conway  Conway Notation transformations
Usage: conway [options] [Conway Notation string] [input_file]
Conway Notation uses algorithms by George W. Hart (http://www.georgehart.com)
http://www.georgehart.com/virtualpolyhedra/conway_notation.html
Antiprism Extensions: Further operations added. See
http://en.wikipedia.org/wiki/Conway_polyhedron_notation
and
http://antitile.readthedocs.io/en/latest/conway.html
Read a polyhedron from a file in OFF format.
If input_file is not given and no seed polyhedron is given in the notation
string then the program reads from standard input.
Options
h,help this help message (run 'off_util H help' for general help)
version version information
H Conway Notation detailed help. seeds and operator descriptions
s apply Conway Notation string substitutions
g use George Hart algorithms (sets s)
c <op=s> user defined operation strings in the form of op,string
op can be any operation letter not currently in use
two character operations can be as alpha# e.g z#
string can be any operations. More than one <op=s> can be used
Examples: c x=kt,y=tk,a#=dwd or c x=kt c y=tk c a#=dwd
t tile mode. when input is a 2D tiling. unsets g
set if seed of Z is detected
r execute operations in reverse order (left to right)
u make final product be averge unit edge length
v verbose output
i <itrs> maximum planarize iterations. 1 for unlimited (default: 1000)
WARNING: unstable models may not finish unless i is set
l <lim> minimum distance change to terminate planarization, as negative
exponent (default: 12 giving 1e12)
WARNING: high values can cause nonterminal behaviour. Use i
z <nums> number of iterations between status reports (implies termination
check) (0 for final report only, 1 for no report), optionally
followed by a comma and the number of iterations between
termination checks (0 for report checks only) (default: 1,1)
o <file> write output to file (default: write to standard output)
Coloring Options (run 'off_util H color' for help on color formats)
V <col> vertex color (default: gold)
E <col> edge color (default: lightgray)
f <mthd> mthd is face coloring method using color in map (default: n)
keyword: none  sets no color
n  by number of sides
s  symmetric coloring
u  unique coloring
o  newly created faces by operation
w  resolve color indexes (overrides V and E)
C <mthd> colouring method for tiles, method can be
none  do not colour tiles
index  use the path index, colour with index numbers
value  use the path index, colour with colour values (default)
association  colour tiles using corresponding base element
colour, optionally followed by a comma and a letter from V,
E, F, or a colour, to colour local tiles by that element type
(default: F), or colour all local tiles with a single colour
(when f w is set)
T <tran> face transparency. valid range from 0 (invisible) to 255 (opaque)
m <maps> color maps for faces to be tried in turn (default: m1, for g, m2)
keyword m1: red,darkorange1,yellow,darkgreen,cyan,blue,magenta,
white,gray50,black
keyword m2: red,blue,green,yellow,brown,magenta,purple,grue,
gray,orange (from George Hart's original applet)
Make truncated cuboctahedron and pass to poly_form to make unit edges
conway dmO  poly_form a r  antiview
A truncated octahedron passed to canonical
conway tk4Y4  canonical  antiview
A nice spiral pattern in Hart mode using operation substitutions
conway eesD g  antiview v 0.01
A snub pentagrammic antiprism
conway s ant5/2  antiview
A snub geodesic sphere
conway s geo_3  antiview
opposite_lace operation on the dual of a geodesic sphere
conway Gd geo_3  antiview
whirl operation repeated twice on a dodecahedron seed and colored radially
conway w^2D  off_color_radial  antiview x ve
The lace operation subscript 2 applied to triangular faces of a cuboctahedron
conway L_2:3aC  antiview v 0.01
An interesting result with what looks like star polygons
conway L_2gC f w  antiview v 0.01
conway was written by
Roger Kaufman.
It uses algorithms by George W. Hart,
http://www.georgehart.com/.
The Conway Notation algorithms were adapted from the
Javascript on George Hart's
Conway Notation page.
Antiprism Extensions: Further operations added. See
and
Canonicalization and planarization may not always converge on
a convex polyhedron.
George Hart has a page on
canonicalization.
The 'Mathematica' algorithms have been written to follow his
Mathematica implementation
The following extended help for the program may be displayed with
conway H
Conway Notation was described by Mathematician John Conway to George Hart in
the late 1990's for a book they planned to coauthor. Due to an illness the book
never came to fruition and John Conway did not think there was enough a for a
separate publication. Conway gave George Hart permission to present it in
"Sculpture Based on Propellerized Polyhedra in the Proceedings of MOSAIC 2000"
The paper can be viewed here: http://www.georgehart.com/propello/propello.html
The project was expected to encourage more operations to be developed which has
happened in various places including here at Antiprism. (www.antiprism.com)
The following is a description of Conway Notation edited from the Conway
Notation web page by George W. Hart (http://www.georgehart.com)
More detailed information and examples can be found at
http://www.georgehart.com/virtualpolyhedra/conway_notation.html
and at
http://en.wikipedia.org/wiki/Conway_polyhedron_notation
and at
http://antitile.readthedocs.io/en/latest/conway.html
Basics: In this notation, one specifies a "seed" polyhedron with a capital
letter. Operations to perform on any polyhedron are specified with lowercase
letters preceding it. This program contains a small set of seeds and operators
from which an infinite number of derived polyhedra can be generated.
Note: This C++ port of Conway Notation can also operate on OFF files from
standard input if the seed polyhedron is not specified. (Antiprism Extension)
Seeds: The platonic solids are denoted T, O, C, I, and D, according to their
first letter. Other polyhedra which are implemented here include prisms, Pn,
antiprisms, An, and pyramids, Yn, where n is a number (3 or greater) which you
specify to indicate the size of the base you want, e.g., Y3=T, P4=C, and A3=O.
(Antiprism Extension: note that more seeds have since been defined)
Operations: Currently, abdegjkmoprst are defined. They are motivated by the
operations needed to create the Archimedean solids and their duals from the
platonic solids. Try each on a cube:
(Antiprism Extension: note that more operations have since been defined)
a = ambo The ambo operation can be thought of as truncating to the edge
midpoints. It produces a polyhedron, aX, with one vertex for each edge of X.
There is one face for each face of X and one face for each vertex of X.
Notice that for any X, the vertices of aX are all 4fold, and that aX=adX.
If two mutually dual polyhedra are in "dual position", with all edges tangent
to a common sphere, the ambo of either is their intersection. For example
aC=aO is the cuboctahedron.
Note: ambo is also known as "rectifying" the polyhedron, or rectification
b = bevel The bevel operation can be defined by bX=taX. bC is the truncated
cuboctahedron. (Antiprism Extension: or "bn" where n is 1 or greater)
Note: bevel is also known as "omnitruncating" the polyhedron, or omnitruncation
d = dual The dual of a polyhedron has a vertex for each face, and a face for
each vertex, of the original polyhedron, e.g., dC=O. Duality is an operation
of order two, meaning for any polyhedron X, ddX=X, e.g., ddC=dO=C.
e = expand This is Mrs. Stott's expansion operation. Each face of X is
separated from all its neighbors and reconnected with a new 4sided face,
corresponding to an edge of X. An ngon is then added to connect the 4sided
faces at each nfold vertex. For example, eC is the rhombicuboctahedron. It
turns out that eX=aaX and so eX=edX (Antiprism Extension: One subscript as "en"
or "e_n" where n is 0 or greater. Two subscripts as "en_m" or "e_n_m" where
n and m are 0 or greater)
Note: expand is also known as "cantellating" the polyhedron, or cantellation
g = gyro The dual operation to s is g. gX=dsdX=dsX, with all 5sided faces.
The gyrocube, gC=gO="pentagonal icositetrahedron", is dual to the snub cube.
g is like k but with the new edges connecting the face centers to the 1/3
points on the edges rather than the vertices. (Antiprism Extension: or "gn"
where n is 1 or greater)
j = join The join operator is dual to ambo, so jX=dadX=daX. jX is like kX
without the original edges of X. It produces a polyhedron with one 4sided
face for each edge of X. For example, jC=jO is the rhombic dodecahedron.
k = kis All faces are processed or kr = just rsided faces are processed
The kis operation divides each nsided face into n triangles. A new vertex is
added in the center of each face, e.g., the kiscube, kC, has 24 triangular
faces. The k operator is dual to t, meaning kX=dtdX.
m = meta Dual to b, mX=dbX=kjX. mC has 48 triangular faces. m is like k
and o combined; new edges connect new vertices at the face centers to the old
vertices and new vertices at the edge midpoints. mX=mdX. mC is the
"hexakis octahedron". (Antiprism Extension: or "mn" where n is 1 or greater)
o = ortho Dual to e, oX=deX=jjX. oC is the trapezoidal icositetrahedron, with
24 kiteshaped faces. oX has the effect of putting new vertices in the middle
of each face of X and connecting them, with new edges, to the edge midpoints of
X. (Antiprism Extension: One subscript as "on" or "o_n" where n is 0 or greater
Two subscripts as "on_m" or "o_n_m" where n and m are 0 or greater)
p = propeller Makes each ngon face into a "propeller" of an ngon
surrounded by n quadrilaterals, e.g., pT is the tetrahedrally stellated
icosahedron. Try pkD and pt6kT. p is a selfdual operation, i.e., dpdX=pX and
dpX=pdX, and p also commutes with a and j, i.e., paX=apX. (This and the next
are extensions were added by George Hart and not specified by Conway)
r = reflect Changes a lefthanded solid to right handed, or vice versa, but
has no effect on a reflexible solid. So rC=C, but compare sC and rsC.
s = snub The snub operation produces the snub cube, sC, from C. It can be
thought of as eC followed by the operation of slicing each of the new 4fold
faces along a diagonal into two triangles. With a consistent handedness to
these cuts, all the vertices of sX are 5fold. Note that sX=sdX.
(Antiprism Extension: or "sn" where n is 1 or greater)
t = truncate All faces are processed or tr = just rsided faces are processed
Truncating a polyhedron cuts off each vertex, producing a new nsided face for
each nfold vertex. The faces of the original polyhedron still appear, but
have twice as many sides, e.g., the tC has six octagonal sides corresponding to
the six squares of the C, and eight triangles corresponding to the cube's eight
vertices.
Antiprism Extension: Further operations added. Also see
http://en.wikipedia.org/wiki/Conway_polyhedron_notation
c = chamfer New hexagonal faces are added in place of edges
B = bowtie Bowtie like triangles divide pentagonal faces
E = ethyl like expand but triangles are divided into 3 kites
G = oppositelace Similar to lace, except with new quad faces split opposite
L_1. (has also been referred to as L_1, not supported)
J = joinedmedial Like medial but new rhombic faces in place of original edges
K = stake Subdivide faces with central quads, and triangles
All faces processed or can be "Kr" where r is 3 or greater
L_0 = joinedlace Similar to lace, except with new quad faces produced in L_1
are not split
L = lace An augmentation of each face by an antiprism, adding a twist
smaller copy of each face, and triangles between
Subscript as "L_n" where n is 0 or greater
All faces processed or can be "L:r" where r is 3 or greater
Both may be specified as "L_n:r"
l = loft An augmentation of each face by prism, adding a smaller copy of
each face with trapezoids between the inner and outer ones
Subscript as "l_n" where n is 0 or greater
All faces processed or can be "l:r" where r is 3 or greater
Both may be specified as "l_n:r"
M = medial Similar to meta except no diagonal edges added, creating quad
faces. Can be "Mn" where n is 1 or greater
n = needle Dual of truncation, triangulate with 2 triangles across every
edge. This bisect faces across all vertices and edges, while
removing original edges
q = quinto ortho followed by truncation of vertices centered on original
faces. This create 2 new pentagons for every original edge
it effectively lines the original faces with pentagons
Lei Williams called this operation "Pental"
S = seed Seed form
u = subdivide Ambo while retaining original vertices. Similar to Loop
subdivision surface for triangle face
One subscript as "un" or "u_n" where n is 1 or greater
Two subscript as "un_m" or "u_n_m" where n and m are 1 or greater
W = waffle Truncation on all vertices and then all faces split into sections
w = whirl Gyro followed by truncation of vertices centered on original
faces. This create 2 new hexagons for every original edge
it effectively lines the original faces with hexagons
X = cross Combination of kis and subdivide operation. Original edges are
divided in half, with triangle and quad faces
Can be "Xn" where n is 1 or greater
z = zip Dual of kis or truncation of the dual. This create new edges
perpendicular to original edges, a truncation beyond "ambo" with
new edges "zipped" between original faces. It is also called
bitruncation
Orientation of the input model will have an effect on chiral operations such as
snub or whirl. The orientation mode is set to positive by default. Operations
have been added to control orientation mode. The mode will remain until changed
+ (plus sign) = positive orientation  (minus sign) = negative orientation
Changing orientation mode can be placed anywhere in the operation string
Summary of operators which can take n as subscript or r as face/vertex number
b  n may be 1 or greater (default: 2)
e  n,m n and m may be 0 or greater except for _0 and _0_0 (default: _2_0)
g  n may be 1 or greater (default: 1)
K  r may be 3 or greater representing face sides
k  r may be 3 or greater representing face sides
L  n,r n may be 0 or greater, r may be 3 or greater
l  n,r n may be 0 or greater, r may be 3 or greater
both n and r may be used together as L_n:r or l_n:r (L_0 may not have r)
without delimiters Lr and lr, r is face sides and subscript default to 1
M  n may be 1 or greater (default: 2)
m  n may be 1 or greater (default: 2)
o  n,m n and m may be 0 or greater except for _0 and _0_0 (default: _2_0)
s  n may be 1 or greater (default: 2)
t  r may be 2 or greater representing vertex connections (2 in tiles)
u  n,m n and m may be 0 or greater except for _0 and _0_0 (default: _2_0)
X  n may be 1 or greater (default: 2)
Antiprism Extension: any operation can be repeated N time by following it with
the superscript symbol ^ and a number greater than 0. Examples: a^3C M0^2T
Seeds which require a number n, 3 or greater
P  Prism
A  Antiprism
Y  Pyramid
Z  Polygon (Antiprism Extension)
R  Random Convex Polyhedron (Antiprism Extension)
Note: Antiprism Extensions will work on tilings. Hart algorithms (d) will not
e.g.: unitile2d 3  conway p t  antiview v 0.1 (t for tile mode)
Regular 2D tilings can be constructed from base polygons. The basic tilings are
One Layer Two Layers Three Layers...
Square: oZ4 o2Z4 o3Z4
Hexagonal: tkZ6 ctkZ6 cctkZ6
Triangular: ktkZ6 kctkZ6 kcctkZ6 (kis operation on Hexagonal)
Name Vertex Fig Op String Dual Name String
Square 4,4,4,4 oZ4 Square do2Z4
Truncated Square 4,8,8 trunc toZ4 Tetrakis Square dto2Z4
Snub Square 3,3,4,3,4 snub soZ4 Cairo Pentagonal dso2Z4
Triangular 3,3,3,3,3,3 kis ktkZ6 Hexagonal ddctkZ6
Hexagonal 6,6,6 tkZ6 Triangular dctkZ6
Trihexagonal 3,6,3,6 ambo atkZ6 Rhombille dactkZ6
Snub Trihexagonal 3,3,3,3,6 snub stkZ6 Floret Pentagonal dsctkZ6
Truncated Hexagonal 3,12,12 trunc ttkZ6 Triakis triangular dtctkZ6
Rhombitrihexagonal 3,4,6,4 expand etkZ6 Deltoidal Trihexagonal dectkZ6
Truncated Trihexagonal 4,6,12 bevel btkZ6 Kisrhombille dbctkZ6
Elongated Triangular 3,3,3,4,4 NonWythoffian Prismatic Triangular none
Substitutions used by George Hart algorithms
P4 > C o > jj dd > gT > D dD > I
A3 > O m > kj ad > a aT > O aO > aC
Y3 > T t > dk gd > g dC > O aI > aD
e > aa j > dad aY > A dO > C gO > gC
b > ta s > dgd dT > T dI > D gI > gD
Equivalent Operations (Antiprism)
b1 = z e1 = d M1 = o m1 = k o1 = S
b2 = b e2 = e M2 = M m2 = m o2 = o
s1 = d u1 = S X1 = k
s2 = s u2 = u X2 = X
Equal but opposite handed: e_n_m = e_m_n, o_n_m = o_m_n, u_n_m = u_m_n
Next:
wythoff  Wythoffstyle constructions
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Programs and Documentation
