Don't pick sticks. Pick sides. Sides of a box. Every time you pick a direction with a stick, also set up two hyperplanes spaced appart. Effectively trying to bu

Don't pick sticks. Pick sides. Sides of a box. Every time you pick a direction with a stick, also set up two hyperplanes spaced appart. Effectively trying to build a hypercube. Once you have 1000 directions chosen (i.e. 2000 sides) you will have fully enclosed a volume. You probably won't have a perfect cube, but this trick works as long as the set of vectors you picked 'are in general position' which has probability 1.