Vicsek fractal

In mathematics the Vicsek fractal, also known as Vicsek snowflake or box fractal, is a fractal arising from a construction similar to that of the Sierpinski carpet, proposed by Tamás Vicsek. It has applications including as compact antennas, particularly in cellular phones.

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. The Hausdorff dimension of this fractal is ${\displaystyle \textstyle {\frac {\log(5)}{\log(3)}}}$ ≈ 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.

Self-similarities I — removing corner squares.

Self-similarities II — keeping corner squares.

The Vicsek fractal has the surprising property that it has zero area yet an infinite perimeter, due to its non-integer dimension. At each iteration, four squares are removed for every five retained, meaning that at iteration n the area is ${\displaystyle \textstyle {({\frac {5}{9}})^{n}}}$ (assuming an initial square of side length 1). When n approached infinity, the area approaches zero. The perimeter however is ${\displaystyle \textstyle {4({\frac {5}{3}})^{n}}}$, 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 as n increases.

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