Inmathematics theVicsek fractal, also known asVicsek snowflake orbox fractal,[1][2] is afractal arising from a construction similar to that of theSierpiński carpet, proposed byTamás Vicsek. It has applications including as compactantennas, particularly in cellular phones.
Box fractal also refers to various iterated fractals created by asquare orrectangular grid 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. TheSierpinski triangle may be approximated by a2 × 2 box fractal with one corner removed. TheSierpinski carpet is a3 × 3 box fractal with the middle square removed.
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 limit of this procedure. TheHausdorff dimension of this fractal is ≈ 1.46497.
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.
The Vicsek fractal has the surprising property that it has zero area yet an infiniteperimeter, due to its non-integer dimension. At each iteration, four squares are removed for every five retained, meaning that at iterationn the area is (assuming an initial square of side length 1). Whenn approached infinity, the area approaches zero. The perimeter however is, because each side is divided into three parts and the center one is replaced with three sides, yielding an increase of three to five. The perimeter approaches infinity asn increases.
The boundary of the Vicsek fractal is theType 1 quadratic Koch curve.
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 ≈ 1.7712.
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 at iterationn, which approaches zero asn approaches infinity.
There exist an infinite number ofcross sections which yield the two-dimensional Vicsek fractal.
