IFS Attractors: Parallelograms

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.

Iterated Function System constructions that produce only rectangles are described here.

Examples

square
(monomino)
square rep-4 square rep-9 square rep-16 square rep-25 square
rectangle
(domino)
domino rep-4 domino rep-9 domino rep-16 domino rep-25 domino
rhombus
(diamond)
diamond rep-4 diamond rep-9 diamond rep-16 diamond rep-25 diamond
parallelogram
(diabolo)
diabolo rep-4 diabolo rep-9 diabolo rep-16 diabolo rep-25 diabolo
parallelogram
(didrafter)
didrafter rep-4 didrafter rep-9 didrafter rep-16 didrafter rep-25 didrafter

IFS

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

Trs: 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.

Trs: 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 additionally be reflected in either or both of two axes, giving 8n2 different IFSs. In the case of squares any element may also be rotated by 90° or 270°, instead of 0° or 180°, giving 16n2 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.

rep-n rectangle √2:1 (rep-2) rectangle √3:1 (rep-3) rectangle 2:1 (rep-4) rectangle √5:1 (rep-5) rectangle √6:1 (rep-6) rectangle
rep-2n rectangle (rep-4) square √3:2 (rep-6) rectangle √2:1 (rep-8) rectangle √5:2 (rep-10) rectangle √6:2 (rep-12) rectangle
rep-3n rectangle √3:2 (rep-6) rectangle (rep-9) square √4:3 (rep-12) recangle √5:3 (rep-15) rectangle √6:3 (rep-18) rectangle

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

Trs: p → peiπ/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 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

Trs: p → p̄eiπ/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-n rectangle √2:1 (rep-2) rectangle √3:1 (rep-3) rectangle 2:1 (rep-4) rectangle √5:1 (rep-5) rectangle √6:1 (rep-6) rectangle
rep-2n rectangle (rep-4) square √3:2 (rep-6) rectangle √2:1 (rep-8) rectangle √5:2 (rep-10) rectangle √6:2 (rep-12) rectangle
rep-3n rectangle √3:2 (rep-6) rectangle (rep-9) square √2:1 (rep-8) rectangle √5:2 (rep-10) rectangle √6:2 (rep-12) rectangle

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.

rep-2 parallelgram rep-2 pi/3 parallelogram tetratan
rep-3 parallelgram rep-2 pi/3 parallelogram
rep-4 parallelgram rep-2 pi/3 parallelogram
rep-4 parallelgram rep-4 pi/3 parallelogram
rep-6 parallelgram rep-6 pi/3 parallelogram

Irrep-tile Parallelograms

There are many ways in which irrep-tile parallograms can be constructed.

The simplest to take a rep-tile, and dissect one or more of the copies further.

A greater variety can be generated from the rep-n2 parallelograms (n>2). A block of k2 (k<n) parallelograms can be replaced by a single parallelogram. This can be repeated while blocks of m2 parallelograms (m≥2) remain.

n=3 square domino diamond diabolo didrafter
n=3 √3:1 rectangle √3:1 rectangle √3:1 rectangle rectangle √3:1 rectangle
n=5 rectangle rectangle
n=5 √5:1 rectangle rectangle rectangle rectangle
n=5 √5:1 rectangle rectangle rectangle rectangle

Several examples can be found on Erich Friedman's Math Magic pages.

Sources:

References

© 2007, 2009, 2014 Stewart R. Hinsley