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. Golombcoined 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 anyk > 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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[edit]Examples
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. 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 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. Anisosceles right triangle is also rep-2.
[edit]Rep-tiles as Fractals
Rep-tiles can be used to create fractals, or shapes that are self-similar at smaller and smaller scales. Rep-tiles that are fully subdivided create simple fractals: for example, an equilateral triangle fully divided into four copies of itself, each of which is fully divided into four copies, and so on. However, more complex fractals can be created by discarding sub-copies at each stage of the subdivision.
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. Golombcoined 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 anyk > 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.
Contents[hide] |
[edit]Examples
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. 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 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. Anisosceles right triangle is also rep-2.
[edit]Rep-tiles as Fractals
Rep-tiles can be used to create fractals, or shapes that are self-similar at smaller and smaller scales. Rep-tiles that are fully subdivided create simple fractals: for example, an equilateral triangle fully divided into four copies of itself, each of which is fully divided into four copies, and so on. However, more complex fractals can be created by discarding sub-copies at each stage of the subdivision.
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