# Rep-tile

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*.^{[1]} In 2012 a generalization of rep-tiles called self-tiling tile sets was introduced by Lee Sallows in *Mathematics Magazine*.^{[2]}

## Contents

## Terminology[edit]

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 an 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* − *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.^{[3]}

## Examples[edit]

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 √5.

The international standard ISO 216 defines sizes of paper sheets using the √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 √*n*:1. An isosceles right triangle is also rep-2.

## Rep-tiles and symmetry[edit]

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[edit]

Some rep-tiles are based on polyforms like polyiamonds and polyominoes, or shapes created by laying equilateral triangles and squares edge-to-edge.

### Squares[edit]

If a polyomino is rectifiable, that is, able to tile a rectangle, then it will also be a rep-tile, because the rectangle will then tile a square; this can be seen clearly 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.

Four copies of some nonominoes and nonakings will tile a square, therefore these polyforms are also rep-36 rep-tiles.

### Equilateral triangles[edit]

Similarly, if a polyiamond tiles an equilateral triangle, it will also be a rep-tile.

### Right triangles[edit]

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[edit]

Polyforms based on isosceles right triangles, with sides in the ratio 1 : 1 : √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^{n} 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.

#### 30°-60°-90° triangles[edit]

Polyforms based on 30°-60°-90° right triangles, with sides in the ratio 1 : √3 : 2, are known as polydrafters. Some are identical to polyminoes and polyiamonds, others are distinct.^{[4]}

## Multiple and variant rep-tilings[edit]

Many of the common rep-tiles are rep-*n*^{2} for all positive integer values of 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.^{[5]} 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.

## Rep-tiles with infinite sides[edit]

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[edit]

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 the German/New-Zealand mathematician Karl Scherer and the American mathematician George Sicherman have found more examples, including a double-pyramid and an elongated version of the sphinx; these pentagonal rep-tiles are illustrated on the Math Magic pages overseen by the American mathematician Erich Friedman.^{[6]} However, the sphinx and its extended versions are the only known pentagons that can be rep-tiled with equal copies.
See Clarke's Reptile pages.

## Rep-tiles and fractals[edit]

### Rep-tiles as fractals[edit]

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.

### Fractals as rep-tiles[edit]

Because fractals are 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.

## Infinite tiling[edit]

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[edit]

## Notes[edit]

## References[edit]

- Gardner, M. (2001), "Rep-Tiles",
*The Colossal Book of Mathematics: Classic Puzzles, Paradoxes, and Problems*, New York: W. W. Norton, pp. 46–58 - Gardner, M. (1991), "Chapter 19: Rep-Tiles, Replicating Figures on the Plane",
*The Unexpected Hanging and Other Mathematical Diversions*, Chicago, IL: Chicago University Press, pp. 222–233 - Langford, C. D. (1940), "Uses of a Geometric Puzzle",
*The Mathematical Gazette*,**24**(260): 209–211, doi:10.2307/3605717 - Niţică, Viorel (2003), "Rep-tiles revisited",
*MASS selecta*, Providence, RI: American Mathematical Society, pp. 205–217, MR 2027179 - Sallows, Lee (2012), "On self-tiling tile sets",
*Mathematics Magazine*,**85**(5): 323–333, doi:10.4169/math.mag.85.5.323, MR 3007213 - Scherer, Karl (1987),
*A Puzzling Journey to the Reptiles and Related Animals* - Wells, D. (1991),
*The Penguin Dictionary of Curious and Interesting Geometry*, London: Penguin, pp. 213–214

## External links[edit]

### Rep-tiles[edit]

Wikimedia Commons has media related to .Rep-tiles |

- Mathematics Centre Sphinx Album: http://mathematicscentre.com/taskcentre/sphinx.htm
- Clarke, A. L. "Reptiles." http://www.recmath.com/PolyPages/PolyPages/Reptiles.htm.
- Weisstein, Eric W. "Rep-Tile".
*MathWorld*. - http://www.uwgb.edu/dutchs/symmetry/reptile1.htm (1999)
- IFStile - program for finding rep-tiles: https://ifstile.com