Visual illusions such as the Necker Cube or the Penrose Impossible Triangle or the Hollow Mask illusion demonstrate that the 'reality' we see consists of constrained models constructed in the brain. The Necker Cube's two-dimensional pattern of lines on paper is compatible with two alternative constructions of a three-dimensional cube, and the brain adopts the two models in turn: the alternation is palpable and its frequency can even be measured. The Penrose Triangle's lines on paper are incompatible with any real-world object. These illusions tease the brain's model-construction software, thereby revealing its existence.