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Introduction

This page will describe how EMAN2 defines the asymmetric unit, and its associated mirror portion, for each of the symmetries. This is mostly important in the context the the projection tools (EMAN1: project3d, EMAN2: e2project3d.py) which project quasi-uniformly over the asymmetric unit for each of the symmetries, generating the reference images for particle classification.

A symmetric object is often defined by its symmetric axes and how they are arranged with respect to one another. For each symmetry one may define a set of affine transformations (i.e. rotations), which when applied to the symmetric object yield it in exactly equivalent (but unique) orientation. These will be referred to as the set of rotational symmetry operations, the total in number of which is the same as the overall symmetry of the object. For instance the 60-fold symmetric icosahedral symmetry has a total of 60 rotational symmetry operations (including the identity). Similarly the 14-fold symmetric D7 symmetry has 14 and so on.

Similar to the rotational symmetry operations, the total number of asymmetric units will match the overall symmetry of the object. The asymmetric unit of a symmetric 3D object can be defined in many ways. In one regard, it is the 3D subvolume of the entire 3D object that an asymmetric object can exist inside of, in each of the asymmetric units, in equivalent orientation, without destroying the overall symmetry of the object, that is, applying a rotational symmetry must yield the 3D object in indentical conformation. However, in cryo-EM is mostly

In EMAN1 it was imperative that the mirror portion