A Self-Similar Set
More fun with fractals as Richard Green explains how to construct the Sierpinski triangle.
Originally shared by Richard Green
The Sierpinski triangle is a fractal named after the Polish mathematician Wacław Sierpiński (1882-1969), who described it as early as 1915.
The fractal can be constructed using an iterative procedure starting with a solid equilateral triangle. The triangle is split evenly into four equal smaller equilateral triangles, and then the middle of the four triangles is removed from the shape. The procedure is then repeated on each of the three remaining triangles, splitting them into three triangles each, and so on to infinity. An animation of this procedure can be found on Wikipedia (http://en.wikipedia.org/wiki/Sierpinski_triangle).
The page http://www.oftenpaper.net/sierpinski.htm by Antonio Marquez-Raygoza contains as many variations on the Sierpinski triangle as anyone could want, and some more. One of these is the Sierpinski fish shown in the picture. You might think that this would be complicated to generate, but the page gives the complete source code, and it is only about ten (fairly short) lines long.
It is very easy to spend an hour skim-reading this single web page, which I found via Jason Bandlow. Also depicted on the page are the Sierpinski Stiletto of Triangular Destruction and the Sierpinski Butterfly of Poisonous Death. The page describes itself as the Sierpinski triangle page to end most Sierpinski triangle pages, and I don’t think this is an exaggeration.