Editing Sierpiński triangle

{{short description|Fractal composed of triangles}}
[[File:Sierpinski triangle.svg|thumb|Sierpiński triangle]]
[[File:Random Sierpinski Triangle animation.gif|thumb|Generated using a random algorithm]]
[[File:Variadic logical AND.svg|thumb|Sierpiński triangle in logic: The first 16 [[Logical conjunction|conjunctions]] of [[Lexicographical order|lexicographically]] ordered arguments. The columns interpreted as binary numbers give 1, 3, 5, 15, 17, 51... {{OEIS|A001317}}]]
The '''Sierpiński triangle''', also called the '''Sierpiński gasket''' or '''Sierpiński sieve''',  is a [[fractal]] with the overall shape of an [[equilateral triangle]], subdivided [[recursion|recursively]] into smaller equilateral triangles. Originally constructed as a curve, this is one of the basic examples of [[self-similarity|self-similar]] sets—that is, it is a mathematically generated pattern 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.

== 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>

==Properties==
For integer number of dimensions <math>d</math>, when doubling a side of an object, <math>2^d</math> copies of it are created, i.e. 2 copies for 1-dimensional object, 4 copies for 2-dimensional object and 8 copies for 3-dimensional object. For the Sierpiński triangle, doubling its side creates 3 copies of itself. Thus the Sierpiński triangle has [[Hausdorff dimension]] <math>\tfrac{\log3}{\log2}\approx 1.585</math>, which follows from solving <math>2^d=3</math> for <math>d</math>.<ref name=FFG120>{{cite book | zbl=0689.28003 | last=Falconer | first=Kenneth | title=Fractal geometry: mathematical foundations and applications | url=https://archive.org/details/fractalgeometrym0000falc | url-access=registration | location=Chichester | publisher=John Wiley | year=1990 | isbn=978-0-471-92287-2 | page=[https://archive.org/details/fractalgeometrym0000falc/page/120 120] }}</ref>

The area of a Sierpiński triangle is zero (in [[Lebesgue measure]]). The area remaining after each iteration is <math>\tfrac34</math> of the area from the previous iteration, and an infinite number of iterations results in an area approaching zero.<ref>{{citation|title=Getting Acquainted with Fractals|first=Gilbert|last=Helmberg|publisher=Walter de Gruyter|year=2007|isbn=9783110190922|page=41|url=https://books.google.com/books?id=PbrlYO83Oq8C&pg=PA41}}.</ref>

The points of a Sierpiński triangle have a simple characterization in [[Barycentric coordinates (mathematics)#Barycentric coordinates on triangles|barycentric coordinates]].<ref>{{Cite web | url=http://www.cut-the-knot.org/ctk/Sierpinski.shtml | title=Many ways to form the Sierpinski gasket}}</ref> If a point has barycentric coordinates <math>(0.u_1u_2u_3\dots,0.v_1v_2v_3\dots,0.w_1w_2w_3\dots)</math>, expressed as [[binary numeral]]s, then the point is in Sierpiński's triangle if and only if <math>u_i+v_i+w_i=1</math> for {{nowrap|all <math>i</math>.}}

==Generalization to other moduli==
A generalization of the Sierpiński triangle can also be generated using [[Pascal's triangle]] if a different modulus <math>P</math> is used. Iteration <math>n</math> can be generated by taking a [[Pascal's triangle]] with <math>P^n</math> rows and coloring numbers by their value modulo <math>P</math>. As <math>n</math> approaches infinity, a fractal is generated.

The same fractal can be achieved by dividing a triangle into a tessellation of <math>P^2</math> similar triangles and removing the triangles that are upside-down from the original, then iterating this step with each smaller triangle.

Conversely, the fractal can also be generated by beginning with a triangle and duplicating it and arranging <math>\tfrac{n(n+1)}{2}</math> of the new figures in the same orientation into a larger similar triangle with the vertices of the previous figures touching, then iterating that step.<ref>{{cite journal|last1=Shannon|first1=Kathleen M.|last2=Bardzell|first2=Michael J.|title=Patterns in Pascal's Triangle – with a Twist|url=https://www.maa.org/press/periodicals/loci/joma/patterns-in-pascals-triangle-with-a-twist|journal=Convergence|publisher=[[Mathematical Association of America]]|date=November 2003|access-date=29 March 2015|archive-date=7 September 2015|archive-url=https://web.archive.org/web/20150907235707/http://www.maa.org/press/periodicals/loci/joma/patterns-in-pascals-triangle-with-a-twist|url-status=live}}</ref>

==Analogues in higher dimensions==
[[File:Rendering von Seite Sierpiński-Pyramide 20230513 002 RGB16.png|thumb|Sierpiński pyramid recursion (8 steps)]]<!-- This section is linked from [[Menger sponge]] -->
The '''Sierpiński tetrahedron''' or '''tetrix''' is the three-dimensional analogue of the Sierpiński triangle, formed by repeatedly shrinking a regular [[tetrahedron]] to one half its original height, putting together four copies of this tetrahedron with corners touching, and then repeating the process.

A tetrix constructed from an initial tetrahedron of side-length <math>L</math> has the property that the total surface area remains constant with each iteration. The initial surface area of the (iteration-0) tetrahedron of side-length <math>L</math> is <math>L^2\sqrt3</math>.  The next iteration consists of four copies with side length <math>\tfrac{L}{2}</math>, so the total area is <math display=inline>4\bigl(\tfrac{L}{2}\bigr)^2\sqrt3=L^2\sqrt3</math> again. Subsequent iterations again quadruple the number of copies and halve the side length, preserving the overall area. Meanwhile, the volume of the construction is halved at every step and therefore approaches zero. The limit of this process has neither volume nor surface but, like the Sierpiński gasket, is an intricately connected curve. Its [[Hausdorff dimension]] is <math display=inline>\tfrac{\log4}{\log2}=2</math>; here "log" denotes the [[natural logarithm]], the numerator is the logarithm of the number of copies of the shape formed from each copy of the previous iteration, and the denominator is the logarithm of the factor by which these copies are scaled down from the previous iteration. If all points are projected onto a plane that is parallel to two of the outer edges, they exactly fill a square of side length <math>\tfrac{L}{\sqrt2}</math> without overlap.<ref>{{citation
 | last1 = Jones | first1 = Huw
 | last2 = Campa | first2 = Aurelio
 | editor1-last = Thalmann | editor1-first = N. M.
 | editor2-last = Thalmann | editor2-first = D.
 | contribution = Abstract and natural forms from iterated function systems
 | doi = 10.1007/978-4-431-68456-5_27
 | location = Tokyo
 | pages = 332–344
 | publisher = Springer
 | series = CGS CG International Series
 | title = Communicating with Virtual Worlds
 | year = 1993| isbn = 978-4-431-68458-9
 }}</ref>

[[File:tetrix_projection_fill_plane.gif|thumb|link={{filepath:tetrix_projection_fill_plane.svg}}|Animation of a rotating level-4 tetrix showing how some [[orthographic projection]]s of a tetrix can fill a plane &ndash; in [http://upload.wikimedia.org/wikipedia/commons/1/19/Tetrix_projection_fill_plane.svg this&nbsp;interactive&nbsp;SVG], move left and right over the tetrix to rotate the 3D model]]

==History==
[[Wacław Sierpiński]] described the Sierpiński triangle in 1915. However, similar patterns appear already as a common motif of 13th-century [[Cosmatesque]] inlay stonework.<ref>{{cite journal | last = Williams | first = Kim | author-link = Kim Williams (architect) | editor-last = Stewart | editor-first = Ian | editor-link = Ian Stewart (mathematician) | date = December 1997 | department = The Mathematical Tourist | doi = 10.1007/bf03024339 | issue = 1 | journal = [[The Mathematical Intelligencer]] | pages = 41–45 | title = The pavements of the Cosmati | volume = 19| s2cid = 189885713 }}</ref>

The [[Apollonian gasket]], named for [[Apollonius of Perga]] (3rd century BC), was first described by  [[Gottfried Leibniz]] (17th century) and is a curved precursor of the 20th-century Sierpiński triangle.<ref>{{cite book| author = Mandelbrot B| year = 1983| title = The Fractal Geometry of Nature| publisher = W. H. Freeman| location = New York| isbn = 978-0-7167-1186-5| page = [https://archive.org/details/fractalgeometryo00beno/page/170 170]| url-access = registration| url = https://archive.org/details/fractalgeometryo00beno/page/170| author-link = Benoit Mandelbrot}}</ref><ref>{{cite book| author = Aste T, [[Denis Weaire|Weaire D]]| year = 2008| title = The Pursuit of Perfect Packing |title-link= The Pursuit of Perfect Packing | edition = 2nd| publisher = Taylor and Francis| location = New York| isbn = 978-1-4200-6817-7| pages = 131–138}}</ref><ref>{{cite book|author-link=Alexandre Kirillov|author=A.A. Kirillov|title=A Tale of Two Fractals|publisher=Birkhauser|year=2013}}</ref>

==Etymology==

The usage of the word "gasket" to refer to the Sierpiński triangle refers to [[gasket]]s such as are found in [[motor]]s, and which sometimes feature a series of holes of decreasing size, similar to the fractal; this usage was coined by [[Benoit Mandelbrot]], who thought the fractal looked similar to "the part that prevents leaks in motors".<ref>{{cite book|last1=Benedetto|first1=John|last2=Wojciech|first2=Czaja|title=Integration and Modern Analysis|page=408}}</ref>

==See also==
* [[Apollonian gasket]], a set of mutually tangent circles with the same combinatorial structure as the Sierpiński triangle
* [[List of fractals by Hausdorff dimension]]
* [[Sierpiński carpet]], another fractal named after Sierpiński and formed by repeatedly removing squares from a larger square
* [[Triforce]], a relic in the ''[[Legend of Zelda]]'' series

==References==
<references/>

==External links==
* {{springer|title=Sierpinski gasket|id=p/s130310}}
* {{MathWorld|title=Sierpinski Sieve|urlname=SierpinskiSieve}}
* {{cite journal|first1=Paul W. K. |last1=Rothemund |first2=Nick |last2=Papadakis |first3=Erik |last3=Winfree |doi=10.1371/journal.pbio.0020424 |title=Algorithmic Self-Assembly of DNA Sierpinski Triangles |journal=PLOS Biology |volume=2 |issue=12 |date=2004 |pmid=15583715 |pmc=534809 |pages=e424 |doi-access=free }}
* [http://www.cut-the-knot.org/Curriculum/Geometry/Tremas.shtml Sierpinski Gasket by Trema Removal] at [[cut-the-knot]]
* [http://www.cut-the-knot.org/triangle/Hanoi.shtml Sierpinski Gasket and Tower of Hanoi] at [[cut-the-knot]]
* [http://www.kevs3d.co.uk/dev/shaders/fractal.html Real-time GPU generated Sierpinski Triangle in 3D]
* [https://books.google.com/books?id=bhc2ngEACAAJ Pythagorean triangles], Waclaw Sierpinski, Courier Corporation, 2003
* [https://oeis.org/A067771 A067771&nbsp;&nbsp;&nbsp;&nbsp;Number of vertices in Sierpiński triangle of order n.] ''at'' [[On-Line Encyclopedia of Integer Sequences|OEIS]]
* [https://scratch.mit.edu/projects/170702288/#player Interactive version of the chaos game]

{{Fractals}}

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