Z-Pentamers

There are 5 element IFSs which are not known to be related to any others. They bear a similar relationship to the Z and S pentominos that the cyclotetramermonomer (red cross dragon, Mandelbrot quintet) bears to the X pentomino, i.e. the centres of the elements correspond to the centres of the pentomino. They also share with the cyclotetramermonomer the property that the elements are rotated by arctan(0.5) or 90+arctan(0.5).

They have c2 symmetry. Both figures have a similarity dimension of 2, and tile the plane.

The trans technique, does not, in this case, result in additional fractals with a uniform measure.

The paired element technique can be used to generate additional fractals from the above. The technique can be applied to either of both of the two available pairs. When applied to both pairs the resulting pairs can be one the same or opposite sides of the centre. This gives 4 possibilities, which is multiplied by 2 as the central element can be left unrotated, or rotated by 180°, and by 2 again for the 2 different base fractals. Of the 16 possibilities 7 are not connected, leaving 9 new 2D polymers. However all 16 possibilities can be combined with themselves to regenerate the parent figure, and therefore tile the plane.

Source: All are independently discovered. They were discovered while investigating the cyclotetramermonomer, as a result of a fortuitious bug in the IFS display program being used.

References:

© 2000 Stewart R. Hinsley