The Sierpinski triangle (also with the original orthography Sierpiński), also called the Sierpinski gasket or the Sierpinski Sieve, is a fractal and attractive fixed set named after the Polish mathematician Wacław Sierpiński who described it in 1915.
Originally constructed as a curve, this is one of the basic examples of selfsimilar sets, i.e. it is a mathematically generated pattern that can be reproducible at any magnification or reduction.
Comparing the Sierpinski triangle or the Sierpinski carpet to equivalent repetitive tiling arrangements, it is evident that similar structures can be built into any reptile arrangements.
Contents
Construction
An algorithm for obtaining arbitrarily close approximations to the Sierpinski triangle is as follows:
Note: each removed triangle (a trema) is topologically an open set.^{[1]}
Note that 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 Sierpinski triangle. Michael Barnsley used an image of a fish to illustrate this in his paper "Vvariable fractals and superfractals."^{[2]}
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_{a} note the dilation by a factor of ½ about a point a, then the Sierpinski triangle with corners a, b, and c is the fixed set of the transformation d_{a} U d_{b} U d_{c}.
This is an attractive fixed set, so that when the operation is applied to any other set repeatedly, the images converge on the Sierpinski triangle. This is what is happening with the triangle above, but any other set would suffice.
If one takes a point and applies each of the transformations d_{a}, d_{b}, and d_{c} to it randomly, the resulting points will be dense in the Sierpinski triangle, so the following algorithm will again generate arbitrarily close approximations to it:
Start by labelling p_{1}, p_{2} and p_{3} as the corners of the Sierpinski triangle, and a random point v_{1}. Set v_{n+1} = ½ ( v_{n} + p_{rn} ), where r_{n} is a random number 1, 2 or 3. Draw the points v_{1} to v_{∞}. If the first point v_{1} was a point on the Sierpiński triangle, then all the points v_{n} lie on the Sierpinski triangle. If the first point v_{1} to lie within the perimeter of the triangle is not a point on the Sierpinski triangle, none of the points v_{n} will lie on the Sierpinski triangle, however they will converge on the triangle. If v_{1} is outside the triangle, the only way v_{n} will land on the actual triangle, is if v_{n} is on what would be part of the triangle, if the triangle was infinitely large.
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