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 <elms> 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
-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
-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
Models may be coloured using the colours from the base model, in this example
the faces of a dodecahedron are coloured randomly, the model is processed
with Conway operations p and o4, and coloured by element
association. The final model shows patches of colour corresponding to
the original coloured faces.
off_color -f U dod | wythoff -c p -C a | wythoff -c o_4 -C a | off_util -S | antiview -x ev
The associated element colouring associates local tiles to the
corresponding face element by default, but they can also be
associated with vertices, edges or faces, or coloured with a colour
off_color cube -v S,Ci -e grey | wythoff -c o_4 -C a,v | antiview
off_color cube -e S,Ci | wythoff -c o_4 -C a,e | antiview
off_color cube -f S,Ci -e grey | wythoff -c o_4 -C a,f | antiview
off_color cube -f S,Ci -e grey | wythoff -c o_4 -C a,white | 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
Two colouring types can be selected for the final model with
option -C. A pattern-based colouring is the default:
vertices are coloured according to which elements
are involved in their 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 the position of their corresponding path in the pattern.
The other colouring is by associating final vertices and faces
with elements in the base model (see below), and inheriting their colours.
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
When the pattern paths are repeated to generate final faces, these
faces may be associated with elements in the base model. A path may:
cycle a base vertex, edge centre or face centre;
not cycle(the path is the whole face);
repeat by reflection; or "wander" around
the model (the path doesn't cycle locally). The behaviour can be determined
from the path specifier.
To analyse a path, write its reflections as a sequence, and remove
sequential repeated reflections. It will then be in
the form (step)association(step_back), where step is a
series of mirrors (maybe none) and step_back is the same
series of mirrors in reverse. The series of mirrors in association
(which may be none) then determines the kind of element associated with the
path, for eample it may be a rotation about an element of a type, etc.
The step is the reflections to move from from the starting
meta triangle to the meta triangle that contains the associated element.
As a example, the patternd fo truncate is [VE]0V0E,0V,0E.
Write down the first path, remove the point specifiers, convert to
reflections, and remove sequential repeats:
0V0E -> VE = (ef)(fv) = effv = ev this gives no
step, and the association is ev, which is a rotation
about a face centre (a reverse direction F).
The final face will therefore cycle the face element of its starting
meta triangle. Similarly, the second path will cycle the vertex element
on the starting meta triangle, and the third path will cycle the
For the purposes of colouring, vertices are associated with
base elements as follows: a final vertex
is associated with a coinciding base vertex, or an edge if it coincides
with an edge centre, otherwise the face element of the meta triangle it
Repalcement for to_nfold
The to_nfold program has some similarities with wythoff,
and automatically infers the tiling pattern from the base model,
but only works for cyclic models. Its handing of oblique polygons
that wind the axis will not allow it to produce series like the
On the other hand, there is no way for a wythoff pattern to specify
a cycle like 0(N/2 steps of)V1(N/2-1 steps of)V0V, which could
draw a triangle that cycles any axis at an oblique angle, because the
steps that need to be specified are relative to the N of the axis.
A solution is to produce a wythoff pattern that is parameterised
by N/D, and apply this to a schwarz_N_2_2p meta tiling.
The following example shows how this technique can be applied to
generate the crossed antiprisms.
Display a 5/2 crossed antiprism, along with a suitable schwarz tiling to
represent the base meta-tiling
polygon ant 5/3 | off_trans -S 2 | antiview - schwarz_5_2_2p
Work out a pattern that generates the crossed antiprism for the
meta-tiling (the -r option is to make the vertex elements lie on
the axis, which is perhaps easier to follow)
wythoff -M schwarz_5_2_2p -r FVE -p [VEF]0VV,0VVE0EVVE0EV | antiview
The first cycle is a fixed number of V's for the polygon step (2):
VV = (D)V
In the second cycle: the first step is relative to N and D (slant "down")
VVE = ((N-D+1)/2)V
; the second step is relative to D (horizontal)
EVVE = E(D)VE
; the third step is relative to N and D (slant "up")
EV = E((N-D-1)/2)V .
The full parametrised pattern is:
To generate an 11/4 crossed antiprism, which has Dv symmetry like the
5/2 crossed antiprism, N=11 and D=4, giving the following pattern
[VEF]0(4)V,0((11-4+1)/2)VE0E(4)VE,0((11-4-1)/2)V = [VEF]0VVVV,0VVVVE0EVVVVE0VVV
Construct the 11/4 crossed antiprism
wythoff -M schwarz_11_2_2p -r FVE -p [VEF]0VVVV,0VVVVE0EVVVVE0EVVV | antiview
For crossed antiprisms with Dh symmetry the parameterised pattern is
different, and could be determined to be, e.g.
Here is the pattern for the 13/5 crossed antiprism, found by plugging N=13
and D=5 into the parametrised pattern above
[VF]0(5),0(13-5)/2 + 1)VE0E(5)VE0E((13-5)/2 - 1)V = [VF]0VVVVV,0VVVVVE0EVVVVVE0EVVV
Construct the 13/5 crossed antiprism
wythoff -M schwarz_13_2_2p -r FVE -p [VF]0VVVVV,0VVVVVE0EVVVVVE0EVVV | antiview
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