Understanding path notation in wythoff

[Andy Baker]

Just grappling with this at the moment and I started with kis.

wythoff -c list gives the recipe as:

[F,V]0_1v1v,1E

this appears to be equivalent to:

[F,V]0_1v1vF_

Is this the case? The latter seems more intuitive to me. Why start a new path for one edge?

[Adrian Rossiter]

Hi Andy

You can output one (example) face for each path, with wythoff -u

wythoff -p [F,V]0_1v1vF_ -u -a cube | antiview

You can see that the with your pattern the example face has entirely
covered a original cube face. Looking at the OFF output

wythoff -p [F,V]0_1v1vF_ -u cube

You can see that this face has 12 sides!

Compare with the wythoff pattern for kis

wythoff -p [F,V]0_1v1v,1E -u -a cube | antiview

This has created a triangle face (which is what is expected for the
Conway operator kis), and has also created a digon on an original cube
edge.

The second path in this pattern, 1E, which made the digon, isn't
especially useful when the pattern is applied to a polyhedron, where
the E elements are all "2-fold". However, if the E elements are not
2-fold, then there will be holes in the final tiling.

For example

Relabel the face and edge elements, with and without the extra path

wythoff -p [F,V]0_1v1v,1E -u -a -r EVF cube | antiview
wythoff -p [F,V]0_1v1v,1E -r EVF cube | antiview

wythoff -p [F,V]0_1v1v -u -a -r EVF cube | antiview
wythoff -p [F,V]0_1v1v -r EVF cube | antiview

Similarly, the input to wythoff might be a meta-tiling that does not
correspond to a polyhedron, and the result will not be a closed tiling
without the extra path (notice wythoff -M, to indicate that the input
should be treated as a meta-tiling)

unitile2d -s t -l 12 -w 6 2 | antiview
unitile2d -s t -l 12 -w 6 2 | wythoff -p [F,V]0_1v1v -M | antiview
unitile2d -s t -l 12 -w 6 2 | wythoff -p [F,V]0_1v1v,1E -M | antiview

Adrian.
--
Adrian Rossiter - adr...@antiprism.com
http://www.instagram.com/adrian_rossiter
http://antiprism.com/adrian

[Andy Baker]

Thanks Adrian, that helps a lot.

And knowing the -u is going to be very useful in debugging.