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UBC Theses and Dissertations

On rigidity of unit-bar frameworks White, Ethan Patrick


A framework in Euclidean space consists of a set of points called joints, and line segments connecting pairs of joints called bars. A framework is flexible if there exists a continuous motion of its joints such that all pairs of joints with a bar remain at a constant distance, but between at least one pair of joints not joined by a bar, the distance changes. For example, a square in the plane is not rigid since it can be deformed into a family of rhombi. This thesis is mainly concerned with infinitesimal motions. Loosely speaking, a framework is infinitesimally rigid if it does not wobble. One example is a motion of a single joint, where all other joints are unmoving, such that the movement of the one joint is perpendicular to all bars attached to it. The distances in an infinitesimal motion are preserved in the initial instant of motion. Infinitesimally rigid frameworks are rigid, and is an easier quality to verify, thereby making it a popular notion of rigidity to study among engineerings, architects, and mathematicians. We present infinitesimally rigid bipartite unit-bar frameworks in ℝ^n, and infinitesimally rigid bipartite frameworks in the plane with girth up to 12. Our constructions make use of the knight's graph; a graph such that vertices (joints) are squares of a chessboard and edges (bars) represent legal moves of the knight. We show that copies of the knight's graph can be assembled to create infinitesimally rigid frameworks in any dimension. Our constructions answer questions of Hiroshi Maehara.

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