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A rep-tile is a plane figure which tiles the plane and can be divided into several smaller copies of itself. From this definition it can be seen that the simply connected attractor of any self-similar IFS with a uniform measure and a similarity dimension of 2 is a rep-tile, and all rep-tiles are attractors of IFSs.
The number of rep-tiles can be demonstrated to be infinite, as the cardinality of subsets of the set of reptiles (e.g. the homeolineomers) can be demonstrated to be infinite. There are a number of sets of rep-tiles (e.g. triangles, parallellograms) which are continuously deformable, with one or two degrees of freedom, while remaining rep-tiles. These sets are uncountably infinite. However, if these sets are considered as single rep-tiles, then I cannot demonstrate that the full set of rep-tiles is uncountably infinite. I conjecture that that set is countably infinite.
An alternative introduction to rep-tiles is given by Steven Dutch of the University of Wisconsin at Green Bay.
It would be nice if either a matrix or hierarchical classification of rep-tiles existed; if only to simplify the design of these web pages. However I have not been able to identify a non-arbitrary classification of either type.
There are several properties which can be used to classify rep-tiles
1. The rep- number, which can be any integer greater than 1. This is not a useful property for classification, except that for small rep- numbers there is a greater hope of being able to implement an exhaustive survey of the rep-tiles with that rep- number. The number of rep-tiles with a particular rep- number is conjectured to be finite.
2. Whether the boundary is fractal or non-fractal, and in the former case whether the number of sides is countable or uncountable.
Theorem R1: The number of rep-tiles with a non-fractal boundary is infinite.
Proof: There is a construction to create an IFS for an arbitrary L-polyomino. As the set of L-polyominoes is countably infinite, and this set is a subset of the rep-tiles with non-fractal boundaries, then the set of rep-tiles with non-fractal boundaries must be at least countably infinite.
Theorem R2: The number of rep-tiles with a fractal boundary is infinite.
Proof: As the set of cis-homeolineomers is countably infinite, and this set is a subset of the rep-tiles with non-fractal boundaries, then the set of rep-tiles with non-fractal boundaries must be at least countably infinite.
Theorem R3: The number of rep-tiles with an uncountable number of sides is infinite.
Proof: As the set of cis-homeolineomers is countably infinite, and this set is a subset of the rep-tiles with an uncountable number of sides, then the set of rep-tiles with an uncountable number of sides must be at least countably infinite.
Theorem R4: The number of rep-tiles with a countable number of sides is infinite.
Proof: As the set of trans-homeolineomers (rep-k parallelograms) is countably infinite, and a fractal with a countable number of sides can be constructed from each member of this set, then the set of rep-tiles with an uncountable number of sides must be at least countably infinite.
3. Whether the boundary has a finite or infinite length. All rep-tiles with non-fractal boundaries have boundaries of finite length, but rep-tiles with fractal boundaries can have boundaries of finite or infinite length.
4. Whether the boundary is non-self-intersecting or self-intersecting; equivalently whether the rep-tile is simply connected or not.
Theorem R5: The number of rep-tiles with a non-self-intersecting boundary is infinite.
Proof: There is a construction to create an IFS for an arbitrary L-polyomino. As the set of L-polyominoes is countably infinite, and this set is a subset of the rep-tiles with non-self-intersecting boundaries, then the set of rep-tiles with non-fractal boundaries must be at least countably infinite.
5. Symmetry group: rep-tiles can have d1, c2, d2, c3, d3, c4, d4 or c6 symmetry, or no non-translational symmetries.
6. Whether the copies have identical (homeopolymers) or varying (heteropolymers) orientations.
7. Whether all copies are directly similar to the rep-tile, or all copies are inversely similar to the rep-tile, or there is a mixture of directly and inversely similar copies.
8. By the layout of the elements, e.g. in a line (lineomers), or a circle (cyclomers), or in various triangular, tetragonal or hexagonal arrays. The utility of this property is limited by the fact that the number of different, and often irregular, arrays increases as the rep- number increases.
9. By the nature of the grid on which the copies are laid out. This allows us to identify a number of rich sets of rep-tiles, including the fractominoes and pletals, which are laid out on a square grid (each fractomino corresponds to a polyomino; each pletal corresponds to a polyplet), and the hextals, which are laid out on a hexagonal grid (each hextal corresponds to a polyhex). I have not demonstrated that iamondtals (corresponding to polyiamonds) are distinct from hextals.
Non-fractal rep-tiles including the following
There are continuous deformable series of IFSs linking some polyominos with corresponding polydiamonds, polydiabolos and polydidrafters.
The following figures are not rep-tiles: any regular polygon other than the equilateral triangle or square; any polyhex; the Z-tetromino; the X-pentomino.
There is a construction by which an IFS can be constructed for any polyomino in which m copies can be put together to form a rectangle. Several copies of this rectangle can be combined to form a square, and then n squares combined to form the polyomino. This construction applies to, inter alia, all I- and L-polyominoes. However it does not always generate the lowest rep- number implementation of that polyomino, e.g. the L-triomino has a rep-4 implementation, and the construction gives a rep-36 implementation.
Similar constructions apply to polykings which can be combined to form a rectangle, and polyiamonds which can be combined to form an equilateral triangle. It is worth noting that a similar construction applies to polydiamonds which combine to form a diamond as these can be constructed by analogy with polyominoes which combine to form a square. The same holds for polydiabolos which combine to form a diabolo, and for polydominos which combine to form a domino.
These are the simplest constructions for rep-tiles. They are also productive points to commence an investigation of rep-tiles.
The angle by which copies are rotated can be solved geometrically for homeolineomers. These angles, and the corresponding tiling vectors, can be applied to other layouts.
The cyclomeric construction is not particularly constructive, resulting in only 11 rep-tiles. However if introduces several of the simpler hextals and pletals, and gives a lead into some of the other cyclically symmetric hextals and pletals.
These are fractals whose space group is that of the hexagon. The elements of a n-hextal can be placed in one-to-on correspondence with those of an n-hex.
These are fractals whose space group is that of the square. The elements of a n-pletal can be placed in one-to-on correspondence with those of an n-plet.
There is a technique, which conserves the similarity dimension, for generating additional fractal IFS attractors from an attractor which has rotational or reflectional symmetry. This technique can be applied to rep-tiles with rotational or reflectional symmetry, to generate more fractals with a similarity dimension of 2. Not all the resulting fractals are singly connected, so not all are rep-tiles. A large proportion, however, are.
The resulting figure occupies some proper fraction of the area of the original figure, and can be tiled to recreate the original figure.
Among the figures this technique can be applied to are homelineomers, c2-symmetric heterolineomers, cyclomers, rep-k2 equilateral triangles, rep-k2 parallelograms, the triamond (demihexagon, isosceles trapezium) and L-triomino.
There are 9 dimers. (I am investigating another 4 possible dimers.)
The number of trimers is larger, and includes 2 cyclotrimers, 5 homeolineotrimers, 1 concertina trimer, 7 caterpillar trimers, paired element trimers (9 fractional cyclotrimers, 8 fractional homelineotrimers, ~25 other fractional lineotrimers, etc) and a number of other figures including cyclodimermonomers (terdragon, terlock, terbolt, w-terjig, z-terjig, tergriffin, etc, other heterolineotrimers. The set is under investigation. Trimers which are not otherwise classified are covered by the assorted trimers page.
© 2001 Stewart R. Hinsley