1607:
2369:
1644:
1449:
199:
3969:
20:
31:, where after each iteration, all original line segments are replaced with four, each a self-similar copy that is 1/3 the length of the original. One formalism of the Hausdorff dimension uses the scale factor (S = 3) and the number of self-similar objects (N = 4) to calculate the dimension, D, after the first iteration to be D = (log N)/(log S) = (log 4)/(log 3) ≈ 1.26.
703:
1363:
173:
triangle that points outward, and this base segment is then deleted to leave a final object from the iteration of unit length of 4. That is, after the first iteration, each original line segment has been replaced with N=4, where each self-similar copy is 1/S = 1/3 as long as the original. Stated
2318:
533:
397:, sets with noninteger Hausdorff dimensions, are found everywhere in nature. He observed that the proper idealization of most rough shapes you see around you is not in terms of smooth idealized shapes, but in terms of fractal idealized shapes:
2203:
286:
Every space-filling curve hits some points multiple times and does not have a continuous inverse. It is impossible to map two dimensions onto one in a way that is continuous and continuously invertible. The topological dimension, also called
2523:
1901:
1194:
174:
another way, we have taken an object with
Euclidean dimension, D, and reduced its linear scale by 1/3 in each direction, so that its length increases to N=S. This equation is easily solved for D, yielding the ratio of logarithms (or
1029:
310:
But topological dimension is a very crude measure of the local size of a space (size near a point). A curve that is almost space-filling can still have topological dimension one, even if it fills up most of the area of a region. A
1611:
1818:
1572:
is a union of three copies of itself, each copy shrunk by a factor of 1/2; this yields a
Hausdorff dimension of ln(3)/ln(2) ≈ 1.58. These Hausdorff dimensions are related to the "critical exponent" of the
833:
168:
shown at right is constructed from an equilateral triangle; in each iteration, its component line segments are divided into 3 segments of unit length, the newly created middle segment is used as the base of a new
3036:
2943:
2742:
389:
For shapes that are smooth, or shapes with a small number of corners, the shapes of traditional geometry and science, the
Hausdorff dimension is an integer agreeing with the topological dimension. But
2225:
2618:
909:
698:{\displaystyle H_{\delta }^{d}(S)=\inf \left\{\sum _{i=1}^{\infty }(\operatorname {diam} U_{i})^{d}:\bigcup _{i=1}^{\infty }U_{i}\supseteq S,\operatorname {diam} U_{i}<\delta \right\},}
2852:
139:
1437:
74:
is 3. That is, for sets of points that define a smooth shape or a shape that has a small number of corners—the shapes of traditional geometry and science—the
Hausdorff dimension is an
2110:
1440:). The Hausdorff measure and the Hausdorff content can both be used to determine the dimension of a set, but if the measure of the set is non-zero, their actual values may disagree.
1971:
points in has
Hausdorff dimension zero and Minkowski dimension one. There are also compact sets for which the Minkowski dimension is strictly larger than the Hausdorff dimension.
255:
is the number of independent parameters one needs to pick out a unique point inside. However, any point specified by two parameters can be instead specified by one, because the
1501:
1074:
525:
1407:
2118:
487:
82:. However, formulas have also been developed that allow calculation of the dimension of other less simple objects, where, solely on the basis of their properties of
1548:
1358:{\displaystyle C_{H}^{d}(S):=H_{\infty }^{d}(S)=\inf \left\{\sum _{k=1}^{\infty }(\operatorname {diam} U_{k})^{d}:\bigcup _{k=1}^{\infty }U_{k}\supseteq S\right\}}
1568:, is a union of two copies of itself, each copy shrunk by a factor 1/3; hence, it can be shown that its Hausdorff dimension is ln(2)/ln(3) ≈ 0.63. The
2446:
1829:
1521:
1186:
1162:
1134:
1114:
1094:
929:
857:
750:
730:
457:
937:
2323:
This inequality can be strict. It is possible to find two sets of dimension 0 whose product has dimension 1. In the opposite direction, it is known that when
141:, as opposed to the more intuitive notion of dimension, which is not associated to general metric spaces, and only takes values in the non-negative integers.
1621:
performed detailed experiments to measure the approximate
Hausdorff dimension for various coastlines. His results have varied from 1.02 for the coastline of
1409:
has the construction of the
Hausdorff measure where the covering sets are allowed to have arbitrarily large sizes (Here, we use the standard convention that
3139:
Gneiting, Tilmann; Ševčíková, Hana; Percival, Donald B. (2012). "Estimators of
Fractal Dimension: Assessing the Roughness of Time Series and Spatial Data".
413:
is yet another similar notion which gives the same value for many shapes, but there are well-documented exceptions where all these dimensions differ.
303: + 1 balls overlap. For example, when one covers a line with short open intervals, some points must be covered twice, giving dimension
1745:
755:
401:
Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.
2954:
3900:
2870:
315:
has an integer topological dimension, but in terms of the amount of space it takes up, it behaves like a higher-dimensional space.
3202:(workshop), Society for Chaos Theory in Psychology and the Life Sciences annual meeting, June 28, 1996, Berkeley, California, see
3076:
1574:
2681:
3253:
2313:{\displaystyle \dim _{\operatorname {Haus} }(X\times Y)\geq \dim _{\operatorname {Haus} }(X)+\dim _{\operatorname {Haus} }(Y).}
386:
is a critical boundary between growth rates that are insufficient to cover the space, and growth rates that are overabundant.
3677:
1967:
is similar to, and at least as large as, the
Hausdorff dimension, and they are equal in many situations. However, the set of
98:
allowing computation of dimensions for highly irregular or "rough" sets, this dimension is also commonly referred to as the
2386:
1661:
271:
involving interweaving the digits of two numbers to yield a single number encoding the same information). The example of a
216:
160:. This underlies the earlier statement that the Hausdorff dimension of a point is zero, of a line is one, etc., and that
2557:
4003:
3953:
3758:
3558:
3469:
3236:
3183:
Larry Riddle, 2014, "Classic
Iterated Function Systems: Koch Snowflake", Agnes Scott College e-Academy (online), see
2408:
1683:
873:
406:
379:
238:
182:
178:) appearing in the figures, and giving—in the Koch and other fractal cases—non-integer dimensions for these objects.
2421:
Many sets defined by a self-similarity condition have dimensions which can be determined explicitly. Roughly, a set
2798:
318:
The Hausdorff dimension measures the local size of a space taking into account the distance between points, the
115:
3893:
3773:
3203:
2390:
1665:
1413:
220:
95:
2066:
3278:
Farkas, Abel; Fraser, Jonathan (30 July 2015). "On the equality of Hausdorff measure and Hausdorff content".
268:
3523:
Dodson, M. Maurice; Kristensen, Simon (June 12, 2003). "Hausdorff Dimension and Diophantine Approximation".
3085:, another variation of fractal dimension that, like Hausdorff dimension, is defined using coverings by balls
279:(taking one real number into a pair of real numbers in a way so that all pairs of numbers are covered) and
3988:
3127:
2629:
288:
3669:
3379:
1477:
1041:
492:
4202:
3886:
109:
108:, i.e. a set where the distances between all members are defined. The dimension is drawn from the
2198:{\displaystyle \dim _{\operatorname {Haus} }(X)=\sup _{i\in I}\dim _{\operatorname {Haus} }(X_{i}).}
4197:
3923:
3863:
427:
The formal definition of the Hausdorff dimension is arrived at by defining first the d-dimensional
3392:
This Knowledge article also discusses further useful characterizations of the Hausdorff dimension.
3067:. Under the same conditions as the previous theorem, the unique fixed point of ψ is self-similar.
1371:
4131:
4126:
4106:
3872:
2781:
2379:
1654:
209:
144:
In mathematical terms, the Hausdorff dimension generalizes the notion of the dimension of a real
466:
4116:
4111:
4091:
1582:
1565:
181:
The Hausdorff dimension is a successor to the simpler, but usually equivalent, box-counting or
94:—have non-integer Hausdorff dimensions. Because of the significant technical advances made by
4121:
4101:
4096:
3692:
3184:
1980:
1557:
3633:
3538:
3411:
1618:
1526:
2784:
around some point. Then the unique fixed point of ψ is a set whose Hausdorff dimension is
2518:{\displaystyle \psi _{i}:\mathbf {R} ^{n}\rightarrow \mathbf {R} ^{n},\quad i=1,\ldots ,m}
1896:{\displaystyle \inf _{Y}\dim _{\operatorname {Haus} }(Y)=\dim _{\operatorname {ind} }(X),}
8:
3993:
3848:
3360:
3302:
Schleicher, Dierk (June 2007). "Hausdorff Dimension, Its Properties, and Its Surprises".
3088:
2529:
2052:
1964:
1714:
1588:
1578:
1569:
1457:
1024:{\displaystyle \dim _{\operatorname {H} }{(X)}:=\inf\{d\geq 0:{\mathcal {H}}^{d}(X)=0\}.}
276:
272:
153:
79:
3637:
3542:
3415:
4172:
4013:
3968:
3804:
3709:
3649:
3564:
3528:
3427:
3337:
3311:
3279:
3166:
3148:
2649:
2637:
1506:
1171:
1147:
1119:
1099:
1079:
914:
842:
836:
735:
715:
442:
331:
83:
3584:
4192:
4008:
3808:
3754:
3721:
3713:
3687:
3673:
3653:
3594:
3568:
3554:
3465:
3431:
3329:
3232:
3225:
3220:
3098:
3093:
3082:
3058:
3054:
2425:
is self-similar if it is the fixed point of a set-valued transformation ψ, that is ψ(
2348:
2219:
are non-empty metric spaces, then the Hausdorff dimension of their product satisfies
428:
422:
410:
390:
175:
104:
More specifically, the Hausdorff dimension is a dimensional number associated with a
48:
3550:
3170:
283:, so that a one-dimensional object completely fills up a higher-dimensional object.
3938:
3829:
3796:
3788:
3737:
3701:
3641:
3610:
3546:
3498:
3419:
3341:
3325:
3321:
3158:
1952:
432:
59:
2948:
is self-similar if and only if the intersections satisfy the following condition:
3983:
3928:
3576:
1968:
1706:
1599:
1472:
87:
55:
1721:
which is defined recursively. It is always an integer (or +∞) and is denoted dim
4065:
4050:
3834:
3817:
3741:
3661:
3527:. Proceedings of Symposia in Pure Mathematics. Vol. 72. pp. 305–347.
3503:
3486:
1813:{\displaystyle \dim _{\mathrm {Haus} }(X)\geq \dim _{\operatorname {ind} }(X).}
165:
28:
3800:
3645:
3423:
2857:
The contraction coefficient of a similitude is the magnitude of the dilation.
4186:
4055:
3580:
3333:
2748:
2625:
1984:
1626:
1465:
436:
52:
828:{\displaystyle {\mathcal {H}}^{d}(S)=\lim _{\delta \to 0}H_{\delta }^{d}(S)}
4075:
4040:
3933:
3769:
3725:
1911:
1622:
460:
319:
145:
105:
63:
3615:
3598:
2754:
The open set condition is a separation condition that ensures the images ψ
1606:
4160:
3943:
1592:
256:
170:
36:
3122:
MacGregor Campbell, 2013, "5.6 Scaling and the Hausdorff Dimension," at
4155:
4035:
3792:
3705:
2393: in this section. Unsourced material may be challenged and removed.
1668: in this section. Unsourced material may be challenged and removed.
1561:
260:
223: in this section. Unsourced material may be challenged and removed.
3867:
3162:
3061:(the intersections are just points), but is also true more generally:
4136:
4045:
3958:
3909:
3533:
3316:
2632:
applied to the complete metric space of non-empty compact subsets of
264:
24:
3624:
Marstrand, J. M. (1954). "The dimension of cartesian product sets".
3402:
Marstrand, J. M. (1954). "The dimension of Cartesian product sets".
2368:
1643:
1602:
in dimension 2 and above is conjectured to be Hausdorff dimension 2.
198:
4060:
4023:
3948:
2777:
1710:
1035:
3284:
3153:
3031:{\displaystyle H^{s}\left(\psi _{i}(E)\cap \psi _{j}(E)\right)=0,}
1448:
4070:
3525:
Fractal Geometry and Applications: A Jubilee of Benoît Mandelbrot
1553:
1453:
1410:
709:
394:
312:
161:
91:
90:, one is led to the conclusion that particular objects—including
75:
3079:
Examples of deterministic fractals, random and natural fractals.
2938:{\displaystyle A\mapsto \psi (A)=\bigcup _{i=1}^{m}\psi _{i}(A)}
1556:
often are spaces whose Hausdorff dimension strictly exceeds the
752:. The Hausdorff d-dimensional outer measure is then defined as
67:
1958:
1696:
78:
agreeing with the usual sense of dimension, also known as the
4027:
3720:
3690:(1929). "On Linear Sets of Points of Fractional Dimensions".
2658:(in certain cases), we need a technical condition called the
1974:
3878:
275:
shows that one can even map the real line to the real plane
164:
can have noninteger Hausdorff dimensions. For instance, the
19:
3845:
Fractal Geometry: Mathematical Foundations and Applications
3447:
Fractal geometry. Mathematical foundations and applications
3357:
Fractal Geometry: Mathematical Foundations and Applications
3198:
Keith Clayton, 1996, "Fractals and the Fractal Dimension,"
1460:, an object with Hausdorff dimension of log(3)/log(2)≈1.58.
71:
2737:{\displaystyle \bigcup _{i=1}^{m}\psi _{i}(V)\subseteq V,}
1955:(1907–1976), e.g., see Hurewicz and Wallman, Chapter VII.
3138:
1595:
have the same Hausdorff dimension as the space they fill.
1116:
is infinite (except that when this latter set of numbers
251:
The intuitive concept of dimension of a geometric object
291:, explains why. This dimension is the greatest integer
2058:
1926:
have the same underlying set of points and the metric
382:, which equals the Hausdorff dimension when the value
299:
by small open balls there is at least one point where
3747:
Several selections from this volume are reprinted in
3118:
3116:
3114:
2957:
2873:
2801:
2684:
2560:
2449:
2228:
2121:
2069:
1832:
1748:
1529:
1509:
1480:
1416:
1374:
1197:
1174:
1150:
1122:
1102:
1082:
1044:
940:
917:
876:
845:
758:
738:
718:
536:
495:
469:
445:
405:
For fractals that occur in nature, the Hausdorff and
118:
3231:. Lecture notes in mathematics 1358. W. H. Freeman.
2343:
is bounded from above by the Hausdorff dimension of
1633:
58:. For instance, the Hausdorff dimension of a single
3449:. John Wiley & Sons, Inc., Hoboken, New Jersey.
2654:To determine the dimension of the self-similar set
2208:This can be verified directly from the definition.
3224:
3111:
3030:
2937:
2846:
2736:
2612:
2517:
2312:
2197:
2104:
1895:
1812:
1542:
1515:
1495:
1431:
1401:
1357:
1180:
1156:
1128:
1108:
1088:
1068:
1023:
923:
903:
851:
827:
744:
724:
697:
519:
481:
451:
378:approaches zero. More precisely, this defines the
133:
23:Example of non-integer dimensions. The first four
3686:
3666:Geometry of sets and measures in Euclidean spaces
3194:
3192:
2770:. Suppose the open set condition holds and each ψ
2747:where the sets in union on the left are pairwise
2613:{\displaystyle A=\bigcup _{i=1}^{m}\psi _{i}(A).}
4184:
3748:
3522:
3401:
2433:, although the exact definition is given below.
2148:
1834:
1417:
1252:
968:
786:
564:
3575:
3459:
2355:. These facts are discussed in Mattila (1995).
904:{\displaystyle \dim _{\operatorname {H} }{(X)}}
362:, the Hausdorff dimension is the unique number
3200:Basic Concepts in Nonlinear Dynamics and Chaos
3189:
3894:
3464:. Cambridge, UK: Cambridge University Press.
3252:Briggs, Jimmy; Tyree, Tim (3 December 2016).
2776:is a similitude, that is a composition of an
1951:These results were originally established by
3768:
3593:
3297:
3295:
3277:
3215:
3213:
3211:
1015:
971:
3251:
2847:{\displaystyle \sum _{i=1}^{m}r_{i}^{s}=1.}
1959:Hausdorff dimension and Minkowski dimension
1697:Hausdorff dimension and inductive dimension
1136:is empty the Hausdorff dimension is zero).
3901:
3887:
3815:
3730:Journal of the London Mathematical Society
3484:
3301:
3219:
2864:which is carried onto itself by a mapping
1975:Hausdorff dimensions and Frostman measures
1610:Estimating the Hausdorff dimension of the
134:{\displaystyle {\overline {\mathbb {R} }}}
3833:
3728:(1937). "Sets of Fractional Dimensions".
3623:
3614:
3532:
3502:
3315:
3292:
3283:
3208:
3152:
3124:Annenberg Learner:MATHematics illuminated
2643:
2409:Learn how and when to remove this message
1684:Learn how and when to remove this message
1483:
1432:{\displaystyle \inf \varnothing =\infty }
431:, a fractional-dimension analogue of the
239:Learn how and when to remove this message
148:. That is, the Hausdorff dimension of an
122:
3842:
3444:
3354:
2105:{\displaystyle X=\bigcup _{i\in I}X_{i}}
1605:
1447:
835:, and the restriction of the mapping to
18:
3660:
3460:Falconer, K. J. (1985). "Theorem 8.3".
3373:
3077:List of fractals by Hausdorff dimension
2662:(OSC) on the sequence of contractions ψ
2630:contractive mapping fixed point theorem
4185:
862:
839:justifies it as a measure, called the
16:Invariant measure of fractal dimension
3882:
2112:is a finite or countable union, then
2391:adding citations to reliable sources
2362:
2358:
2051:. A partial converse is provided by
1666:adding citations to reliable sources
1637:
1139:
416:
221:adding citations to reliable sources
192:
3057:. This is clear in the case of the
2059:Behaviour under unions and products
712:is taken over all countable covers
358:. For a sufficiently well-behaved
263:is equal to the cardinality of the
13:
3515:
1764:
1761:
1758:
1755:
1566:zero-dimensional topological space
1426:
1329:
1276:
1230:
1096:-dimensional Hausdorff measure of
1060:
989:
946:
882:
762:
641:
588:
511:
14:
4214:
3857:
3304:The American Mathematical Monthly
1634:Properties of Hausdorff dimension
1420:
51:, that was introduced in 1918 by
3967:
2675:with compact closure, such that
2480:
2465:
2367:
1642:
1496:{\displaystyle \mathbb {R} ^{n}}
1069:{\displaystyle d\in [0,\infty )}
859:-dimensional Hausdorff Measure.
520:{\displaystyle d\in [0,\infty )}
197:
100:Hausdorff–Besicovitch dimension.
3478:
3453:
3438:
3395:
3386:
3367:
2543:< 1. Then there is a unique
2493:
2378:needs additional citations for
1939:is topologically equivalent to
1653:needs additional citations for
295:such that in every covering of
208:needs additional citations for
3875:at Encyclopedia of Mathematics
3818:"Fractals and self similarity"
3487:"Fractals and self similarity"
3348:
3326:10.1080/00029890.2007.11920440
3271:
3245:
3227:The Fractal Geometry of Nature
3177:
3132:
3045:is the Hausdorff dimension of
3011:
3005:
2989:
2983:
2932:
2926:
2889:
2883:
2877:
2722:
2716:
2604:
2598:
2475:
2304:
2298:
2279:
2273:
2254:
2242:
2189:
2176:
2141:
2135:
1887:
1881:
1862:
1856:
1804:
1798:
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1773:
1625:to 1.25 for the west coast of
1396:
1390:
1301:
1281:
1246:
1240:
1219:
1213:
1063:
1051:
1006:
1000:
961:
955:
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891:
822:
816:
793:
779:
773:
613:
593:
558:
552:
514:
502:
96:Abram Samoilovitch Besicovitch
1:
3908:
3589:. Princeton University Press.
3445:Falconer, Kenneth J. (2003).
3104:
2764:) do not overlap "too much".
2335:, the Hausdorff dimension of
183:Minkowski–Bouligand dimension
3816:Hutchinson, John E. (1981).
3485:Hutchinson, John E. (1981).
3462:The Geometry of Fractal Sets
1402:{\displaystyle C_{H}^{d}(S)}
354:) grows polynomially with 1/
188:
126:
7:
3868:Encyclopedia of Mathematics
3774:"Dimension und äußeres Maß"
3626:Proc. Cambridge Philos. Soc
3599:"La dimension et la mesure"
3551:10.1090/pspum/072.1/2112110
3404:Proc. Cambridge Philos. Soc
3070:
1468:have Hausdorff dimension 0.
1443:
1166:unlimited Hausdorff content
289:Lebesgue covering dimension
10:
4219:
3843:Falconer, Kenneth (2003).
3835:10.1512/iumj.1981.30.30055
3753:. Boston: Addison-Wesley.
3670:Cambridge University Press
3504:10.1512/iumj.1981.30.30055
3380:Cambridge University Press
3355:Falconer, Kenneth (2003).
3259:. University of Washington
2792:is the unique solution of
2647:
2536:with contraction constant
2023:> 0 and for every ball
1987:subsets of a metric space
1910:ranges over metric spaces
1550:has Hausdorff dimension 1.
482:{\displaystyle S\subset X}
420:
4169:
4148:
4084:
4022:
3976:
3965:
3916:
3749:Edgar, Gerald A. (1993).
3646:10.1017/S0305004100029236
3424:10.1017/S0305004100029236
2624:The theorem follows from
1709:metric space. There is a
3742:10.1112/jlms/s1-12.45.18
3205:, accessed 5 March 2015.
3186:, accessed 5 March 2015.
3129:, accessed 5 March 2015.
2019:holds for some constant
1503:has Hausdorff dimension
1034:This is the same as the
267:(this can be seen by an
47:, or more specifically,
3603:Fundamenta Mathematicae
3374:Morters, Peres (2010).
2349:upper packing dimension
1452:Dimension of a further
3053:denotes s-dimensional
3032:
2939:
2915:
2848:
2822:
2738:
2705:
2644:The open set condition
2622:
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1612:coast of Great Britain
1583:analysis of algorithms
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1025:
925:
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853:
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699:
645:
592:
521:
483:
453:
407:box-counting dimension
403:
380:box-counting dimension
322:. Consider the number
135:
32:
3822:Indiana Univ. Math. J
3781:Mathematische Annalen
3693:Mathematische Annalen
3616:10.4064/fm-28-1-81-89
3491:Indiana Univ. Math. J
3033:
2940:
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2849:
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2739:
2685:
2671:There is an open set
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2567:
2520:
2435:
2331:are Borel subsets of
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2200:
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1898:
1815:
1609:
1558:topological dimension
1545:
1543:{\displaystyle S^{1}}
1518:
1498:
1451:
1434:
1404:
1360:
1313:
1260:
1183:
1159:
1131:
1111:
1091:
1071:
1026:
926:
906:
854:
830:
747:
727:
700:
625:
572:
522:
484:
454:
399:
136:
110:extended real numbers
80:topological dimension
22:
4085:Dimensions by number
3765:See chapters 9,10,11
3751:Classics on fractals
2955:
2871:
2799:
2682:
2558:
2447:
2387:improve this article
2226:
2119:
2067:
1830:
1746:
1739:is non-empty. Then
1662:improve this article
1619:Lewis Fry Richardson
1589:Space-filling curves
1579:recurrence relations
1527:
1507:
1478:
1414:
1372:
1195:
1172:
1148:
1120:
1100:
1080:
1042:
938:
915:
874:
843:
756:
736:
716:
534:
493:
467:
443:
439:is constructed: Let
217:improve this article
116:
3864:Hausdorff dimension
3849:John Wiley and Sons
3638:1954PCPS...50..198M
3543:2003math......5399D
3416:1954PCPS...50..198M
3361:John Wiley and Sons
3254:"Hausdorff Measure"
3141:Statistical Science
3089:Intrinsic dimension
2837:
1965:Minkowski dimension
1715:inductive dimension
1570:Sierpinski triangle
1560:. For example, the
1458:Sierpinski triangle
1389:
1239:
1212:
869:Hausdorff dimension
863:Hausdorff dimension
815:
551:
273:space-filling curve
154:inner product space
41:Hausdorff dimension
4014:Degrees of freedom
3917:Dimensional spaces
3801:10338.dmlcz/100363
3793:10.1007/BF01457179
3706:10.1007/BF01454831
3221:Mandelbrot, Benoît
3028:
2935:
2860:In general, a set
2844:
2823:
2734:
2660:open set condition
2650:Open set condition
2638:Hausdorff distance
2610:
2515:
2310:
2195:
2162:
2102:
2091:
1918:. In other words,
1893:
1842:
1810:
1615:
1598:The trajectory of
1540:
1513:
1493:
1462:
1429:
1399:
1375:
1355:
1225:
1198:
1178:
1154:
1126:
1106:
1086:
1066:
1021:
921:
901:
849:
825:
801:
800:
742:
722:
695:
537:
517:
479:
449:
338:required to cover
334:of radius at most
176:natural logarithms
131:
33:
4180:
4179:
3989:Lebesgue covering
3954:Algebraic variety
3873:Hausdorff measure
3722:A. S. Besicovitch
3688:A. S. Besicovitch
3679:978-0-521-65595-8
3163:10.1214/11-STS370
3099:Fractal dimension
3094:Packing dimension
3083:Assouad dimension
3059:Sierpinski gasket
3055:Hausdorff measure
2419:
2418:
2411:
2359:Self-similar sets
2147:
2076:
1833:
1694:
1693:
1686:
1523:, and the circle
1516:{\displaystyle n}
1181:{\displaystyle S}
1157:{\displaystyle d}
1140:Hausdorff content
1129:{\displaystyle d}
1109:{\displaystyle X}
1089:{\displaystyle d}
924:{\displaystyle X}
852:{\displaystyle d}
785:
745:{\displaystyle S}
725:{\displaystyle U}
452:{\displaystyle X}
429:Hausdorff measure
423:Hausdorff measure
417:Formal definition
411:packing dimension
391:Benoit Mandelbrot
342:completely. When
249:
248:
241:
129:
49:fractal dimension
4210:
4203:Dimension theory
3977:Other dimensions
3971:
3939:Projective space
3903:
3896:
3889:
3880:
3879:
3852:
3847:(2nd ed.).
3839:
3837:
3812:
3787:(1–2): 157–179.
3778:
3764:
3745:
3717:
3683:
3657:
3620:
3618:
3590:
3586:Dimension Theory
3577:Hurewicz, Witold
3572:
3536:
3509:
3508:
3506:
3482:
3476:
3475:
3457:
3451:
3450:
3442:
3436:
3435:
3399:
3393:
3390:
3384:
3383:
3371:
3365:
3364:
3359:(2nd ed.).
3352:
3346:
3345:
3319:
3299:
3290:
3289:
3287:
3275:
3269:
3268:
3266:
3264:
3258:
3249:
3243:
3242:
3230:
3217:
3206:
3196:
3187:
3181:
3175:
3174:
3156:
3136:
3130:
3120:
3037:
3035:
3034:
3029:
3018:
3014:
3004:
3003:
2982:
2981:
2967:
2966:
2944:
2942:
2941:
2936:
2925:
2924:
2914:
2909:
2853:
2851:
2850:
2845:
2836:
2831:
2821:
2816:
2743:
2741:
2740:
2735:
2715:
2714:
2704:
2699:
2619:
2617:
2616:
2611:
2597:
2596:
2586:
2581:
2524:
2522:
2521:
2516:
2489:
2488:
2483:
2474:
2473:
2468:
2459:
2458:
2414:
2407:
2403:
2400:
2394:
2371:
2363:
2319:
2317:
2316:
2311:
2294:
2293:
2269:
2268:
2238:
2237:
2204:
2202:
2201:
2196:
2188:
2187:
2172:
2171:
2161:
2131:
2130:
2111:
2109:
2108:
2103:
2101:
2100:
2090:
2053:Frostman's lemma
1953:Edward Szpilrajn
1902:
1900:
1899:
1894:
1877:
1876:
1852:
1851:
1841:
1819:
1817:
1816:
1811:
1794:
1793:
1769:
1768:
1767:
1705:be an arbitrary
1689:
1682:
1678:
1675:
1669:
1646:
1638:
1549:
1547:
1546:
1541:
1539:
1538:
1522:
1520:
1519:
1514:
1502:
1500:
1499:
1494:
1492:
1491:
1486:
1438:
1436:
1435:
1430:
1408:
1406:
1405:
1400:
1388:
1383:
1368:In other words,
1364:
1362:
1361:
1356:
1354:
1350:
1343:
1342:
1332:
1327:
1309:
1308:
1299:
1298:
1279:
1274:
1238:
1233:
1211:
1206:
1187:
1185:
1184:
1179:
1163:
1161:
1160:
1155:
1135:
1133:
1132:
1127:
1115:
1113:
1112:
1107:
1095:
1093:
1092:
1087:
1075:
1073:
1072:
1067:
1030:
1028:
1027:
1022:
999:
998:
993:
992:
964:
950:
949:
930:
928:
927:
922:
910:
908:
907:
902:
900:
886:
885:
858:
856:
855:
850:
834:
832:
831:
826:
814:
809:
799:
772:
771:
766:
765:
751:
749:
748:
743:
731:
729:
728:
723:
704:
702:
701:
696:
691:
687:
680:
679:
655:
654:
644:
639:
621:
620:
611:
610:
591:
586:
550:
545:
526:
524:
523:
518:
488:
486:
485:
480:
458:
456:
455:
450:
433:Lebesgue measure
307: = 1.
244:
237:
233:
230:
224:
201:
193:
140:
138:
137:
132:
130:
125:
120:
43:is a measure of
4218:
4217:
4213:
4212:
4211:
4209:
4208:
4207:
4198:Metric geometry
4183:
4182:
4181:
4176:
4165:
4144:
4080:
4018:
3972:
3963:
3929:Euclidean space
3912:
3907:
3860:
3855:
3776:
3761:
3746:
3680:
3662:Mattila, Pertti
3561:
3518:
3516:Further reading
3513:
3512:
3483:
3479:
3472:
3458:
3454:
3443:
3439:
3400:
3396:
3391:
3387:
3376:Brownian Motion
3372:
3368:
3353:
3349:
3300:
3293:
3276:
3272:
3262:
3260:
3256:
3250:
3246:
3239:
3218:
3209:
3197:
3190:
3182:
3178:
3137:
3133:
3121:
3112:
3107:
3073:
2999:
2995:
2977:
2973:
2972:
2968:
2962:
2958:
2956:
2953:
2952:
2920:
2916:
2910:
2899:
2872:
2869:
2868:
2832:
2827:
2817:
2806:
2800:
2797:
2796:
2775:
2759:
2710:
2706:
2700:
2689:
2683:
2680:
2679:
2667:
2652:
2646:
2592:
2588:
2582:
2571:
2559:
2556:
2555:
2541:
2484:
2479:
2478:
2469:
2464:
2463:
2454:
2450:
2448:
2445:
2444:
2415:
2404:
2398:
2395:
2384:
2372:
2361:
2289:
2285:
2264:
2260:
2233:
2229:
2227:
2224:
2223:
2183:
2179:
2167:
2163:
2151:
2126:
2122:
2120:
2117:
2116:
2096:
2092:
2080:
2068:
2065:
2064:
2061:
2042:
1977:
1961:
1947:
1934:
1872:
1868:
1847:
1843:
1837:
1831:
1828:
1827:
1789:
1785:
1754:
1753:
1749:
1747:
1744:
1743:
1724:
1699:
1690:
1679:
1673:
1670:
1659:
1647:
1636:
1600:Brownian motion
1534:
1530:
1528:
1525:
1524:
1508:
1505:
1504:
1487:
1482:
1481:
1479:
1476:
1475:
1473:Euclidean space
1446:
1415:
1412:
1411:
1384:
1379:
1373:
1370:
1369:
1338:
1334:
1328:
1317:
1304:
1300:
1294:
1290:
1275:
1264:
1259:
1255:
1234:
1229:
1207:
1202:
1196:
1193:
1192:
1173:
1170:
1169:
1149:
1146:
1145:
1142:
1121:
1118:
1117:
1101:
1098:
1097:
1081:
1078:
1077:
1043:
1040:
1039:
994:
988:
987:
986:
954:
945:
941:
939:
936:
935:
916:
913:
912:
890:
881:
877:
875:
872:
871:
865:
844:
841:
840:
837:measurable sets
810:
805:
789:
767:
761:
760:
759:
757:
754:
753:
737:
734:
733:
717:
714:
713:
675:
671:
650:
646:
640:
629:
616:
612:
606:
602:
587:
576:
571:
567:
546:
541:
535:
532:
531:
494:
491:
490:
468:
465:
464:
444:
441:
440:
425:
419:
346:is very small,
245:
234:
228:
225:
214:
202:
191:
121:
119:
117:
114:
113:
88:self-similarity
70:is 2, and of a
56:Felix Hausdorff
17:
12:
11:
5:
4216:
4206:
4205:
4200:
4195:
4178:
4177:
4170:
4167:
4166:
4164:
4163:
4158:
4152:
4150:
4146:
4145:
4143:
4142:
4134:
4129:
4124:
4119:
4114:
4109:
4104:
4099:
4094:
4088:
4086:
4082:
4081:
4079:
4078:
4073:
4068:
4066:Cross-polytope
4063:
4058:
4053:
4051:Hyperrectangle
4048:
4043:
4038:
4032:
4030:
4020:
4019:
4017:
4016:
4011:
4006:
4001:
3996:
3991:
3986:
3980:
3978:
3974:
3973:
3966:
3964:
3962:
3961:
3956:
3951:
3946:
3941:
3936:
3931:
3926:
3920:
3918:
3914:
3913:
3906:
3905:
3898:
3891:
3883:
3877:
3876:
3870:
3859:
3858:External links
3856:
3854:
3853:
3840:
3828:(5): 713–747.
3813:
3772:(March 1919).
3766:
3759:
3718:
3700:(1): 161–193.
3684:
3678:
3658:
3632:(3): 198–202.
3621:
3591:
3581:Wallman, Henry
3573:
3559:
3519:
3517:
3514:
3511:
3510:
3497:(5): 713–747.
3477:
3470:
3452:
3437:
3410:(3): 198–202.
3394:
3385:
3366:
3347:
3310:(6): 509–528.
3291:
3270:
3244:
3237:
3207:
3188:
3176:
3147:(2): 247–277.
3131:
3109:
3108:
3106:
3103:
3102:
3101:
3096:
3091:
3086:
3080:
3072:
3069:
3039:
3038:
3027:
3024:
3021:
3017:
3013:
3010:
3007:
3002:
2998:
2994:
2991:
2988:
2985:
2980:
2976:
2971:
2965:
2961:
2946:
2945:
2934:
2931:
2928:
2923:
2919:
2913:
2908:
2905:
2902:
2898:
2894:
2891:
2888:
2885:
2882:
2879:
2876:
2855:
2854:
2843:
2840:
2835:
2830:
2826:
2820:
2815:
2812:
2809:
2805:
2771:
2755:
2745:
2744:
2733:
2730:
2727:
2724:
2721:
2718:
2713:
2709:
2703:
2698:
2695:
2692:
2688:
2663:
2648:Main article:
2645:
2642:
2621:
2620:
2609:
2606:
2603:
2600:
2595:
2591:
2585:
2580:
2577:
2574:
2570:
2566:
2563:
2539:
2526:
2525:
2514:
2511:
2508:
2505:
2502:
2499:
2496:
2492:
2487:
2482:
2477:
2472:
2467:
2462:
2457:
2453:
2417:
2416:
2375:
2373:
2366:
2360:
2357:
2321:
2320:
2309:
2306:
2303:
2300:
2297:
2292:
2288:
2284:
2281:
2278:
2275:
2272:
2267:
2263:
2259:
2256:
2253:
2250:
2247:
2244:
2241:
2236:
2232:
2206:
2205:
2194:
2191:
2186:
2182:
2178:
2175:
2170:
2166:
2160:
2157:
2154:
2150:
2146:
2143:
2140:
2137:
2134:
2129:
2125:
2099:
2095:
2089:
2086:
2083:
2079:
2075:
2072:
2060:
2057:
2040:
1979:If there is a
1976:
1973:
1960:
1957:
1943:
1930:
1904:
1903:
1892:
1889:
1886:
1883:
1880:
1875:
1871:
1867:
1864:
1861:
1858:
1855:
1850:
1846:
1840:
1836:
1821:
1820:
1809:
1806:
1803:
1800:
1797:
1792:
1788:
1784:
1781:
1778:
1775:
1772:
1766:
1763:
1760:
1757:
1752:
1722:
1698:
1695:
1692:
1691:
1650:
1648:
1641:
1635:
1632:
1631:
1630:
1604:
1603:
1596:
1586:
1575:Master theorem
1551:
1537:
1533:
1512:
1490:
1485:
1469:
1466:Countable sets
1445:
1442:
1428:
1425:
1422:
1419:
1398:
1395:
1392:
1387:
1382:
1378:
1366:
1365:
1353:
1349:
1346:
1341:
1337:
1331:
1326:
1323:
1320:
1316:
1312:
1307:
1303:
1297:
1293:
1289:
1286:
1283:
1278:
1273:
1270:
1267:
1263:
1258:
1254:
1251:
1248:
1245:
1242:
1237:
1232:
1228:
1224:
1221:
1218:
1215:
1210:
1205:
1201:
1188:is defined by
1177:
1153:
1141:
1138:
1125:
1105:
1085:
1076:such that the
1065:
1062:
1059:
1056:
1053:
1050:
1047:
1038:of the set of
1032:
1031:
1020:
1017:
1014:
1011:
1008:
1005:
1002:
997:
991:
985:
982:
979:
976:
973:
970:
967:
963:
960:
957:
953:
948:
944:
931:is defined by
920:
899:
896:
893:
889:
884:
880:
864:
861:
848:
824:
821:
818:
813:
808:
804:
798:
795:
792:
788:
784:
781:
778:
775:
770:
764:
741:
721:
706:
705:
694:
690:
686:
683:
678:
674:
670:
667:
664:
661:
658:
653:
649:
643:
638:
635:
632:
628:
624:
619:
615:
609:
605:
601:
598:
595:
590:
585:
582:
579:
575:
570:
566:
563:
560:
557:
554:
549:
544:
540:
516:
513:
510:
507:
504:
501:
498:
478:
475:
472:
448:
421:Main article:
418:
415:
409:coincide. The
393:observed that
247:
246:
205:
203:
196:
190:
187:
166:Koch snowflake
162:irregular sets
128:
124:
62:is zero, of a
15:
9:
6:
4:
3:
2:
4215:
4204:
4201:
4199:
4196:
4194:
4191:
4190:
4188:
4175:
4174:
4168:
4162:
4159:
4157:
4154:
4153:
4151:
4147:
4141:
4139:
4135:
4133:
4130:
4128:
4125:
4123:
4120:
4118:
4115:
4113:
4110:
4108:
4105:
4103:
4100:
4098:
4095:
4093:
4090:
4089:
4087:
4083:
4077:
4074:
4072:
4069:
4067:
4064:
4062:
4059:
4057:
4056:Demihypercube
4054:
4052:
4049:
4047:
4044:
4042:
4039:
4037:
4034:
4033:
4031:
4029:
4025:
4021:
4015:
4012:
4010:
4007:
4005:
4002:
4000:
3997:
3995:
3992:
3990:
3987:
3985:
3982:
3981:
3979:
3975:
3970:
3960:
3957:
3955:
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3760:0-201-58701-7
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3560:9780821836378
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3313:
3309:
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3298:
3296:
3286:
3281:
3274:
3255:
3248:
3240:
3238:0-7167-1186-9
3234:
3229:
3228:
3222:
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3214:
3212:
3204:
3201:
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3193:
3185:
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3000:
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2949:
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2795:
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2707:
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2696:
2693:
2690:
2686:
2678:
2677:
2676:
2674:
2669:
2666:
2661:
2657:
2651:
2641:
2639:
2635:
2631:
2627:
2626:Stefan Banach
2607:
2601:
2593:
2589:
2583:
2578:
2575:
2572:
2568:
2564:
2561:
2554:
2553:
2552:
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2542:
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2500:
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2439:
2434:
2432:
2428:
2424:
2413:
2410:
2402:
2392:
2388:
2382:
2381:
2376:This section
2374:
2370:
2365:
2364:
2356:
2354:
2350:
2346:
2342:
2338:
2334:
2330:
2326:
2307:
2301:
2295:
2290:
2286:
2282:
2276:
2270:
2265:
2261:
2257:
2251:
2248:
2245:
2239:
2234:
2230:
2222:
2221:
2220:
2218:
2214:
2209:
2192:
2184:
2180:
2173:
2168:
2164:
2158:
2155:
2152:
2144:
2138:
2132:
2127:
2123:
2115:
2114:
2113:
2097:
2093:
2087:
2084:
2081:
2077:
2073:
2070:
2056:
2054:
2050:
2046:
2038:
2034:
2030:
2026:
2022:
2018:
2014:
2010:
2006:
2002:
1999:) > 0 and
1998:
1994:
1990:
1986:
1983:μ defined on
1982:
1972:
1970:
1966:
1956:
1954:
1949:
1946:
1942:
1938:
1933:
1929:
1925:
1921:
1917:
1913:
1909:
1890:
1884:
1878:
1873:
1869:
1865:
1859:
1853:
1848:
1844:
1838:
1826:
1825:
1824:
1807:
1801:
1795:
1790:
1786:
1782:
1776:
1770:
1750:
1742:
1741:
1740:
1738:
1734:
1730:
1728:
1720:
1716:
1712:
1708:
1704:
1688:
1685:
1677:
1667:
1663:
1657:
1656:
1651:This section
1649:
1645:
1640:
1639:
1628:
1627:Great Britain
1624:
1620:
1617:
1616:
1613:
1608:
1601:
1597:
1594:
1590:
1587:
1584:
1580:
1576:
1571:
1567:
1563:
1559:
1555:
1552:
1535:
1531:
1510:
1488:
1474:
1470:
1467:
1464:
1463:
1459:
1456:example. The
1455:
1450:
1441:
1439:
1423:
1393:
1385:
1380:
1376:
1351:
1347:
1344:
1339:
1335:
1324:
1321:
1318:
1314:
1310:
1305:
1295:
1291:
1287:
1284:
1271:
1268:
1265:
1261:
1256:
1249:
1243:
1235:
1226:
1222:
1216:
1208:
1203:
1199:
1191:
1190:
1189:
1175:
1167:
1164:-dimensional
1151:
1137:
1123:
1103:
1083:
1057:
1054:
1048:
1045:
1037:
1018:
1012:
1009:
1003:
995:
983:
980:
977:
974:
965:
958:
951:
942:
934:
933:
932:
918:
894:
887:
878:
870:
860:
846:
838:
819:
811:
806:
802:
796:
790:
782:
776:
768:
739:
719:
711:
692:
688:
684:
681:
676:
672:
668:
665:
662:
659:
656:
651:
647:
636:
633:
630:
626:
622:
617:
607:
603:
599:
596:
583:
580:
577:
573:
568:
561:
555:
547:
542:
538:
530:
529:
528:
508:
505:
499:
496:
476:
473:
470:
462:
446:
438:
437:outer measure
434:
430:
424:
414:
412:
408:
402:
398:
396:
392:
387:
385:
381:
377:
373:
370:) grows as 1/
369:
365:
361:
357:
353:
349:
345:
341:
337:
333:
329:
325:
321:
316:
314:
308:
306:
302:
298:
294:
290:
284:
282:
278:
274:
270:
266:
262:
258:
254:
243:
240:
232:
222:
218:
212:
211:
206:This section
204:
200:
195:
194:
186:
184:
179:
177:
172:
167:
163:
159:
155:
152:-dimensional
151:
147:
142:
111:
107:
102:
101:
97:
93:
89:
85:
81:
77:
73:
69:
65:
61:
57:
54:
53:mathematician
50:
46:
42:
38:
30:
26:
21:
4171:
4137:
4076:Hyperpyramid
4041:Hypersurface
3998:
3934:Affine space
3924:Vector space
3844:
3825:
3821:
3784:
3780:
3770:F. Hausdorff
3750:
3736:(1): 18–25.
3733:
3729:
3726:H. D. Ursell
3697:
3691:
3665:
3629:
3625:
3606:
3602:
3595:E. Szpilrajn
3585:
3534:math/0305399
3524:
3494:
3490:
3480:
3461:
3455:
3446:
3440:
3407:
3403:
3397:
3388:
3375:
3369:
3356:
3350:
3317:math/0505099
3307:
3303:
3273:
3261:. Retrieved
3247:
3226:
3199:
3179:
3144:
3140:
3134:
3123:
3064:
3063:
3050:
3046:
3042:
3040:
2947:
2861:
2859:
2856:
2789:
2785:
2772:
2767:
2766:
2761:
2756:
2753:
2746:
2672:
2670:
2664:
2659:
2655:
2653:
2633:
2623:
2548:
2547:compact set
2544:
2537:
2533:
2527:
2437:
2436:
2430:
2426:
2422:
2420:
2405:
2396:
2385:Please help
2380:verification
2377:
2352:
2344:
2340:
2336:
2332:
2328:
2324:
2322:
2216:
2212:
2210:
2207:
2062:
2048:
2044:
2036:
2032:
2028:
2024:
2020:
2016:
2012:
2008:
2004:
2000:
1996:
1992:
1988:
1978:
1962:
1950:
1944:
1940:
1936:
1931:
1927:
1923:
1919:
1915:
1912:homeomorphic
1907:
1905:
1822:
1736:
1732:
1731:
1726:
1718:
1702:
1700:
1680:
1671:
1660:Please help
1655:verification
1652:
1623:South Africa
1577:for solving
1367:
1165:
1143:
1033:
868:
866:
707:
461:metric space
435:. First, an
426:
404:
400:
388:
383:
375:
371:
367:
366:such that N(
363:
359:
355:
351:
347:
343:
339:
335:
327:
323:
317:
309:
304:
300:
296:
292:
285:
281:continuously
280:
277:surjectively
252:
250:
235:
226:
215:Please help
210:verification
207:
180:
157:
149:
146:vector space
143:
106:metric space
103:
99:
64:line segment
44:
40:
34:
4161:Codimension
4140:-dimensions
4061:Hypersphere
3944:Free module
2532:mapping on
2530:contraction
2528:are each a
1711:topological
1593:Peano curve
257:cardinality
171:equilateral
66:is 1, of a
37:mathematics
4187:Categories
4156:Hyperspace
4036:Hyperplane
3263:3 February
3105:References
2551:such that
2440:. Suppose
2399:March 2015
2039:, then dim
1991:such that
1823:Moreover,
1735:. Suppose
1713:notion of
1674:March 2015
1562:Cantor set
708:where the
261:real plane
229:March 2015
29:Koch curve
25:iterations
4046:Hypercube
4024:Polytopes
4004:Minkowski
3999:Hausdorff
3994:Inductive
3959:Spacetime
3910:Dimension
3809:122001234
3714:125368661
3654:122475292
3569:119613948
3432:122475292
3334:0002-9890
3285:1411.0867
3154:1101.1444
2997:ψ
2993:∩
2975:ψ
2918:ψ
2897:⋃
2881:ψ
2878:↦
2804:∑
2726:⊆
2708:ψ
2687:⋃
2636:with the
2590:ψ
2569:⋃
2545:non-empty
2507:…
2476:→
2452:ψ
2347:plus the
2296:
2271:
2258:≥
2249:×
2240:
2174:
2156:∈
2133:
2085:∈
2078:⋃
1879:
1854:
1796:
1783:≥
1771:
1707:separable
1591:like the
1427:∞
1421:∅
1345:⊇
1330:∞
1315:⋃
1288:
1277:∞
1262:∑
1231:∞
1061:∞
1049:∈
978:≥
952:
888:
807:δ
794:→
791:δ
685:δ
669:
657:⊇
642:∞
627:⋃
600:
589:∞
574:∑
543:δ
512:∞
500:∈
474:⊂
265:real line
189:Intuition
127:¯
45:roughness
4193:Fractals
4173:Category
4149:See also
3949:Manifold
3664:(1995).
3609:: 81–9.
3597:(1937).
3583:(1948).
3223:(1982).
3171:88512325
3071:See also
2782:dilation
2778:isometry
2749:disjoint
1969:rational
1554:Fractals
1444:Examples
1036:supremum
395:fractals
269:argument
92:fractals
4071:Simplex
4009:Fractal
3634:Bibcode
3539:Bibcode
3412:Bibcode
3342:9811750
3065:Theorem
2768:Theorem
2438:Theorem
1981:measure
1733:Theorem
1581:in the
1454:fractal
710:infimum
313:fractal
259:of the
156:equals
84:scaling
76:integer
27:of the
4028:shapes
3807:
3757:
3712:
3676:
3652:
3567:
3557:
3468:
3430:
3340:
3332:
3235:
3169:
3126:, see
3041:where
2788:where
2780:and a
1906:where
320:metric
68:square
4132:Eight
4127:Seven
4107:Three
3984:Krull
3805:S2CID
3777:(PDF)
3710:S2CID
3650:S2CID
3565:S2CID
3529:arXiv
3428:S2CID
3338:S2CID
3312:arXiv
3280:arXiv
3257:(PDF)
3167:S2CID
3149:arXiv
2035:) in
2015:)) ≤
1985:Borel
463:. If
459:be a
332:balls
330:) of
60:point
4117:Five
4112:Four
4092:Zero
4026:and
3755:ISBN
3674:ISBN
3555:ISBN
3466:ISBN
3330:ISSN
3265:2022
3233:ISBN
3049:and
2429:) =
2327:and
2291:Haus
2266:Haus
2235:Haus
2215:and
2169:Haus
2128:Haus
2047:) ≥
2041:Haus
1963:The
1922:and
1849:Haus
1717:for
1701:Let
1564:, a
1471:The
1285:diam
1144:The
867:The
682:<
666:diam
597:diam
489:and
86:and
72:cube
4122:Six
4102:Two
4097:One
3866:at
3830:doi
3797:hdl
3789:doi
3738:doi
3702:doi
3698:101
3642:doi
3611:doi
3547:doi
3499:doi
3420:doi
3322:doi
3308:114
3159:doi
2628:'s
2389:by
2351:of
2287:dim
2262:dim
2231:dim
2211:If
2165:dim
2149:sup
2124:dim
2063:If
1935:of
1914:to
1874:ind
1870:dim
1845:dim
1835:inf
1791:ind
1787:dim
1751:dim
1729:).
1723:ind
1664:by
1418:inf
1253:inf
1168:of
969:inf
943:dim
911:of
879:dim
787:lim
732:of
565:inf
374:as
219:by
35:In
4189::
3826:30
3824:.
3820:.
3803:.
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3785:79
3783:.
3779:.
3734:12
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3724:;
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3696:.
3672:.
3668:.
3648:.
3640:.
3630:50
3628:.
3607:28
3605:.
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3579:;
3563:.
3553:.
3545:.
3537:.
3495:30
3493:.
3489:.
3426:.
3418:.
3408:50
3406:.
3378:.
3336:.
3328:.
3320:.
3306:.
3294:^
3210:^
3191:^
3165:.
3157:.
3145:27
3143:.
3113:^
2842:1.
2751:.
2668:.
2640:.
2339:×
2055:.
2031:,
2011:,
1948:.
1223::=
966::=
527:,
185:.
112:,
39:,
4138:n
3902:e
3895:t
3888:v
3851:.
3838:.
3832::
3811:.
3799::
3791::
3763:.
3744:.
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3716:.
3704::
3682:.
3656:.
3644::
3636::
3619:.
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3571:.
3549::
3541::
3531::
3507:.
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3474:.
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3414::
3382:.
3363:.
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3288:.
3282::
3267:.
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3043:s
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2786:s
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2760:(
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2732:,
2729:V
2723:)
2720:V
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2691:i
2673:V
2665:i
2656:A
2634:R
2608:.
2605:)
2602:A
2599:(
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2584:m
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2573:i
2565:=
2562:A
2549:A
2540:i
2538:r
2534:R
2513:m
2510:,
2504:,
2501:1
2498:=
2495:i
2491:,
2486:n
2481:R
2471:n
2466:R
2461::
2456:i
2431:E
2427:E
2423:E
2412:)
2406:(
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2397:(
2383:.
2353:Y
2345:X
2341:Y
2337:X
2333:R
2329:Y
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2308:.
2305:)
2302:Y
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2283:+
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2277:X
2274:(
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2243:(
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2193:.
2190:)
2185:i
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2153:i
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2139:X
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2071:X
2049:s
2045:X
2043:(
2037:X
2033:r
2029:x
2027:(
2025:B
2021:s
2017:r
2013:r
2009:x
2007:(
2005:B
2003:(
2001:μ
1997:X
1995:(
1993:μ
1989:X
1945:X
1941:d
1937:Y
1932:Y
1928:d
1924:Y
1920:X
1916:X
1908:Y
1891:,
1888:)
1885:X
1882:(
1866:=
1863:)
1860:Y
1857:(
1839:Y
1808:.
1805:)
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1799:(
1780:)
1777:X
1774:(
1765:s
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1759:a
1756:H
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