Sierpiński carpet

In three dimensions, a similar construction based on cubes is known as the Menger sponge.

The same procedure is then applied recursively to the remaining 8 subsquares, ad infinitum.

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

For example, the disjoint union of a Sierpiński carpet and a circle is also a universal plane curve.

Recall that a continuum is a nonempty connected compact metric space.

The topic of Brownian motion on the Sierpiński carpet has attracted interest in recent years.

[7] Mobile phone and Wi-Fi fractal antennas have been produced in the form of few iterations of the Sierpiński carpet.

Due to their self-similarity and scale invariance, 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.