Editing Sierpiński triangle (section)

=== Shrinking and duplication ===
<!-- [[image:Animated construction of Sierpinski Triangle.gif|166px|right|thumb|Animated construction. Click to enlarge.]] -->
The same sequence of shapes, converging to the Sierpiński triangle, can alternatively be generated by the following steps:
#Start with any triangle in a plane (any closed, bounded region in the plane will actually work).  The canonical Sierpiński triangle uses an [[equilateral triangle]] with a base parallel to the horizontal axis (first image).
#Shrink the triangle to {{sfrac|1|2}} height and {{sfrac|1|2}} width, make three copies, and position the three shrunken triangles so that each triangle touches the two other triangles at a corner (image 2). Note the emergence of the central hole—because the three shrunken triangles can between them cover only {{sfrac|3|4}} of the area of the original. (Holes are an important feature of Sierpiński's triangle.)
#Repeat step 2 with each of the smaller triangles (image 3 and so on).

This infinite process is not dependent upon the starting shape being a triangle—it is just clearer that way. The first few steps starting, for example, from a square also tend towards a Sierpiński triangle. [[Michael Barnsley]] used an image of a fish to illustrate this in his paper "V-variable fractals and superfractals."<ref>{{cite arXiv |author=Michael Barnsley |author-link=Michael Barnsley |display-authors=etal |year=2003 |title=V-variable fractals and superfractals |eprint=math/0312314 |mode=cs2}}</ref><ref>NOVA (public television program). The Strange New Science of Chaos (episode). Public television station WGBH Boston. Aired 31 January 1989.</ref>

[[File:Sierpinski triangle evolution square.svg|thumb|upright=2.2|Iterating from a square|center]]
The actual fractal is what would be obtained after an infinite number of iterations.  More formally, one describes it in terms of functions on closed sets of points.  If we let ''d''<sub>A</sub> denote the dilation by a factor of {{sfrac|1|2}} about a point A, then the Sierpiński triangle with corners A, B, and C is the fixed set of the transformation {{tmath|d_\mathrm{A} \cup d_\mathrm{B} \cup d_\mathrm{C} }}.

This is an [[attractive fixed set]], so that when the operation is applied to any other set repeatedly, the images converge on the Sierpiński triangle.  This is what is happening with the triangle above, but any other set would suffice.