to_nfold  change the rotational symmetry of a polyhedron
Usage: to_nfold [options] fraction [input_file]
Generalise an axial model by changing its rotational symmetry. Read a model,
in OFF format, with an mfold rotational axis on the zaxis, and create a
new model, generally nonplanar, with the same relative connections, but
with an nfold axis instead. fraction is given as n, or n/d (n and d
integers). Vertices of a face originally separated by x/m of a turn around
the zaxis will be separated by xd/n of a turn in the final model. If
input_file is not given the program reads from standard input. to_nfold is based
on an idea by Bruce R. Gilson.
Options
h,help this help message (run 'off_util H help' for general help)
version version information
o <file> write output to file (default: write to standard output)
Increase rotational symmetry, e.g. pentagonal prism to hexagonal prism
to_nfold 6 pri5  antiview
Decrease rotational symmetry, e.g. pentagonal prism to triangular prism
to_nfold 3 pri5  antiview
Change rotational symmetry with a fraction, the denominator introduces a
step, e.g. pentagonal prism to {7/2}prism
to_nfold 7/2 pri5  antiview
Change the basic step of a model, e.g. pentagramic dipyramid to pentagonal
pyramid (2*3 = 1 mod(5))
to_nfold 5/3 dip5/2  antiview
Open an edge, taken as a digon, e.g. convert snub disphenoid to (nonJohnson)
snub square antiprism
to_nfold 4 j_snub_disphenoid  antiview
Close a polygon down to a digon, e.g. convert snub square antiprism to
(nonJohnson) snub disphenoid
to_nfold 2 j_snub_square_antiprism  antiview
Convert a polyhedral torus
unitile2d s t  off_trans R 90,0,0  to_nfold 11/2  antiview
Algorithm
First the cyclic symmetry of the model, C_{n_from}, is found.
An arbitrary sector of angular width 2π / n_from is considered, and
this sector is scaled by rotation to have angular width
2π · d_to / n_to.
This determines the new vertex positions unambiguously.
The faces are determined by considering each set of faces which are equivalent
under C_{n_from} symmetry, and then choosing a minimal chain
of edges that will repeat by this symmtry to recreate the edges of this
set of faces. The chain is transformed, edge by edge, to match the
final symmetry and step. It is then
repeated by the final symmetry into one or more circuits, which are
the final faces.
To transform each edge, the original central angle is considered
positive if the symmetry axis lies in the sweep of this edge and the last,
otherwise negative, but if the edge intersects the symmetry axis (180
degree edge) then its sign is ambiguous. In this case, if the sum of
angles with the adjoining edges is positive then the sign is taken as
negative, and vice versa. If the sum is zero, then the sign is chosen,
arbitrarily.
Issues
 If a model has faces that meet at a 180 degree edge
in the base model then they may not meet in the final model.
 Some polygons may have a face that shares the axial symmetry but
will not divide into valid edge chains, e.g. a rectangular bowtie
quadrilateral has 4 edges and 2fold symmetry, but a chain of two
of its edges cannot be repeated by 2fold symmetry to recreate the
original bow tie.
 Some models have an axial fractional polygon, or a form that can
be identified with one (e.g. a polyhedral tube might follow the
path of pentagram edges). The step of a polygon tends to be
preserved when applying to_nfold, and it may be that the
apparent final model step will divide the final n_to. Such
models are not constructible, and will lead to a broken final model,
likely with edges from a vertex onto itself,
e.g. to_nfold 3 pri7/3.
 Faces that slant and are intersected by the axis may lead to
skew polygons that wrap the axis several times. This may not
be the desired result, however, it is correct for the
algorithm. e.g. to_nfold 7 ant5/3.
to_nfold is based on an
idea by Bruce R. Gilson.
Next:
symmetro  polyhedra with regular polygons at symmetry axes
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