Editing Sierpiński triangle (section)

=== Chaos game ===
[[File:Triângulo de Sierpinski.gif|thumb|Animated creation of a Sierpiński triangle using the chaos game]]
If one takes a point and applies each of the transformations ''d''<sub>A</sub>, ''d''<sub>B</sub>, and ''d''<sub>C</sub> to it randomly, the resulting points will be dense in the Sierpiński triangle, so the following algorithm will again generate arbitrarily close approximations to it:<ref>{{citation|title=Chaos and Fractals: An Elementary Introduction|first=David P.|last=Feldman|publisher=Oxford University Press|year=2012|isbn=9780199566440|contribution=17.4 The chaos game|pages=178–180|contribution-url=https://books.google.com/books?id=exnWM_ZHK0MC&pg=PA178}}.</ref>

Start by labeling '''p'''<sub>1</sub>, '''p'''<sub>2</sub> and '''p'''<sub>3</sub> as the corners of the Sierpiński triangle, and a random point '''v'''<sub>1</sub>. Set {{math|1='''v'''<sub>''n''+1</sub> = {{sfrac|1|2}}('''v'''<sub>''n''</sub> + '''p'''<sub>''r<sub>n</sub>''</sub>)}}, where ''r<sub>n</sub>'' is a random number 1, 2 or 3. Draw the points '''v'''<sub>1</sub> to '''v'''<sub>∞</sub>. If the first point '''v'''<sub>1</sub> was a point on the Sierpiński triangle, then all the points '''v'''<sub>''n''</sub> lie on the Sierpiński triangle. If the first point '''v'''<sub>1</sub> to lie within the perimeter of the triangle is not a point on the Sierpiński triangle, none of the points '''v'''<sub>''n''</sub> will lie on the Sierpiński triangle, however they will converge on the triangle. If '''v'''<sub>1</sub> is outside the triangle, the only way '''v'''<sub>''n''</sub> will land on the actual triangle, is if '''v'''<sub>''n''</sub> is on what would be part of the triangle, if the triangle was infinitely large.

Or more simply:
# Take three points in a plane to form a triangle.
# Randomly select any point inside the triangle and consider that your current position.
# Randomly select any one of the three vertex points.
# Move half the distance from your current position to the selected vertex.
# Plot the current position.
# Repeat from step 3.

This method is also called the [[chaos game]], and is an example of an [[iterated function system]]. You can start from any point outside or inside the triangle, and it would eventually form the Sierpiński Gasket with a few leftover points (if the starting point lies on the outline of the triangle, there are no leftover points). With pencil and paper, a brief outline is formed after placing approximately one hundred points, and detail begins to appear after a few hundred.