Editing Sierpiński triangle (section)

=== 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>