Infinitesimally Rigid Construction of the Algebraic Numbers

In my thesis I will generalize a previous result of M. Maehara. In “Distances in a rigid unit-distance graph in the plane”1 he proved that the distances that occur between vertices in planar rigid unit-distance graphs are precisely the positive real algebraic numbers. A unit-distance graph is a framework of equal length bars which are connected in a flexible way at their endpoints. Such a framework is called rigid if it cannot be deformed without changing the length of the bars. There is also a stronger notion of rigidity which is called infinitesimal rigidity. This stronger notion asks that the framework cannot even be deformed infinitesimally without an infinitesimal change of the lengths of the bars. The picture on this page shows an example of a framework which is rigid but not infinitesimally rigid. The point with the arrows, cannot really move relative to the rest of the construction, although you can do this infinitesimally in the direction indicated by the arrows. A yet...Show moreIn my thesis I will generalize a previous result of M. Maehara. In “Distances in a rigid unit-distance graph in the plane”1 he proved that the distances that occur between vertices in planar rigid unit-distance graphs are precisely the positive real algebraic numbers. A unit-distance graph is a framework of equal length bars which are connected in a flexible way at their endpoints. Such a framework is called rigid if it cannot be deformed without changing the length of the bars. There is also a stronger notion of rigidity which is called infinitesimal rigidity. This stronger notion asks that the framework cannot even be deformed infinitesimally without an infinitesimal change of the lengths of the bars. The picture on this page shows an example of a framework which is rigid but not infinitesimally rigid. The point with the arrows, cannot really move relative to the rest of the construction, although you can do this infinitesimally in the direction indicated by the arrows. A yet unanswered question was whether Maehara’s result also holds for infinitesimally rigid frameworks. It turned out to be true even with this stronger notion of rigidity. I will prove this in my thesis by showing that Maehara’s construction is infinitesimally rigid in most cases and give a different construction for the cases where Maehara’s construction isn’t infinitesimally rigid.Show less

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Infinitesimally Rigid Construction of the Algebraic Numbers