n-flake
2273:Packing
2256:Higuchi
2246:Assouad
1527:,
1326:, ...,
961:packing
851:packing
820:packing
704:maximal
702:is the
679:packing
593:minimal
591:is the
2670:People
2620:Random
2527:Filled
2495:H tree
2414:String
2302:system
2149:
2139:
2104:
1872:, has
1767:) for
1744:strict
1531:where
160:French
150:Polish
42:, the
2746:Other
2133:38–47
1272:from
621:cover
529:below
482:limit
191:cover
81:in a
58:of a
2137:ISBN
2102:ISBN
1852:Haus
1696:Haus
1539:and
1511:and
1070:and
1058:The
807:and
492:and
2147:Zbl
1943:box
1939:dim
1885:box
1881:dim
1848:dim
1718:dim
1705:dim
1692:dim
1623:dim
1598:dim
1567:dim
1442:dim
1404:dim
1397:max
1347:dim
1311:).
1175:log
1153:vol
1146:log
1128:lim
1101:box
1097:dim
512:or
324:log
304:log
286:lim
265:box
261:dim
60:set
50:or
38:In
2823::
2756:"
2165:.
2145:.
2135:.
2096:.
1782:10
1519:+
1082:.
1002:.
777:,
750:,
535:.
520:.
508:,
504:,
500:,
477:.
282::=
166:.
2752:"
2219:e
2212:t
2205:v
2171:.
2153:.
2110:.
2028:.
2023:2
2020:1
2015:=
2011:}
2004:,
1999:4
1996:1
1991:,
1986:3
1983:1
1978:,
1973:2
1970:1
1965:,
1962:1
1959:,
1956:0
1952:{
1914:R
1893:1
1890:=
1860:0
1857:=
1826:Q
1812:n
1794:n
1790:2
1778:=
1765:ε
1763:(
1761:N
1752:n
1727:.
1644:.
1641:)
1638:B
1635:(
1619:+
1616:)
1613:A
1610:(
1591:)
1588:B
1585:+
1582:A
1579:(
1553:b
1549:a
1545:B
1541:b
1537:A
1533:a
1529:b
1525:a
1521:B
1517:A
1513:B
1509:A
1473:.
1470:}
1467:)
1462:n
1458:A
1454:(
1438:,
1432:,
1429:)
1424:1
1420:A
1416:(
1400:{
1394:=
1391:)
1386:n
1382:A
1367:1
1363:A
1359:(
1332:n
1328:A
1324:1
1321:A
1309:S
1305:r
1289:r
1285:S
1274:S
1270:r
1254:n
1250:R
1239:S
1235:r
1219:r
1215:S
1204:r
1187:,
1181:r
1170:)
1165:r
1161:S
1157:(
1138:0
1132:r
1121:n
1118:=
1115:)
1112:S
1109:(
1080:ε
1076:ε
1053:ε
1051:(
1049:N
1039:N
1035:S
1031:S
1023:S
1015:S
983:.
980:)
977:4
973:/
966:(
957:N
950:)
947:2
943:/
936:(
924:N
917:)
914:2
910:/
903:(
894:N
887:)
881:(
869:N
862:)
856:(
847:N
816:N
786:N
759:N
738:N
690:)
684:(
675:N
664:S
650:)
644:(
632:N
579:)
573:(
564:N
465:d
441:S
419:d
408:C
402:)
396:(
393:N
373:d
350:.
344:)
337:/
333:1
330:(
319:)
313:(
310:N
296:0
279:)
276:S
273:(
217:)
211:(
208:N
177:S
136:)
133:d
130:,
127:X
124:(
99:n
94:R
69:S
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
