Sierpinski carpet

The construction of the Sierpinski carpet begins with a square.

It is a three-dimensional extension of the Cantor set and Sierpinski carpet.

In a similar way, the Sierpinski carpet is a universal curve for all curves that can be drawn on the two-dimensional plane.

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

Wacław Sierpiński gives the first example of an absolutely normal number and describes the Sierpinski carpet.

In this case what develops is a fractal pattern, a constantly changing with the same fractal dimension as a "Sierpinski carpet".

The Sierpinski carpet is a plane fractal first described by Wacław Sierpiński in 1916.

The Sierpinski carpet can also be created by iterating every pixel in a square and using the following algorithm to decide if the pixel is filled.

The iterative processes used in creating the Cantor set and the Sierpinski carpet are examples of finite subdivision rules, as is barycentric subdivision.

A different 2D analogue of the Cantor set is the Sierpinski carpet, where a square is divided up into nine smaller squares, and the middle one removed.

Comparing the Sierpinski triangle or the Sierpinski carpet to equivalent repetitive tiling arrangements, it is evident that similar structures can be built into any rep-tile arrangements.

Sierpinski carpet, also known as the Sierpinski universal curve, is a one-dimensional planar Peano continuum that contains a homeomorphic image of any one-dimensional planar continuum.

Martin Barlow and Richard Bass have shown that a random walk on the Sierpinski carpet diffuses at a slower rate than an unrestricted random walk in the plane.

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.

Sierpiński demonstrated that this fractal is a universal curve, in that any possible one-dimensional graph, projected onto the two-dimensional plane, is homeomorphic to a subset of the Sierpinski carpet.

Three well-known fractals are named after him (the Sierpinski triangle, the Sierpinski carpet and the Sierpinski curve), as are Sierpinski numbers and the associated Sierpiński problem.

Each face of the Menger sponge is a Sierpinski carpet; furthermore, any intersection of the Menger sponge with a diagonal or medium of the initial cube M is a Cantor set.