Rohan requested a continuous map from a 2D surface onto a non-porous patch of positive 3D volume. There's one under our noses, but maybe not where you think. We certainly have continuous maps from 1D onto 3D, so it is tempting to compose such a volume-filling function with some sort of (necessarily multivalued) 1D inverse of a planefilling function. But such inverses are wildly discontinuous, no matter which values you choose as the definition. Although 1/2+i/2 is "merely" triple point, the Hilbert "curve" is dense with quadruple points. The preimage of a sufficiently small neighborhood of such a point will comprise at least four disjoint pieces. Could the 1D to 3D map possibly smooth everything out? Maybe, but there's a much cleaner solution. SPOILER: gosper.org/3dsno.png gives a continuous mapping of an equilateral triangle onto a standard 3 simplex. —rwg