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This "universality" of the Sierpiński carpet is not a true universal property in the sense of category theory: it does not uniquely characterize this space up to homeomorphism. For example, the disjoint union of a Sierpiński carpet and a circle is also a universal plane curve. However, in 1958
602:, starts in the same way, by subdividing the unit square into nine smaller squares and removing the middle of them. At the next level of subdivision, it subdivides each of the squares into 25 smaller squares and removes the middle one, and it continues at the
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108:. It can be realised as the set of points in the unit square whose coordinates written in base three do not both have a digit '1' in the same position, using the infinitesimal number representation of
357:
675:, they easily accommodate multiple frequencies. They are also easy to fabricate and smaller than conventional antennas of similar performance, thus being optimal for pocket-sized mobile phones.
956:
N. A. Saidatul, A. A. H. Azremi, R. B. Ahmad, P. J. Soh and F. Malek, "A development of
Fractal PIFA (planar inverted F antenna) with bandwidth enhancement for mobile phone applications," 2009
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can be extended to other shapes. For instance, subdividing an equilateral triangle into four equilateral triangles, removing the middle triangle, and recursing leads to the
1650:
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T. Kalaimani, P. M. Venkatesh, R. Mohanamurali and T. Shanmuganantham, "A modified
Sierpinski carpet fractal antenna for wireless applications," 2013
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on the Sierpiński carpet diffuses at a slower rate than an unrestricted random walk in the plane. The latter reaches a mean distance proportional to
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Sierpiński demonstrated that his carpet is a universal plane curve. That is: the
Sierpinski carpet is a compact subset of the plane with
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W. -L. Chen, G. -M. Wang and C. -X. Zhang, "Small-Size
Microstrip Patch Antennas Combining Koch and Sierpinski Fractal-Shapes," in
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which is entirely contained in the carpet. This square contains a smaller square whose coordinates are multiples of
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on the Sierpiński carpet has attracted interest in recent years. Martin Barlow and
Richard Bass have shown that a
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is a continuum embedded in the plane. Suppose its complement in the plane has countably many connected components
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797:(1916). "Sur une courbe cantorienne qui contient une image biunivoque et continue de toute courbe donnée".
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without satisfying the parabolic one. The existence of such an example was an open problem for many years.
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steps, but the random walk on the discrete Sierpiński carpet reaches only a mean distance proportional to
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subsquares in a 3-by-3 grid, and the central subsquare is removed. The same procedure is then applied
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In the same paper
Whyburn gave another characterization of the Sierpiński carpet. Recall that a
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have been produced in the form of few iterations of the Sierpiński carpet. Due to their
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inequalities (so called "sub-Gaussian inequalities") and that it satisfies the elliptic
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and has no 'local cut-points' is homeomorphic to the
Sierpinski carpet. Here a
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80:. In three dimensions, a similar construction based on cubes is known as the
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973:, Melmaruvathur, India, 2013, pp. 722-725, doi: 10.1109/iccsp.2013.6577150.
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uniquely characterized the Sierpiński carpet as follows: any curve that is
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765:. Oxford Mathematical Monographs. Oxford University Press. p. 31.
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in the interior of the carpet. Then there is a square centered at
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International
Conference on Communication and Signal Processing
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The process of recursively removing squares is an example of a
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Rummler, Hansklaus (1993). "Squaring the circle with holes".
986:, vol. 7, pp. 738-741, 2008, doi: 10.1109/LAWP.2008.2002808.
574:. They also showed that this random walk satisfies stronger
302:, so it cannot be contained in the carpet – a contradiction.
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Brownian motion and harmonic analysis on Sierpiński carpets
30:"Sierpinski snowflake" redirects here. For other uses, see
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Automatic
Sequences: Theory, Applications, Generalizations
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1, and every subset of the plane with these properties is
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Sierpinski Carpet solved by means of modular arithmetics
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The construction of the Sierpiński carpet begins with a
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with the middle line erased creates a Sierpiński carpet
837:"Topological chcracterization of the Sierpinski curve"
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of two sets of real numbers has this property, so its
618:) smaller squares and removing the middle one. By the
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is a nonempty connected compact metric space. Suppose
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to two dimensions; another such generalization is the
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598:A variation of the Sierpiński carpet, called the
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265:Suppose by contradiction that there is a point
57:in 1916. The carpet is a generalization of the
984:IEEE Antennas and Wireless Propagation Letters
72:, removing one or more copies, and continuing
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164:The area of the carpet is zero (in standard
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525:is homeomorphic to the Sierpiński carpet.
898:On-Line Encyclopedia of Integer Sequences
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370:to some subset of the Sierpiński carpet.
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606:th step by subdividing each square into
529:Brownian motion on the Sierpiński carpet
506:the union of the boundaries of the sets
1596:List of fractals by Hausdorff dimension
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685:List of fractals by Hausdorff dimension
390:for which some connected neighborhood
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1000:Variations on the Theme of Tremas II
763:Some Novel Types of Fractal Geometry
622:, the area of the resulting set is
594:Third iteration of the Wallis sieve
499:is a simple closed curve for each
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1651:Iterated function system fractals
1578:How Long Is the Coast of Britain?
915:The American Mathematical Monthly
1666:Science and technology in Poland
130:{\displaystyle 0.1111\dots =0.2}
27:Plane fractal built from squares
868:Barlow, Martin; Bass, Richard,
104:to the remaining 8 subsquares,
41:6 steps of a Sierpinski carpet.
1602:The Fractal Geometry of Nature
1:
1618:Chaos: Making a New Science
364:Lebesgue covering dimension
96:. The square is cut into 9
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888:Sloane, N. J. A.
723:Cambridge University Press
29:
1010:Sierpinski Carpet Project
258:of the carpet is empty.
854:10.4064/fm-45-1-320-324
142:finite subdivision rule
1610:The Beauty of Fractals
799:C. R. Acad. Sci. Paris
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1551:Hamid Naderi Yeganeh
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1314:Space-filling curve
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1174:Space-filling curve
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308:Hausdorff dimension
78:Sierpiński triangle
53:first described by
1661:Topological spaces
1541:Aleksandr Lyapunov
1521:Desmond Paul Henry
1485:Self-avoiding walk
1480:Percolation theory
1124:Iterated function
1065:Fractal dimensions
1005:Sierpiński Cookies
901:. OEIS Foundation.
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1378:Misiurewicz point
1283:Strange attractor
1164:Apollonian gasket
1154:Sierpinski carpet
736:978-0-521-82332-6
660:Mobile phone and
646:Cartesian product
380:locally connected
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310:of the carpet is
250:goes to infinity.
68:The technique of
55:Wacław Sierpiński
47:Sierpiński carpet
18:Sierpinski carpet
16:(Redirected from
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39:
33:
19:
1630:Chaos theory
1625:Kaleidoscope
1616:
1608:
1600:
1526:Gaston Julia
1506:Georg Cantor
1331:Escape-time
1263:Gosper curve
1211:Lévy C curve
1196:Dragon curve
1075:Box-counting
983:
978:
970:
965:
957:
952:
919:
913:
907:
895:
882:
870:
863:
844:
840:
827:
802:
798:
789:
762:
753:
716:
704:
659:
656:Applications
609:
600:Wallis sieve
599:
597:
586:Wallis sieve
569:
561:
546:
532:
520:
513:is dense in
485:
481:
457:
436:
429:
422:
411:
404:
400:
383:
372:
368:homeomorphic
361:
305:
297:
262:
253:
222:
213:
192:
188:
172:
163:
139:
106:ad infinitum
105:
91:
88:Construction
67:
46:
44:
1621:(1987 book)
1613:(1986 book)
1605:(1982 book)
1591:Fractal art
1511:Bill Gosper
1475:Lévy flight
1221:Peano curve
1216:Moore curve
1102:Topological
1087:Correlation
847:: 320–324.
805:: 629–632.
725:. pp.
616:odd squares
539:random walk
386:is a point
158:Peano curve
102:recursively
74:recursively
63:Cantor dust
49:is a plane
1645:Categories
1429:Orbit trap
1424:Buddhabrot
1417:techniques
1405:Mandelbulb
1206:Koch curve
1139:Cantor set
841:Fund. Math
819:46.0295.02
781:0970.28001
745:1086.11015
696:References
175:Denote as
148:Properties
59:Cantor set
1536:Paul Lévy
1415:Rendering
1400:Mandelbox
1346:Julia set
1258:Hexaflake
1189:Minkowski
1109:Recursion
1092:Hausdorff
811:0001-4036
652:is zero.
567:for some
414:continuum
344:≈
335:
324:
291:for some
119:⋯
98:congruent
1446:fractals
1333:fractals
1301:L-system
1243:T-square
1051:Fractals
835:(1958).
761:(2001).
713:(2003).
679:See also
256:interior
1395:Tricorn
1248:n-flake
1097:Packing
1080:Higuchi
1070:Assouad
944:1247533
936:2324662
890:(ed.).
638:
624:
559:√
544:√
288:
276:
242:
230:
211:
199:
186:. Then
51:fractal
1656:Curves
1494:People
1444:Random
1351:Filled
1319:H tree
1238:String
1126:system
942:
934:
817:
809:
779:
769:
743:
733:
729:–406.
572:> 2
552:after
347:1.8928
263:Proof:
173:Proof:
116:0.1111
94:square
1570:Other
932:JSTOR
875:(PDF)
662:Wi-Fi
614:(the
521:Then
442:, ...
220:. So
896:The
807:ISSN
767:ISBN
731:ISBN
671:and
612:+ 1)
306:The
254:The
45:The
924:doi
920:100
849:doi
815:JFM
803:162
777:Zbl
741:Zbl
727:405
460:→ ∞
403:− {
394:of
332:log
321:log
300:+ 1
228:= (
195:+ 1
168:).
125:0.2
1647::
1580:"
940:MR
938:.
930:.
918:.
894:.
845:45
843:.
839:.
813:.
775:.
739:.
721:.
608:(2
484:≠
435:,
428:,
407:}
359:.
197:=
144:.
137:.
84:.
65:.
1576:"
1043:e
1036:t
1029:v
946:.
926::
857:.
851::
821:.
783:.
747:.
635:4
632:/
628:π
610:i
604:i
570:β
562:n
554:n
547:n
523:X
517:.
515:X
510:i
508:C
503:;
501:i
496:i
494:C
489:;
486:j
482:i
476:j
474:C
469:i
467:C
462:;
458:i
452:i
450:C
440:3
437:C
433:2
430:C
426:1
423:C
418:X
405:p
401:U
396:p
392:U
388:p
338:3
327:8
298:k
293:k
285:3
282:/
279:1
271:P
267:P
248:i
244:)
239:9
236:/
233:8
225:i
223:a
216:i
214:a
208:9
205:/
202:8
193:i
189:a
184:i
179:i
177:a
122:=
34:.
20:)
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