A Truchet tiling is a tiling made of the same square tile repeated over and
over in a square grid with different orientations.
See
Wikipedia article. There are very many Truchet pages on the web.
Here is an example made from the tile below left.
The sequence of tiles shows a sequence of explorations leading up to
the fractal construction explained later.
Tile number and description: 0, as in
Truchet's original
investigation
change number of tiles in
tiling=
Click or on the tiling to change the orientations of the tiles in
the tiling.
Space Filling curves
The Smith Truchet tiling of quarter circles is sort of space filling - it
fills a lot of space.
But
for a mathematical space filling curve, we need a limiting process
to pass from one level of iteration to another, as for example in the
case of the Hilbert curve:
Example: Hilbert curve
Hinged tiling
The iterative process I use to get a mathematically space filling curve from
a Truchet tiling is to use a hinged tiling.
Example Hinged tiling
Hinged tiling applied to Truchet tiling
Corresponding L-system
L-systems, (Wikipedia page)
describe fractal like structres with words (strings of symbols)
describing some fractal process
and rules to transform them.
For the hinged tiling, we can label the directions of the paths L or
R for whether they turn left or right, and label h and v for crossing
horizontal or vertical tile boundaries.
Truchet tiling as image filter
The number of iterations of the process depend on the darkness
of an image. It's best to use simple images with strong contrast,
e.g., just black and white.
Use the mouse to move the control dots round the screen, or else use the
following buttons for different effects.
Reset state of center of tile (this may not produce a noticable effect
if there are
many iterations)
mouse:
Variable itteration on other deterministic fractals: Sierpinski's triangle