Sierpiński gasket

Let $S_0$ be a triangular area, and define $S_{n+1}$ to be obtained from $S_n$ by replacing each triangular area in $S_n$ with three similar and similarly oriented triangular areas each intersecting with each of the other two at exactly one vertex, each one half the linear scale of the original in size. The limiting set as $n\to \mathrm{\infty }$ (alternately the intersection of all these sets) is a Sierpiński gasket, also known as a Sierpiński triangle.

Figure 1: Sierpiński gasket stage 0, a single triangle, and at stage 1, three triangles

Figure 2: Stage 2, nine triangles, and stage $n$, $3^n$ triangles

Title	Sierpiński gasket
Canonical name	SierpinskiGasket
Date of creation	2013-05-18 22:53:46
Last modified on	2013-05-18 22:53:46
Owner	mathwizard (128)
Last modified by	unlord (1)
Numerical id	41
Author	mathwizard (1)
Entry type	Definition
Classification	msc 28A80
Synonym	Sierpinski triangle
Synonym	Sierpinski gasket
Synonym	Sierpiński triangle
Related topic	MengerSponge