The Quaternion Group as a Symmetry Group

Abstract

We briefly review the distinction between abstract groups and symmetry groups of objects, and discuss the question of which groups have appeared as the symmetry groups of physical objects. To our knowledge, the quaternion group (a beautiful group with eight elements) has not appeared in this fashion. We describe the quaternion group, both formally and intuitively, and give our strategy for representing the quaternion group as the symmetry group of a physical sculpture. arXiv:1404.6596v1 [math.HO]

Quaternionic Assessment of EEG Traces on Nervous Multidimensional Hyperspheres

“…Because the mathematical procedures have been thoroughly described by many authors, here we favour an approach for non technical readers, based on plainillustrations.The website https://quaternions.online/provides an introduction to 3D quaternion graphics. See also, for an affordable treatment of quaternions, Hart (2014). In an EEG trace (Figure 1A), every point inside the plot displaying amplitude versus time can be described by three vectors lying each one on a different spatial coordinate: x, y or z (say, e.g., green, red and blue arrows).…”

“…Quaternion trajectories in three dimensions can be depicted as arcs moving inside a three-dimensional frame encompassing the axes x, y, and z ( Figure 1C). The movement of the arc is dictated by the constraints of the axis rotation and the Euler angle (Hart, 2014). Every single point of the EEG trace can be projected to a hyperspheric manifold, either using punctiform ( Figure 1D), or uniform ( Figure 1F) orthographic projections.…”