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 `{T _{i}}` of shape conserving transforms such
that the distributed union of the set

The set `{U _{i}}` is an example of an
iterated function system (IFS).

Much investigation of tiling focuses on the space `ℝ ^{2}`, 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

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 n_{i}i^{2}=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. c^{2} + mc=n

**b.** cubic, i.e. c^{3} + kc^{2} + mc=n

**c**. quartic, i.e. c^{4} + jc^{3} + kc^{2} +
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**
(poly**plet** frac**tals**) and **hextals** (poly**hex**
frac**tals**).

**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.

- Perron Tiles (notation)
- Quadratic Tiles
- by Perron number
- Reptiles
- Pletals
- Dipletals
- Tetrapletals
- Pentapletals
- Octapletals
- Enneapletals
- Decapletals

- Hextals
- Trihextals
- linear trihextals and derivatives
- fudgeflake and derivatives
- other trihextals

- Tetrahextals
- cyclic tetrahextals and derivatives
- linear tetraahextals and derivatives

- Heptahextals
- flowsnake (Gosper curve) and derivatives
- spiral heptahextals and derivatives
- linear heptahextals and derivatives
- dragon heptahextals and derivatives
- other heptahextals and derivatives

- Enneahextals
- Dodecahextals

- Trihextals
- Other Reptiles

- Pletals
- Irreptiles
- Dipletal derivatives
- Trihextal derivatives
- Tetrahextal derivatives
- Derivatives of other reptiles
- fractional tame twindragons

- by construction
- by attractor

- Reptiles
- Cubic Tiles
- by Perron number
- Plastic Tiles
- 2nd Cubic Tiles
- Doaudy Tiles
- Tribonacci Tiles
- 6th Cubic Tiles
- 8th Cubic Tiles
- 12th Cubic Tiles

- by construction

- by Perron number
- Quartic Tiles
- Golden Tiles
- golden bee and derivatives

- Silver Tiles

- Golden Tiles

- by Perron number
- Pseudo-Perron Tiles
- Non-Perron Tiles
- Free Tiles
- order 2 right angled triangle

- Fixed Tiles

- Free Tiles
- Rauzy Tiles
- Self-Affine Tiles

- by construction
- mirror tiles
- trans tiles
- dissection
- partial post(auto)composition
- partial post(allo)composition
- grouped element tiles
- co-cell tiles

- tiles by order
- order 2 tiles
- order 3 tiles
- order 4 tiles

- tiles by shape
- Triangles
- Parallelograms
- Kites
- Trapezia
- Wedge Trapezia
- Isoceles Trapezia

- Simple Rectilinear Hexagons (Els)
- Complex Rectilinear Hexagons
- Polyforms

- Quadratic Tiles

- Mathematical results
- at least a countable infinity of self-similar tiles
- at least a countable infinity of self-similar (countablegon) fractal tiles

© 2007, 2008, 2016 Stewart R. Hinsley