

Several programs have features involving symmetry:
 antiview, off2vrml, off2pov 
display symmetry elements
 poly_kscope 
repeats a model symmetrically, used to make compounds
 off_align 
repeats a model symmetrically, used to augment polyhedra
 off_trans 
aligns a model according to symmetry
 off_report 
prints symmetry information for a model
 off_color 
colours by symmetry orbit
The program option parameters are organised around the following ideas
 Symmetry group
 A set of (Euclidean) transformations that carry a
polyhedron onto itself, described in general form using
Schoenflies notation (see below) e.g. Oh, D3v.
 Full symmetry group of a polyhedron
 The set of all (Euclidean) transformations that carry a
polyhedron onto itself.
 Symmetry subgroup, or subsymmetry
 A set of transformations from a symmetry group which, considered
alone, also form a symmetry group, e.g. a cube has Oh symmetry
and has a 3fold axis corresponding to a C3v subgroup.
 Symmetry orbit of an element
 A set of equivalent elements, those elements that this element is
carried on to by a symmetry or subsymmetry of the model.
 Standard alignment of a symmetry
 A symmetry group could be aligned anywhere in the coordinate
system, but there are particular alignments that fit nicely
with the coordinate axes, and these are used as the 'standard'
alignments in Antiprism. They are a way of associating a symbol
like D3v with a fixed set of transformations.
 Conjugation subtype of a subsymmetry
 This is an integer used to distinguish subgroups which are
not carried onto each other (by conjugation) by the
transformations of the parent symmetry group. For example a
cube has a 2 fold axis through midedge and a 2fold axis
through a face centre. There is no symmetry of the cube that
carries one onto the other and so they will have different
subtype numbers. Geometrically, they look different in the cube.
 Symmetry realignment
 If you align a polyhedron with the standard set of symmetries
for its full symmetry group there is often more than one
distinct way to achieve this (a transformation not in
the symmetry group that transforms the symmetry group onto
itself). For example, if you align a cereal boxlike cuboid
naturally with the coordinate axes there are 6 possibilities
i.e. the centres of the three rectangle types can lie on any
of the axes, with 3x2x1 = 6. Possibilities for some polyhedra
are infinite e.g. the symmetry group of a pyramid does not change
when it is translated along its principal axis. The realignment
is given by a series of colon separated numbers, the first
number selects from a finite set of realignments, and the
following numbers are decimals to control rotations and
translations as follows:
 axial rotation:
1 number  degrees around principle axis
 full rotation:
3 numbers  degrees around x, y and z axes
 axial translation:i
1 number  distance to translate along principal axis
 plane translation:
2 numbers  distance to translate along two
orthogonal directions in (mirror) plane
 full translation:
3 numbers  distance to translate along x, y and z axes
 Schoenflies notation
 Used to specify symmetry groups. The standard alignments have,
preferentially, a centre (fixed point) on the origin a principal
rotational axes on the zaxis, a dihedral axis on the xaxis, a
mirror normal on the yaxis (except Cs has a mirror normal on the
zaxis). The polyhedral symmetry types have a 3fold axis on (1,1,1).
In the following list of symbols, when a type contains 'n' this must
be replaced by an integer (giving an nfold axis), and for S this
integer must be even.
 C1  identity
 Cs  mirror
 Ci  inversion
 Cn  cyclic rotational
 Cnv  cyclic rotational with vertical mirror
 Cnh  cyclic rotational with horizontal mirror
 Dn  dihedral rotational
 Dnv  dihedral rotational with vertical mirror
 Dnh  dihedral rotational with horizontal mirror
 Sn  cyclic rotational (n/2fold) with rotationreflection
 T  tetrahedral rotational
 Td  full tetrahedral
 Th  tetrahedral with inversion (pyritohedral)
 O  octahedral rotational
 Oh  full octahedral
 I  icosahedral rotational
 Ih  full icosahedral
There are two symmetry reports. off_report S s gives a general listing
of the full symmetry, subsymmetries with numer of types, realignment
possibilities and the axes with their number
off_report S s rh_cubo
The subsymmetry and realignment possibilities indicate what will be
valid in the symmetry options.
There is also a list of orbits, with the total number of orbits for
each element type, and the number of elements in each orbit.
off_report C O rh_cubo
A value of 1 for verts, edges, faces indicates the polyhedron is
repsepctively isogonal, isotoxal, isohedral.
The orbits can also be calculated for a subgroup, using the
y option
off_report y D2h C O rh_cubo
off_report y D2h,1 C O rh_cubo
These are the same orbits that are used for colouring.
The first
number indicates the number of colours that will be used, and the
second numbers are the number of elements having each colour
off_color f S rh_cubo  antiview
off_color f S,D2h rh_cubo  antiview
off_color f S,D2h,1 rh_cubo  antiview
