Dimers (2D)

Introduction

A dimer is defined as a 2-element polymer, that is the self-similar attractor of a linear zeroth order IFS with a uniform measure. Those dimers with a similarity dimension of 2 are rep-tiles, i.e. they also tile the plane. Of all rep-tiles the dimers are the easiest to enumerate, as there are just 4 independent parameters, the angles by which each element is rotated with respect to the whole figure, and whether each element is directly or inversely similar to the whole figure.

Therefore these parameters can be used to name dimers, as a,b-dimers, where a and b are the angles. Those in which both elements are directly similar to the whole figure are the a,b-cis-dimers, and those in which both elements are inversely similar are the a,b-trans-dimers. Where the two elements are rotated by the same angle specification of the 2^nd can be omitted. More generally an apostrophe is appended to the angle to indicate that the corresponding element is inversely similar to the whole figure. Most rep-tile dimers already also have trivial names, and I introduce trivial names for the remainder.

In addition to the rep-2 parallelograms, of which they are an uncountably infinite number with a single degree of freedom (the angle between sides), 9 distinct dimers have been identified. The rep-2 rectangle occurs as a special case of the rep-2 parallelogram, but can also be constructed in a manner parallelling that of the diursus and twindragon, rather than that of the rep-2 parallelogram.

Image	Name(s)	Min. # in Unit Cell	degeneracy
	rep-2 rectangle sqrt(2):1 rectangle z0-homeolineodimer 90°(,90°)-(cis-)dimer	1	16
	diursus tame twindragon z1-homeolineodimer	1	8
	twindragon z2-homeolineodimer 45°(,45°)-(cis-)dimer	1	8
	rep-2 parallelogram trans-homeolineodimer	1	16
	Harter-Heighway dragon Heighway dragon tiling dragon Jurassic Park fractal Dragon fractal (z-)45°,135°-(cis-)dimer	2	4
	Cesaro triangle Cesaro sweep Cesaro curve rep-2 right-angled triangle right isoceles triangle (z-)135°,225°-(cis-)dimer	2	8
	scorpion (w-)45°',315°-dimer	2	4
	arachnodragon (w-)135°',225°-dimer	2	4
	Lévy Curve Lévy fractal Lévy Dragon (w-)45°,315°-(cis-)dimer	4	8
	dibolt (w-)45°',135°'-(trans-)dimer	4	4

Construction

2D dimers are the attractors of 2 element IFSs in which the two transforms are constructed as follows.

1. a contraction by a factor of 2^½ towards the origin.
2. optional reflection in the X-axis
3. a rotation about the origin
4. translation by vectors (1,0) or (-1,0)

How many dimers?

Even with the restricted phase space of the set of dimers an exhaustive search would be time consuming, so investigation of the set of dimers has to be based on theoretical and heuristic grounds. A set of cis-homeolineomers has been identified, of which the rep-2 rectangle, diursus and twindragon are dimers, whereas the trans-homeolineomers for a continous series of parallelograms. This gives 3 characteristic angles for dimers. Details of the construction of homeolineomers can be found on the page for these fractals.

The twindragon is a fractomino, that is a fractal rep-tile corresponding to a polyomino, in this case the domino. As such there is an underlying 4-fold symmetry. As it has an underlying 4-fold symmetry figures application of one of the operations of the D₄ symmetry group to either element is a promising method of discovering new 2 dimensional figures. There are 64 possible combinations, but symmetry reduces this to 12 distinct figures, These are listed below, together with the multiple rotations and angles that generate them.

Of the 12 distinct figures, 8 are amongst the 10 figures identified as dimers. The remaining 4 have not been demonstrated to have a uniform measure, and hence to be rep-tiles. They are presented below for completeness. If anyone can demonstrate that any of these figures tile the plane please let me know at webmaster@meden.demon.co.uk.

Similar attempts to derive new fractals from the diursus and rep-2 rectangle are unproductive. There are however multiple IFSs which generate these figures. There are 4 IFSs which generate the diursus, as each element can independently be rotated by 180°, plus another 4 IFSs generating its mirror image. There are 8 IFSs which generate the rep-2 triangle, as each element can independently be rotated by 180° and/or reflected in the X-axis.

If anyone is aware of any other dimers, or of figures which are composed of two similar elements of different sizes and which tile the plane, please let me know at webmaster@meden.demon.co.uk.

These include 2 well-known figures with non-intersecting boundaries, the Harter-Heighway dragon and the Cesaro triangle. These are obviously plane-tiling figures with a uniform measure.

The remaining figures have self-intersecting boundaries, and therefore contain voids within their shape. Of these one is a well known fractal, Lévy Curve. Two other figures, for which I introduce the names scorpion and arachnodragon, have also been demonstrated to tile the plane.

The diursus is not a fractomino, and only has an underlying 2-fold symmetry. Applying the operations of the D₂ symmetry group to the elements of the diursus does not generate any new rep-tiles.

Looking at the detail of the area in which the 2 components mingles it doesn't look as if has a uniform measure. However experimental investigation using a program mode which plots the measure shows a measure which is uniform to within the limits of the program and the eye. That the scorpionis 2 dimensional can be seen more clearly by putting togerther 4 copies, in 4 different orientations (reflected in the X- and Y-axes). This produces a figure with a filled core and a ragged edge, which visually appears would tile with plane.

Properties

The 10 attractors given here include 3 which are non-fractal (rep-2 rectangle, rep-2 parallelogram and Cesaro triangle). They include 3 with fractal but non-self-intersecting boundaries (twindragon, diursus and Harter-Heighway dragon) and 4 with fractal and self-intersecting boundaries (Lévy curve, scorpion, arachnodragon and dibolt).

The non-homeomeric attractors occur in pairs - the Harter-Heighway dragon and Cesaro triangle, the Lévy Curve and dibolt, and the scorpion and arachnodragon. Each pair has the same minimum number of copies in a unit cell, and parallel constructions for tilings of the plane.

Tiling

All 10 figures tile the plane. For the homeolineomers (twindragon, diursus, rep-2 rectangle and rep-2 parallelogram) the simplest unit cell contains 1 copy of the figure. For the Cesaro triangle and Harter-Heighway dragon the simplest unit cell contains 2 copies of the figure (making a rep-2 parallelogram or twindragon respectively). For the remaining figures (Lévy curve, scorpion, arachnodragon and dibolt) the simplest unit cell contains 4 copies of the figure.

The diursus and twindragon each tile the plane in many different ways. There are 3 regular tilings in which the unit cell is a single copy of the figure. In these tiles the vectors joining corresponding points are (4,0)(8^½cosa,8^½sina), (4,0)(-2+8^½cosa,8^½sina) and (4-2^½cosa,2^½cosa)(8^½cosa,8^½sina), where a is the angle by which the elements are rotated.

More generally a z_m-homeolineo-n-mer, m <> 0, tiles the plane with single copy unit cells in 2n-1 ways.

Tilings composed of diursiand twindragons are always periodic in one direction, with single row unit cells. They can be aperiodic or periodic (with arbitrarily large unit cells) in the other direction.

The 'natural' tiling for the rep-2 rectangle has vectors (4,0)(0,8^½).

Because of the non-fractal, parallel, edges, row or columns can be slid by an arbitrary amount, giving vectors (4,y)(0,8^½) and (4,0)(x,8^½), and an uncountably infinite number of possible tilings with single copy unit cells. Additional aperiodic and periodic tilings are also possible, when x or y is not fixed but varies aperiodically or periodically.

It is also possible to create tilings in which the tiling is composed of elements in 2 different orientations. Because the ratio of the sides of the rep-2 rectangle is irrational these tilings are all aperiodic.

The tilings of the rep-2 parallelogram in which all the copies have the same orientation parallel those of the rep-2 rectangle.

The rep-2 parallelogram also has tilings in which the copies have two different orientations. These are always periodic in one direction, and may be periodic or aperiodic in the other direction. The smallest unit cell consists of two copies of the figure and gives a herringbone tiling.

There are also an additional set of herringbone tilings of the 45°-rep-2 parallelogram.

Two copies of Cesaro trianglecan be combined to form a square or 45°-rep-2 parallelogram. From these combinations tilings of the plane by the Cesaro triangle can be generated from tilings by the square or 45°-rep-2 parallelogram. Note that as two Cesaro triangles can be combined to form a square in two different ways aperiodic tilings, based on periodic tilings of the square, can be obtained.

Two copies of the Harter-Heighway dragon can be combined to from a twindragon, thus demonstrating that the Harter-Heighway dragon tiles the plane. Four copies can also be combined to form a twindragon, in two different ways, giving an alternative tiling of the plane by the Harter-Heighway dragon. Consequently an infinite number of tilings of the plane by the Harter-Heighway dragon, differing in the number and arrangement of copies of these two alternative tilings within the unit cell, exist. Non-periodic tilings also exist.

Tilings of the plane by the remaining four figures are less obvious. That they tile the plane can be seen if one zooms in far enough, but this doesn't necessarily help ascertain the tilings.

A construction for the tiling of the plane by Lévy Curve was obtained at "Tiling the Plane with the Lévy Dragon". The unit cell consists of 4 copies rotated about the origin by 0°, 90°, 180° and 270°. As the Lévy Curveis symmetrical, rotation by 180° can be replaced by reflection in the X-axis.

An alternative construction, based on reflection in none, either or both of the line crossing the origin at 45° to the X- and Y-axes gives the same result.

The unit cell doesn't show the plane-filling nature of the Lévy Curve clearly. This can be shown by drawing 4 unit cells.

Other visually interesting figures can be obtained from combinations of the Lévy Curve. Three are shown, which may be called the Lévy Barbell, Lévy Carpetand Lévy Octopus respectively. The Barbel and Carpet are both half of a unit cell from the tiling.

The unit cell for the dibolt is generated in the same way as for the Lévy Curve. In this case the figure is not symmetrical, and reflection in the X-axis is not an acceptable subsitute for rotation by 180°. The reflectional construction does not appear to work.

Again the unit cell doesn't show the plane-filling nature of the figure clearly. In this case 9 unit cells are required.

The tiling of the plane by the scorpion is the clearest of the tilings of the plane by any of the 4 figures with self-intersectioning boundaries. There is a tiling in which the unit cell consists of 2 copies of the figure, one inverted in the origin. It is not immediately clear on inspection that this is a tiling unit cell, but drawing 4 unit cells clarifies this.

A different unit cell, composed of 4 copies of the figure, can be produced by reflecting each element of the 2 copy unit cell can in the X-axis. This is equivalent to reflection in none, either or both of the X- and Y-axes, and produces a unit cell which self-evidently tiles the plane.

Another unit cell again consists of 4 copies of the figure, reflected in none, either or both of the lines crossing the origin at 45° to the X- and Y-axes. This is equivalent to half of the preceding (the images below are rotated by 45° for ease of comparison).

Two or four copies of the arachnodragon tile the plane, in the same way as for the scorpion. In this case the 2^nd image is rotated -45° for ease of comparison.

Related fractals - homeolineomers, trimers, meta-dimers, fractominoes

Sources: the dibolt, scorpion and arachnodragon are independent inventions, for which are am unaware of any other sources. The diursus is also an independent invention, which I have since seen under the name of tame twindragon in a paper by Yang et al. The rectangle and parallelogram were also independent inventions, probably only because they're so boring that no-one had bothered to write them up. The Cesaro triangle was originally reverse engineered from an L-system, as were the twindragon and Harter-Heighway dragon. The Lévy Curve is also a well-known figure.

The tiling of the Lévy Curve is taken from Larry Riddle; other tilings are independent inventions.

data file dimer.xil; generator dimer.pl; images generated at 6x5 per screen at 1280x1024

References:

• Rep-tiles (Steven Dutch, University of Wisconsin - Green Bay) - mentions the Cesaro triangle, rectangle and parallelogram.
• The Cesaro triangle is given (as the Cesaro Sweep) at ThinkQuest.
• Lévy Dragon - Larry Riddle
• Tiling the Plane with the Lévy Dragon - Larry Riddle

© 2001 Stewart R. Hinsley