Rep-tile

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.

The order of a shape, whether using rep-tiles or irrep-tiles is the smallest possible number of tiles which will suffice.

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.

Some rep-tiles, like the square and equilateral triangle, are symmetrical and remain identical when reflected in a mirror.

Dissection of the sphinx and some other asymmetric rep-tiles requires use of both the original shape and its mirror-image.

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

[4] Many of the common rep-tiles are rep-n2 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.

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.

However, if the fractal has an empty interior, this decomposition may not lead to a tiling of the entire plane.

By construction, any fractal defined by an iterated function system of n contracting maps of the same ratio is rep-n.

Among regular polygons, only the triangle and square can be dissected into smaller equally sized copies of themselves.

The "sphinx" polyiamond rep-tile. Four copies of the sphinx can be put together as shown to make a larger sphinx.
A selection of rep-tiles, including the sphinx , two fish and the 5-triangle
The chair substitution (left) and a portion of a chair tiling (right)
Defining an aperiodic tiling (the pinwheel tiling ) by repeatedly dissecting and inflating a rep-tile.
Rep-tiles based on rectifiable octominoes
Rep-tiles created from rectifiable nonominoes and 9-polykings (nonakings)
Rep-tiles created from equilateral triangles
A fish-like rep-tile based on three equilateral triangles
A rocket-like rep-tile created from a dodeciamond, or twelve equilateral triangles laid edge-to-edge (and corner-to-corner)
Rep-tiles based on right triangles
A fish-like rep-tile based on four isosceles right triangles
A tridrafter, or shape created by three triangles of 30°-60°-90°
The same tridrafter as a reptile
A tetradrafter, or shape created from four 30°-60°-90° triangles
The same tetradrafter as a reptile
A hexadrafter, or shape created by six 30°-60°-90° triangles
The same hexadrafter as a reptile
Variant rep-tilings of the rep-9 L-tetromino
Variant rep-tilings of the rep-9 sphinx hexiamond
Horned triangle or teragonic triangle
A pentagonal rep-tile discovered by Karl Scherer
Geometrical dissection of an L-triomino (rep-4)
A fractal based on an L-triomino (rep-4)
Another fractal based on an L-triomino
Another fractal based on an L-triomino
A Sierpinski triangle based on three smaller copies of a Sierpinski triangle
A Sierpinski carpet based on eight smaller copies of a Sierpinski carpet
A dragon curve based on 4 smaller copies of a dragon curve