
The Sierpiński carpet is a plane
fractal
In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as il ...
first described by
Wacław Sierpiński
Wacław Franciszek Sierpiński (; 14 March 1882 – 21 October 1969) was a Polish mathematician. He was known for contributions to set theory (research on the axiom of choice and the continuum hypothesis), number theory, theory of functions, and t ...
in 1916. The carpet is a generalization of the
Cantor set
In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883.
T ...
to two dimensions; another is
Cantor dust.
The technique of
subdividing a shape into smaller copies of itself, removing one or more copies, and continuing
recursively
Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathematic ...
can be extended to other shapes. For instance, subdividing an equilateral triangle into four equilateral triangles, removing the middle triangle, and recursing leads to the
Sierpiński triangle
The Sierpiński triangle (sometimes spelled ''Sierpinski''), also called the Sierpiński gasket or Sierpiński sieve, is a fractal attractive fixed set with the overall shape of an equilateral triangle, subdivided recursively into smaller equi ...
. In three dimensions, a similar construction based on cubes is known as the
Menger sponge
In mathematics, the Menger sponge (also known as the Menger cube, Menger universal curve, Sierpinski cube, or Sierpinski sponge) is a fractal curve. It is a three-dimensional generalization of the one-dimensional Cantor set and two-dimensional S ...
.
Construction
The construction of the Sierpiński carpet begins with a
square
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...
. The square is cut into 9
congruent subsquares in a 3-by-3 grid, and the central subsquare is removed. The same procedure is then applied
recursively
Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathematic ...
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
.
The process of recursively removing squares is an example of a
finite subdivision rule.
Properties

The area of the carpet is zero (in standard
Lebesgue measure
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides ...
).
:Proof: Denote as the area of iteration . Then . So , which tends to 0 as goes to infinity.
The
interior
Interior may refer to:
Arts and media
* ''Interior'' (Degas) (also known as ''The Rape''), painting by Edgar Degas
* ''Interior'' (play), 1895 play by Belgian playwright Maurice Maeterlinck
* ''The Interior'' (novel), by Lisa See
* Interior de ...
of the carpet is empty.
:Proof: Suppose by contradiction that there is a point in the interior of the carpet. Then there is a square centered at which is entirely contained in the carpet. This square contains a smaller square whose coordinates are multiples of for some . But, if this square has not been previously removed, it must have been holed in iteration , so it cannot be contained in the carpet – a contradiction.
The
Hausdorff dimension
In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was first introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, o ...
of the carpet is
.
Sierpiński demonstrated that his carpet is a universal plane curve.
That is: the Sierpinski carpet is a compact subset of the plane with
Lebesgue covering dimension
In mathematics, the Lebesgue covering dimension or topological dimension of a topological space is one of several different ways of defining the dimension of the space in a
topologically invariant way.
Informal discussion
For ordinary Euclidean ...
1, and every subset of the plane with these properties is
homeomorphic to some subset of the Sierpiński carpet.
This "universality" of the Sierpiński carpet is not a true universal property in the sense of category theory: it does not uniquely characterize this space up to homeomorphism. For example, the disjoint union of a Sierpiński carpet and a circle is also a universal plane curve. However, in 1958
Gordon Whyburn uniquely characterized the Sierpiński carpet as follows: any curve that is
locally connected
In topology and other branches of mathematics, a topological space ''X'' is
locally connected if every point admits a neighbourhood basis consisting entirely of open, connected sets.
Background
Throughout the history of topology, connectedne ...
and has no 'local cut-points' is homeomorphic to the Sierpinski carpet. Here a local cut-point is a point for which some connected neighborhood of has the property that is not connected. So, for example, any point of the circle is a local cut point.
In the same paper Whyburn gave another characterization of the Sierpiński carpet. Recall that a
continuum is a nonempty connected compact metric space. Suppose is a continuum embedded in the plane. Suppose its complement in the plane has countably many connected components and suppose:
* the diameter of goes to zero as ;
* the boundary of and the boundary of are disjoint if ;
* the boundary of is a simple closed curve for each ;
* the union of the boundaries of the sets is dense in .
Then is homeomorphic to the Sierpiński carpet.
Brownian motion on the Sierpiński carpet
The topic of
Brownian motion
Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas).
This pattern of motion typically consists of random fluctuations in a particle's position insi ...
on the Sierpiński carpet has attracted interest in recent years.
Martin Barlow and Richard Bass have shown that a
random walk
In mathematics, a random walk is a random process that describes a path that consists of a succession of random steps on some mathematical space.
An elementary example of a random walk is the random walk on the integer number line \mathbb ...
on the Sierpiński carpet diffuses at a slower rate than an unrestricted random walk in the plane. The latter reaches a mean distance proportional to after steps, but the random walk on the discrete Sierpiński carpet reaches only a mean distance proportional to for some . They also showed that this random walk satisfies stronger
large deviation inequalities (so called "sub-Gaussian inequalities") and that it satisfies the elliptic
Harnack inequality In mathematics, Harnack's inequality is an inequality relating the values of a positive harmonic function at two points, introduced by . Harnack's inequality is used to prove Harnack's theorem about the convergence of sequences of harmonic function ...
without satisfying the parabolic one. The existence of such an example was an open problem for many years.
Wallis sieve

A variation of the Sierpiński carpet, called the Wallis sieve, starts in the same way, by subdividing the unit square into nine smaller squares and removing the middle of them. At the next level of subdivision, it subdivides each of the squares into 25 smaller squares and removes the middle one, and it continues at the th step by subdividing each square into (the
odd squares) smaller squares and removing the middle one. By the
Wallis product
In mathematics, the Wallis product for , published in 1656 by John Wallis, states that
:\begin
\frac & = \prod_^ \frac = \prod_^ \left(\frac \cdot \frac\right) \\ pt& = \Big(\frac \cdot \frac\Big) \cdot \Big(\frac \cdot \frac\Big) \cdot \Big(\f ...
, the area of the resulting set is , unlike the standard Sierpiński carpet which has zero limiting area. Although the Wallis sieve has positive
Lebesgue measure
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides ...
, no subset that is a
Cartesian product
In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is
: A\ ...
of two sets of real numbers has this property, so its
Jordan measure In mathematics, the Peano–Jordan measure (also known as the Jordan content) is an extension of the notion of size (length, area, volume) to shapes more complicated than, for example, a triangle, disk, or parallelepiped.
It turns out that for a ...
is zero.
Applications
Mobile phone and
Wi-Fi
Wi-Fi () is a family of wireless network protocols, based on the IEEE 802.11 family of standards, which are commonly used for local area networking of devices and Internet access, allowing nearby digital devices to exchange data by radio w ...
fractal antennas have been produced in the form of few iterations of the Sierpiński carpet. Due to their
self-similarity
__NOTOC__
In mathematics, a self-similar object is exactly or approximately similar to a part of itself (i.e., the whole has the same shape as one or more of the parts). Many objects in the real world, such as coastlines, are statistically s ...
and
scale invariance
In physics, mathematics and statistics, scale invariance is a feature of objects or laws that do not change if scales of length, energy, or other variables, are multiplied by a common factor, and thus represent a universality.
The technical te ...
, 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.
See also
*
List of fractals by Hausdorff dimension
According to Benoit Mandelbrot, "A fractal is by definition a set for which the Hausdorff-Besicovitch dimension strictly exceeds the topological dimension."
Presented here is a list of fractals, ordered by increasing Hausdorff dimension, to illust ...
*
Menger sponge
In mathematics, the Menger sponge (also known as the Menger cube, Menger universal curve, Sierpinski cube, or Sierpinski sponge) is a fractal curve. It is a three-dimensional generalization of the one-dimensional Cantor set and two-dimensional S ...
References
External links
Variations on the Theme of Tremas IISierpiński CookiesSierpinski Carpet Project
{{Fractals
Iterated function system fractals
Curves
Topological spaces
Science and technology in Poland