Dust with Line Shadows in Every Direction

The orange shapes form the first few levels in the construction of a Cantor set. As the level increases, this shape tends to a totally disconnected set, looking like a clump of dust. If you imagine shining light on it, you would expect its shadow to be full of holes. Its extraordinary property is that the shadow continuously fills a line segment; moreover, it does it from every possible direction of illumination from outside the bounding square.