Make a list of angles from 0 to 120 degrees in steps of 30 degrees:

This gives the sequence of points encountered by starting at {0,0} headed east, turning 0°, moving 1 unit, turning 30°, moving 1 unit, turning 60°, moving 1 unit, etc.:

Draw a line through the points along the path. This is sometimes referred to as “turtle geometry”, since you can think of it as the path taken by a turtle that follows a sequence of (turn, move) instructions:

Using a smaller increment gives a path with more segments and whose angle changes more slowly, giving a smooth spiral at the beginning:

Increasing the maximum angle gives a longer path... up to a point. Eventually this path retraces itself:

This is the code to draw a path up to a maximum angle of 1,000:

Make the code interactive by wrapping the expression with Manipulate, replacing the fixed value 1000 with the variable max, and specifying that max varies from 100 to 5000. Drag the slider to change the maximum angle:

Add a second control for the step size, and let it vary from 5 to 6:

Deploy the Manipulate to the Wolfram Cloud where anyone with a browser can use it:

Click on the link in the output to visit the site.

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