1618:
2380:
1655:
1460:
210:
42:, 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) â 1.26.
3980:
31:
714:
1374:
184:
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
2329:
1622:
544:
408:, 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:
2214:
297:
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
2534:
1912:
1205:
185:
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
1040:
321:
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
1829:
1583:
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) â 1.58. These Hausdorff dimensions are related to the "critical exponent" of the
844:
179:
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
3047:
2954:
2753:
400:
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
2236:
2629:
920:
709:{\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\},}
2863:
150:
1448:
85:
is 3. That is, for sets of points that define a smooth shape or a shape that has a small number of cornersâthe shapes of traditional geometry and scienceâthe
Hausdorff dimension is an
2121:
1451:). 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.
1982:
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.
266:
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
1512:
1085:
536:
1418:
2129:
498:
93:. 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
1559:
1369:{\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\}}
1579:, 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) â 0.63. The
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1840:
1532:
1197:
1173:
1145:
1125:
1105:
940:
868:
761:
741:
468:
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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
152:, 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.
1632:
performed detailed experiments to measure the approximate
Hausdorff dimension for various coastlines. His results have varied from 1.02 for the coastline of
1420:
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
3150:
Gneiting, Tilmann; Ć evÄĂkovĂĄ, Hana; Percival, Donald B. (2012). "Estimators of
Fractal Dimension: Assessing the Roughness of Time Series and Spatial Data".
3911:
424:
is yet another similar notion which gives the same value for many shapes, but there are well-documented exceptions where all these dimensions differ.
314: + 1 balls overlap. For example, when one covers a line with short open intervals, some points must be covered twice, giving dimension
1756:
3087:
1585:
766:
412:
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.
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2397:
1672:
227:
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417:
390:
193:
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2419:
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has an integer topological dimension, but in terms of the amount of space it takes up, it behaves like a higher-dimensional space.
3213:(workshop), Society for Chaos Theory in Psychology and the Life Sciences annual meeting, June 28, 1996, Berkeley, California, see
2692:
3264:
2324:{\displaystyle \dim _{\operatorname {Haus} }(X\times Y)\geq \dim _{\operatorname {Haus} }(X)+\dim _{\operatorname {Haus} }(Y).}
397:
is a critical boundary between growth rates that are insufficient to cover the space, and growth rates that are overabundant.
3904:
2401:
1978:
is similar to, and at least as large as, the
Hausdorff dimension, and they are equal in many situations. However, the set of
1676:
231:
106:
109:
allowing computation of dimensions for highly irregular or "rough" sets, this dimension is also commonly referred to as the
282:
involving interweaving the digits of two numbers to yield a single number encoding the same information). The example of a
279:
171:. This underlies the earlier statement that the Hausdorff dimension of a point is zero, of a line is one, etc., and that
2568:
3999:
2640:
299:
3194:
Larry Riddle, 2014, "Classic
Iterated Function Systems: Koch Snowflake", Agnes Scott College e-Academy (online), see
884:
189:) appearing in the figures, and givingâin the Koch and other fractal casesânon-integer dimensions for these objects.
2432:
Many sets defined by a self-similarity condition have dimensions which can be determined explicitly. Roughly, a set
3680:
3390:
2390:
1665:
220:
4213:
3897:
2809:
120:
329:
The Hausdorff dimension measures the local size of a space taking into account the distance between points, the
126:
4208:
3934:
3784:
3214:
1424:
2077:
3289:
Farkas, Abel; Fraser, Jonathan (30 July 2015). "On the equality of Hausdorff measure and Hausdorff content".
4142:
4137:
4117:
3534:
Dodson, M. Maurice; Kristensen, Simon (June 12, 2003). "Hausdorff Dimension and Diophantine Approximation".
3096:, another variation of fractal dimension that, like Hausdorff dimension, is defined using coverings by balls
2792:
290:(taking one real number into a pair of real numbers in a way so that all pairs of numbers are covered) and
4127:
4122:
4102:
1593:
1576:
3138:
4132:
4112:
4107:
3703:
3403:
This Knowledge (XXG) article also discusses further useful characterizations of the Hausdorff dimension.
1991:
1568:
1488:
1052:
503:
3644:
3549:
3422:
1629:
119:, i.e. a set where the distances between all members are defined. The dimension is drawn from the
2209:{\displaystyle \dim _{\operatorname {Haus} }(X)=\sup _{i\in I}\dim _{\operatorname {Haus} }(X_{i}).}
4004:
3874:
3859:
3371:
3099:
2540:
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1975:
1725:
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438:
The formal definition of the Hausdorff dimension is arrived at by defining first the d-dimensional
287:
283:
164:
90:
3078:. Under the same conditions as the previous theorem, the unique fixed point of Ï is self-similar.
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4183:
4024:
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3720:
3660:
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3159:
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2648:
847:
342:
155:
In mathematical terms, the Hausdorff dimension generalizes the notion of the dimension of a real
94:
477:
192:
The Hausdorff dimension is a successor to the simpler, but usually equivalent, box-counting or
17:
4203:
4019:
3765:
3732:
3698:
3684:
3605:
3565:
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3340:
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3109:
3104:
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3065:
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439:
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105:âhave non-integer Hausdorff dimensions. Because of the significant technical advances made by
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70:
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around some point. Then the unique fixed point of Ï is a set whose Hausdorff dimension is
2529:{\displaystyle \psi _{i}:\mathbf {R} ^{n}\rightarrow \mathbf {R} ^{n},\quad i=1,\ldots ,m}
1979:
1907:{\displaystyle \inf _{Y}\dim _{\operatorname {Haus} }(Y)=\dim _{\operatorname {ind} }(X),}
1717:
1610:
1483:
98:
66:
3313:
Schleicher, Dierk (June 2007). "Hausdorff Dimension, Its Properties, and Its Surprises".
1035:{\displaystyle \dim _{\operatorname {H} }{(X)}:=\inf\{d\geq 0:{\mathcal {H}}^{d}(X)=0\}.}
3648:
3553:
3426:
1617:
4076:
4061:
3672:
1517:
1182:
1158:
1130:
1110:
1090:
925:
853:
746:
726:
453:
176:
39:
3595:
4197:
4066:
3819:
3724:
3664:
3591:
3579:
3442:
3236:
2759:
2636:
2436:
is self-similar if it is the fixed point of a set-valued transformation Ï, that is Ï(
2230:
are non-empty metric spaces, then the Hausdorff dimension of their product satisfies
1995:
1637:
1476:
447:
115:
More specifically, the Hausdorff dimension is a dimensional number associated with a
63:
3561:
3181:
294:, so that a one-dimensional object completely fills up a higher-dimensional object.
4086:
4051:
3944:
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3736:
3352:
3336:
1922:
1633:
471:
330:
156:
116:
74:
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is self-similar if and only if the intersections satisfy the following condition:
4171:
3954:
2379:
1732:
which is defined recursively. It is always an integer (or +â) and is denoted dim
1654:
1603:
267:
209:
181:
47:
3845:
3828:
3752:
3538:. Proceedings of Symposia in Pure Mathematics. Vol. 72. pp. 305â347.
3514:
3497:
1824:{\displaystyle \dim _{\mathrm {Haus} }(X)\geq \dim _{\operatorname {ind} }(X).}
4166:
4046:
3811:
3656:
3434:
2868:
The contraction coefficient of a similitude is the magnitude of the dilation.
1572:
1459:
271:
3344:
4147:
4056:
3969:
3920:
839:{\displaystyle {\mathcal {H}}^{d}(S)=\lim _{\delta \to 0}H_{\delta }^{d}(S)}
275:
35:
3626:
3609:
2765:
The open set condition is a separation condition that ensures the images Ï
4071:
4034:
3959:
2788:
1721:
1046:
3133:
MacGregor Campbell, 2013, "5.6 Scaling and the Hausdorff Dimension," at
4081:
3803:
3716:
2404: in this section. Unsourced material may be challenged and removed.
1679: in this section. Unsourced material may be challenged and removed.
1564:
1464:
1421:
720:
405:
323:
234: in this section. Unsourced material may be challenged and removed.
172:
102:
86:
3878:
3173:
3072:(the intersections are just points), but is also true more generally:
30:
3544:
3327:
2643:
applied to the complete metric space of non-empty compact subsets of
78:
3635:
Marstrand, J. M. (1954). "The dimension of cartesian product sets".
3413:
Marstrand, J. M. (1954). "The dimension of Cartesian product sets".
1613:
in dimension 2 and above is conjectured to be Hausdorff dimension 2.
4038:
3295:
3164:
3042:{\displaystyle H^{s}\left(\psi _{i}(E)\cap \psi _{j}(E)\right)=0,}
1616:
1458:
29:
3536:
Fractal Geometry and Applications: A Jubilee of BenoĂźt Mandelbrot
101:, one is led to the conclusion that particular objectsâincluding
3090:
Examples of deterministic fractals, random and natural fractals.
2949:{\displaystyle A\mapsto \psi (A)=\bigcup _{i=1}^{m}\psi _{i}(A)}
1567:
often are spaces whose Hausdorff dimension strictly exceeds the
82:
3893:
763:. The Hausdorff d-dimensional outer measure is then defined as
2373:
1648:
203:
89:
agreeing with the usual sense of dimension, also known as the
3701:(1929). "On Linear Sets of Points of Fractional Dimensions".
2669:(in certain cases), we need a technical condition called the
3889:
1000:
773:
286:
shows that one can even map the real line to the real plane
175:
can have noninteger Hausdorff dimensions. For instance, the
3856:
Fractal Geometry: Mathematical Foundations and Applications
3458:
Fractal geometry. Mathematical foundations and applications
3368:
Fractal Geometry: Mathematical Foundations and Applications
3209:
Keith Clayton, 1996, "Fractals and the Fractal Dimension,"
1471:, an object with Hausdorff dimension of log(3)/log(2)â1.58.
2748:{\displaystyle \bigcup _{i=1}^{m}\psi _{i}(V)\subseteq V,}
1966:(1907â1976), e.g., see Hurewicz and Wallman, Chapter VII.
1606:
have the same Hausdorff dimension as the space they fill.
1127:
is infinite (except that when this latter set of numbers
262:
The intuitive concept of dimension of a geometric object
302:, explains why. This dimension is the greatest integer
1937:
have the same underlying set of points and the metric
393:, which equals the Hausdorff dimension when the value
310:
by small open balls there is at least one point where
3758:
Several selections from this volume are reprinted in
3129:
3127:
3125:
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2812:
2695:
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2239:
2132:
2080:
1843:
1759:
1540:
1520:
1491:
1427:
1385:
1208:
1185:
1161:
1133:
1113:
1093:
1055:
951:
928:
887:
856:
769:
749:
729:
547:
506:
480:
456:
416:
For fractals that occur in nature, the Hausdorff and
129:
3242:. Lecture notes in mathematics 1358. W. H. Freeman.
2354:
is bounded from above by the Hausdorff dimension of
69:. For instance, the Hausdorff dimension of a single
4159:
4095:
4033:
3987:
3927:
3460:. John Wiley & Sons, Inc., Hoboken, New Jersey.
2665:To determine the dimension of the self-similar set
2219:This can be verified directly from the definition.
3235:
3041:
2948:
2857:
2747:
2623:
2528:
2323:
2208:
2115:
1906:
1823:
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1442:
1412:
1368:
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1079:
1034:
934:
914:
862:
838:
755:
735:
708:
530:
492:
462:
389:approaches zero. More precisely, this defines the
144:
34:Example of non-integer dimensions. The first four
3677:Geometry of sets and measures in Euclidean spaces
3205:
3203:
2781:. Suppose the open set condition holds and each Ï
2758:where the sets in union on the left are pairwise
2624:{\displaystyle A=\bigcup _{i=1}^{m}\psi _{i}(A).}
2444:, although the exact definition is given below.
2159:
1845:
1428:
1263:
979:
797:
575:
2366:. These facts are discussed in Mattila (1995).
915:{\displaystyle \dim _{\operatorname {H} }{(X)}}
373:, the Hausdorff dimension is the unique number
3211:Basic Concepts in Nonlinear Dynamics and Chaos
3905:
3475:. Cambridge, UK: Cambridge University Press.
3263:Briggs, Jimmy; Tyree, Tim (3 December 2016).
2787:is a similitude, that is a composition of an
1962:These results were originally established by
8:
3308:
3306:
3226:
3224:
3222:
1026:
982:
2858:{\displaystyle \sum _{i=1}^{m}r_{i}^{s}=1.}
1970:Hausdorff dimension and Minkowski dimension
1708:Hausdorff dimension and inductive dimension
1147:is empty the Hausdorff dimension is zero).
3912:
3898:
3890:
3741:Journal of the London Mathematical Society
2875:which is carried onto itself by a mapping
1986:Hausdorff dimensions and Frostman measures
1621:Estimating the Hausdorff dimension of the
145:{\displaystyle {\overline {\mathbb {R} }}}
3844:
3739:(1937). "Sets of Fractional Dimensions".
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3135:Annenberg Learner:MATHematics illuminated
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2420:Learn how and when to remove this message
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1443:{\displaystyle \inf \varnothing =\infty }
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442:, a fractional-dimension analogue of the
250:Learn how and when to remove this message
159:. That is, the Hausdorff dimension of an
133:
132:
130:
128:
2116:{\displaystyle X=\bigcup _{i\in I}X_{i}}
846:, and the restriction of the mapping to
3471:Falconer, K. J. (1985). "Theorem 8.3".
3121:
3088:List of fractals by Hausdorff dimension
2673:(OSC) on the sequence of contractions Ï
2641:contractive mapping fixed point theorem
1431:
850:justifies it as a measure, called the
27:Invariant measure of fractal dimension
2123:is a finite or countable union, then
7:
2402:adding citations to reliable sources
2062:. A partial converse is provided by
1677:adding citations to reliable sources
232:adding citations to reliable sources
3068:. This is clear in the case of the
2070:Behaviour under unions and products
723:is taken over all countable covers
369:. For a sufficiently well-behaved
274:is equal to the cardinality of the
1775:
1772:
1769:
1766:
1577:zero-dimensional topological space
1437:
1340:
1287:
1241:
1107:-dimensional Hausdorff measure of
1071:
957:
893:
652:
599:
522:
25:
3315:The American Mathematical Monthly
1645:Properties of Hausdorff dimension
62:, that was introduced in 1918 by
3978:
2686:with compact closure, such that
2491:
2476:
2378:
1653:
1507:{\displaystyle \mathbb {R} ^{n}}
1080:{\displaystyle d\in [0,\infty )}
870:-dimensional Hausdorff Measure.
531:{\displaystyle d\in [0,\infty )}
208:
111:HausdorffâBesicovitch dimension.
2554:< 1. Then there is a unique
2504:
2389:needs additional citations for
1950:is topologically equivalent to
1664:needs additional citations for
306:such that in every covering of
219:needs additional citations for
18:Hausdorff-Besicovitch dimension
3886:at Encyclopedia of Mathematics
3829:"Fractals and self similarity"
3498:"Fractals and self similarity"
3337:10.1080/00029890.2007.11920440
3238:The Fractal Geometry of Nature
3056:is the Hausdorff dimension of
3022:
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3000:
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2187:
2152:
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1898:
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1636:to 1.25 for the west coast of
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107:Abram Samoilovitch Besicovitch
1:
3600:. Princeton University Press.
3456:Falconer, Kenneth J. (2003).
2775:) do not overlap "too much".
2346:, the Hausdorff dimension of
194:MinkowskiâBouligand dimension
3827:Hutchinson, John E. (1981).
3496:Hutchinson, John E. (1981).
3473:The Geometry of Fractal Sets
1413:{\displaystyle C_{H}^{d}(S)}
365:) grows polynomially with 1/
137:
3879:Encyclopedia of Mathematics
3785:"Dimension und Ă€uĂeres MaĂ"
3637:Proc. Cambridge Philos. Soc
3610:"La dimension et la mesure"
3562:10.1090/pspum/072.1/2112110
3415:Proc. Cambridge Philos. Soc
1479:have Hausdorff dimension 0.
1177:unlimited Hausdorff content
300:Lebesgue covering dimension
4230:
3854:Falconer, Kenneth (2003).
3846:10.1512/iumj.1981.30.30055
3764:. Boston: Addison-Wesley.
3681:Cambridge University Press
3515:10.1512/iumj.1981.30.30055
3391:Cambridge University Press
3366:Falconer, Kenneth (2003).
3270:. University of Washington
2803:is the unique solution of
2658:
2547:with contraction constant
2034:> 0 and for every ball
1998:subsets of a metric space
1921:ranges over metric spaces
1561:has Hausdorff dimension 1.
493:{\displaystyle S\subset X}
431:
4180:
3976:
3760:Edgar, Gerald A. (1993).
3657:10.1017/S0305004100029236
3435:10.1017/S0305004100029236
2635:The theorem follows from
1720:metric space. There is a
3753:10.1112/jlms/s1-12.45.18
3216:, accessed 5 March 2015.
3197:, accessed 5 March 2015.
3140:, accessed 5 March 2015.
2030:holds for some constant
1514:has Hausdorff dimension
1045:This is the same as the
278:(this can be seen by an
58:, or more specifically,
3614:Fundamenta Mathematicae
3385:Morters, Peres (2010).
2360:upper packing dimension
1463:Dimension of a further
3064:denotes s-dimensional
3043:
2950:
2926:
2859:
2833:
2749:
2716:
2655:The open set condition
2633:
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1908:
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1623:coast of Great Britain
1594:analysis of algorithms
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916:
864:
840:
757:
737:
710:
656:
603:
532:
494:
464:
418:box-counting dimension
414:
391:box-counting dimension
333:. Consider the number
146:
43:
3833:Indiana Univ. Math. J
3792:Mathematische Annalen
3704:Mathematische Annalen
3627:10.4064/fm-28-1-81-89
3502:Indiana Univ. Math. J
3044:
2951:
2906:
2860:
2813:
2750:
2696:
2682:There is an open set
2626:
2578:
2531:
2446:
2342:are Borel subsets of
2326:
2211:
2118:
1909:
1826:
1620:
1569:topological dimension
1556:
1554:{\displaystyle S^{1}}
1529:
1509:
1462:
1445:
1415:
1371:
1324:
1271:
1194:
1170:
1142:
1122:
1102:
1082:
1037:
937:
917:
865:
841:
758:
738:
711:
636:
583:
533:
495:
465:
410:
147:
121:extended real numbers
91:topological dimension
33:
4096:Dimensions by number
3776:See chapters 9,10,11
3762:Classics on fractals
2966:
2882:
2810:
2693:
2569:
2458:
2398:improve this article
2237:
2130:
2078:
1841:
1757:
1750:is non-empty. Then
1673:improve this article
1630:Lewis Fry Richardson
1600:Space-filling curves
1590:recurrence relations
1538:
1518:
1489:
1425:
1383:
1206:
1183:
1159:
1131:
1111:
1091:
1053:
949:
926:
885:
854:
767:
747:
727:
545:
504:
478:
454:
450:is constructed: Let
228:improve this article
127:
3875:Hausdorff dimension
3860:John Wiley and Sons
3649:1954PCPS...50..198M
3554:2003math......5399D
3427:1954PCPS...50..198M
3372:John Wiley and Sons
3265:"Hausdorff Measure"
3152:Statistical Science
3100:Intrinsic dimension
2848:
1976:Minkowski dimension
1726:inductive dimension
1581:Sierpinski triangle
1571:. For example, the
1469:Sierpinski triangle
1400:
1250:
1223:
880:Hausdorff dimension
874:Hausdorff dimension
826:
562:
284:space-filling curve
165:inner product space
52:Hausdorff dimension
4025:Degrees of freedom
3928:Dimensional spaces
3812:10338.dmlcz/100363
3804:10.1007/BF01457179
3717:10.1007/BF01454831
3232:Mandelbrot, BenoĂźt
3039:
2946:
2871:In general, a set
2855:
2834:
2745:
2671:open set condition
2661:Open set condition
2649:Hausdorff distance
2621:
2526:
2321:
2206:
2173:
2113:
2102:
1929:. In other words,
1904:
1853:
1821:
1626:
1609:The trajectory of
1551:
1524:
1504:
1473:
1440:
1410:
1386:
1366:
1236:
1209:
1189:
1165:
1137:
1117:
1097:
1077:
1032:
932:
912:
860:
836:
812:
811:
753:
733:
706:
548:
528:
490:
460:
349:required to cover
345:of radius at most
187:natural logarithms
142:
44:
4191:
4190:
4000:Lebesgue covering
3965:Algebraic variety
3884:Hausdorff measure
3733:A. S. Besicovitch
3699:A. S. Besicovitch
3690:978-0-521-65595-8
3174:10.1214/11-STS370
3110:Fractal dimension
3105:Packing dimension
3094:Assouad dimension
3070:Sierpinski gasket
3066:Hausdorff measure
2430:
2429:
2422:
2370:Self-similar sets
2158:
2087:
1844:
1705:
1704:
1697:
1534:, and the circle
1527:{\displaystyle n}
1192:{\displaystyle S}
1168:{\displaystyle d}
1151:Hausdorff content
1140:{\displaystyle d}
1120:{\displaystyle X}
1100:{\displaystyle d}
935:{\displaystyle X}
863:{\displaystyle d}
796:
756:{\displaystyle S}
736:{\displaystyle U}
463:{\displaystyle X}
440:Hausdorff measure
434:Hausdorff measure
428:Formal definition
422:packing dimension
402:Benoit Mandelbrot
353:completely. When
260:
259:
252:
140:
60:fractal dimension
16:(Redirected from
4221:
4214:Dimension theory
3988:Other dimensions
3982:
3950:Projective space
3914:
3907:
3900:
3891:
3863:
3858:(2nd ed.).
3850:
3848:
3823:
3798:(1â2): 157â179.
3789:
3775:
3756:
3728:
3694:
3668:
3631:
3629:
3601:
3597:Dimension Theory
3588:Hurewicz, Witold
3583:
3547:
3520:
3519:
3517:
3493:
3487:
3486:
3468:
3462:
3461:
3453:
3447:
3446:
3410:
3404:
3401:
3395:
3394:
3382:
3376:
3375:
3370:(2nd ed.).
3363:
3357:
3356:
3330:
3310:
3301:
3300:
3298:
3286:
3280:
3279:
3277:
3275:
3269:
3260:
3254:
3253:
3241:
3228:
3217:
3207:
3198:
3192:
3186:
3185:
3167:
3147:
3141:
3131:
3048:
3046:
3045:
3040:
3029:
3025:
3015:
3014:
2993:
2992:
2978:
2977:
2955:
2953:
2952:
2947:
2936:
2935:
2925:
2920:
2864:
2862:
2861:
2856:
2847:
2842:
2832:
2827:
2754:
2752:
2751:
2746:
2726:
2725:
2715:
2710:
2630:
2628:
2627:
2622:
2608:
2607:
2597:
2592:
2535:
2533:
2532:
2527:
2500:
2499:
2494:
2485:
2484:
2479:
2470:
2469:
2425:
2418:
2414:
2411:
2405:
2382:
2374:
2330:
2328:
2327:
2322:
2305:
2304:
2280:
2279:
2249:
2248:
2215:
2213:
2212:
2207:
2199:
2198:
2183:
2182:
2172:
2142:
2141:
2122:
2120:
2119:
2114:
2112:
2111:
2101:
2064:Frostman's lemma
1964:Edward Szpilrajn
1913:
1911:
1910:
1905:
1888:
1887:
1863:
1862:
1852:
1830:
1828:
1827:
1822:
1805:
1804:
1780:
1779:
1778:
1716:be an arbitrary
1700:
1693:
1689:
1686:
1680:
1657:
1649:
1560:
1558:
1557:
1552:
1550:
1549:
1533:
1531:
1530:
1525:
1513:
1511:
1510:
1505:
1503:
1502:
1497:
1449:
1447:
1446:
1441:
1419:
1417:
1416:
1411:
1399:
1394:
1379:In other words,
1375:
1373:
1372:
1367:
1365:
1361:
1354:
1353:
1343:
1338:
1320:
1319:
1310:
1309:
1290:
1285:
1249:
1244:
1222:
1217:
1198:
1196:
1195:
1190:
1174:
1172:
1171:
1166:
1146:
1144:
1143:
1138:
1126:
1124:
1123:
1118:
1106:
1104:
1103:
1098:
1086:
1084:
1083:
1078:
1041:
1039:
1038:
1033:
1010:
1009:
1004:
1003:
975:
961:
960:
941:
939:
938:
933:
921:
919:
918:
913:
911:
897:
896:
869:
867:
866:
861:
845:
843:
842:
837:
825:
820:
810:
783:
782:
777:
776:
762:
760:
759:
754:
742:
740:
739:
734:
715:
713:
712:
707:
702:
698:
691:
690:
666:
665:
655:
650:
632:
631:
622:
621:
602:
597:
561:
556:
537:
535:
534:
529:
499:
497:
496:
491:
469:
467:
466:
461:
444:Lebesgue measure
318: = 1.
255:
248:
244:
241:
235:
212:
204:
151:
149:
148:
143:
141:
136:
131:
54:is a measure of
21:
4229:
4228:
4224:
4223:
4222:
4220:
4219:
4218:
4209:Metric geometry
4194:
4193:
4192:
4187:
4176:
4155:
4091:
4029:
3983:
3974:
3940:Euclidean space
3923:
3918:
3871:
3866:
3853:
3826:
3787:
3779:
3772:
3759:
3757:
3731:
3697:
3691:
3673:Mattila, Pertti
3671:
3634:
3604:
3586:
3572:
3533:
3529:
3527:Further reading
3524:
3523:
3495:
3494:
3490:
3483:
3470:
3469:
3465:
3455:
3454:
3450:
3412:
3411:
3407:
3402:
3398:
3387:Brownian Motion
3384:
3383:
3379:
3365:
3364:
3360:
3312:
3311:
3304:
3288:
3287:
3283:
3273:
3271:
3267:
3262:
3261:
3257:
3250:
3230:
3229:
3220:
3208:
3201:
3193:
3189:
3149:
3148:
3144:
3132:
3123:
3118:
3084:
3006:
2984:
2983:
2979:
2969:
2964:
2963:
2927:
2880:
2879:
2808:
2807:
2786:
2770:
2717:
2691:
2690:
2678:
2663:
2657:
2599:
2567:
2566:
2552:
2489:
2474:
2461:
2456:
2455:
2426:
2415:
2409:
2406:
2395:
2383:
2372:
2296:
2271:
2240:
2235:
2234:
2190:
2174:
2133:
2128:
2127:
2103:
2076:
2075:
2072:
2053:
1988:
1972:
1958:
1945:
1879:
1854:
1839:
1838:
1796:
1760:
1755:
1754:
1735:
1710:
1701:
1690:
1684:
1681:
1670:
1658:
1647:
1611:Brownian motion
1541:
1536:
1535:
1516:
1515:
1492:
1487:
1486:
1484:Euclidean space
1457:
1423:
1422:
1381:
1380:
1345:
1311:
1301:
1270:
1266:
1204:
1203:
1181:
1180:
1157:
1156:
1153:
1129:
1128:
1109:
1108:
1089:
1088:
1051:
1050:
997:
952:
947:
946:
924:
923:
888:
883:
882:
876:
852:
851:
848:measurable sets
770:
765:
764:
745:
744:
725:
724:
682:
657:
623:
613:
582:
578:
543:
542:
502:
501:
476:
475:
452:
451:
436:
430:
357:is very small,
256:
245:
239:
236:
225:
213:
202:
125:
124:
99:self-similarity
81:is 2, and of a
67:Felix Hausdorff
28:
23:
22:
15:
12:
11:
5:
4227:
4225:
4217:
4216:
4211:
4206:
4196:
4195:
4189:
4188:
4181:
4178:
4177:
4175:
4174:
4169:
4163:
4161:
4157:
4156:
4154:
4153:
4145:
4140:
4135:
4130:
4125:
4120:
4115:
4110:
4105:
4099:
4097:
4093:
4092:
4090:
4089:
4084:
4079:
4077:Cross-polytope
4074:
4069:
4064:
4062:Hyperrectangle
4059:
4054:
4049:
4043:
4041:
4031:
4030:
4028:
4027:
4022:
4017:
4012:
4007:
4002:
3997:
3991:
3989:
3985:
3984:
3977:
3975:
3973:
3972:
3967:
3962:
3957:
3952:
3947:
3942:
3937:
3931:
3929:
3925:
3924:
3919:
3917:
3916:
3909:
3902:
3894:
3888:
3887:
3881:
3870:
3869:External links
3867:
3865:
3864:
3851:
3839:(5): 713â747.
3824:
3783:(March 1919).
3777:
3770:
3729:
3711:(1): 161â193.
3695:
3689:
3669:
3643:(3): 198â202.
3632:
3602:
3592:Wallman, Henry
3584:
3570:
3530:
3528:
3525:
3522:
3521:
3508:(5): 713â747.
3488:
3481:
3463:
3448:
3421:(3): 198â202.
3405:
3396:
3377:
3358:
3321:(6): 509â528.
3302:
3281:
3255:
3248:
3218:
3199:
3187:
3158:(2): 247â277.
3142:
3120:
3119:
3117:
3114:
3113:
3112:
3107:
3102:
3097:
3091:
3083:
3080:
3050:
3049:
3038:
3035:
3032:
3028:
3024:
3021:
3018:
3013:
3009:
3005:
3002:
2999:
2996:
2991:
2987:
2982:
2976:
2972:
2957:
2956:
2945:
2942:
2939:
2934:
2930:
2924:
2919:
2916:
2913:
2909:
2905:
2902:
2899:
2896:
2893:
2890:
2887:
2866:
2865:
2854:
2851:
2846:
2841:
2837:
2831:
2826:
2823:
2820:
2816:
2782:
2766:
2756:
2755:
2744:
2741:
2738:
2735:
2732:
2729:
2724:
2720:
2714:
2709:
2706:
2703:
2699:
2674:
2659:Main article:
2656:
2653:
2632:
2631:
2620:
2617:
2614:
2611:
2606:
2602:
2596:
2591:
2588:
2585:
2581:
2577:
2574:
2550:
2537:
2536:
2525:
2522:
2519:
2516:
2513:
2510:
2507:
2503:
2498:
2493:
2488:
2483:
2478:
2473:
2468:
2464:
2428:
2427:
2386:
2384:
2377:
2371:
2368:
2332:
2331:
2320:
2317:
2314:
2311:
2308:
2303:
2299:
2295:
2292:
2289:
2286:
2283:
2278:
2274:
2270:
2267:
2264:
2261:
2258:
2255:
2252:
2247:
2243:
2217:
2216:
2205:
2202:
2197:
2193:
2189:
2186:
2181:
2177:
2171:
2168:
2165:
2161:
2157:
2154:
2151:
2148:
2145:
2140:
2136:
2110:
2106:
2100:
2097:
2094:
2090:
2086:
2083:
2071:
2068:
2051:
1990:If there is a
1987:
1984:
1971:
1968:
1954:
1941:
1915:
1914:
1903:
1900:
1897:
1894:
1891:
1886:
1882:
1878:
1875:
1872:
1869:
1866:
1861:
1857:
1851:
1847:
1832:
1831:
1820:
1817:
1814:
1811:
1808:
1803:
1799:
1795:
1792:
1789:
1786:
1783:
1777:
1774:
1771:
1768:
1763:
1733:
1709:
1706:
1703:
1702:
1661:
1659:
1652:
1646:
1643:
1642:
1641:
1615:
1614:
1607:
1597:
1586:Master theorem
1562:
1548:
1544:
1523:
1501:
1496:
1480:
1477:Countable sets
1456:
1453:
1439:
1436:
1433:
1430:
1409:
1406:
1403:
1398:
1393:
1389:
1377:
1376:
1364:
1360:
1357:
1352:
1348:
1342:
1337:
1334:
1331:
1327:
1323:
1318:
1314:
1308:
1304:
1300:
1297:
1294:
1289:
1284:
1281:
1278:
1274:
1269:
1265:
1262:
1259:
1256:
1253:
1248:
1243:
1239:
1235:
1232:
1229:
1226:
1221:
1216:
1212:
1199:is defined by
1188:
1164:
1152:
1149:
1136:
1116:
1096:
1087:such that the
1076:
1073:
1070:
1067:
1064:
1061:
1058:
1049:of the set of
1043:
1042:
1031:
1028:
1025:
1022:
1019:
1016:
1013:
1008:
1002:
996:
993:
990:
987:
984:
981:
978:
974:
971:
968:
964:
959:
955:
942:is defined by
931:
910:
907:
904:
900:
895:
891:
875:
872:
859:
835:
832:
829:
824:
819:
815:
809:
806:
803:
799:
795:
792:
789:
786:
781:
775:
752:
732:
717:
716:
705:
701:
697:
694:
689:
685:
681:
678:
675:
672:
669:
664:
660:
654:
649:
646:
643:
639:
635:
630:
626:
620:
616:
612:
609:
606:
601:
596:
593:
590:
586:
581:
577:
574:
571:
568:
565:
560:
555:
551:
527:
524:
521:
518:
515:
512:
509:
489:
486:
483:
459:
432:Main article:
429:
426:
420:coincide. The
404:observed that
258:
257:
216:
214:
207:
201:
198:
177:Koch snowflake
173:irregular sets
139:
135:
73:is zero, of a
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4226:
4215:
4212:
4210:
4207:
4205:
4202:
4201:
4199:
4186:
4185:
4179:
4173:
4170:
4168:
4165:
4164:
4162:
4158:
4152:
4150:
4146:
4144:
4141:
4139:
4136:
4134:
4131:
4129:
4126:
4124:
4121:
4119:
4116:
4114:
4111:
4109:
4106:
4104:
4101:
4100:
4098:
4094:
4088:
4085:
4083:
4080:
4078:
4075:
4073:
4070:
4068:
4067:Demihypercube
4065:
4063:
4060:
4058:
4055:
4053:
4050:
4048:
4045:
4044:
4042:
4040:
4036:
4032:
4026:
4023:
4021:
4018:
4016:
4013:
4011:
4008:
4006:
4003:
4001:
3998:
3996:
3993:
3992:
3990:
3986:
3981:
3971:
3968:
3966:
3963:
3961:
3958:
3956:
3953:
3951:
3948:
3946:
3943:
3941:
3938:
3936:
3933:
3932:
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3771:0-201-58701-7
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3571:9780821836378
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3482:0-521-25694-1
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3297:
3292:
3285:
3282:
3266:
3259:
3256:
3251:
3249:0-7167-1186-9
3245:
3240:
3239:
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3227:
3225:
3223:
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2707:
2704:
2701:
2697:
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2687:
2685:
2680:
2677:
2672:
2668:
2662:
2654:
2652:
2650:
2646:
2642:
2638:
2637:Stefan Banach
2618:
2612:
2604:
2600:
2594:
2589:
2586:
2583:
2579:
2575:
2572:
2565:
2564:
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2445:
2443:
2439:
2435:
2424:
2421:
2413:
2403:
2399:
2393:
2392:
2387:This section
2385:
2381:
2376:
2375:
2369:
2367:
2365:
2361:
2357:
2353:
2349:
2345:
2341:
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2306:
2301:
2297:
2293:
2287:
2281:
2276:
2272:
2268:
2262:
2259:
2256:
2250:
2245:
2241:
2233:
2232:
2231:
2229:
2225:
2220:
2203:
2195:
2191:
2184:
2179:
2175:
2169:
2166:
2163:
2155:
2149:
2143:
2138:
2134:
2126:
2125:
2124:
2108:
2104:
2098:
2095:
2092:
2088:
2084:
2081:
2069:
2067:
2065:
2061:
2057:
2049:
2045:
2041:
2037:
2033:
2029:
2025:
2021:
2017:
2013:
2010:) > 0 and
2009:
2005:
2001:
1997:
1994:Ό defined on
1993:
1985:
1983:
1981:
1977:
1969:
1967:
1965:
1960:
1957:
1953:
1949:
1944:
1940:
1936:
1932:
1928:
1924:
1920:
1901:
1895:
1889:
1884:
1880:
1876:
1870:
1864:
1859:
1855:
1849:
1837:
1836:
1835:
1818:
1812:
1806:
1801:
1797:
1793:
1787:
1781:
1761:
1753:
1752:
1751:
1749:
1745:
1741:
1739:
1731:
1727:
1723:
1719:
1715:
1707:
1699:
1696:
1688:
1678:
1674:
1668:
1667:
1662:This section
1660:
1656:
1651:
1650:
1644:
1639:
1638:Great Britain
1635:
1631:
1628:
1627:
1624:
1619:
1612:
1608:
1605:
1601:
1598:
1595:
1591:
1587:
1582:
1578:
1574:
1570:
1566:
1563:
1546:
1542:
1521:
1499:
1485:
1481:
1478:
1475:
1474:
1470:
1467:example. The
1466:
1461:
1454:
1452:
1450:
1434:
1404:
1396:
1391:
1387:
1362:
1358:
1355:
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1306:
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1298:
1295:
1282:
1279:
1276:
1272:
1267:
1260:
1254:
1246:
1237:
1233:
1227:
1219:
1214:
1210:
1202:
1201:
1200:
1186:
1178:
1175:-dimensional
1162:
1150:
1148:
1134:
1114:
1094:
1068:
1065:
1059:
1056:
1048:
1029:
1023:
1020:
1014:
1006:
994:
991:
988:
985:
976:
969:
962:
953:
945:
944:
943:
929:
905:
898:
889:
881:
873:
871:
857:
849:
830:
822:
817:
813:
807:
801:
793:
787:
779:
750:
730:
722:
703:
699:
695:
692:
687:
683:
679:
676:
673:
670:
667:
662:
658:
647:
644:
641:
637:
633:
628:
618:
614:
610:
607:
594:
591:
588:
584:
579:
572:
566:
558:
553:
549:
541:
540:
539:
519:
516:
510:
507:
487:
484:
481:
473:
457:
449:
448:outer measure
445:
441:
435:
427:
425:
423:
419:
413:
409:
407:
403:
398:
396:
392:
388:
384:
381:) grows as 1/
380:
376:
372:
368:
364:
360:
356:
352:
348:
344:
340:
336:
332:
327:
325:
319:
317:
313:
309:
305:
301:
295:
293:
289:
285:
281:
277:
273:
269:
265:
254:
251:
243:
233:
229:
223:
222:
217:This section
215:
211:
206:
205:
199:
197:
195:
190:
188:
183:
178:
174:
170:
166:
163:-dimensional
162:
158:
153:
122:
118:
113:
112:
108:
104:
100:
96:
92:
88:
84:
80:
76:
72:
68:
65:
64:mathematician
61:
57:
53:
49:
41:
37:
32:
19:
4182:
4148:
4087:Hyperpyramid
4052:Hypersurface
4009:
3945:Affine space
3935:Vector space
3855:
3836:
3832:
3795:
3791:
3781:F. Hausdorff
3761:
3747:(1): 18â25.
3744:
3740:
3737:H. D. Ursell
3708:
3702:
3676:
3640:
3636:
3617:
3613:
3606:E. Szpilrajn
3596:
3545:math/0305399
3535:
3505:
3501:
3491:
3472:
3466:
3457:
3451:
3418:
3414:
3408:
3399:
3386:
3380:
3367:
3361:
3328:math/0505099
3318:
3314:
3284:
3272:. Retrieved
3258:
3237:
3210:
3190:
3155:
3151:
3145:
3134:
3075:
3074:
3061:
3057:
3053:
3051:
2958:
2872:
2870:
2867:
2800:
2796:
2783:
2778:
2777:
2772:
2767:
2764:
2757:
2683:
2681:
2675:
2670:
2666:
2664:
2644:
2634:
2559:
2558:compact set
2555:
2548:
2544:
2538:
2448:
2447:
2441:
2437:
2433:
2431:
2416:
2407:
2396:Please help
2391:verification
2388:
2363:
2355:
2351:
2347:
2343:
2339:
2335:
2333:
2227:
2223:
2221:
2218:
2073:
2059:
2055:
2047:
2043:
2039:
2035:
2031:
2027:
2023:
2019:
2015:
2011:
2007:
2003:
1999:
1989:
1973:
1961:
1955:
1951:
1947:
1942:
1938:
1934:
1930:
1926:
1923:homeomorphic
1918:
1916:
1833:
1747:
1743:
1742:
1737:
1729:
1713:
1711:
1691:
1682:
1671:Please help
1666:verification
1663:
1634:South Africa
1588:for solving
1378:
1176:
1154:
1044:
879:
877:
718:
472:metric space
446:. First, an
437:
415:
411:
399:
394:
386:
382:
378:
377:such that N(
374:
370:
366:
362:
358:
354:
350:
346:
338:
334:
328:
320:
315:
311:
307:
303:
296:
292:continuously
291:
288:surjectively
263:
261:
246:
237:
226:Please help
221:verification
218:
191:
168:
160:
157:vector space
154:
117:metric space
114:
110:
75:line segment
55:
51:
45:
4172:Codimension
4151:-dimensions
4072:Hypersphere
3955:Free module
2543:mapping on
2541:contraction
2539:are each a
1722:topological
1604:Peano curve
268:cardinality
182:equilateral
77:is 1, of a
48:mathematics
4198:Categories
4167:Hyperspace
4047:Hyperplane
3274:3 February
3116:References
2562:such that
2451:. Suppose
2410:March 2015
2050:, then dim
2002:such that
1834:Moreover,
1746:. Suppose
1724:notion of
1685:March 2015
1573:Cantor set
719:where the
272:real plane
240:March 2015
40:Koch curve
36:iterations
4057:Hypercube
4035:Polytopes
4015:Minkowski
4010:Hausdorff
4005:Inductive
3970:Spacetime
3921:Dimension
3820:122001234
3725:125368661
3665:122475292
3580:119613948
3443:122475292
3345:0002-9890
3296:1411.0867
3165:1101.1444
3008:ψ
3004:∩
2986:ψ
2929:ψ
2908:⋃
2892:ψ
2889:↦
2815:∑
2737:⊆
2719:ψ
2698:⋃
2647:with the
2601:ψ
2580:⋃
2556:non-empty
2518:…
2487:→
2463:ψ
2358:plus the
2307:
2282:
2269:≥
2260:×
2251:
2185:
2167:∈
2144:
2096:∈
2089:⋃
1890:
1865:
1807:
1794:≥
1782:
1718:separable
1602:like the
1438:∞
1432:∅
1356:⊇
1341:∞
1326:⋃
1299:
1288:∞
1273:∑
1242:∞
1072:∞
1060:∈
989:≥
963:
899:
818:δ
805:→
802:δ
696:δ
680:
668:⊇
653:∞
638:⋃
611:
600:∞
585:∑
554:δ
523:∞
511:∈
485:⊂
276:real line
200:Intuition
138:¯
56:roughness
4204:Fractals
4184:Category
4160:See also
3960:Manifold
3675:(1995).
3620:: 81â9.
3608:(1937).
3594:(1948).
3234:(1982).
3182:88512325
3082:See also
2793:dilation
2789:isometry
2760:disjoint
1980:rational
1565:Fractals
1455:Examples
1047:supremum
406:fractals
280:argument
103:fractals
4082:Simplex
4020:Fractal
3645:Bibcode
3550:Bibcode
3423:Bibcode
3353:9811750
3076:Theorem
2779:Theorem
2449:Theorem
1992:measure
1744:Theorem
1592:in the
1465:fractal
721:infimum
324:fractal
270:of the
167:equals
95:scaling
87:integer
38:of the
4039:shapes
3818:
3768:
3723:
3687:
3663:
3578:
3568:
3479:
3441:
3351:
3343:
3246:
3180:
3137:, see
3052:where
2799:where
2791:and a
1917:where
331:metric
79:square
4143:Eight
4138:Seven
4118:Three
3995:Krull
3816:S2CID
3788:(PDF)
3721:S2CID
3661:S2CID
3576:S2CID
3540:arXiv
3439:S2CID
3349:S2CID
3323:arXiv
3291:arXiv
3268:(PDF)
3178:S2CID
3160:arXiv
2046:) in
2026:)) â€
1996:Borel
474:. If
470:be a
343:balls
341:) of
71:point
4128:Five
4123:Four
4103:Zero
4037:and
3766:ISBN
3685:ISBN
3566:ISBN
3477:ISBN
3341:ISSN
3276:2022
3244:ISBN
3060:and
2440:) =
2338:and
2302:Haus
2277:Haus
2246:Haus
2226:and
2180:Haus
2139:Haus
2058:) â„
2052:Haus
1974:The
1933:and
1860:Haus
1728:for
1712:Let
1575:, a
1482:The
1296:diam
1155:The
878:The
693:<
677:diam
608:diam
500:and
97:and
83:cube
4133:Six
4113:Two
4108:One
3877:at
3841:doi
3808:hdl
3800:doi
3749:doi
3713:doi
3709:101
3653:doi
3622:doi
3558:doi
3510:doi
3431:doi
3333:doi
3319:114
3170:doi
2639:'s
2400:by
2362:of
2298:dim
2273:dim
2242:dim
2222:If
2176:dim
2160:sup
2135:dim
2074:If
1946:of
1925:to
1885:ind
1881:dim
1856:dim
1846:inf
1802:ind
1798:dim
1762:dim
1740:).
1734:ind
1675:by
1429:inf
1264:inf
1179:of
980:inf
954:dim
922:of
890:dim
798:lim
743:of
576:inf
385:as
230:by
46:In
4200::
3837:30
3835:.
3831:.
3814:.
3806:.
3796:79
3794:.
3790:.
3745:12
3743:.
3735:;
3719:.
3707:.
3683:.
3679:.
3659:.
3651:.
3641:50
3639:.
3618:28
3616:.
3612:.
3590:;
3574:.
3564:.
3556:.
3548:.
3506:30
3504:.
3500:.
3437:.
3429:.
3419:50
3417:.
3389:.
3347:.
3339:.
3331:.
3317:.
3305:^
3221:^
3202:^
3176:.
3168:.
3156:27
3154:.
3124:^
2853:1.
2762:.
2679:.
2651:.
2350:Ă
2066:.
2042:,
2022:,
1959:.
1234::=
977::=
538:,
196:.
123:,
50:,
4149:n
3913:e
3906:t
3899:v
3862:.
3849:.
3843::
3822:.
3810::
3802::
3774:.
3755:.
3751::
3727:.
3715::
3693:.
3667:.
3655::
3647::
3630:.
3624::
3582:.
3560::
3552::
3542::
3518:.
3512::
3485:.
3445:.
3433::
3425::
3393:.
3374:.
3355:.
3335::
3325::
3299:.
3293::
3278:.
3252:.
3184:.
3172::
3162::
3062:H
3058:E
3054:s
3037:,
3034:0
3031:=
3027:)
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3017:(
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2998:E
2995:(
2990:i
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2845:s
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2836:r
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2797:s
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2773:V
2771:(
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2743:,
2740:V
2734:)
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2728:(
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2702:i
2684:V
2676:i
2667:A
2645:R
2619:.
2616:)
2613:A
2610:(
2605:i
2595:m
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2587:=
2584:i
2576:=
2573:A
2560:A
2551:i
2549:r
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2521:,
2515:,
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2509:=
2506:i
2502:,
2497:n
2492:R
2482:n
2477:R
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2467:i
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2417:(
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2364:Y
2356:X
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2319:.
2316:)
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2294:+
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2285:(
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2201:)
2196:i
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2164:i
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2150:X
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2082:X
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2054:(
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2044:r
2040:x
2038:(
2036:B
2032:s
2028:r
2024:r
2020:x
2018:(
2016:B
2014:(
2012:Ό
2008:X
2006:(
2004:Ό
2000:X
1956:X
1952:d
1948:Y
1943:Y
1939:d
1935:Y
1931:X
1927:X
1919:Y
1902:,
1899:)
1896:X
1893:(
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247:(
242:)
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224:.
169:n
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20:)
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