Definition
Figure and ground is a gestalt-psychology distinction between the foregrounded object of attention (the figure) and the unattended surroundings (the ground). Escher's tessellations exploit the distinction: in Mosaic II and Tiling of the Plane with Birds, the negative space is itself a figure. In Gödel, Escher, Bach Topic 3, Hofstadter introduces a mathematical analogue: when can the complement of a recursively enumerable set itself be recursively enumerable?
Why it matters
How it works
In vision, the figure/ground assignment depends on attention, prior expectations, and contour orientation. In mathematics, a set's figure/ground status is determined by whether it can be generated positively. If you have a rule that builds members directly, the set is the figure. If members are defined only as "not in a different set," the set is the ground. A set whose complement is also generatable is recursive (decidable); one whose complement is not is r.e. but undecidable.