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The basic square is decomposed into nine smaller squares in the 3-by-3 grid. The four squares at the corners and the middle square are left, the other squares being removed. The process is repeated recursively for each of the five remaining subsquares. The Vicsek fractal is the set obtained at the
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An alternative construction (shown below in the left image) is to remove the four corner squares and leave the middle square and the squares above, below, left and right of it. The two constructions produce identical limiting curves, but one is rotated by 45 degrees with respect to the other.
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There is a three-dimensional analogue of the Vicsek fractal. It is constructed by subdividing each cube into 27 smaller ones, and removing all but the "center cross", the central cube and the six cubes touching the center of each face. Its
Hausdorff dimension is
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with various boxes removed or absent and, at each iteration, those present and/or those absent have the previous image scaled down and drawn within them. The
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Similarly to the two-dimensional Vicsek fractal, this figure has zero volume. Each iteration retains 7 cubes for every 27, resulting in a volume of
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Four iterations of the saltire form of the fractal (top) and the cross form of the fractal (bottom).
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ICMMT 4th
International Conference on, Proceedings Microwave and Millimeter Wave Technology, 2004
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Shan Fuqi; Gu
Hongming; Gao Baoxin (2004). "Analysis of a vicsek fractal patch antenna".
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where the jump=2/3 randomly towards either the center or one of the vertices of a square
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The Vicsek fractal has the surprising property that it has zero area yet an infinite
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approached infinity, the area approaches zero. The perimeter however is
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Animation of the 3D analogue of the Vicsek fractal (third iteration)
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also refers to various iterated fractals created by a
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463:{\displaystyle \textstyle {\frac {\log(7)}{\log(3)}}}
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Self-similarities II — keeping corner squares.4
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317:(assuming an initial square of side length 1). When
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Self-similarities I — removing corner squares.
44:arising from a construction similar to that of the
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361:{\displaystyle \textstyle {4({\frac {5}{3}})^{n}}}
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310:{\displaystyle \textstyle {({\frac {5}{9}})^{n}}}
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532:which yield the two-dimensional Vicsek fractal.
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117:box fractal with the middle square removed.
20:Vicsek fractal (5th iteration of cross form)
52:. It has applications including as compact
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375:The boundary of the Vicsek fractal is the
593:. Beijing, China: IEEE. pp. 98–101.
109:box fractal with one corner removed. The
408:Flight to and around a 3D Vicsek fractal
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1269:List of fractals by Hausdorff dimension
552:List of fractals by Hausdorff dimension
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56:, particularly in cellular phones.
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528:There exist an infinite number of
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1251:How Long Is the Coast of Britain?
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1275:The Fractal Geometry of Nature
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383:Analogues in higher dimensions
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71:6 steps of a Sierpinski carpet
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126:limit of this procedure. The
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1291:Chaos: Making a New Science
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521:, which approaches zero as
377:Type 1 quadratic Koch curve
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1334:Eponymous geometric shapes
599:10.1109/ICMMT.2004.1411469
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105:may be approximated by a
1283:The Beauty of Fractals
542:Box-counting dimension
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252:Approximation by the
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82:fractal built from a
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1229:Lewis Fry Richardson
1224:Hamid Naderi Yeganeh
1014:Burning Ship fractal
946:Weierstrass function
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987:Space-filling curve
964:Multifractal system
847:Space-filling curve
832:Sierpinski triangle
562:Sierpinski triangle
244:Cross-stitch island
130:of this fractal is
128:Hausdorff dimension
103:Sierpinski triangle
1214:Aleksandr Lyapunov
1194:Desmond Paul Henry
1158:Self-avoiding walk
1153:Percolation theory
797:Iterated function
738:Fractal dimensions
632:Weisstein, Eric W.
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1234:Wacław Sierpiński
1219:Benoit Mandelbrot
1143:Fractal landscape
1051:Misiurewicz point
956:Strange attractor
837:Apollonian gasket
827:Sierpinski carpet
557:Sierpinski carpet
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111:Sierpinski carpet
46:Sierpiński carpet
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1174:Michael Barnsley
1041:Lyapunov fractal
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852:Blancmange curve
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32:, also known as
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656:"Box Fractals"
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678:Wolfram Alpha
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674:"Box Fractal"
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658:. 2014-01-03.
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635:"Box Fractal"
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517:at iteration
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1303:Chaos theory
1298:Kaleidoscope
1289:
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1273:
1199:Gaston Julia
1179:Georg Cantor
1004:Escape-time
936:Gosper curve
925:
884:Lévy C curve
869:Dragon curve
748:Box-counting
682:. Retrieved
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273:the area is
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121:Construction
90:
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50:Tamás Vicsek
37:
33:
29:
23:
1294:(1987 book)
1286:(1986 book)
1278:(1982 book)
1264:Fractal art
1184:Bill Gosper
1148:Lévy flight
894:Peano curve
889:Moore curve
775:Topological
760:Correlation
684:21 February
372:increases.
187:≈ 1.46497.
91:Box fractal
80:Self-affine
38:box fractal
26:mathematics
1318:Categories
1102:Orbit trap
1097:Buddhabrot
1090:techniques
1078:Mandelbulb
879:Koch curve
812:Cantor set
576:References
470:≈ 1.7712.
261:Properties
254:chaos game
1329:L-systems
1209:Paul Lévy
1088:Rendering
1073:Mandelbox
1019:Julia set
931:Hexaflake
862:Minkowski
782:Recursion
765:Hausdorff
640:MathWorld
445:
428:
267:perimeter
162:
145:
1324:Fractals
1119:fractals
1006:fractals
974:L-system
916:T-square
724:Fractals
617:44047788
536:See also
54:antennas
1068:Tricorn
921:n-flake
770:Packing
753:Higuchi
743:Assouad
63:Variant
42:fractal
40:, is a
1167:People
1117:Random
1024:Filled
992:H tree
911:String
799:system
615:
605:
570:-flake
95:square
1243:Other
613:S2CID
115:3 × 3
113:is a
107:2 × 2
84:3 × 2
686:2019
680:Site
603:ISBN
232:Anti
86:grid
28:the
595:doi
442:log
425:log
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142:log
97:or
36:or
24:In
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Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
