WORK IN PROGRESS PLEASE IGNORE FOR A WHILE
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.
In single particle reconstruction we must project over this asymmetric unit as accurately as possible. To do this we often think in terms of the unit sphere and imagine it as encapsulating the symmetric object. We isolate a single asymmetric unit and define the surface of the unit sphere we must uniformly project through to ensure the sampling of the single asymmetric unit in all of its unique orientations. To do this we could choose any of the asymmetric units, but ultimately we only need one.
This seems simple enough, but depending on the approach, we must also consider the influence of mirror or reflective symmetry which occurs when we generate a projection from the opposite side of the hemispher.
In EMAN1 it was imperative that the mirror portion