{{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 rep-tile. Four copies of the sphinx can be put together as shown to make a larger sphinx.]] In the geometry of tessellations, a '''rep-tile''' or '''reptile''' is a shape that can be dissected into smaller copies of the same shape. The term was coined as a pun on animal reptiles by recreational mathematician Solomon W. Golomb and popularized by Martin Gardner in his "Mathematical Games" column in the May 1963 issue of ''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 sets was introduced by Lee Sallows in ''Mathematics Magazine''.{{sfnp|Sallows|2012}}
[[File:A selection of rep-tiles.gif|thumb|400px|right|A selection of rep-tiles, including the sphinx, two fish and the 5-triangle]]
==Terminology== [[File:L_substitution_tiling.svg|thumb|The chair substitution (left) and a portion of a chair tiling (right)]] A rep-tile is labelled rep-''n'' if the dissection uses ''n'' copies. Such a shape necessarily forms the prototile for a tiling of the plane, in many cases a nonperiodic 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'' − ''k'' + ''n'') for any ''k'' > 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) by repeatedly dissecting and inflating a rep-tile.]] Every square, rectangle, parallelogram, rhombus, or triangle is rep-4. The sphinx hexiamond (illustrated above) is rep-4 and rep-9, and is one of few known self-replicating pentagons. The Gosper island is rep-7. The 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 with side lengths in the ratio 1:2 is rep-5, and its rep-5 dissection forms the basis of the aperiodic pinwheel tiling. By Pythagoras' theorem, the hypotenuse, or sloping side of the rep-5 triangle, has a length of {{radic|5}}.
The international standard 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 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 is {{radic|''n''}}:1. An isosceles right triangle is also rep-2.
==Rep-tiles and symmetry==
Some rep-tiles, like the square and equilateral triangle, are symmetrical and remain identical when reflected in a mirror. Others, like the sphinx, are asymmetrical and exist in two distinct forms 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 polyforms like polyiamonds and polyominoes, or shapes created by laying equilateral triangles and squares edge-to-edge.
===Squares===
If a polyomino is rectifiable, that is, able to tile a 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. This can be seen in the octominoes, 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 octominoes]]
Four copies of some nonominoes and nonakings will tile a square, therefore these polyforms are also rep-36 rep-tiles.
[[File:Nonominoes.gif|700px|thumb|none|Rep-tiles created from rectifiable nonominoes and 9-polykings (nonakings)]]
===Equilateral triangles===
Similarly, if a 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 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 right triangles, with sides in the ratio 1 : 1 : {{sqrt|2}}, are known as polyabolos. 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. 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 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 triangles.
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 : {{sqrt|3}} : 2, are known as polydrafters. Some are identical to polyiamonds.<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 {{mvar|n}}. In particular this is true for three trapezoids 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 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 was widely believed to be the only example known, but Karl Scherer and George Sicherman 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. {{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 fractals, or shapes that are self-similar 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. For instance, the Sierpinski carpet is formed in this way from a rep-tiling of a square into 27 smaller squares, and the 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 can be used to create four fractals, two of which are identical except for orientation. {| |- 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 fractals 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, this decomposition may not lead to a tiling of the entire plane. For example, the Sierpinski triangle is rep-3, tiled with three copies of itself, and the 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 is a space-filling curve with a non-empty interior; it is rep-4, and does form a tiling. Similarly, the Gosper island 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 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 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 of the hexagon with hexagons. The hexagon is therefore an irrep-∞ 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