It looks like a maze, and it is. Basically, it's a 6-D hypercube, but that isn't really necessary to solve it. The path from the start to the goal is:
East, East, In, Zig, Zig, In, Down, Forth, West, West, Forth, Zag, Back, East, East, Up, Out, Out, South, West, In, In, East, Up, West, Back, West, Forth, Zag, Forth, East, East, Down, South, Out, West, West, Out (38 steps)
As the text says, getting to the goal isn't the important part. Instead, if you map out the motions made by the black dots on the bottom part of the applet, you will find the following pattern:
+-+-+ +-+-+ +-+-+ +-+-+ +-+-+ | | | | | | | | + + + +-+-+ . + . +-+-+ +-+-+ | | | | | | | | + + + +-+-+ . + . +-+-+ + +-+
So the answer is METER.
Archivist's note: This puzzle had also had a trained answer, as described in the training page. The flavor text mentions "knitting her brow", so we apply the answer from Knitting, USE THIS SCRIPT, to get NEXUS.