Tracks

Draws fractal curves point by point on a black screen. You enter a complex constant lambda as two numbers (real part, then imaginary part) and a scale; the larger the scale number, the smaller the picture. The program then runs the map z → λz(1−z) backwards, picking one of the two square roots at random each step, and plots every point it lands on. The points settle onto the boundary of the map’s Julia set, so a dragon- or ring-shaped outline slowly fills in. It runs until you press BREAK; the REMs warn that a picture can take an hour.

The program is saved as “Tracks”. Its REMs explain the prompts and say that the suggested input 3, 0, 3 “does make a circle as stated in the article in SYNTAX”, so it is a type-in from an article in the SYNTAX newsletter; the author is not named. This copy comes from a Larken disk of magazine type-ins.

Program Analysis

Structure

• Lines 5–8: REMs explaining the inputs.
• Lines 10–180: set the screen centre (127, 88) and a starting point, read the inputs, discard the first ten points, then loop forever: plot, iterate.
• Line 1000: complex square root of x+jy.
• Line 2000: replaces lambda with 4/lambda once, at input time.
• Line 3000: complex multiply.
• Line 4000: one backward step: multiply by 4/lambda, subtract from 1, take the root, choose its sign at random (IF RND<0.5), and halve 1 minus the root.
• Lines 7000–7060: a leftover random-number test that is never called.

Notable Techniques

• Complex arithmetic is done in separate real and imaginary variables with small subroutines, a common approach in BASIC fractal programs of the time.
• The random choice of root is the standard way to draw a Julia set by inverse iteration: the two branches together cover the whole boundary.

Potential Issues

• A small scale number with a large set can put points off the screen, which appears to stop the program with an “Integer out of range” error at PLOT.
• Lines 4020 and 4050 form 1−z as x=1-x without changing the sign of y, so for complex lambda the steps strictly work on the complex conjugate. The pictures are still Julia-type curves, but they may be mirror images of the textbook ones.
• There is no way out of the plotting loop except BREAK.

This program came from a Larken disk and was converted to run from tape: the LKDOS save and load lines and the LOGICALL load were removed, and it is saved without variables. It was tested in ZEsarUX with lambda 3, 0 at scale 3 and lambda 1, 1.5 at scale 2.