![]() The first few steps starting, for example, from a square also tend towards a Sierpinski triangle. Note that this infinite process is not dependent upon the starting shape being a triangle-it is just clearer that way. Repeat step 2 with each of the smaller triangles (image 3 and so on).(Holes are an important feature of Sierpinski's triangle.) Note the emergence of the central hole-because the three shrunken triangles can between them cover only 3 / 4 of the area of the original. Shrink the triangle to 1 / 2 height and 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).The canonical Sierpinski triangle uses an equilateral triangle with a base parallel to the horizontal axis (first image). Start with any triangle in a plane (any closed, bounded region in the plane will actually work).The same sequence of shapes, converging to the Sierpinski triangle, can alternatively be generated by the following steps: This process of recursively removing triangles is an example of a finite subdivision rule. Repeat step 2 with each of the remaining smaller triangles infinitely.Įach removed triangle (a trema) is topologically an open set. ![]() Subdivide it into four smaller congruent equilateral triangles and remove the central triangle.The Sierpinski triangle may be constructed from an equilateral triangle by repeated removal of triangular subsets:
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