Fractal Space Filling Curves
Exquisitely convoluted, space filling curves, are created by very simple
recursive procedures.
These two famous examples were invented by
David Hilbert (left, the Hilbert curve), and Wraclaw Sierpinski (right).
The shapes these curves fill can be subdivided into a pattern of tiles,
where each tile is similar in shape to the whole.
Each tile subdivides again until you get down to the most basic shape,
which is just a few lines linked so as to traverse the pattern of tiles.
In this example, each more complicated version of the curve is made by
replacing each colored portion with a copy of the whole curve
(perhaps mirrored or rotated).
These three curves are based on the same lattice, or pattern of tiles, as
the one above.
The first (left) has long strait lines and is not a very smooth fill.
The second (middle) is full of holes and so is more of a gasket than a
fill.
The last (right) is a true fill curve.
Square tiles fit together to form a rectangular lattice.
Five tiles fit together to form a cross, these cross shapes also form a
rectangular lattice.
Five crosses fit together to form a larger cross like shape, these cross
like shapes also form a rectangular lattice.
Five cross like shapes fit together to form another cross like shape...
The fill curve based on the above pattern should be familiar by now,
its the background pattern for all of these pages.
One of my favorites.
This right triangle subdivides into a pattern of four smaller triangles.
Another right triangle this one divides into nine tiles.
Here you can see how higher order curves are created by inserting the next
lower order curve between each line segment of the lowest order curve.
Another favorite, this one by Gosper is one of the most convoluted,
yet one of the smoothest fills I know of.
This fill is based on hexagonal tiles in a honeycomb lattice.
Seven hexagons together form a six pointed shape, these in turn fit
together to form a honeycomb lattice...
The Sierpinski curve is based on a triangle divided into two tiles.
distorting it and making one tile smaller than the other,
makes an unusual adjustable fill curve.
I also have an interactive version of this
Mutant Sierpinski Curve
This trapezoid filling curve is based on two interleaved tile connection
patterns.
The curve on the right starts and ends on opposite corners.
The one on the left starts and ends on the same corner.
Each is made of three from the right (green, yellow & blue tiles),
and one from the left (red tiles).
These are just a few of the many such curves known,
many more are no doubt possible.
All of these images were created using turtle graphics in
PowerLogo.
Most can also be made using L-systems,
the
lsystems
directory in the
Spanky Fractal Database
contains several Fractint L-systems files,
including many interesting space filling curves.
William McWorter and I converted many of my fractal curves to L-systems for
Fractint, these can be found in the file
TEACHOUT.L.
Also at Spanky is a
Fractint L-Systems Tutorial
by William McWorter, including a section on
Space Filling Curves.