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In order to give an operational meaning to the property of self-similarity, we are necessarily restricted to dealing with finite approximations of the limit figure. This is done using the method which we will call box self-similarity where measurements are made on finite stages of the figure using
777:
is an organizational model with an affine self-similar hierarchy, where a given viable system is one element of the System One of a viable system one recursive level higher up, and for whom the elements of its System One are viable systems one recursive level lower down.
198:. The idea is just an extension of the idea of similarity of two triangles. Note that two triangles are similar if the numerical values of their sides are different however the corresponding dimensionless quantities, such as their angles, coincide.
2178:
860:, self-similarity commonly refers to the fact that music often consists of parts that are repeated in time. In other words, music is self-similar under temporal translation, rather than (or in addition to) under scaling.
77:
and scale-invariant; it can be continually magnified 3x without changing shape. The non-trivial similarity evident in fractals is distinguished by their fine structure, or detail on arbitrarily small scales. As a
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by different amounts in the x- and y-directions. This means that to appreciate the self similarity of these fractal objects, they have to be rescaled using an
1275:
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Self-similarity has important consequences for the design of computer networks, as typical network traffic has self-similar properties. For example, in
1435:
1113:
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if the figure can be decomposed into parts which are exact replicas of the whole. Any arbitrary part contains an exact replica of the whole figure.
1188:
Dayeen F. R., Hassan M. K. (2016). "Multi-multifractality, dynamic scaling and neighbourhood statistics in weighted planar stochastic lattice".
223:
Since mathematically, a fractal may show self-similarity under indefinite magnification, it is impossible to recreate this physically. Peitgen
57:, are statistically self-similar: parts of them show the same statistical properties at many scales. Self-similarity is a typical property of
1417:
916:
2196:
1060:
Hassan M. K., Hassan M. Z., Pavel N. I. (2011). "Dynamic scaling, data-collapseand Self-similarity in
Barabasi-Albert networks".
804:
Self-similarity can be found in nature, as well. To the right is a mathematically generated, perfectly self-similar image of a
53:
to a part of itself (i.e., the whole has the same shape as one or more of the parts). Many objects in the real world, such as
1448:
709:
are inaccurate, and networks designed without taking self-similarity into account are likely to function in unexpected ways.
515:
1276:
https://www.larecherche.fr/math%C3%A9matiques-histoire-des-sciences/%C2%AB-comment-jai-d%C3%A9couvert-les-fractales-%C2%BB
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A time developing phenomenon is said to exhibit self-similarity if the numerical value of certain observable quantity
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Peitgen, Heinz-Otto; Jürgens, Hartmut; Saupe, Dietmar; Maletsky, Evan; Perciante, Terry; and Yunker, Lee (1991).
365:
1333:
2202:
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is an exact form of self-similarity where at any magnification there is a smaller piece of the object that is
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2015:
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data traffic patterns seem to be statistically self-similar. This property means that simple models using a
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measured at different times are different but the corresponding dimensionless quantity at given value of
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Hassan M. K., Hassan M. Z. (2009). "Emergence of fractal behavior in condensation-driven aggregation".
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1443:. International Semiotics Institute at Imatra; Semiotic Society of Finland. p. 240.
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Foote, Jonathan (30 October 1999). "Visualizing music and audio using self-similarity".
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1007:
990:"How long is the coast of Britain? Statistical self-similarity and fractional dimension"
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If parts of a figure are small replicas of the whole, then the figure is called
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Proceedings of the seventh ACM international conference on
Multimedia (Part 1)
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1985:
1961:
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Gaussian Self-Affinity and
Fractals: Globality, the Earth, 1/F Noise, and R/S
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808:, which bears a marked resemblance to natural ferns. Other plants, such as
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display various types and amounts of self-similarity, as do sections of
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shown by zooming in on the
Feigenbaum point at (−1.401155189..., 0)
2044:
2029:
2024:
2005:
1739:
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are a powerful technique for building self-similar sets, including the
309:
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1307:
2000:
1946:
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1709:
896:
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54:
720:, i.e. they appear self-similar when transformed via an appropriate
600:. The composition of functions creates the algebraic structure of a
1901:
1493:
1437:
On
Musical Self-Similarity: Intersemiosis as Synecdoche and Analogy
1289:"On the self-similar nature of Ethernet traffic (extended version)"
843:
1202:
1141:
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248:
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has an infinitely repeating self-similarity when it is magnified.
16:
Whole of an object being mathematically similar to part of itself
1919:
601:
457:
1550:
1620:
1234:
Fractals for the
Classroom: Strategic Activities Volume One
805:
1286:
1187:
1126:
1551:
Sapozhnikov, Victor; Foufoula-Georgiou, Efi (May 1996).
1334:"How Fractals Can Explain What's Wrong with Wall Street"
86:
may resemble the whole, further detail is not revealed.
1287:
Leland, W.E.; Taqqu, M.S.; et al. (January 1995).
836:
is self-similar in the frequency or wavelength domains.
981:
728:
describes stock market log return self-similarity in
578:{\displaystyle {\mathfrak {L}}=(X,S,\{f_{s}:s\in S\})}
227:
suggest studying self-similarity using approximations:
1496:— New articles about Self-Similarity. Waltz Algorithm
987:
612:. The dyadic monoid can be visualized as an infinite
518:
470:
429:
368:
320:
165:
130:
95:
1601:
1354:
of
Financial Markets ", Princeton University Press!
1270:Comment j'ai découvert les fractales, Interview de
624:elements, then the monoid may be represented as a
608:has only two elements, the monoid is known as the
577:
501:
441:
412:
351:
186:
151:
116:
1328:
853:named the 'infinity series' in much of his music.
2243:
757:. The complement of the large circles becomes a
646:A more general notion than self-similarity is
1636:
159:remain invariant. It happens if the quantity
1112:: CS1 maint: multiple names: authors list (
569:
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496:
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413:{\displaystyle X=\bigcup _{s\in S}f_{s}(X)}
1643:
1629:
1506:
69:to the whole. For instance, a side of the
1385:
1201:
1140:
1073:
1255:
917:Self-similarity of network data analysis
785:
748:
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247:
32:
20:
2197:List of fractals by Hausdorff dimension
1433:
753:A triangle subdivided repeatedly using
639:; the automorphisms can be pictured as
464:such that the equation above holds for
2244:
716:movements are described as displaying
1624:
1509:"Self-affinity and fractal dimension"
1370:
724:for the level of detail being shown.
1350:Campbell, Lo and MacKinlay (1991) "
988:Mandelbrot, Benoit B. (5 May 1967).
1490:— a self-similar fractal zoom movie
1423:from the original on 9 August 2017.
1296:IEEE/ACM Transactions on Networking
1236:, p.21. Springer-Verlag, New York.
521:
49:object is exactly or approximately
37:Standard (trivial) self-similarity.
13:
1500:
812:, exhibit strong self-similarity.
301:is self-similar if there exists a
236:This vocabulary was introduced by
14:
2278:
2179:How Long Is the Coast of Britain?
1589:from the original on 30 July 2018
1553:"Self-Affinity in Braided Rivers"
1481:
764:
502:{\displaystyle \{f_{s}:s\in S\}}
352:{\displaystyle \{f_{s}:s\in S\}}
243:
1434:Pareyon, Gabriel (April 2011).
1427:
1364:
1344:
1322:
849:has made use of a self-similar
453:self-similar if it is the only
2203:The Fractal Geometry of Nature
1507:Mandelbrot, Benoit B. (1985).
1280:
1264:
1226:
1190:Chaos, Solitons & Fractals
1181:
1120:
1092:10.1088/1751-8113/44/17/175101
1053:
968:The Fractal Geometry of Nature
966:Mandelbrot, Benoit B. (1982).
960:
572:
529:
407:
401:
181:
169:
111:
99:
1:
1602:Benoît B. Mandelbrot (2002).
1261:Peitgen, et al (1991), p.2-3.
953:
616:; more generally, if the set
286:
207:explain the concept as such:
1650:
1016:10.1126/science.156.3775.636
781:
690:is also self-similar around
635:of the dyadic monoid is the
592:. The homeomorphisms may be
7:
2219:Chaos: Making a New Science
1220:10.1016/j.chaos.2016.06.006
892:Non-well-founded set theory
864:
858:music information retrieval
815:
653:
252:A self-affine fractal with
82:, whereas any portion of a
10:
2283:
1536:10.1088/0031-8949/32/4/001
1159:10.1103/physreve.79.021406
797:
442:{\displaystyle X\subset Y}
2170:
2094:
2043:
2014:
1930:
1900:
1882:
1723:
1658:
856:In the research field of
1560:Water Resources Research
736:Finite subdivision rules
598:iterated function system
1062:J. Phys. A: Math. Theor
755:barycentric subdivision
699:teletraffic engineering
662:Self-similarity in the
232:grids of various sizes.
152:{\displaystyle x/t^{z}}
2211:The Beauty of Fractals
1488:"Copperplate Chevrons"
795:
761:
683:
667:
590:self-similar structure
579:
503:
443:
414:
353:
308:indexing a set of non-
257:
234:
221:
188:
187:{\displaystyle f(x,t)}
153:
118:
117:{\displaystyle f(x,t)}
38:
30:
1396:10.1145/319463.319472
937:Tweedie distributions
887:Long-range dependency
798:Further information:
789:
752:
722:affine transformation
673:
661:
580:
504:
444:
415:
354:
281:affine transformation
251:
229:
217:strictly self-similar
209:
189:
154:
119:
36:
24:
2157:Lewis Fry Richardson
2152:Hamid Naderi Yeganeh
1942:Burning Ship fractal
1874:Weierstrass function
922:Self-similar process
707:Poisson distribution
643:of the binary tree.
641:hyperbolic rotations
516:
468:
427:
366:
318:
163:
128:
93:
1915:Space-filling curve
1892:Multifractal system
1775:Space-filling curve
1760:Sierpinski triangle
1572:1996WRR....32.1429S
1528:1985PhyS...32..257M
1338:Scientific American
1212:2016CSF....91..228D
1151:2009PhRvE..79b1406H
1084:2011JPhA...44q5101K
1008:1967Sci...156..636M
771:viable system model
744:Sierpinski triangle
254:Hausdorff dimension
2257:Scaling symmetries
2142:Aleksandr Lyapunov
2122:Desmond Paul Henry
2086:Self-avoiding walk
2081:Percolation theory
1725:Iterated function
1666:Fractal dimensions
1460:on 8 February 2017
1380:. pp. 77–80.
1038:on 19 October 2021
902:Self-dissimilarity
882:Logarithmic spiral
810:Romanesco broccoli
800:Patterns in nature
796:
792:Romanesco broccoli
762:
692:Misiurewicz points
684:
668:
596:, resulting in an
575:
499:
439:
410:
390:
349:
268:is a feature of a
258:
184:
149:
114:
39:
31:
2239:
2238:
2185:Coastline paradox
2162:Wacław Sierpiński
2147:Benoit Mandelbrot
2071:Fractal landscape
1979:Misiurewicz point
1884:Strange attractor
1765:Apollonian gasket
1755:Sierpinski carpet
1580:10.1029/96wr00490
1494:"Self-Similarity"
1450:978-952-5431-32-2
1332:(February 1999).
1330:Benoit Mandelbrot
1308:10.1109/90.282603
1272:Benoit Mandelbrot
1002:(3775): 636–638.
759:Sierpinski carpet
375:
296:topological space
272:whose pieces are
238:Benoit Mandelbrot
2274:
2102:Michael Barnsley
1969:Lyapunov fractal
1827:Sierpiński curve
1780:Blancmange curve
1645:
1638:
1631:
1622:
1621:
1617:
1598:
1596:
1594:
1588:
1566:(5): 1429–1439.
1557:
1547:
1513:
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1459:
1453:. Archived from
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1077:
1057:
1051:
1047:
1045:
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1034:. Archived from
985:
979:
964:
912:Self-replication
851:integer sequence
674:An image of the
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215:....A figure is
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63:Scale invariance
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2117:Felix Hausdorff
2090:
2054:Brownian motion
2039:
2010:
1933:
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1869:Pythagoras tree
1726:
1719:
1715:Self-similarity
1659:Characteristics
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1592:
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1516:Physica Scripta
1511:
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1274:, La Recherche
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703:packet switched
682:self-similarity
678:which exhibits
656:
604:. When the set
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196:dynamic scaling
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2267:Self-reference
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2262:Homeomorphisms
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2132:Helge von Koch
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2064:Brownian motor
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2035:Pickover stalk
2032:
2027:
2021:
2019:
2012:
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1998:
1993:
1991:Newton fractal
1988:
1983:
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1974:Mandelbrot set
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1957:Newton fractal
1954:
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1910:Fractal canopy
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1854:Vicsek fractal
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1770:Fibonacci word
1767:
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1745:Koch snowflake
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1619:
1618:
1613:978-0387989938
1612:
1599:
1548:
1522:(4): 257–260.
1502:
1499:
1498:
1497:
1491:
1483:
1482:External links
1480:
1477:
1476:
1449:
1426:
1405:978-1581131512
1404:
1387:10.1.1.223.194
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1360:978-0691043012
1343:
1321:
1279:
1263:
1254:
1225:
1180:
1119:
1068:(17): 175101.
1052:
998:. New Series.
980:
976:978-0716711865
958:
957:
955:
952:
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949:
944:
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929:
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907:Self-reference
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837:
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790:Close-up of a
783:
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766:
765:In cybernetics
763:
688:Mandelbrot set
664:Mandelbrot set
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80:counterexample
71:Koch snowflake
27:Koch snowflake
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2255:
2253:
2250:
2249:
2247:
2232:
2229:
2227:
2224:
2221:
2220:
2216:
2213:
2212:
2208:
2205:
2204:
2200:
2198:
2195:
2193:
2190:
2186:
2183:
2182:
2180:
2176:
2175:
2173:
2169:
2163:
2160:
2158:
2155:
2153:
2150:
2148:
2145:
2143:
2140:
2138:
2135:
2133:
2130:
2128:
2125:
2123:
2120:
2118:
2115:
2113:
2110:
2108:
2105:
2103:
2100:
2099:
2097:
2093:
2087:
2084:
2082:
2079:
2077:
2074:
2072:
2069:
2065:
2062:
2060:
2059:Brownian tree
2057:
2056:
2055:
2052:
2051:
2049:
2046:
2042:
2036:
2033:
2031:
2028:
2026:
2023:
2022:
2020:
2017:
2013:
2007:
2004:
2002:
1999:
1997:
1994:
1992:
1989:
1987:
1986:Multibrot set
1984:
1980:
1977:
1976:
1975:
1972:
1970:
1967:
1963:
1962:Douady rabbit
1960:
1958:
1955:
1953:
1950:
1949:
1948:
1945:
1943:
1940:
1939:
1937:
1935:
1929:
1921:
1918:
1917:
1916:
1913:
1911:
1908:
1907:
1905:
1903:
1899:
1893:
1890:
1889:
1887:
1885:
1881:
1875:
1872:
1870:
1867:
1865:
1862:
1860:
1857:
1855:
1852:
1850:
1847:
1845:
1842:
1840:
1837:
1833:
1832:Z-order curve
1830:
1828:
1825:
1823:
1820:
1818:
1815:
1813:
1810:
1808:
1805:
1803:
1802:Hilbert curve
1800:
1798:
1795:
1791:
1788:
1787:
1786:
1785:De Rham curve
1783:
1781:
1778:
1777:
1776:
1773:
1771:
1768:
1766:
1763:
1761:
1758:
1756:
1753:
1751:
1750:Menger sponge
1748:
1746:
1743:
1741:
1738:
1736:
1735:Barnsley fern
1733:
1732:
1730:
1728:
1722:
1716:
1713:
1711:
1708:
1704:
1701:
1699:
1696:
1694:
1691:
1689:
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1679:
1678:
1677:
1674:
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1669:
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1664:
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1657:
1653:
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1627:
1626:
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1615:
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1585:
1581:
1577:
1573:
1569:
1565:
1561:
1554:
1549:
1545:
1541:
1537:
1533:
1529:
1525:
1521:
1517:
1510:
1505:
1504:
1501:Self-affinity
1495:
1492:
1489:
1486:
1485:
1473:
1456:
1452:
1446:
1439:
1438:
1430:
1419:
1415:
1411:
1407:
1401:
1397:
1393:
1388:
1383:
1376:
1375:
1367:
1361:
1357:
1353:
1347:
1339:
1335:
1331:
1325:
1317:
1313:
1309:
1305:
1301:
1297:
1290:
1283:
1277:
1273:
1267:
1258:
1251:
1250:3-540-97346-X
1247:
1243:
1242:0-387-97346-X
1239:
1235:
1229:
1221:
1217:
1213:
1209:
1204:
1199:
1195:
1191:
1184:
1176:
1172:
1168:
1164:
1160:
1156:
1152:
1148:
1143:
1138:
1135:(2): 021406.
1134:
1130:
1123:
1115:
1109:
1101:
1097:
1093:
1089:
1085:
1081:
1076:
1071:
1067:
1063:
1056:
1050:
1037:
1033:
1029:
1025:
1021:
1017:
1013:
1009:
1005:
1001:
997:
996:
991:
984:
977:
973:
969:
963:
959:
948:
945:
943:
940:
938:
935:
933:
930:
928:
925:
923:
920:
918:
915:
913:
910:
908:
905:
903:
900:
898:
895:
893:
890:
888:
885:
883:
880:
878:
875:
873:
872:Droste effect
870:
869:
859:
855:
852:
848:
845:
842:
838:
835:
831:
828:
824:
820:
819:
813:
811:
807:
801:
793:
788:
779:
776:
775:Stafford Beer
772:
760:
756:
751:
747:
745:
741:
737:
733:
731:
727:
723:
719:
718:self-affinity
715:
710:
708:
704:
700:
695:
693:
689:
681:
677:
676:Barnsley fern
672:
665:
660:
651:
649:
648:Self-affinity
644:
642:
638:
637:modular group
634:
633:automorphisms
629:
627:
623:
619:
615:
611:
610:dyadic monoid
607:
603:
599:
595:
591:
566:
563:
560:
557:
552:
548:
541:
538:
535:
532:
526:
512:
511:
510:
493:
490:
487:
484:
479:
475:
463:
459:
456:
452:
436:
433:
430:
404:
396:
392:
386:
383:
380:
376:
372:
369:
362:
361:
360:
343:
340:
337:
334:
329:
325:
314:
311:
307:
304:
300:
297:
294:
284:
282:
279:
275:
271:
267:
266:self-affinity
263:
255:
250:
244:Self-affinity
241:
239:
233:
228:
226:
220:
218:
214:
208:
206:
203:
199:
197:
178:
175:
172:
166:
144:
140:
135:
131:
108:
105:
102:
96:
87:
85:
84:straight line
81:
76:
72:
68:
64:
60:
56:
52:
48:
44:
35:
28:
23:
19:
2231:Chaos theory
2226:Kaleidoscope
2217:
2209:
2201:
2127:Gaston Julia
2107:Georg Cantor
1932:Escape-time
1864:Gosper curve
1812:Lévy C curve
1797:Dragon curve
1714:
1676:Box-counting
1606:. Springer.
1603:
1591:. Retrieved
1563:
1559:
1519:
1515:
1472:Google Books
1462:. Retrieved
1455:the original
1436:
1429:
1373:
1366:
1352:Econometrics
1346:
1337:
1324:
1299:
1295:
1282:
1266:
1257:
1233:
1228:
1193:
1189:
1183:
1132:
1129:Phys. Rev. E
1128:
1122:
1108:cite journal
1065:
1061:
1055:
1040:. Retrieved
1036:the original
999:
993:
983:
967:
962:
932:Tessellation
877:Golden ratio
834:Shepard tone
803:
768:
734:
730:econometrics
714:stock market
711:
696:
685:
645:
630:
621:
617:
605:
589:
587:
461:
450:
422:
305:
298:
290:
265:
259:
235:
230:
224:
222:
216:
213:self-similar
212:
210:
204:
200:
88:
47:self-similar
46:
40:
18:
2222:(1987 book)
2214:(1986 book)
2206:(1982 book)
2192:Fractal art
2112:Bill Gosper
2076:Lévy flight
1822:Peano curve
1817:Moore curve
1703:Topological
1688:Correlation
1302:(1): 1–15.
1042:12 November
847:Per Nørgård
712:Similarly,
614:binary tree
278:anisotropic
262:mathematics
75:symmetrical
43:mathematics
2246:Categories
2030:Orbit trap
2025:Buddhabrot
2018:techniques
2006:Mandelbulb
1807:Koch curve
1740:Cantor set
1470:(Also see
954:References
942:Zipf's law
740:Cantor set
509:. We call
449:, we call
359:for which
310:surjective
303:finite set
287:Definition
55:coastlines
2137:Paul Lévy
2016:Rendering
2001:Mandelbox
1947:Julia set
1859:Hexaflake
1790:Minkowski
1710:Recursion
1693:Hausdorff
1544:250815596
1382:CiteSeerX
1203:1409.7928
1142:0901.2761
1075:1101.4730
897:Recursion
782:In nature
726:Andrew Lo
564:∈
491:∈
455:non-empty
434:⊂
384:∈
377:⋃
341:∈
240:in 1964.
194:exhibits
2252:Fractals
2047:fractals
1934:fractals
1902:L-system
1844:T-square
1652:Fractals
1584:Archived
1418:Archived
1175:26023004
1167:19391746
1100:15700641
1032:15662830
1024:17837158
970:, p.44.
865:See also
844:composer
816:In music
742:and the
654:Examples
594:iterated
256:=1.8272.
73:is both
59:fractals
1996:Tricorn
1849:n-flake
1698:Packing
1681:Higuchi
1671:Assouad
1593:30 July
1568:Bibcode
1524:Bibcode
1464:30 July
1414:3329298
1316:6011907
1208:Bibcode
1196:: 228.
1147:Bibcode
1080:Bibcode
1004:Bibcode
995:Science
947:Fractal
927:Teragon
821:Strict
293:compact
270:fractal
202:Peitgen
67:similar
51:similar
2095:People
2045:Random
1952:Filled
1920:H tree
1839:String
1727:system
1610:
1542:
1447:
1412:
1402:
1384:
1358:
1314:
1248:
1240:
1173:
1165:
1098:
1030:
1022:
974:
841:Danish
827:fugues
823:canons
680:affine
628:tree.
626:p-adic
602:monoid
458:subset
274:scaled
225:et al.
205:et al.
2171:Other
1587:(PDF)
1556:(PDF)
1540:S2CID
1512:(PDF)
1458:(PDF)
1441:(PDF)
1421:(PDF)
1410:S2CID
1378:(PDF)
1312:S2CID
1292:(PDF)
1198:arXiv
1171:S2CID
1137:arXiv
1096:S2CID
1070:arXiv
1028:S2CID
1608:ISBN
1595:2018
1466:2018
1445:ISBN
1400:ISBN
1356:ISBN
1246:ISBN
1244:and
1238:ISBN
1163:PMID
1114:link
1044:2020
1020:PMID
972:ISBN
839:The
806:fern
769:The
686:The
631:The
620:has
45:, a
1576:doi
1532:doi
1392:doi
1304:doi
1216:doi
1155:doi
1088:doi
1049:PDF
1012:doi
1000:156
773:of
460:of
423:If
260:In
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