Editing Sierpiński triangle (section)

== Constructions ==
There are many different ways of constructing the Sierpiński triangle.

=== Removing triangles ===
The Sierpiński triangle may be constructed from an [[equilateral triangle]] by repeated removal of triangular subsets:
# Start with an equilateral triangle.
# Subdivide it into four smaller congruent equilateral triangles and remove the central triangle.
# Repeat step 2 with each of the remaining smaller triangles infinitely.
[[File:Sierpinski triangle evolution.svg|thumb|upright=2.2|The evolution of the Sierpiński triangle|center]]Each removed triangle (a ''trema'') is [[topology|topologically]] an [[open set]].<ref>{{cite web| url = http://www.cut-the-knot.org/triangle/Tremas.shtml| title = "Sierpinski Gasket by Trema Removal"}}</ref> This process of recursively removing triangles is an example of a [[finite subdivision rule]].

=== Shrinking and duplication ===
<!-- [[image:Animated construction of Sierpinski Triangle.gif|166px|right|thumb|Animated construction. Click to enlarge.]] -->
The same sequence of shapes, converging to the Sierpiński triangle, can alternatively be generated by the following steps:
#Start with any triangle in a plane (any closed, bounded region in the plane will actually work).  The canonical Sierpiński triangle uses an [[equilateral triangle]] with a base parallel to the horizontal axis (first image).
#Shrink the triangle to {{sfrac|1|2}} height and {{sfrac|1|2}} width, make three copies, and position the three shrunken triangles so that each triangle touches the two other triangles at a corner (image 2). Note the emergence of the central hole—because the three shrunken triangles can between them cover only {{sfrac|3|4}} of the area of the original. (Holes are an important feature of Sierpiński's triangle.)
#Repeat step 2 with each of the smaller triangles (image 3 and so on).

This infinite process is not dependent upon the starting shape being a triangle—it is just clearer that way. The first few steps starting, for example, from a square also tend towards a Sierpiński triangle. [[Michael Barnsley]] used an image of a fish to illustrate this in his paper "V-variable fractals and superfractals."<ref>{{cite arXiv |author=Michael Barnsley |author-link=Michael Barnsley |display-authors=etal |year=2003 |title=V-variable fractals and superfractals |eprint=math/0312314 |mode=cs2}}</ref><ref>NOVA (public television program). The Strange New Science of Chaos (episode). Public television station WGBH Boston. Aired 31 January 1989.</ref>

[[File:Sierpinski triangle evolution square.svg|thumb|upright=2.2|Iterating from a square|center]]
The actual fractal is what would be obtained after an infinite number of iterations.  More formally, one describes it in terms of functions on closed sets of points.  If we let ''d''<sub>A</sub> denote the dilation by a factor of {{sfrac|1|2}} about a point A, then the Sierpiński triangle with corners A, B, and C is the fixed set of the transformation {{tmath|d_\mathrm{A} \cup d_\mathrm{B} \cup d_\mathrm{C} }}.

This is an [[attractive fixed set]], so that when the operation is applied to any other set repeatedly, the images converge on the Sierpiński triangle.  This is what is happening with the triangle above, but any other set would suffice.

=== 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.

=== Arrowhead construction of Sierpiński gasket ===
[[File:Arrowhead curve 1 through 6.png|thumb|Arrowhead construction of the Sierpiński gasket]]
Another construction for the Sierpiński gasket shows that it can be constructed as a [[curve]] in the plane. It is formed by a process of repeated modification of simpler curves, analogous to the construction of the [[Koch snowflake]]:
# Start with a single line segment in the plane
# Repeatedly replace each line segment of the curve with three shorter segments, forming 120° angles at each junction between two consecutive segments, with the first and last segments of the curve either parallel to the original line segment or forming a 60° angle with it.

At every iteration, this construction gives a continuous curve. In the limit, these approach a curve that traces out the Sierpiński triangle by a single continuous directed (infinitely wiggly) path, which is called the [[Sierpiński arrowhead curve|Sierpiński arrowhead]].<ref>{{citation|first=P.|last=Prusinkiewicz|contribution=Graphical applications of L-systems|title=Proceedings of Graphics Interface '86 / Vision Interface '86|pages=247–253|year=1986|contribution-url=https://blog.itu.dk/mpgg-e2012/files/2012/09/graphicalgi86.pdf|access-date=2014-03-19|archive-date=2014-03-20|archive-url=https://web.archive.org/web/20140320011732/https://blog.itu.dk/mpgg-e2012/files/2012/09/graphicalgi86.pdf|url-status=dead}}.</ref> In fact, the aim of Sierpiński's original article in 1915 was to show an example of a curve (a Cantorian curve), as the title of the article itself declares.<ref>{{Cite journal|last=Sierpiński|first=Waclaw|date=1915|title=Sur une courbe dont tout point est un point de ramification|journal=Compt. Rend. Acad. Sci. Paris|volume=160|pages=302–305|url=https://gallica.bnf.fr/ark:/12148/bpt6k31131|archive-date=2020-08-06|access-date=2020-04-21|archive-url=https://web.archive.org/web/20200806202128/https://gallica.bnf.fr/ark:/12148/bpt6k31131|url-status=live}}</ref><ref name=":0">{{Citation|last1=Brunori|first1=Paola|title=Imperial Porphiry and Golden Leaf: Sierpinski Triangle in a Medieval Roman Cloister|url=https://www.researchgate.net/publication/326251830|date=2018-07-07|pages=595–609|publisher=Springer International Publishing|language=en|doi=10.1007/978-3-319-95588-9_49|isbn=9783319955872|last2=Magrone|first2=Paola|last3=Lalli|first3=Laura Tedeschini|series=Advances in Intelligent Systems and Computing |volume=809 |s2cid=125313277}}</ref>

=== Cellular automata ===
The Sierpiński triangle also appears in certain [[cellular automata]] (such as [[Rule 90]]), including those relating to [[Conway's Game of Life]]. For instance, the [[Life-like cellular automaton]] B1/S12 when applied to a single cell will generate four approximations of the Sierpiński triangle.<ref>{{citation|contribution-url=http://cmc11.uni-jena.de/proceedings/rumpf.pdf|first=Thomas|last=Rumpf|contribution=Conway's Game of Life accelerated with OpenCL|pages=459–462|title=Proceedings of the Eleventh International Conference on Membrane Computing (CMC 11)|year=2010|access-date=2014-03-19|archive-date=2016-07-29|archive-url=https://web.archive.org/web/20160729210324/http://cmc11.uni-jena.de/proceedings/rumpf.pdf|url-status=live}}.</ref> A very long, one cell–thick line in standard life will create two mirrored Sierpiński triangles. The time-space diagram of a replicator pattern in a cellular automaton also often resembles a Sierpiński triangle, such as that of the common replicator in HighLife.<ref>{{citation|title=Emergent patterning phenomena in 2D cellular automata|first1=Eleonora|last1=Bilotta|author1-link=Eleonora Bilotta|first2=Pietro|last2=Pantano|journal=Artificial Life|date=Summer 2005|volume=11|issue=3|pages=339–362|doi=10.1162/1064546054407167|pmid=16053574|s2cid=7842605}}.</ref> The Sierpiński triangle can also be found in the [[Ulam-Warburton automaton]] and the Hex-Ulam-Warburton automaton.<ref>{{citation|arxiv=1408.5937|last1=Khovanova|first1=Tanya|last2=Nie|first2=Eric|last3=Puranik|first3=Alok|title=The Sierpinski Triangle and the Ulam-Warburton Automaton|journal=Math Horizons|volume=23|issue=1|pages=5–9|year=2014|doi=10.4169/mathhorizons.23.1.5|s2cid=125503155}}</ref>

===Pascal's triangle===
[[File:Sierpinski_Pascal_triangle.svg|thumb|A level-5 approximation to a Sierpiński triangle obtained by shading the first 2<sup>5</sup> (32) levels of a Pascal's triangle white if the binomial coefficient is even and black otherwise]]
If one takes [[Pascal's triangle]] with <math>2^n</math> rows and colors the even numbers white, and the odd numbers black, the result is an approximation to the Sierpiński triangle.  More precisely, the [[limit of a sequence|limit]] as {{mvar|n}} approaches infinity of this [[Parity (mathematics)|parity]]-colored <math>2^n</math>-row Pascal triangle is the Sierpiński triangle.<ref>{{citation|title=How to Cut a Cake: And other mathematical conundrums|first=Ian|last=Stewart|publisher=Oxford University Press|year=2006|isbn=9780191500718|page=145|url=https://books.google.com/books?id=theofRmeg0oC&pg=PT145}}.</ref>

As the proportion of black numbers tends to zero with increasing ''n'', a corollary is that the proportion of odd binomial coefficients tends to zero as ''n'' tends to infinity.<ref>Ian Stewart, "How to Cut a Cake", Oxford University Press, page 180</ref>

===Towers of Hanoi===
The [[Towers of Hanoi]] puzzle involves moving disks of different sizes between three pegs, maintaining the property that no disk is ever placed on top of a smaller disk. The states of an {{mvar|n}}-disk puzzle, and the allowable moves from one state to another, form an [[undirected graph]], the [[Hanoi graph]], that can be represented geometrically as the [[intersection graph]] of the set of triangles remaining after the {{mvar|n}}th step in the construction of the Sierpiński triangle. Thus, in the limit as {{mvar|n}} goes to infinity, this sequence of graphs can be interpreted as a discrete analogue of the Sierpiński triangle.<ref>{{citation
 | last = Romik | first = Dan
 | arxiv = math.CO/0310109
 | doi = 10.1137/050628660
 | issue = 3
 | journal = SIAM Journal on Discrete Mathematics
 | mr = 2272218
 | pages = 610–62
 | title = Shortest paths in the Tower of Hanoi graph and finite automata
 | volume = 20
 | year = 2006| s2cid = 8342396
 }}.</ref>