This paper seeks to bring mathematical clarity to the modelling of rigid motion origami involving quaternions and dual quaternions. We illustrate with an analysis of a single vertex Muria-ori pattern utilizing quaternions and an analysis of a multi-vertex Muria-ori pattern utilizing dual quaternions. We construct a complete lists of formulas that determine the edge vectors, normal vectors and fold angles throughout the folding transformations. In addition we provide explicit functional representations of the resulting folding transformations.
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