The relative displacements of the walls of many veins, joints, and dikes demonstrate that these structures are dilatant cracks. We infer that dilatant cracks propagate in a principal stress plane, normal to the maximum tensile or least compressive stress. Arrays of echelon crack segments appear to emerge from the peripheries of some dilatant cracks. Breakdown of a parent crack into an echelon array may be initiated by a spatial or temporal rotation of the remote principal stresses about an axis parallel to the crack propagation direction. Near the parent-crack tip, a rotation of the local principal stresses is induced in the same sense, but not necessarily through the same angle. Incipient echelon cracks form at the parent-crack tip normal to the local maximum tensile stress. Further longitudinal growth along surfaces that twist about axes parallel to the propagation direction realigns each echelon crack into a remote principal stress plane. The walls of these twisted cracks may be idealized as helicoidal surfaces. An array of helicoidal cracks sweeps out less surface area than one parent crack twisting through the same angle. Thus, many echelon cracks grow from a single parent because the work done in creating the array, as measured by its surface area, decreases as the number of cracks increases. In cross sections perpendicular to the propagation direction, echelon cracks grow laterally, each crack overlapping its neighbors, until the mechanical interaction of adjacent cracks limits this growth. Dilation of each crack pinches the tips of adjacent cracks into an asymmetrical form and introduces local stresses that can cause lateral growth along a curving, sigmoidal path. Sigmoidal echelon cracks may link at tip-to-plane intersections, leaving a step in the through-going crack wall. The geometry of dilatant echelon cracks may be used to infer spatial or temporal changes in the orientation of principal stresses in the Earth.

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