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oftenpaper.net Throughout my years playing around with fractals, the Sierpinski triangle has been a consistent staple. The triangle is named after Wacław Sierpiński and as fractals are wont the pattern appears in many places, so there are many different ways of constructing the triangle on a computer. All of the methods are fundamentally iterative. The most obvious method is probably the triangle-in-triangle approach. We start with one triangle, and at every step we replace each triangle with 3 subtriangles: axiom = Polygon [{{ 0 , 0 }, { 1 , Sqrt [ 3 ]} / 2 , { 1 , 0 }}]; next [ prev_ ] := prev /. Polygon [{ p1_ , p2_ , p3_ }] :> { Polygon [{p1, (p1 + p2) / 2 , (p1 + p3) / 2 }], Polygon [{p2, (p2 + p3) / 2 , (p1 + p2) / 2 }], Polygon [{p3, (p1 + p3) / 2 , (p2 + p3) / 2 }]}; draw [ n_ ] := Graphics [{ EdgeForm [ Black ], Nest [next, N @ axiom, n]}]; This tr...