To help interpret the polygonal patterned ground on Mars, we present recent findings about a similar form of patterned ground in a particularly cold and arid region on Earth, the Dry Valleys of Antarctica. In this region, distinct arrays of interconnected polygons, which we refer to herein simply as patterned ground, characterize many surfaces, reflecting a subsurface network of interconnected, subvertical wedges of sand that grow incrementally as sand progressively fills soil fractures. The fractures form initially as thermoelastic stresses arise during periods of rapid cooling of frozen ground, and they continue to open and close in response to thermal cycles. We describe the initiation and maturation of the patterned ground using data for the growth of sand wedges and for the evolution of crack patterns and microrelief over time scales ranging up to 106 years.