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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 3-fold 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 mid-edge and a 2-fold 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 box-like 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 z-axis, a dihedral axis on the x-axis, a
mirror normal on the y-axis (except Cs has a mirror normal on the
z-axis). The polyhedral symmetry types have a 3-fold 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 n-fold 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/2-fold) with rotation-reflection
- 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
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