It is aesthetically pleasing to construct homeolineomers such that they are centred on the origin. However if this is not necessary, and an identical homeolineo-n-mer can be constructed by contraction by n½ towards the origin, rotation about the origin, (optional) reflection in the X-axis, and translation by vectors corresponding to any set of n equally spaced collinear points.
When constructed centred on the origin homeolineomers are degenerate, and any element can be rotated an additional kp. The degeneracy is broken when constructed with a different set of translations. When the translations are by vectors (0,0) ... (n-1,0) distinct rep-tiles are generated by rotating all but the first element by an additional p. I introduce the term caterpillar polymer for those rep-tiles corresponding to the cis-homeolineomers and concertina polymer for those corresponding to the trans-homeolineomers.
There are no additional rep-tiles for n=2; the construction above generates rotated versions of the homeolineodimers.
The concertina polymers share with the trans-homeolineomers the property that 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 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. The concertina polymers have two copies per unit cell, the 2nd copy being rotated by p with respect ot the 1st. A more compact unit cell can be obtained by adding translation by (n,0) after the rotation.
The unit cell tiling vectors are (n,0) and (n½.cosa,n½.sina). These vectors are the same as for the corresponding trans-homeolineomers, so we can deduce that, as the unit cell for a concertina polymercontains two copies of the rep-tile a concertina polymer has half the area of the corresponding trans-homeolineomers.
In the same way as a concertina polymer corresponds to a trans-homeolineomer there are caterpillar polymers corresponding to cis-homeolineomers. As the caterpillar polymers do not have c2-symmetry the caterpillar polymer corresponding to the z-m-cis-homeolineomer is not the same as that corresponding to the zm-cis-homeolineomer. However, although the potential caterpillar-m polymers do tile the plane, they are generally not simply connected; the exceptions are the caterpillar-3 trimer, caterpillar-2 trimer, caterpillar-1 polymers. (It is only conjectured that caterpillar polymers are simply connected for m>-2 for arbitrary n, but it has been shown experimentally for n=36).
Most caterpillar polymers have self-intersecting boundaries. The exceptions are the caterpillar2 trimer, caterpillar3 trimer, caterpillar2 tetramer, caterpillar3 tetramer and caterpillar4 pentamer. It is conjectured that all others have self-intersecting boundaries.
All caterpillarm-polymers, for n=3 and n=4 are shown below (with 1 and 4 unit cells).
The number of caterpillar-polymers increases with increasing n, and is usually one greater than the number of cis-homeolineo-n-mers.
© 2001 Stewart R. Hinsley