The Sierpiński triangle, also called the Sierpiński gasket or Sierpiński sieve, is a fractal with the overall shape of an equilateral triangle, subdivided recursively into smaller equilateral triangles. Originally constructed as a Sierpiński curve, this is one of the basic examples of self-similar sets—that is, it is a mathematically generated pattern reproducible at any magnification or reduction. It is named after the Polish mathematician Wacław Sierpiński but appeared as a decorative pattern many centuries before the work of Sierpiński.
Constructions There are many different ways of constructing the Sierpiński triangle.
Removing triangles The Sierpiński triangle may be constructed from an equilateral triangle by repeated removal of triangular subsets:
Start with an equilateral triangle. Subdivide it into four smaller congruent equilateral triangles and remove the central triangle. Repeat step 2 with each of the remaining smaller triangles infinitely. Each removed triangle (a trema) is topologically an open set. This process of recursively removing triangles is an example of a finite subdivision rule.
Shrinking and duplication 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 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). 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. (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 (as illustrated below), and Michael Barnsley used an image of a fish to illustrate this in his paper "V-variable fractals and superfractals."
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 dA denote the dilation by a factor of 1/2 about a point A, then the Sierpiński triangle with corners A, B, and C is the fixed set of the transformation d A ∪ d B ∪ d C {\displaystyle 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 compact set repeatedly, the images converge (in Hausdorff metric) to the Sierpiński triangle. This is what is happening with the triangle above, but any other compact set would suffice.
Chaos game
If one takes a point and applies each of the transformations dA, dB, and dC to it randomly, the resulting points will be dense in the Sierpiński triangle, so the following algorithm will again generate arbitrarily close approximations to it: Start by labeling p1, p2 and p3 as the corners of the Sierpiński triangle, and a random point v1. Set vn+1 = 1/2(vn + prn), where rn is a random number 1, 2 or 3. Draw the points v1 to v∞. If the first point v1 was a point on the Sierpiński triangle, then all the points vn lie on the Sierpiński triangle. If the first point v1 to lie within the perimeter of the triangle is not a point on the Sierpiński triangle, none of the points vn will lie on the Sierpiński triangle, however they will converge on the triangle. If v1 is outside the triangle, the only way vn will land on the actual triangle, is if vn is on what would be part of the triangle, if the triangle were infinitely large. Or more simply:
Take three points in a plane to form a triangle. Randomly select any point inside the triangle and consider that your current position. Randomly select any one of the three vertex points. Move half the distance from your current position to the selected vertex. Plot the current position. Repeat from step 3. This method is also called the chaos game, and is an example of an iterated function system. You can start from any point outside or inside the triangle, and it would eventually form the Sierpiński Gasket with a few leftover points (if the starting point lies on the outline of the triangle, there are no leftover points). With pencil and paper, a brief outline is formed after placing approximately one hundred points, and detail begins to appear after a few hundred.
Arrowhead construction of Sierpiński gasket
Another construction for the Sierpiński gasket shows that it can be constructed as a curve in the plane. It is formed by a process of repeated modification of simpler curves, analogous to the construction of the Koch snowflake:
Start with a single line segment in the plane Repeatedly replace each line segment of the curve with three shorter segments, forming 120° angles at each junction between two consecutive segments, with the first and last segments of the curve either parallel to the original line segment or forming a 60° angle with it. At every iteration, this construction gives a continuous curve. In the limit, these approach a curve that traces out the Sierpiński triangle by a single continuous directed (infinitely wiggly) path, which is called the Sierpiński arrowhead. In fact, the aim of Sierpiński's original article in 1915 was to show an example of a curve (a Cantorian curve), as the title of the article itself declares.
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