# Sierpiński triangle

**The Sierpinski triangle is an attractive fixed set with the overall shape of an equilateral triangle, subdivided recursively into smaller equilateral triangles. Originally constructed as a curve, this is one of the basic examples of self-similar sets. It is named after the Polish mathematician Wacław Sierpiński.**

### About Sierpiński triangle in brief

The Sierpinski triangle is an attractive fixed set with the overall shape of an equilateral triangle, subdivided recursively into smaller equilateral triangles. Originally constructed as a curve, this is one of the basic examples of self-similar sets. It is named after the Polish mathematician Wacław Sierpiński, but appeared as a decorative pattern many centuries before his work. There are many different ways of constructing the SierPinski triangle, including the chaos game, which is an example of an iterated function system. The time-space replicator also resembles a pattern in a cellular automaton, such as Conway’s Life-like cellular automata B12. In fact, the aim of the original article by Spinski of 1915, was to show an example of a curve, as the title of the article declares itself to be a Sier pinski triangle. In the limit, these approach a curve that traces out a single continuous directed path, which is called the Sierski arrowhead.

The Sierspinski arrowhead can also be seen in the Life-S1 automaton when applied to a single cell in a standard life-space, which will create two mirrored Sier Pinski triangles in the same time-Space replicator that replicator replicator will generate four approximations of the triangle in a single line. The same sequence of shapes can alternatively be generated by the following steps: Note that this infinite process is not dependent upon the starting shape being a triangle—it is just clearer that way. If the first point 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 was infinitely large. Or more simply: This method is also called the chaosgame, and is anexample of an iteration system. You can start from any point outside or inside the triangle and it would eventually form a few leftover points.

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This page is based on the article Sierpiński triangle published in Wikipedia (as of Nov. 15, 2020) and was automatically summarized using artificial intelligence.