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The goal of this paper is to investigate a new shape analysis method based on randomized cuts of 3D surface meshes. The general strategy is to generate a random set of mesh segmentations and then to measure how often each edge of the mesh lies on a segmentation boundary in the randomized set. The resulting "partition function" defined on edges provides a continuous measure of where natural part boundaries occur in a mesh, and the set of "most consistent cuts" provides a stable list of global shape features. The paper describes methods for generating random distributions of mesh segmentations, studies sensitivity of the resulting partition functions to noise, tessellation, pose, and intra-class shape variations, and investigates applications in mesh visualization, segmentation, deformation, and registration.