# Menger sponge

PMlinkescapeworddivisions

A *Sierpinski carpet ^{}* is the set of all points $(x,y)$ such that $x$ or $y$ is in
the Cantor set

^{}. An equivalent

^{}and perhaps simpler definition is:

Let ${S}_{0}$ be a unit square. Let ${S}_{n+1}$ be ${S}_{n}$, with each square divided into ninths, by being divided into thirds horizontally and vertically, and the central resulting square removed, and the other resulting squares treated separately in further divisions. The limit as $n\to \mathrm{\infty}$ of ${S}_{n}$ is a Sierpinski carpet. An approximation is shown in figure 1.

The *Menger sponge* is a fractal^{} embedded in 3-dimensional space. It can be seen as a 3-d generalization^{} of the Sierpinski carpet, which is itself a 2-dimensional generalization of the Cantor set. The Menger sponge is almost always represented as being constructed from Cantor sets using the “middle third” rule.

The Menger sponge consists of all points $(x,y,z)$ such that $(x,y)$, $(y,z)$, and $(x,z)$ are all in Sierpinski carpets. Each “face” is a Sierpinski carpet.

Similarily to the Sierpinski carpet the Menger sponge can be constructed in the following way:

Start with a unit cube and split it into 27 smaller cubes of equal size. Remove the central cube and the ones joining a face with it. Then start over with the remaining smaller cubes.

Title | Menger sponge |
---|---|

Canonical name | MengerSponge |

Date of creation | 2013-03-22 14:27:05 |

Last modified on | 2013-03-22 14:27:05 |

Owner | mathwizard (128) |

Last modified by | mathwizard (128) |

Numerical id | 23 |

Author | mathwizard (128) |

Entry type | Definition |

Classification | msc 28A80 |

Related topic | Fractal |

Related topic | CantorSet |

Related topic | SierpinskiGasket |

Defines | Sierpinski carpet |