IFS Attractors: Parallelograms

Introduction

In some ways parallelograms are among the least interesting of self-similar tiles. A single construction generates all parallelograms as order 4 rep-tiles (which are Perron tiles) which tile the plane with a single copy in the unit cell, and one might consider this sufficient.

However there are 3 parallelograms which have lower order. The √2:1 rectangle is an order 2 Perron rep-tile, the √3:1 rectangle is an order 3 Perron rep-tile, and the golden rectangle is an order 3 Perron irrep-tile. The corresponding parallelograms are pseudo-Perron tiles.

All of these are the lowest order members of larger classes of contructions: the order 4 tiles of the rep-n²-parallograms, √2:1 rectangle and √3:1 rectangles of the rep-mn-parallelograms, and the golden tile of the metallic parallelograms.

There are many other ways of dissecting parallelograms into similar parallelograms. A subset of these, in which a square of integer side is dissected into smaller squares also of integer side have been investigated at Erich Friedman's Math Magic pages. (When generalised to parallelograms these these become ir-reptiles with IFS { p →k_i/np + v_i : k_i, n ∈ ℤ}.) They lose the property that sides are of integer size (except from rhombi), but the ratio of areas remains the ratio of a pair of integer squares.)

• Integer Square Tilings:
• Counting Square Tilings:
• Square Strip Tilings:
• Partridge Packings:
• Hanoi and Neighbourly Square Tilings:
• No Touch Tilings:
• Anti-Partridge Tilings:
• Decreasing Square Paths:

Rep-n² Parallelograms

Any parallelogram is a rep-n² rep-tile, and tiles the plane with one copy per unit cell.

Specific classes of parallelograms are squares (all angles and sides equal), rectangles (all angles equal) and rhombi (all sides equal). Several classes of polyforms include parallelograms, including polyominos, polyplets, polyabolos (aka polytans), polyiamonds and polydrafters.

Examples

IFS

In the simplest construction of these figures all the transforms are n-fold contractions combined with translations. This may be written

T_rs: p → p/n + ra + sb, r,s ∈0..n-1, where a and b are non-colinear vectors.

These IFSs have two degrees of freedom, which means that their attractors include an uncountably infinite number of dissimilar parallelograms. The degrees of freedom may be parameterised as stretch and skew. If the archetypal IFS is made that of the square with a = 1 and b = i, then the generalised IFS can be written as a = t and b = k + i, i.e.

T_rs: p → p/n + rt + sk + si, r,s ∈0..n-1

Degeneracy

However, because of the symmetry of the attractors, the attractors are degenerate, i.e. any particular parallelogram can be produced by more than one IFS. In all cases any element of the parallelogram may be rotated by 180°, giving 2ⁿ² different IFSs. In the case of rectangles and rhombi any element may be reflected in either or both of two axes, giving 4ⁿ² different IFSs. (Reflection in both axes is the same as rotation by 180°.) In the case of squares any element may also be rotated by 90° or 270°, instead of 0° or 180°, giving 8ⁿ² different IFSs.

Furthermore some particular figures have additional IFSs; for example in the domino pairs of elements form a square, and any or all these can collectively by rotated by 90°. Thus the rep-4 domino is 16384 (4.8⁴) rather than 4096 (8⁴) -fold degenerate.

This degeneracy becomes significant when applying various algorithmic methods of generating new reptiles and irreptiles from a given reptile or irreptile.

Rep-mn Rectangles

There are also rep-mn rectangles, including two with rep- numbers less than 4.

The rep-nn rectangle is square (one of the degenerate alternatives to the rep-n² square above). The rep-nm rectangle is the same as the rep-mn rectangle, but rotated by 90°.

IFS

The simplest construction is a combination of an anticlockwise rotation by 90°, combined with contraction by √(mn) and a translation, which may be written as

T_rs: p → pe^iπ/2/√(mn) + r1 + (s-1)√(m/n)i, r ∈1..n, s ∈1..m, where 1 and i are orthogonal unit vectors.

These IFSs have no degrees of freedom, their attractors being the √n:m rectangles. These can be placed in a one-to-one correspondence with the rational numbers (m/n), and hence form a countably infinite set of attractors.

Degeneracy

Rep-mn rectangles are in general 4^mn degenerate. Some specific instances (e.g. square and domino - see above) have greater degrees of degeneracy.

Rep-mn Parallelograms

The √n:m rectangles can also be generated by a combination of reflection in the x-axis, anticlockwise rotation by 90°, contraction by √(mn) and a translation, which may be written as

T_rs: p → p̄e^iπ/2/√(mn) + (r-1)1 + (s-1)√(m/n)i, r ∈1..n, s ∈1..m, where 1 and i are orthogonal unit vectors.

These are a special case of the rep-mn parallelograms, which are sets of attractors with a single degree of freedom, which can be parameterised as the angle of the lower left corner. The IFS transforms then become

T_rs: p → p̄e^ω/2/√(mn) + (r-1)1 + (s-1)√(m/n)i, r ∈1..n, s ∈1..m, where 1 and i are orthogonal unit vectors.

rep-2 parallelogram rep-3 parallelogram rep-4 parallelogram rep-4 parallelogram rep-6 parallelogram

Mixed Orientation Parallelograms

It is also possible to generate rectangles composed of alternating columns (or rows) of rotated and unrotated elements.