Editing Rectangle (section)

==Squared, perfect, and other tiled rectangles==
[[File:Perfektes_Rechteck.svg|thumb|right|A perfect rectangle of order 9]]
[[File:smallest_perfect_squared_squares.svg|thumb|Lowest-order perfect squared square (1) and the three smallest perfect squared squares (2&ndash;4) &ndash; {{nowrap|all are simple squared squares}}]]
A rectangle tiled by squares, rectangles, or triangles is said to be a "squared", "rectangled", or "triangulated" (or "triangled") rectangle respectively. The tiled rectangle is ''perfect''<ref name="BSST"/><ref>{{cite journal |author=J.D. Skinner II |author2=C.A.B. Smith |author3=W.T. Tutte |name-list-style=amp |date=November 2000 |title=On the Dissection of Rectangles into Right-Angled Isosceles Triangles |journal=[[Journal of Combinatorial Theory, Series B]] |volume=80 |issue=2 |pages=277–319 |doi=10.1006/jctb.2000.1987|doi-access=free }}</ref> if the tiles are [[Similarity (geometry)|similar]] and finite in number and no two tiles are the same size. If two such tiles are the same size, the tiling is ''imperfect''. In a perfect (or imperfect) triangled rectangle the triangles must be [[right triangle]]s. A database of all known perfect rectangles, perfect squares and related shapes can be found at [http://www.squaring.net/ squaring.net]. The lowest number of squares need for a perfect tiling of a rectangle is 9<ref>{{cite OEIS|A219766|Number of nonsquare simple perfect squared rectangles of order n up to symmetry}}</ref> and the lowest number needed for a [[squaring the square|perfect tilling a square]] is 21, found in 1978 by computer search.<ref>{{cite web|access-date=2021-09-26|title=Squared Squares; Perfect Simples, Perfect Compounds and Imperfect Simples|url=http://www.squaring.net/sq/ss/spss/o21/spsso21.html|website=www.squaring.net}}</ref>

A rectangle has [[Commensurability (mathematics)|commensurable]] sides if and only if it is tileable by a finite number of unequal squares.<ref name="BSST">{{cite journal |author=R.L. Brooks |author2=C.A.B. Smith |author3=A.H. Stone |author4=W.T. Tutte |name-list-style=amp |year=1940 |title=The dissection of rectangles into squares |journal=[[Duke Mathematical Journal|Duke Math. J.]] |volume=7 |issue=1 |pages=312–340 |doi=10.1215/S0012-7094-40-00718-9 |url=http://projecteuclid.org/euclid.dmj/1077492259}}</ref><ref>{{cite journal |author=R. Sprague |year=1940 |title=Ũber die Zerlegung von Rechtecken in lauter verschiedene Quadrate |journal=[[Crelle's Journal|Journal für die reine und angewandte Mathematik]] |language=de |volume=1940 |issue=182 |pages=60–64 |doi=10.1515/crll.1940.182.60 |s2cid=118088887}}</ref> The same is true if the tiles are unequal isosceles [[wikt:right triangle|right triangles]].

The tilings of rectangles by other tiles which have attracted the most attention are those by congruent non-rectangular [[polyomino]]es, allowing all rotations and reflections. There are also tilings by congruent [[polyabolo]]es.