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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
591:, 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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97:. 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
346:
664:, 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.
945:
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
1639:
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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 Sierpiński 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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786:(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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is not connected. So, for example, any point of the circle is a local cut point.
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and has no 'local cut-points' is homeomorphic to the Sierpiński carpet. Here a
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69:. In three dimensions, a similar construction based on cubes is known as the
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962:, 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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754:. Oxford Mathematical Monographs. Oxford University Press. p. 31.
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949:, Loughborough, UK, 2009, pp. 113-116, doi: 10.1109/LAPC.2009.5352584.
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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".
975:, vol. 7, pp. 738-741, 2008, doi: 10.1109/LAWP.2008.2002808.
563:. They also showed that this random walk satisfies stronger
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Brownian motion and harmonic analysis on Sierpiński carpets
19:"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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Sierpiński 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
826:"Topological chcracterization of the Sierpinski curve"
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of two sets of real numbers has this property, so its
607:) 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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587:A variation of the Sierpiński carpet, called the
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254:Suppose by contradiction that there is a point
46:in 1916. The carpet is a generalization of the
973:IEEE Antennas and Wireless Propagation Letters
61:, removing one or more copies, and continuing
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153:The area of the carpet is zero (in standard
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514:is homeomorphic to the Sierpiński carpet.
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359:to some subset of the Sierpiński carpet.
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595:th step by subdividing each square into
518:Brownian motion on the Sierpiński carpet
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1585:List of fractals by Hausdorff dimension
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674:List of fractals by Hausdorff dimension
379:for which some connected neighborhood
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989:Variations on the Theme of Tremas II
752:Some Novel Types of Fractal Geometry
611:, the area of the resulting set is
583:Third iteration of the Wallis sieve
488:is a simple closed curve for each
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1640:Iterated function system fractals
1567:How Long Is the Coast of Britain?
904:The American Mathematical Monthly
1655:Science and technology in Poland
119:{\displaystyle 0.1111\dots =0.2}
16:Plane fractal built from squares
857:Barlow, Martin; Bass, Richard,
93:to the remaining 8 subsquares,
30:6 steps of a Sierpiński carpet.
1591:The Fractal Geometry of Nature
1:
1607:Chaos: Making a New Science
353:Lebesgue covering dimension
85:. The square is cut into 9
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877:Sloane, N. J. A.
712:Cambridge University Press
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999:Sierpiński Carpet Project
247:of the carpet is empty.
843:10.4064/fm-45-1-320-324
131:finite subdivision rule
1599:The Beauty of Fractals
788:C. R. Acad. Sci. Paris
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1163:Space-filling curve
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297:Hausdorff dimension
67:Sierpiński triangle
42:first described by
1650:Topological spaces
1530:Aleksandr Lyapunov
1510:Desmond Paul Henry
1474:Self-avoiding walk
1469:Percolation theory
1113:Iterated function
1054:Fractal dimensions
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890:. OEIS Foundation.
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1367:Misiurewicz point
1272:Strange attractor
1153:Apollonian gasket
1143:Sierpinski carpet
725:978-0-521-82332-6
649:Mobile phone and
635:Cartesian product
369:locally connected
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299:of the carpet is
239:goes to infinity.
57:The technique of
44:Wacław Sierpiński
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1589:
1515:Gaston Julia
1495:Georg Cantor
1320:Escape-time
1252:Gosper curve
1200:Lévy C curve
1185:Dragon curve
1064:Box-counting
972:
967:
959:
954:
946:
941:
908:
902:
896:
884:
871:
859:
852:
833:
829:
816:
791:
787:
778:
751:
742:
705:
693:
648:
645:Applications
598:
589:Wallis sieve
588:
586:
575:Wallis sieve
558:
550:
535:
521:
509:
502:is dense in
474:
470:
446:
425:
418:
411:
400:
393:
389:
372:
361:
357:homeomorphic
350:
294:
286:
251:
242:
211:
202:
181:
177:
161:
152:
128:
95:ad infinitum
94:
80:
77:Construction
56:
35:
33:
1610:(1987 book)
1602:(1986 book)
1594:(1982 book)
1580:Fractal art
1500:Bill Gosper
1464:Lévy flight
1210:Peano curve
1205:Moore curve
1091:Topological
1076:Correlation
836:: 320–324.
794:: 629–632.
714:. pp.
605:odd squares
528:random walk
375:is a point
147:Peano curve
91:recursively
63:recursively
52:Cantor dust
38:is a plane
1634:Categories
1418:Orbit trap
1413:Buddhabrot
1406:techniques
1394:Mandelbulb
1195:Koch curve
1128:Cantor set
830:Fund. Math
808:46.0295.02
770:0970.28001
734:1086.11015
685:References
164:Denote as
137:Properties
48:Cantor set
1525:Paul Lévy
1404:Rendering
1389:Mandelbox
1335:Julia set
1247:Hexaflake
1178:Minkowski
1098:Recursion
1081:Hausdorff
800:0001-4036
641:is zero.
556:for some
403:continuum
333:≈
324:
313:
280:for some
108:⋯
87:congruent
1435:fractals
1322:fractals
1290:L-system
1232:T-square
1040:Fractals
824:(1958).
750:(2001).
702:(2003).
668:See also
245:interior
1384:Tricorn
1237:n-flake
1086:Packing
1069:Higuchi
1059:Assouad
933:1247533
925:2324662
879:(ed.).
627:
613:
548:√
533:√
277:
265:
231:
219:
200:
188:
175:. Then
40:fractal
1645:Curves
1483:People
1433:Random
1340:Filled
1308:H tree
1227:String
1115:system
931:
923:
806:
798:
768:
758:
732:
722:
718:–406.
561:> 2
541:after
336:1.8928
252:Proof:
162:Proof:
105:0.1111
83:square
1559:Other
921:JSTOR
864:(PDF)
651:Wi-Fi
603:(the
510:Then
431:, ...
209:. So
885:The
796:ISSN
756:ISBN
720:ISBN
660:and
601:+ 1)
295:The
243:The
34:The
913:doi
909:100
838:doi
804:JFM
792:162
766:Zbl
730:Zbl
716:405
449:→ ∞
392:− {
383:of
321:log
310:log
289:+ 1
217:= (
184:+ 1
157:).
114:0.2
1636::
1569:"
929:MR
927:.
919:.
907:.
883:.
834:45
832:.
828:.
802:.
764:.
728:.
710:.
597:(2
473:≠
424:,
417:,
396:}
348:.
186:=
133:.
126:.
73:.
54:.
1565:"
1032:e
1025:t
1018:v
935:.
915::
846:.
840::
810:.
772:.
736:.
624:4
621:/
617:π
599:i
593:i
559:β
551:n
543:n
536:n
512:X
506:.
504:X
499:i
497:C
492:;
490:i
485:i
483:C
478:;
475:j
471:i
465:j
463:C
458:i
456:C
451:;
447:i
441:i
439:C
429:3
426:C
422:2
419:C
415:1
412:C
407:X
394:p
390:U
385:p
381:U
377:p
327:3
316:8
287:k
282:k
274:3
271:/
268:1
260:P
256:P
237:i
233:)
228:9
225:/
222:8
214:i
212:a
205:i
203:a
197:9
194:/
191:8
182:i
178:a
173:i
168:i
166:a
111:=
23:.
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