Homeolineomers are 2 dimensional plane-tiling self-similar fractal rep-tiles in which the elements are placed at regular intervals along a straight line, and in which the elements all have the same orientation. This can be divided into the cis-homeolineomers, in which the elements are directly similar to the figure, and the trans-homeolineomers, in which the elements are inversely similar to the figure.
They may be constructed using IFSs in which the affine transformations correspond to contraction by n½ (where n is the number of elements), rotation by a characteristic angle and translation by a vector corresponding to the position of the element, preceded in the case of trans-homeolineomers by reflection in the X-axis.
The trans-homeolineomers are the rep-n parallelograms. For each integer number of elements (greater than 1) there is an uncountably infinite set of attractors with a single degree of freedom, which can be taken as the angle between the sides of the triangle. This angle is also the angle by which the elements are rotated with respect to the figure. This angle can take any value except for jp, but the attractors for -a and p-a are similar to the attractor for a.
There is but a finite number of cis-homeolineomers with any particular number of elements.
Theorem: the angle by which the components are rotated, for a cis-homolineo-n-mer, is arccos((n2.c4-1-m2.c2)/(2mc)) where c is n-0.5 (which simplifies to arccos(mc/2)).
Proof: a triangle can be drawn on the fractal connecting corresponding points. The lengths of the sides of the triangle are in the ratios nc2:mc:1. Take the equation relating the lengths of sides of a triangle and solve for the angle.
As the resulting angles are in general irrational in any units it is both inconvenient and inaccurate to use the angle to distinguish between the different cis-homolineo-n-mers. Therefore I identify the attractors by the connectivity class and index, m. For m=0 the attractor is a sqrt(n):1 or rep-n rectangle, identical in form to the 90° rep-n parallelogram, which may also be denoted as the h-homelineo-n-mer. The remaining attractors have non-intersecting fractal boundaries and are denoted as the zm-homolineo-n-mers.
All zm-homeolineo-n-mers, for n=2 to n=7 are shown below.
The number of z-homeolineomers increases with increasing n. The numbers for some values of n are given in the table below.
| n | # | n | # | n | # |
| 2 | 2 | 11 | 6 | 20 | 8 |
| 3 | 3 | 12 | 6 | 21 | 9 |
| 4 | 3 | 13 | 7 | 22 | 9 |
| 5 | 4 | 14 | 7 | 23 | 9 |
| 6 | 4 | 15 | 7 | 24 | 9 |
| 7 | 5 | 16 | 7 | 25 | 9 |
| 8 | 5 | 17 | 8 | 27 | 10 |
| 9 | 5 | 18 | 8 | 30 | 10 |
| 10 | 6 | 19 | 8 | 32 | 11 |
Conjecture: there are no additional homeolineomers.
The z2-homeolineodimer is the well known twindragon. I introduce the name diursus (think of a chorus line of dancing polar bears) for the z1-homeolineodimer.
© 2000, 2001 Stewart R. Hinsley