What is it ? :

This transformation produce tilings based on the regular tesselation of the plane by squares.

The standard square tiling is rather uninteresting. This transform can replace each sides of the tiles with a broken line. This broken line is mirrored or flipped in some definite way and quite nice tilings appear. It is exactly this idea Escher used in his well-known tesselations.
All the periodic tilings consist of blocks of four tiles repeating regularily. In one block, the tiles may differ from one another. I think I exhausted all the tilings of this type (I found 25 of them).
An "quasiperiodic" mode has also been added. Some other features are available.
 

How to start ? :

Load "Pixel" in mt.ufm. As a coloring, choose a gradient-like or fbm-like one. Load the transform. Choose "Mode" : "Drop Points". You should see a square. This is the base tile. You can now replace its sides by broken lines. To do it, select "Point 1", take the eyedropper and click near the bottom side of the square (between the two corners). To avoid problems, you shouldn't put the point too high or too low (its imaginary coordinate should be between -0.2 and 0.2. The shape should have changed. The bottom side (0,0)-(1,0) has been transformed into (0,0)-Point 1-(1,0) and the other sides have been modified accordingly. You can add more points by choosing "Point 2", then "Point 3", ... You're not obliged to use all the points (usually, two or three should be sufficient). Take care not to put them too far from the bottom side of the tile. If you want to "erase" the latest point you put, just reset it to (1,0). Once you're satisfied, go back to the "Tiling" mode, zoom out a bit and play with "Longitudinal Mode" and "Transverse Mode" to see the different tilings. You may want to use a mask (Checkerboard, for instance) in order to see the tiling better. Note that you won't get anything interesting from this transform if you leave all the points to (1,0).
 

The Parameters :

Mode : 
- Tiling : The usual mode.
- Drop Points : Use it to define the parameters "Point 1", "Point 2", ... , "Point 10".

Longitudinal Mode : 
Describes on which sides of the tiles the broken line will be mirrored. Use it to produce different tilings. The "quasiperiodic" choose this randomly and produce an quasiperiodic tiling. The tiling obtained is not periodic, but consists of a finite number of different tiles.

Transverse Mode : 
Describes on which sides of the tiles the broken line will be flipped. Works the same as "Longitudinal Mode".
The combinaison of the five periodic modes give you 25 different tesselations.

Rotation Mode : Tells how the image mapped in each shape will be rotated.

Mask :
Allow you to mask some tiles, ie assign them solid color. It's useful to give them another color or texture, for instance. There are two "strange" masks : "Selected Tiles" and "Unselected Tiles". To use them, choose the tiles you want to be set solid (or not solid) with the "Selected Tile" parameters (there are four of them). Use the eyedropper and just click or the tile you want to select.

Mapping Center\Rotation\Magnification : Center\Rotation\Magnification of the image that will be mapped on the tiles.

Shape Mask : A shape made of tiles will be assigned solid color. The shape is defined by "Shape Mask p1\p2". See how the "Pattern Rotation" parameter will affect the shape...

Shape Mask p1\p2 : To choose the size  and the location of the shape mask. Try them with the eyedropper.
 

Example :

rectiling
 

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