# Rep-tile

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{{Short description|Shape subdivided into copies of itself}}{{For|the class of animals|Reptile}}[[File:Self-replication of sphynx hexidiamonds.svg|thumb|200px|The "sphinx" [polyiamond](/source/polyiamond) rep-tile. Four copies of the [sphinx](/source/sphinx_tiling) can be put together as shown to make a larger sphinx.]]
In the [geometry](/source/geometry) of [tessellation](/source/tessellation)s, a '''rep-tile''' or '''reptile''' is a shape that can be [dissected](/source/dissection_(geometry)) into smaller copies of the same shape. The term was coined as a [pun](/source/pun) on animal [reptiles](/source/Reptilia) by [recreational mathematician](/source/recreational_mathematician) [Solomon W. Golomb](/source/Solomon_W._Golomb) and popularized by [Martin Gardner](/source/Martin_Gardner) in his "[Mathematical Games](/source/Mathematical_Games_(column))" column in the May 1963 issue of ''[Scientific American](/source/Scientific_American)''.<ref>[http://www.martin-gardner.org/SciAm12.html A Gardner's Dozen—Martin's Scientific American Cover Stories]</ref> In 2012 a generalization of rep-tiles called [self-tiling tile set](/source/self-tiling_tile_set)s was introduced by [Lee Sallows](/source/Lee_Sallows) in ''[Mathematics Magazine](/source/Mathematics_Magazine)''.{{sfnp|Sallows|2012}}

[[File:A selection of rep-tiles.gif|thumb|400px|right|A selection of rep-tiles, including the [sphinx](/source/sphinx_tiling), two fish and the 5-triangle]]

==Terminology==
[[File:L_substitution_tiling.svg|thumb|The chair substitution (left) and a portion of a [chair tiling](/source/chair_tiling) (right)]]
A rep-tile is labelled rep-''n'' if the dissection uses ''n'' copies. Such a shape necessarily forms the [prototile](/source/prototile) for a tiling of the plane, in many cases a [nonperiodic tiling](/source/aperiodic_tiling). 
A rep-tile dissection using different sizes of the original shape is called an irregular rep-tile or irreptile. If the dissection uses ''n'' copies, the shape is said to be irrep-''n''. If all these sub-tiles are of different sizes then the tiling is additionally described as perfect. A shape that is rep-''n'' or irrep-''n'' is trivially also irrep-(''kn''&nbsp;&minus;&nbsp;''k''&nbsp;+&nbsp;''n'') for any ''k''&nbsp;>&nbsp;1, by replacing the smallest tile in the rep-''n'' dissection by ''n'' even smaller tiles. The order of a shape, whether using rep-tiles or irrep-tiles is the smallest possible number of tiles which will suffice.{{sfnp|Gardner|2001}}

==Examples==
[[File:pinwheel 2.gif|250px|thumb|right|Defining an aperiodic tiling (the [pinwheel tiling](/source/pinwheel_tiling)) by repeatedly dissecting and inflating a rep-tile.]]
Every [square](/source/square), [rectangle](/source/rectangle), [parallelogram](/source/parallelogram), [rhombus](/source/rhombus), or [triangle](/source/triangle) is rep-4. The [sphinx](/source/sphinx_tiling) [hexiamond](/source/polyiamond) (illustrated above) is rep-4 and rep-9, and is one of few known self-replicating pentagons. The [Gosper island](/source/Gosper_curve) is rep-7. The [Koch snowflake](/source/Koch_snowflake) is irrep-7: six small snowflakes of the same size, together with another snowflake with three times the area of the smaller ones, can combine to form a single larger snowflake.

A [right triangle](/source/right_triangle) with side lengths in the ratio 1:2 is rep-5, and its rep-5 dissection forms the basis of the aperiodic [pinwheel tiling](/source/pinwheel_tiling). By [Pythagoras' theorem](/source/Pythagoras'_theorem), the [hypotenuse](/source/hypotenuse), or sloping side of the rep-5 triangle, has a length of {{radic|5}}.

The international standard [ISO 216](/source/ISO_216) defines sizes of paper sheets using the {{math|{{sqrt|2}}}}, in which the long side of a rectangular sheet of paper is the [square root of two](/source/square_root_of_two) times the short side of the paper. Rectangles in this shape are rep-2. A rectangle (or parallelogram) is rep-''n'' if its [aspect ratio](/source/aspect_ratio) is {{radic|''n''}}:1. An [isosceles](/source/isosceles_triangle) right triangle is also rep-2.

==Rep-tiles and symmetry==

Some rep-tiles, like the [square](/source/square) and [equilateral triangle](/source/equilateral_triangle), are [symmetrical](/source/symmetry) and remain identical when [reflected in a mirror](/source/mirror_reflection). Others, like the [sphinx](/source/sphinx_tiling), are [asymmetrical](/source/asymmetry) and exist in [two distinct forms](/source/chirality) related by mirror-reflection. Dissection of the sphinx and some other asymmetric rep-tiles requires use of both the original shape and its mirror-image.

==Rep-tiles and polyforms==

Some rep-tiles are based on [polyform](/source/polyform)s like [polyiamond](/source/polyiamond)s and [polyomino](/source/polyomino)es, or shapes created by laying [equilateral triangle](/source/equilateral_triangle)s and [square](/source/square)s edge-to-edge.

===Squares===

If a polyomino is rectifiable, that is, able to tile a [rectangle](/source/rectangle), then it will also be a rep-tile, because the rectangle will have an integer side length ratio and will thus tile a [square](/source/square). This can be seen in the [octomino](/source/octomino)es, which are created from eight squares. Two copies of some octominoes will tile a square; therefore these octominoes are also rep-16 rep-tiles.

[[File:Rep-tiles constructed from rectifiable octominoes.gif|thumb|350px|none|Rep-tiles based on rectifiable [octomino](/source/octomino)es]]

Four copies of some [nonomino](/source/nonomino)es and [nonakings](/source/polyking) will tile a square, therefore these [polyform](/source/polyform)s are also rep-36 rep-tiles.

[[File:Nonominoes.gif|700px|thumb|none|Rep-tiles created from rectifiable [nonomino](/source/nonomino)es and [9-polykings](/source/polyking) (nonakings)]]

===Equilateral triangles===

Similarly, if a [polyiamond](/source/polyiamond) tiles an equilateral triangle, it will also be a rep-tile.

thumb|700px|none|Rep-tiles created from equilateral triangles

{|
|- valign=top
| thumb|285px|none|A fish-like rep-tile based on three equilateral triangles
| thumb|none|A rocket-like rep-tile created from a dodeciamond, or twelve equilateral triangles laid edge-to-edge (and corner-to-corner)
|}

===Right triangles===

A [right triangle](/source/right_triangle) is a triangle containing one right angle of 90°. Two particular forms of right triangle have attracted the attention of rep-tile researchers, the 45°-90°-45° triangle and the 30°-60°-90° triangle.

====45°-90°-45° triangles====

Polyforms based on [isosceles](/source/isosceles_triangle) [right triangle](/source/right_triangle)s, with sides in the ratio 1&nbsp;:&nbsp;1&nbsp;:&nbsp;[{{sqrt|2}}](/source/Square_root_of_2), are known as [polyabolo](/source/polyabolo)s. An infinite number of them are rep-tiles. Indeed, the simplest of all rep-tiles is a single isosceles right triangle. It is rep-2 when divided by a single line bisecting the right angle to the [hypotenuse](/source/hypotenuse). Rep-2 rep-tiles are also rep-2<sup>n</sup> and the rep-4,8,16+ triangles yield further rep-tiles. These are found by discarding half of the sub-copies and permutating the remainder until they are [mirror-symmetrical](/source/mirror_image) within a right triangle. In other words, two copies will tile a right triangle. One of these new rep-tiles is reminiscent of the fish formed from three [equilateral triangle](/source/equilateral_triangle)s.

thumb|none|600px|Rep-tiles based on right triangles

thumb|220px|none|A fish-like rep-tile based on four isosceles right triangles

====30°-60°-90° triangles====

Polyforms based on 30°-60°-90° right triangles, with sides in the ratio 1&nbsp;:&nbsp;[{{sqrt|3}}](/source/Square_root_of_3)&nbsp;:&nbsp;2, are known as [polydrafter](/source/polydrafter)s. Some are identical to [polyiamond](/source/polyiamond)s.<ref name=polydrafters>[http://www.recmath.org/PolyCur/drirrep/index.html Polydrafter Irreptiling]</ref>

{|
|- valign=top
|thumb|A tridrafter, or shape created by three triangles of 30°-60°-90°
|thumb|The same tridrafter as a reptile
|} 

{|
|- valign=top
|thumb|
A tetradrafter, or shape created from four 30°-60°-90° triangles
|thumb|The same tetradrafter as a reptile
|} 

{|
|- valign=top
|thumb|A hexadrafter, or shape created by six 30°-60°-90° triangles
|thumb|The same hexadrafter as a reptile
|}

==Multiple and variant rep-tilings==

Many of the common rep-tiles are rep-{{math|''n''<sup>2</sup>}} for all positive integer values of&nbsp;{{mvar|n}}. In particular this is true for three [trapezoid](/source/trapezoid)s including the one formed from three equilateral triangles, for three axis-parallel hexagons (the L-tromino, L-tetromino, and P-pentomino), and the sphinx hexiamond.{{sfnp|Niţică|2003}} In addition, many rep-tiles, particularly those with higher rep-''n'', can be self-tiled in different ways. For example, the rep-9 L-tetramino has at least fourteen different rep-tilings. The rep-9 sphinx hexiamond can also be tiled in different ways. 

{|
|- valign=top
|
thumb|500px|Variant rep-tilings of the rep-9 L-tetromino
|thumb|300px|Variant rep-tilings of the rep-9 sphinx hexiamond
|}

==Rep-tiles with infinite sides==
thumb|200px|Horned triangle or teragonic triangle

The most familiar rep-tiles are polygons with a finite number of sides, but some shapes with an infinite number of sides can also be rep-tiles. For example, the [teragonic](/source/teragon) triangle, or horned triangle, is rep-4. It is also an example of a fractal rep-tile.

==Pentagonal rep-tiles==

Triangular and quadrilateral (four-sided) rep-tiles are common, but pentagonal rep-tiles are rare. For a long time, the [sphinx](/source/sphinx_tiling) was widely believed to be the only example known, but Karl Scherer and [George Sicherman](/source/Sicherman_dice) have found more examples, including a double-pyramid and an elongated version of the sphinx.<ref>These pentagonal rep-tiles are illustrated on the [https://erich-friedman.github.io/mathmagic/1002.html Math Magic] pages overseen by [Erich Friedman](/source/Friedman_number). {{Cite web |url=https://erich-friedman.github.io/mathmagic/1002.html |title=Math Magic, Problem of the Month (October 2002) |access-date=2026-03-09 }}</ref> However, the sphinx and its extended versions are the only known pentagons that can be rep-tiled with equal copies.<ref>See Clarke's [http://www.recmath.com/PolyPages/PolyPages/Reptiles.htm Reptile pages].</ref>

{|
|- valign=top
|thumb|A pentagonal rep-tile discovered by Karl Scherer
|}
{{clear}}

==Rep-tiles and fractals==

===Rep-tiles as fractals===

Rep-tiles can be used to create [fractal](/source/fractal)s, or shapes that are [self-similar](/source/self-similarity) at smaller and smaller scales. A rep-tile fractal is formed by subdividing the rep-tile, removing one or more copies of the subdivided shape, and then continuing [recursively](/source/recursion). For instance, the [Sierpinski carpet](/source/Sierpinski_carpet) is formed in this way from a rep-tiling of a square into 27 smaller squares, and the [Sierpinski triangle](/source/Sierpinski_triangle) is formed  from a rep-tiling of an equilateral triangle into four smaller triangles. When one sub-copy is discarded, a rep-4 L-[triomino](/source/triomino) can be used to create four fractals, two of which are identical except for [orientation](/source/orientation_(geometry)).
{|
|- valign=top
| 250px|thumb|left|Geometrical dissection of an L-triomino (rep-4) 
| 250px|thumb|right|A fractal based on an L-triomino (rep-4)
|- valign=top
| 250px|thumb|left|Another fractal based on an L-triomino 
| 250px|thumb|right|Another fractal based on an L-triomino
|}

===Fractals as rep-tiles===

Because [fractal](/source/fractal)s are often self-similar on smaller and smaller scales, many may be decomposed into copies of themselves like a rep-tile. However, if the fractal has an empty [interior](/source/Interior_(topology)), this decomposition may not lead to a tiling of the entire plane. For example, the [Sierpinski triangle](/source/Sierpinski_triangle) is rep-3, tiled with three copies of itself, and the [Sierpinski carpet](/source/Sierpinski_carpet) is rep-8, tiled with eight copies of itself, but repetition of these decompositions does not form a tiling. On the other hand, the [dragon curve](/source/dragon_curve) is a [space-filling curve](/source/space-filling_curve) with a non-empty interior; it is rep-4, and does form a tiling. Similarly, the [Gosper island](/source/Gosper_curve) is rep-7, formed from the space-filling Gosper curve, and again forms a tiling.

By construction, any fractal defined by an [iterated function system](/source/iterated_function_system) of n contracting maps of the same ratio is rep-n.
{|
|- valign=top
| thumb|left|A Sierpinski triangle based on three smaller copies of a Sierpinski triangle
| thumb|right|A Sierpinski carpet based on eight smaller copies of a Sierpinski carpet
| thumb|right|A dragon curve based on 4 smaller copies of a dragon curve
|}

==Infinite tiling==

Among regular polygons, only the triangle and square can be dissected into smaller equally sized copies of themselves. However, a regular [hexagon](/source/hexagon) can be dissected into six equilateral triangles, each of which can be dissected into a regular hexagon and three more equilateral triangles. This is the basis for an infinite [tiling](/source/tessellation) of the hexagon with hexagons. The hexagon is therefore an irrep-[∞](/source/Infinity_symbol) or irrep-infinity irreptile.

==See also==
* {{annotated link|Mosaic}}
* {{annotated link|Self-replication}}
* {{annotated link|Self-tiling tile set}}
* {{annotated link|Reptiles (M. C. Escher)}}

==Notes==
{{reflist}}

==References==
*{{citation
 | last = Gardner | first = M. | author-link = Martin Gardner
 | contribution = Rep-Tiles
 | location = New York
 | pages = 46–58
 | publisher = W. W. Norton
 | title = The Colossal Book of Mathematics: Classic Puzzles, Paradoxes, and Problems
 | year = 2001}}
*{{citation|last=Gardner|first=M.|authorlink=Martin Gardner|contribution=Chapter 19: Rep-Tiles, Replicating Figures on the Plane|title=The Unexpected Hanging and Other Mathematical Diversions|location=Chicago, IL|publisher=Chicago University Press|pages=222–233|year=1991}} 
* {{citation |last1=Langford |first1=C. D. |title=Uses of a Geometric Puzzle |journal=The Mathematical Gazette |date=1940 |volume=24 |issue=260 |pages=209–211 |doi=10.2307/3605717|jstor=3605717 }}
*{{citation
 | last = Niţică | first = Viorel
 | contribution = Rep-tiles revisited
 | location = Providence, RI
 | mr = 2027179
 | pages = 205–217
 | publisher = American Mathematical Society
 | title = MASS selecta
 | year = 2003}}
*{{citation
 | last = Sallows | first = Lee
 | doi = 10.4169/math.mag.85.5.323
 | issue = 5
 | journal = Mathematics Magazine
 | mr = 3007213
 | pages = 323–333
 | title = On self-tiling tile sets
 | volume = 85
 | year = 2012}}
*{{citation|last=Scherer|first=Karl|title=A Puzzling Journey to the Reptiles and Related Animals|year=1987}}
*{{citation|last=Wells|first=D.|title=The Penguin Dictionary of Curious and Interesting Geometry|location=London|publisher=Penguin|pages=213–214|year=1991}}

==External links==

=== Rep-tiles ===
{{Commons category|Rep-tiles}}
*Mathematics Centre Sphinx Album: http://mathematicscentre.com/taskcentre/sphinx.htm
* Clarke, A. L. "Reptiles." http://www.recmath.com/PolyPages/PolyPages/Reptiles.htm. 
*{{mathworld|title=Rep-Tile|urlname=Rep-Tile}}
*http://www.uwgb.edu/dutchs/symmetry/reptile1.htm {{Webarchive|url=https://web.archive.org/web/20111027142835/http://www.uwgb.edu/dutchs/SYMMETRY/reptile1.htm |date=2011-10-27 }}  (1999)
*IFStile - program for finding rep-tiles: https://ifstile.com

=== Irrep-tiles ===
*[https://erich-friedman.github.io/mathmagic/1002.html Math Magic - Problem of the Month 10/2002] 
*[http://blog.tanyakhovanova.com/?p=226 Tanya Khovanova - L-Irreptiles]

{{Tessellation}}

Category:Tessellation
Category:Fractals

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Adapted from the Wikipedia article [Rep-tile](https://en.wikipedia.org/wiki/Rep-tile) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Rep-tile?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.