A lineotrimer is a fractal consisting of 3 congruent elements, laid out in a line. There are 5 homeolineotrimers, in which the 3 elements have identical orientation. Other lineotrimers, i.e. heterolineotrimers, do not have all elements with the same orientation.
Unlike the homeolineotrimers I do not have a construction which allows all heterolineotrimers to be identified directly.
Conjecture: All heterolineotrimers are bar hextals, implying that the elements are rotated by one or more of 30°, 90°, 150°, 210°, 270° and 330°.
There are 216 distinct combinations of angle of rotation for bar hextals. Each element can also be directly or inversely, similar to the fractal. This gives 1728 distinct IFSs to be considered. Symmetry considerations reduce this fourfold to 432. The number of distinct bar hextals is less that this, as those fractals in which all 3 elements are identically oriented, or in which the outer two elements are identically oriented, have greater than fourfold degeneracy.
The rep-tiles shown on this page have been identified by visual inspection. This is not a reliable means of identifying rep-tiles, as in some case it is necessary to zoom in to a large degree to identify that the figure is a rep-tile. (The cases where the figure has a non-intersecting boundary are clear, but many of those with intersecting boundaries are not.)
The best known heterolineotrimer is Knuth's terdragon, which is the 30,90,30-lineotrimer. The terdragon is 1/3rd of the fudgeflake, so the plane can be tiled with the terdragon, by using 3 terdragons to make a fudgeflake, and then tiling the plane with the fudgeflake.
The 90',30',90'-lineotrimer, to which is I give the trivial name of z-terjig, bears the same relationship to the trans-fudgeflake as the terdragon does to the cis-fudgeflake.. This also tiles the plane, in the same way.
There are other c2-symmetric bar hextals. Reflecting the outside elements of the terdragon in the X-axis results in the 30',90,30'-lineotrimer, for which I introduce the trivial name of short terbolt (it is reminiscent of a lightning bolt). This tiles the plane, with 2 copies in the unit cell.
Similarly reflecting the outside elements of the z-terjig in the X-axis results in the 90,30',90-lineotrimer, for which I introduce the trivial name of w-terjig. This also tiles the plane, also with 2 copies in the unit cell.
Increasing the rotation of the central element of the terdragon to 150° results in the 30,150,30-lineotrimer, for which I introduce the trivial name of tergriffin. This tiles the plane, with 3 copies in a unit cell, with hexagonal symmetry. (It is not clear on inspection that the unit cell contains 3 copies, but plotting 7 unit cells arranged hexagonally clarifies this.)
Reflecting the central element of the tergriffin in the X-axis results in the 30,150,30-lineotrimer, for which I introduce the trivial name of terlock (with connotations of morlock - it's pretty monstrous, or should that be monstrously pretty - and padlock). This also tiles with plane, with 6 copies in the unit cell.
The other two fractals with angles of 30° and 150° are the 30',150',30'-lineotrimer (long terbolt) and 30',150,30'-lineotrimer (scroll trimer). The long terbolt tiles the plane with 3 copies per unit cell, though it is necessary to plot several unit cells for this to be visually convincing.
The scroll trimer also tiles the plane, but I have not identified the unit cell. It is possible to lay out a cluster of 6 copies with a common origin, but this cluster, although it fills in a region of the plane at the centre of the cluster, does not seem to form a unit cell.
It is estimated that there are another 13 bar hextals, without any rotational or reflectional symmetry.
The 30,210,330-lineotrimer (z-half terdragon) is the only "half" lineotrimer which occurs as a bar hextal. As it is a half of a terdragon and a terdragon is a half of a fudgeflake, it tiles the plane with 6 copies in the unit cell.
The 30,30,150-lineotrimer (filigree trimer I) also tiles the plane. I have not identified the unit cell, but suspect that it contains 4 or 8 copies.
Sources: the terdragon was reverse engineered from an image found on a web site. The remainder were independently discovered. R. William Gosper has independently discovered the z-terjig (which he calls the hourglass) and the zhalf-terdragon
data file trihextal.xil; generator hextal.pl; images generated at 6x5 per screen at 1280x1024
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