Sierpiński triangle

The Sierpiński triangle, also called the Sierpiński gasket or Sierpiński sieve, is a fractal 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—that is, it is a mathematically generated pattern that is 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.

The Sierpinski triangle may be constructed from an equilateral triangle by repeated removal of triangular subsets:

1. Start with an equilateral triangle.
2. Subdivide it into four smaller congruent equilateral triangles and remove the central triangle.
3. Repeat step 2 with each of the remaining smaller triangles infinitely.

An example of the result of the algorithm

• try it on codepen.io.

References

1. Conversano, Elisa; Tedeschini-Lalli, Laura (2011), "Sierpinski Triangles in Stone on Medieval Floors in Rome" (PDF), APLIMAT Journal of Applied Mathematics, 4: 114, 122

2. Brunori, Paola; Magrone, Paola; Lalli, Laura Tedeschini (2018-07-07), "Imperial Porphiry and Golden Leaf: Sierpinski Triangle in a Medieval Roman Cloister", Advances in Intelligent Systems and Computing, Springer International Publishing, pp. 595–609,

3. "Sierpinski Gasket by Trema Removal"

4. Michael Barnsley; et al. (2003), "V-variable fractals and superfractals", arXiv:math/0312314

5. Peitgen, Heinz-Otto; Jürgens, Hartmut; Saupe, Dietmar; Maletsky, Evan; Perciante, Terry; and Yunker, Lee (1991). Fractals for the Classroom: Strategic Activities Volume One, p.39. Springer-Verlag, New York. ISBN 0-387-97346-X and ISBN 3-540-97346-X.