Editing Prototile

{{Short description|Basic shape(s) used in a tessellation}}
[[File:Penrose Tiling (Rhombi).svg|thumb|upright=1.35|This form of the [[aperiodic tiling|aperiodic]] [[Penrose tiling]] has two prototiles, a thick [[rhombus]] (shown blue in the figure) and a thin rhombus (green).]]
In [[mathematics]], a '''prototile''' is one of the shapes of a tile in a [[tessellation]].<ref>{{citation|page=174|title=A Course in Modern Geometries|series=[[Undergraduate Texts in Mathematics]]|first=Judith N.|last=Cederberg|edition=2nd|publisher=Springer-Verlag|year=2001|isbn=978-0-387-98972-3|url=https://books.google.com/books?id=Fo9tqL99jdMC&pg=PA174}}.</ref>

==Definition==
A tessellation of the plane or of any other space is a cover of the space by [[closed set|closed]] shapes, called tiles, that have [[disjoint sets|disjoint]] [[interior (topology)|interiors]]. Some of the tiles may be [[congruence (geometry)|congruent]] to one or more others. If {{mvar|S}} is the set of tiles in a tessellation, a set {{mvar|R}} of shapes is called a set of prototiles if no two shapes in {{mvar|R}} are congruent to each other, and every tile in {{mvar|S}} is congruent to one of the shapes in {{mvar|R}}.<ref>{{citation|page=7|title=Introductory Tiling Theory for Computer Graphics|series=Synthesis Lectures on Computer Graphics and Animation|first=Craig S.|last=Kaplan|publisher=Morgan & Claypool Publishers|year=2009|isbn=978-1-60845-017-6|url=https://books.google.com/books?id=OPtQtnNXRMMC&pg=PA7}}.</ref>

It is possible to choose many different sets of prototiles for a tiling: translating or rotating any one of the prototiles produces another valid set of prototiles. However, every set of prototiles has the same [[cardinality]], so the number of prototiles is well defined. A tessellation is said to be ''monohedral'' if it has exactly one prototile.

==Aperiodicity==
[[File:Smith aperiodic monotiling.svg|thumb|A tiling that does not repeat and uses only one shape, discovered by [[David Smith (hobbyist)|David Smith]]]]
A set of prototiles is said to be aperiodic if every tiling with those prototiles is an [[aperiodic tiling]].  In March 2023, four researchers, [[Chaim Goodman-Strauss]], [[David Smith (hobbyist)|David Smith]], Joseph Samuel Myers and Craig S. Kaplan, announced the discovery of an aperiodic monohedral prototile (monotile) and a proof that the tile discovered by David Smith is an aperiodic monotile, i.e. a solution to a longstanding open [[einstein problem]].<ref>{{Cite news |last=Roberts |first=Siobhan |date=2023-03-28 |title=Elusive 'Einstein' Solves a Longstanding Math Problem |language=en-US |work=The New York Times |url=https://www.nytimes.com/2023/03/28/science/mathematics-tiling-einstein.html |access-date=2023-06-02 |issn=0362-4331}}</ref><ref>{{cite journal | arxiv=2303.10798 | last1=Smith | first1=David | author2=Joseph Samuel Myers | last3=Kaplan | first3=Craig S. | last4=Goodman-Strauss | first4=Chaim | title=An aperiodic monotile | journal=Combinatorial Theory | year=2024 | volume=4 | doi=10.5070/C64163843 }}</ref>

In higher dimensions, the problem had been solved earlier: the [[Schmitt-Conway-Danzer tile]] is the prototile of a monohedral aperiodic tiling of three-dimensional [[Euclidean space]], and cannot tile space periodically.

==References==
{{reflist}}

{{Tessellation}}

[[Category:Tessellation]]