Abstract
In addition to the usual symmetries by reflections and rotations, abstract polytopes can have external symmetries such as self-duality and self-Petriality. In this dissertation, we describe and study a way of measuring the invariance of abstract polytopes under such external operations. We then present methods for constructing abstract polytopes with specified external symmetries. In particular, we describe how to construct polyhedra that are self-dual and self-Petrie, and how to construct polytopes that are self-dual and chiral.
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