Self-Similar Tiles and Related Figures

For other material on fractals see my old site.

A tile A is a subset of a space S such that there is a set {Ti} of shape conserving transforms such that the distributed union of the set {A.Ti} is S, and the intersection of A.Ti and A.Tj, ∀i,j, i ≠ j, is the empty set, or more informally a geometric figure copies of which can fill the whole of the space, with no gaps and no overlaps. (Opinion differs as to whether a tile must be simply connected.) A self-similar tile is a tile for which there is a set of restricted affine transforms {Ui} such that the distributed union of the set {A.Ui} is A, and the the intersection of A.Ui and A.Uj, ∀i,j, i ≠ j, is the empty set is the empty set. The restriction is to contractions, rotations, translations (and depending on the definition of self-similarity) reflections and combinations thereof.

The set {Ui} is an example of an iterated function system (IFS).

Much investigation of tiling focuses on the space ℝ2, i.e. the Euclidian plane, as does this site.

For other material on self-similar tiles see my old site.

Shape conserving transforms are contractions, rotations and translations, and combinations of these. (Opinion differs on whether reflections should also be included.)

Complex numbers provide a convenient notation for such transforms. A transform T (not involving reflection) of a point p can be defined as T:p → ceiωp + x. When reflection is involved the point on the right hand side of the equation is replaced by its complex conjugate, i.e. the transform becomes T:p → ceiωp + x.

There are many possible criteria for classifying tiles.

A. Whether the tile is a polygon (with a finite number of sides) a countablegon (with a countable number of sides) or a teragon (with an uncountable number of sides).

B. Whether the tile is simple (with a non-intersecting boundary) or complex (with an intersecting boundary), and in the case of a polygon whether it is convex or concave. (I conjecture that all self-similar countablegon and teragon tiles are concave). The convex polygon self-similar tiles are all triangles, all paralellograms, and a wide variety of other quadrilaterals. Examples of concave polygons which are self-similar tiles include (some) rectilinear hexagons (both simple and complex), and those rectifiable polyominoes which aren't convex.

C. By the minimum number of similar, or congruent, elements in a self-similar dissection. Tiles with a dissection into congruent shapes are rep-tiles, and tiles with a dissection into similar shapes are irrep-tiles. The minimum number of elements in a dissection into similar elements is commonly smaller than in a dissection into congruent elements.

For other material on rep- tiles see my old site.

D. Whether the elements of the dissections are directly (without reflection) or inversely (with reflection) similar to the tile.

E. By the nature of the equation relating the sizes of the elements to the size of the tile, which may be

i. unconstrained, i.e. a + b=1

ii. Σ nii2=k

iii. a polynomial P(c)=1

In the last case the roots of polynomial are often (always?) related to Perron numbers. We can distinguish between tiles in which the minimal polynomial of the Perron number is

a. quadratic, i.e. c2 + mc=n

b. cubic, i.e. c3 + kc2 + mc=n

c. quartic, i.e. c4 + jc3 + kc2 + mc=n

Presumably there are also tiles related to quintic, sextic, etc Perron numbers, but I don't know of any.

F. By symmetry group of the tiling. There are rich sets with square or hexagonal tilings, for which I coin the names pletals (polyplet fractals) and hextals (polyhex fractals).

G. By the number of degrees of freedom the shape of the tile has. All tiles have 4 degrees of freedom, corresponding to size, orientation, and position (x and y coordinates). Some tiles have one or two additional degrees of freedom which are stretch and skew, or in some tiles with one additional degree of freedom some combination of these two.

H. By topological properties, such as the number of crossing points of the boundary, or the number of holes. Which elements are in contact with which other elements could also be used, but this is a property of a dissection of an attractor, and all attractors have multiple dissections (any order n tile can be replaced by an order 2n-1 tile by dissecting one of its elements). It is still potentially useful to break attractors of a particular order into groups of manageable size for study.

I. By the number of copies of the tile in the minimal unit cell.

J. By the dimension of the boundary.

There is no obvious best way to arrange a classification of tiles.