Much of Utopia Planitia is underlain by polygonal terrane. Most of the troughs defining the polygons are 250 m to 1 km wide; typical diameters of polygons are 4--5 km. These dimensions are orders of magnitude greater than the corresponding dimensions for familiar polygonal structures on Earth; consequently, the Earth structures are not valid mechanical analogues. Crater and superposition relationships indicate that trough formation was coeval with deposition of the polygonal terrane material. Based on crater age correlations with Elysium volcanism and outflow-channel activity, either a volocanic or a sedimentary source for this material is possible. Tensile stresses responsible for the troughs were produced by cooling or dessication shrinkage plus differential compaction over a rough buried surface that is the structurally disrupted remnant of the cratered highlands. Shrinkage alone will produce polygons orders-of-magnitude smaller than those present, but differential compaction will reflect the scale and pattern of the buried topography, and thus control the scale and pattern of the giant polygons of Utopia.