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26R.Wallace/BioSystems103 (2011) 18–26

Two points are in the same orbit if they are similarly placed within their tiles or within the grout pattern.

(2)The isotropy group of x∈B consists of those g in G with

˛(g)=x=ˇ(g).It is trivial for every point except those in 1/2 ∩B,for which it is Z2×Z2,the direct product of integers modulo two with itself.

By contrast,embedding the tiled structure within a larger context permits definition of a much richer structure,i.e.,the iden-tification of local symmetries.

We construct a second groupoid as follows.Consider the plane R2as being decomposed as the disjoint union of P1=B∩X(the grout),P2=B\P1(the complement of P1in B,which is the tiles), and P3=R2/B(the exterior of the tiled room).Let E be the group of all euclidean motions of the plane,and define the local symmetry groupoid G loc as the set of triples(x, ,y)in B×E×B for which x= y, and for which y has a neighborhood u in R2such that (u∩P i)⊆P i for i=1–3.The composition is given by the same formula as for G( , R2).

For this groupoid-in-context there are only afinite number of orbits:

O1=interior points of the tiles.

O2=interior edges of the tiles.

O3=interior crossing points of the grout.

O4=exterior boundary edge points of the tile grout.

O5=boundary‘T’points.

O6=boundary corner points.

The isotropy group structure is,however,now very rich indeed: The isotropy group of a point in O1is now isomorphic to the entire rotation group O2.

It is Z2×Z2for O2.

For O3it is the eight-element dihedral group D4.

For O4,O5and O6it is simply Z2.

These are the‘local symmetries’of the tile-in-context. References

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