D_{3} PascGalois Triangle |
D_{4} PascGalois Triangle |
D_{5} PascGalois Triangle |
D_{6} PascGalois Triangle |
D_{7} PascGalois Triangle |
D_{8} PascGalois Triangle |
You may notice that D_{4} and D_{8} have much
more easily discernable patterns than the others. Unfortunately, the reason
for this is beyond the scope of this paper. However, if we change the coloring
scheme so that all the rotations are colored blue/green and the flips are colored
red/purple, see what happens:
D_{3} PascGalois Triangle |
D_{4} PascGalois Triangle |
D_{5} PascGalois Triangle |
D_{6} PascGalois Triangle |
D_{7} PascGalois Triangle |
D_{8} PascGalois Triangle |
Although the triangles for some are always fuzzier than others (e.g. D_{4}
and D_{8}), all of the triangles exhibit the similarity to the
triangle for Z_{2} when rotations and flips are clearly distinguishable.
Again, this happens because when you lump all the flips together as if they
were the same element and you lump the reflections together as if they were
all the same, the resulting multiplication table is the same as that for Z_{2}
with 0 replaced by the rotations and 1 replaced by the flips.