wythoff - Wythoff-style constructions
Usage: wythoff [options] [input_file]
Read a file in OFF format and apply a specified pattern to generate polygon
tiles. The polyhedron faces are divided by a 'meta' operation into triangles
each having vertices which are a vertex V, edge centre E and face centre F.
A start point is positioned on one of these triangles, the next point is
found by using the pattern to step between triangles, leading to a circuit.
If input_file is not given the program reads from standard input.
-h,--help this help message (run 'off_util -H help' for general help)
--version version information
-p <pat> pattern in form: [Coords0:Coords1:...]Path0,Path1...
Coordinates are barycentric, in form aVbEcF:
VEF element letters, and a,b,c are barycentric coordinates
corresponding to the following element letter. Ommiting a
an element letter and coordinate sets the coordinate to zero.
Ommitting just the coordinate sets the coordinate to 1. E.g
V = (1,0,0), VE = (1,1,0), V2E3F = (1,2,3)
Paths are in the form: TrisPidx0Move0Pidx1Move1...
Tris: one of +-* (default +) indicating that paths should
start for positive, negative or both kinds of triangles.
Pidx: an index number of a point from the coordinates list
Move: an operation for stepping to the next triangle, given
as a series of characters from the following:
_ - no move, stay on the same triangle
v,e,f - step over side opposite V,E,F
V,E,F - step two trianglesi, rotating about V,E,F,
according to: V=ef, E=fv, F=ve
Paths can start with either a move or a point, but cannot both
start and end with a move
-c <op> Conway polyhedron notation operator, or 'list' to list all
available operators with their corresponding patterns
-R reverse pattern, exchanges the signs of the start triangles
-r relabel pattern, exactly three letters VEF written in any order
e.g. EFV relabels the pattern as V->E,v->e,E->F,e->f,F->V,f->v
-M input geometry is a 'meta' tiling, don't apply meta operation
-i tiles are coloured by index number (default, colour by value)
-u output only one example of each type of tile (one per path)
-a add the 'meta'-transformed base
-f <ht> lift the face centres by this height
-q quiet, don't print report
-o <file> write output to file (default: write to standard output)
List the Conway operators for lots of examples of the constructive notation
wythoff -c list
Wythoff equivalents to |pqr, p|qr, pq|r, pqr|, for (2 3 5). The base model
schwarz_2_3_5p is already the 'meta' triangle tiling so use
wythoff -M -p [VEF]0V,0E,0F,0V0E0F schwarz_2_3_5p | antiview
wythoff -M -p [V]0E,0F schwarz_2_3_5p | antiview
wythoff -M -p [VE]0V,0E,0v0e schwarz_2_3_5p | antiview
wythoff -M -p [VEF]0v0e,0e0f,0f0v schwarz_2_3_5p | antiview
By default a polyhedron is processed by the Conway 'meta' operation
before applying the constructive notation. Here is the equivalent
of applying Conway notation 'ambo' to a cube
wythoff -p [E]0V,0F cube | antiview
If the base polyhedron is already like a meta tiling, i.e. 2-colourable
triangle faces and even order vertices, then it may be used as-is by
specifying -M. For example, the following unitle2d triangle-tiled
torus is like a 'meta' tiling, but does not correspond to a polyhedron,
and the snub operation is applied to it
unitile2d -s t -w 8 -l 24 2 | wythoff -c s -M | antiview
wythoff expands on the ideas behind the Wythoff symbol to create
a specified tiling pattern from any suitable triangular tiling.
Triangle tilings are created, by default, from the input model
by the Conway 'meta' operation. This consists of all triangles that join
a face centre to one of its vertices and a neighbouring edge centre. With
regard to starting triangles, the meta triangle (Vn, En, F) is labelled
-/black, and the meta triangle (Vn+1, En, F) is labelled +/white.
Alternatively, with option -M the input will be considered
to be a suitable triangle tiling, and must be 2-colourable and
all the vertices of even order.
The output tiling will not necesarily be planar, or have equal edge lengths.
The output of 'spherical' tilings can be processed with the canonical
In the final model, vertices are coloured according to which elements
are involved in the pattern coordinates: V=0/red, E=1/green, F=2/blue,
VE=3/yellow, EF=4/cyan, FE=5/magenta, VEF=6/grey. Faces are coloured
according to their corresponding position in the pattern.
Comparison with corresponding Conway notation operators
Conway notation is applied to polyhedra, where vertices, edges and faces
are clearly distinguishable. Constructive notation is applied to triangle
tilings, derived from polyhedra by the Conway meta operation, and in
these triangle tilings the three "element" types are equivalent.
Some Conway operators produce an edge with the same centre as an original
edge. A corresponding constructive notation operation must produce a
digon for this edge, because it is a polygon that wraps the element, and
not just the edge of an adjoining face.
As a consequence, the Conway seed corresponds to a constructive notation
'seed' operator which, when applied to a polyhedron, generates the original
polyhedron, but with digons along the edges.
As element types on the base tiling are equivalent, constructive notation
operations that can be transformed into each other by permuting the element
types can also be considered equivalent. Some equivalent Conway notation
operators expressed in constructive notation are: seed/dual/ambo,
truncate/zip/expand, kis/needle/subdivide. Some operators
are symmetric with respect to the element types, for example snub, meta
Replacement for twist program
The twist program created a model connected like a twisted
struct zig-zag tensegrity. The wythoff program can create
equivalently connected models with the following pattern.
wythoff -p [4V2E8F]0feEV0fe0E0V ico | antiview
poly_kscope - polyhedral kaleidoscope
Programs and Documentation