Polymers

The attractor of the average IFS (iterated function system) is visually uninteresting - a dust or a blot. Here I define a subset of the attractors of IFSs which share common geometrical properties and are in most cases visually interesting.

The subset is defined by the following axioms.

  1. The IFS is a Cartesian linear IFS, i.e. it is composed of affine transformations, which in 2-space (R2) have the form
x=ax + by + e
y=cx + dy + f
  1. The IFS is a stateless or zeroth order IFS, i.e. the set of eligible transforms to be used at any particular point is independent of the history.
  2. Each point in the attractor belongs to the same number of components, i.e. the attractor has a uniform measure. I am not aware of any such attractor for which points in the attractor correspond to more than one of the components, though if the 3rd or 4th axions are omitted these can be trivially generated. (An alternative axion is that each point in the attractor belongs to exactly one component).
  3. The attractor is self-similar, i.e. can be broken down into several components (each corresponding to a single transform) each of which is similar to the overall attractor.
  4. Each component is congruent to each other. Other self-similar fractals, including metatrapezia, for which this property does not hold can be mechanically generated from fractals for which the property does hold.
  5. The attractor is simply connected, i.e. is all in one piece.

I introduce the term polymer for a set of the attractors defined by the above axioms.

In order to enumerate the polymers I add the following additional rule.

  1. A set of attractors which can inter-converted by an affine transform are said to represent the same polymer.

There can be more than one set of transforms which generates the same attractor. For example a square can be decomposed in 4, 9, 16 etc components, and each component can be rotated by any of 4 angles, or reflected in any of its symmetry axis. Sets of transforms generating the same attractor can be considerably different, and it seems unreasonable to lose the distinction between them. There I introduce the term Polymer system or P-system to describe an IFS which generates a polymer. For each polymers-generating IFS Ti, for any affine transformation A the attractor for the IFS A-1TiA is obtained by taking the attractor for Ti and applying A to it. By the previous rule the attractors are the same polymers. I define all IFSs related in this way to be members of the same P-system.

The set of polymers does not include all interesting attractors.

If Axiom 2 is omitted, and 1st and higher order IFSs are allowed, additional attractors, including the figures defined by many L-systems, arbitrary regular or irregular polygons (composed of triangles) and arbitrary polyiamonds (composed of equilateral triangles) and polyominoes (composed of squares) are accessible.

If Axiom 3 is omitted, and figures which are not self-similar are allowed, additional P-systems are available for some attractors, e.g. starburst, sunburst and sunrise triangles. For someP- systems (e.g. the didiparallelogram) the sets of self-similar and self-affine fractals are the same; for other P-systems (e.g. the diparallelogram) both sets are infinite (À1 members), but the self-affine set is larger; for other P-systems (e.g. the Harter-Heighway dragon) there is one single self-similar attractor amonst an infinite number of self-affine attractors; and for yet other P-systems (e.g. the starburst triangles) there are no self-similar members. There may also be additional attractors which are self-affine but not self-similar, but I have not discovered any such.

If Axiom 4 is omitted and figures which are decomposable into parts which are similar one to the other, but not congruent one to the other, additional attractors, including the metatrapezoidal snowflake are available. A large set of attractors, for which I introduce the term metapolymer, can be generated mechanically from polymers. For the general class of attractors accessible when Axion 4 is omitted I introduce the term pseudopolymer

If Axiom 5 is omitted polyamids become accessible.

If Axiom 6 is omitted additional attractors are available, some of which have the interesting properties of a similarity dimension of 2, and of a uniform measure.

The number of polymers and P-systems is at least countably infinite. Therefore there is a need for a classification scheme, it being impracticable to give each polymer or P-system an individual (trivial) name.

A P-system can be unambiguously described by its number of components, and their relative position, orientation and size, and whether each components has the shape of the attractor, or of its mirror image. However this does not uniquely describe a P-system as the relative position of the 1st and 2nd components is arbitrary, giving a rotation and rescaling of the polymer. Furthermore any fully general scheme based on this would be little better than a listing of the parameters of the transforms.

Moreover symmetric polymers are associated with multiple P-systems, but it seems unnecessary to distinguish between these, or between P-systems which generate the same attractor, but the components of which have been internally rearranged.

Therefore, although I define systematic names for polymers and P-systems, I do not apply a single system uniformly to all polymers and P-systems

The number of components of a polymer or P-system is an important property. Polymers with 2, 3, 4, etc components are described as di-, tri-, tetra-, etc -mers. When the number of components is implied by the rest of the systematic name this part is omitted. For polymers with large numbers of components the number of components may be used instead, e.g. 48-mer, instead of octotetracontamer.

Polymers in which all components have the same orientation are called homeopolymers; this in which the orientation differs are called heteropolymers.

Polymers in which all components are directly similar to the attractor are called cis-polymers; those in which all components are inversely similar to the attractor are called trans-polymers.

Many polymers come in mirror image forms. These are distinguished as laevo-polymers or l-polymers, and dextro-polymers or d-polymers. I have yet to specify rules which define which of the pair is the l-polymer and which the d-polymer. Note that by the rule above the mirror images are the same polymer.

Many polymers have regularly placed components. The term lineomer is introduced for the case when they are placed in a straight line; the term cyclomer for the case when the are placed in a circle; and the term astromer for the case when they are laid out in lines radiating from a core. Terms will be introduced for polymers with components laid out on a grid .Annulomer is a generic term covering all polymers in which the components are laid out on concentric rings, including cyclomers, astromers, and other categories such as intermers, dendromers amd endomers, which I do not define for the time being. Crystomer is a generic term covering all polymer in which components are laid out on a regular grid.

There are overlaps between these categories.

The orientation of components is specified by listing then angles of rotation as a prefix; where no ambiguity arises repeated values are omitted, e.g. 0-cyclotrimer (trivial name: Sierpinski triangle), 30-cyclotrimer (trivial name: fudgeflake).

In a number of cases there is more than one polymer with the same number, position, and orientation of components, the polymers differing in the size of the components. These polymers could be differentiated by the similarity dimension of the fractal, or the magnitude of the contraction of each component. Instead I allocate to classes based on the connections between the components.

In a polymer with a similarity dimension of 2 at least one pair of components meet at a line, rather than at a set of points.

If the polymer has a similarity dimension of less than 2 then components meet at points only.

For example there is a z-30-cyclotrimer, with a dimension of 2, and an a-30-cyclotrimer, with a smaller dimension.

© 2000, 2001 Stewart R. Hinsley