EXTRA: rotegrity  make rotegrity models
Usage: rotegrity [options] [input_file]
Read a file in OFF format containing a roughly spherical polyhedron, or
previously twisted model, and try to convert into a rotegrity. Units
keep original edge colours. If input_file is not given the program reads
from standard input.
Options
h,help this help message (run 'off_util H help' for general help)
version version information
a <type> model type: rotegrity, nexorade, for nexorade followed
by an optional comma and strut length
f <frac> fraction of length for end sections (default: 1/3)
t input model is already twisted (produced by this program or
'twist' program)
M <mthd> method of conversion from base model  twist, double, joined,
or X (default: t)
O <type> output type for units: full (face), rotegrity (3 short struts),
nexorade (long strut), Nexorade (long strut, direction vertices)
(default: full). Only 'full' output can
be used as input with option t
m <maps> a comma separated list of colour maps used to transform colour
indexes (default: rand), a part consisting of letters from
v, e, f, selects the element types to apply the map list to
(default 'vef').
c <type> colouring type: edge (base model edges), symmetry, unit
(according to shape), none (default: edge)
n <itrs> number of iterations (default 10000)
s <perc> percentage to adjust corrections on iteration (default: 98)
l <lim> minimum change of vertex distance to terminate, as negative
exponent (default: 15 giving 1e15)
z <n> status checking and reporting every n iterations, 1 for no
status (default: 1000)
q quiet, do not print status messages
T reproduce output of former 'twist' program (see Notes), f is
twist factor, O is output type, unused options silently ignored
o <file> write output to file (default: write to standard output)
Rotegrity Examples
See also,
rotegrity examples with images.
Convert a geodesic sphere into a nexorade and colour by symmetry
rotegrity c s O r geo_3_1  antiview v 0.01
Make a rotegrity with double units. It has alternate winding around base
faces and base vertices.
rotegrity c s O r M 2 geo_2_1  antiview v 0.01
Make a rotegrity with triple units. It has alternate winding around base
faces and base vertices.
rotegrity f 0.2 c s O r M 3 geo_2  antiview v 0.01
Add a central white sphere to make models easier to view
off_color v invisible e invisible f white geo_10 o sph.off
rotegrity c s O r M 2 geo_3_1  antiview v 0.01  sph.off
Class I and II geodesic spheres have edges related by mirror symmetry,
and these appear to become rotegrity units of the same length (there are
ten shorter edge lengths, and ten edge orbits in this example). The
colours of the rotegrity here are taken from the edge colours of the
base model
off_color e S geo_3_3  rotegrity  antiview v 0.01
rotegrity geo_3_3  off_report C E
off_report C O geo_3_3
The convex hull of a random collection of points will generally
jam during solution.
repel N 83 n 100000  conv_hull  rotegrity O r  antiview
A previously processsed model can be processed again by specifying by
using full face outout with rotegrity O f and rereading with
rotegrity t
zono P 10  rotegrity c s s 1 n 100 O f  rotegrity t O r  antiview v 0.02
Nexorade Examples
Convert a geodesic sphere into a nexorade and colour by symmetry
rotegrity a n c s O n geo_3_1  antiview v 0.01
Having seen the report written to the screen by the previous command,
recreate the model with the given radius (antiview e 0.0078889)
and a strut length slightly longer than the minimum
(rotegrity a n,0.45)
rotegrity a n,0.45 c s O n geo_3_1  antiview v 0.0078889 e 0.0078889
Make a nexorade with double struts.
rotegrity a n,0.45 c s O n M 2 geo_2_1  antiview v 0.0079 e 0.0079
A previously processsed model can be processed again by specifying by
using full face outout with rotegrity O f and rereading with
rotegrity t
zono P 10  rotegrity a n c s s 1 n 100 O f  rotegrity a n t O n  antiview v 0.02
There are some relevant threads on the
GeodesicHelp group:
Tensegrities, nexorades & rotegrities,
Nexorades/Rotegrities  Frequencies 1 through 3 and
New Nexorade/Rotegrity project.
The program will not solve all base models, and even models which appear
reasonable may scramble, jam or contract to a point.
Try reducing option s if a model scrambles or contracts (e.g.
s 0.1), and use a low number of iterations to see how the model
transforms before failing.
Option M selects different model types, and makes arrangements
of units that replace an original unit. Although, M d produces
attractive models it, along with M X, does not produce especially
unique models. M d is like applying the Conway join operator,
then forming a basic rotegrity, then reversing the openings corresponding
to original face centres. M X is like applying the Conway ortho
operator, then forming a basic rotegrity, then reversing the openings
corresponding to original face centres. Reviewing a number of other unit
replacement schemes revealed they were all like applying a Conway
operator, applying M t or M j, then reversing openings
corresponding to particular "centre" types in the Conway operator pattern.
To produce attractive raytraced images of rotegrity models
with the elements represented by straps, use the off2pov include
file share/pov_inc/rotegrity.inc included in
the Antiprism package. Control the strap width with
off2pov v and the starp thickness
. Example command (must use rotegrity O f
to include faces data and off2pov t no_tri to ensure that
the faces are not triangulated)
off_color e S geo_3_1  rotegrity O f  off2pov t no_tri v 0.02 i rotegrity.inc o rot_geo_3_1.pov
povray +a +p +H600 +W800 rot_geo_3_1.pov
To produce attractive raytraced images of nexorade models
with the elements represented by rods, use the off2pov include
file share/pov_inc/nexorade.inc included in the Antiprism
package. Check the solution report to choose an edge radius
and strut length. Example command, (must use rotegrity O f
to include faces data and off2pov t no_tri to ensure that
the faces are not triangulated).
off_color e S geo_3_1  rotegrity a n O f o nex_geo_3_1.off
off2pov t no_tri v 0.00789 e 0.45 i nexorade.inc o nex_geo_3_1.pov nex_geo_3_1.off
povray +a +p +H600 +W800 nex_geo_3_1.pov
Twist program compatibility
The functionality of the old twist program, now removed from Antiprism,
is temporarily available through option T.
The twisted model is like a zigzag tensegrity, but without the zigzag;
the strut and its string loop lie on a plane.
An edge of a polyhedron has two vertices on the ends and is flanked by
two faces. In the dual these faces are associated with vertices, and so
an edge can be associated with four coordinates: two polyhedron vertices
and two dual vertices.
An edge of a polyhedron can be associated, by the vertices above, with
an edge in the dual. The planes through these two edges and the centre
intersect in a line through the centre. The centre of a twist edge, the
same size as the polyhedron edge can be put on this line and the twist
edge rotated about the line by f*90 degrees. Finally, the twist
edge is translated by the proportion f between the centre of the
polyhedron edge and the centre of the dual edge.
Finally, the twist edge segment is extended to see where it intersects
the planes through its edge neighbours, then the extended edge is
scaled (using the model centre) back to the original size.
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EXTRA: mmop_origami  multimodular polyhedron origami
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