Algorithms and Data Structures

N.Wirth. Algorithms and Data Structures. Oberon version 105
x0 := x0 + (u DIV 2); y0 := y0 + (u DIV 2);
Draw.Set(x0, y0);
A(i)
UNTIL i = n
END Hilbert.
Fig. 3.4. Hilbert curves H 1 … H 5 .
A similar but slightly more complex and aesthetically more sophisticated example is shown in Fig. 3.6.
This pattern is again obtained by superimposing several curves, two of which are shown in Fig. 3.5. S i is
called the Sierpinski curve of order i. What is its recursion scheme? One is tempted to single out the leaf
S 1 as a basic building block, possibly with one edge left off. But this does not lead to a solution. The
principal difference between Sierpinski curves and Hilbert curves is that Sierpinski curves are closed
(without crossovers). This implies that the basic recursion scheme must be an open curve and that the four
parts are connected by links not belonging to the recusion pattern itself. Indeed, these links consist of the
four straight lines in the outermost four corners, drawn with thicker lines in Fig. 3.5. They may be regarded
as belonging to a non-empty initial curve S 0 , which is a square standing on one corner. Now the recursion
schema is readily established. The four constituent patterns are again denoted by A, B, C and D, and the
connecting lines are drawn explicitly. Notice that the four recursion patterns are indeed identical except for
90 degree rotations.