I've always been a fan of tiled designs and my interest in them was reinvigorated when I saw Asao Tokolo's works. So, I set out to create a coloring in Ultra Fractal that could generate similar patterns. My first efforts were with elliptical arcs (generally, quarter- and half-ellipses), as they can easily be made to enter and exit a square tile perpendicular to the edge. This makes the curves smooth over the image.

Above is a real simple example using a single tile (in the center, highlighted). This tile uses four arcs, two quarter-ellipses (quarter-circles, actually, centered on the lower left corner) and two half ellipses (top and right side). By randomly rotating and flipping this pattern to the other tiles, all manners of curves and loops are formed.

Of course, more complex tilings can be created, with a bit of work. This one uses five different tile patterns, each with eight elliptical arcs.

These tilings can be combined to further expand the creative possibilities. Here is another version of the above tiling, made by joining two copies of it.

The next step in the development of the coloring formula is to go beyond ellipses to general Bezier curves. This will allow for far more flexibility in making curves, albeit at a greater cost to the designer.

Above is a real simple example using a single tile (in the center, highlighted). This tile uses four arcs, two quarter-ellipses (quarter-circles, actually, centered on the lower left corner) and two half ellipses (top and right side). By randomly rotating and flipping this pattern to the other tiles, all manners of curves and loops are formed.

Of course, more complex tilings can be created, with a bit of work. This one uses five different tile patterns, each with eight elliptical arcs.

These tilings can be combined to further expand the creative possibilities. Here is another version of the above tiling, made by joining two copies of it.

The next step in the development of the coloring formula is to go beyond ellipses to general Bezier curves. This will allow for far more flexibility in making curves, albeit at a greater cost to the designer.