In the geometry of tessellations, a shape that can be dissected into smaller copies of the same shape is called a reptile or rep-tile. Solomon W. Golomb coined the term for self-replicating tilings. The shape is labelled as 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 shape that tiles itself using different sizes is called an irregular rep-tile or irreptile. If the tiling 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.
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Every square, rectangle, parallelogram, rhombus, or triangle is rep-4. The "sphinx" hexiamond (illustrated) is also rep-4 and is the only known self-replicating pentagon. 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.
The international standard ISO 216 defines sizes of paper sheets using the Lichtenberg ratio, 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 is rep-n if its aspect ratio is √n:1. An isosceles right triangle is also rep-2.