Re: Tilings and groups


Hello-
I may be misunderstanding your question, but would it help to look
at the 'tilings' as being members of one of the 17 symmetry groups of
the plane? In which case, are you asking if all 17 groups are
generated by 2 elements? (or fewer? exactly 2?)
If you are looking at all surfaces- I don't know if there is a a
characterization of symmetry groups for all 2-manifolds or not...

Apologies if I am stating the obvious!

cheers-
Eric


On Feb 16, 4:26 am, Gerard Westendorp <west...@xxxxxxxxx> wrote:
Christopher J. Henrich wrote:

[..]

I was wondering if it is true that every group with 2 generators can be
thought of as tiling of a 2-dimensional surface.

[..]

In your identification of a group with a tiling, do you want a
bijection between the set of elements of the group and the set of
tiles?

yes.
My question was probably not formulated very well. Instead of 'tiling'
I should have written 'Platonic tesselation'. These are tilings that
are generated by the action of a symmetry group.

I have some examples on my own web site:http://www.xs4all.nl/~westy31/Geometry/Geometry.html

All of these tilings can be generated by just 2 generators. This
leads to the question:

"Can all symmetry groups of Platonic tesselations be generated by
just 2 generators?"

And the reverse:

"Can a Platonic tesselation of a 2 dimensional manifold be constructed
for any groups with 2 generators?"

There is also this site, where they generate tilings by specifying
algebraic relations on the generators:http://www.geom.uiuc.edu/apps/unifweb/

Gerard

.

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