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Some of the most striking features of the Magellan radar images of Venus are the intricate and varied deformation patterns on the volcanic plains. Particularly puzzling are remarkably linear features with a regular spacing of about 1 km first observed in the gridded plains of Guinevere Planitia. The very close spacing of these features requires an unreasonably thin lithospheric layer in conventional geophysical models for regular spacing. We have found many sets of parallel, regularly spaced lineations similar to the closely spaced features of the gridded plains. They are composed of parallel, thin, straight lineations whose microwave reflectivity does not appear to depend on radar illumination direction. These sets of parallel lineations, which we interpret to be fractures, typically cover areas with dimensions of hundreds of kilometers. Several examples of these regular lineations are characterized in terms of their average fracture spacing, which is between 1 and 2.5 km; the scatter in individual spacings is about ±1/3 the average. Based on observations, we hypothesize that these features are extension fractures in the brittle upper layers of the volcanic plains material, which formed well after emplacement and solidification of the flows. For parallel extension cracks, models based on the stress distribution around a crack and observations of regular jointing in sedimentary layers predict that the spacing of fractures should be roughly equal to the depth of fracturing, or to the thickness of the jointed layer. However, we observe that fracture sets in diverse locations all have nearly the same spacing, implying either that the fractures always form with a depth extent of about 1 km or that the spacing is independent of the thickness of the layer containing the fractures. We propose a model based on a shear lag mechanism used by materials scientists to explain morphologically similar patterns of regularly spaced tensile cracks on the brittle surfaces of layered composites. In this model, lithospheric extension causes a fracture to form in a surface layer of basalt several hundred meters to a kilometer thick. A horizontal detachment, corresponding to an older plains surface below the fractured layer, allows frictionally resisted lateral displacement of the surface layer in the region near the extension fracture. The model gives a spacing that is independent of layer thickness and is able to reproduce the observed fracture spacing using reasonable values of material constants. It provides a framework for using these sets of lineations to help determine mechanical properties of the volcanic plains and to place constraints on the sequence of their tectonic evolution. Another unresolved problem of the gridded plains has been the presence of an orthogonal set of brighter structures with a markedly different appearance and a somewhat larger scale of spacing. We utilize the observed morphology and self-similarity of feature lengths to determine the reasons for their contrasting morphology. It is concluded that they are primarily extensional structures which penetrate the entire 2–4 km thick brittle crustal layer and which are likely to have accommodated some shear deformation.