• A question about self-similar tiles

    From Stewart Robert Hinsley@21:1/5 to All on Tue Apr 26 13:55:22 2016
    My nephew came up with a question (or two) about self-similar tiles

    1) Given a patch of a tiling can a edge tile be removed by a continuous movement, or is it locked in place?
    2) Given a self-similar tile can the copies that make up a tile be
    separated by a continuous movement, or are they locked in place?

    The crudest answer is that some tiles are locked and others aren't, but
    can anything more general be said.

    Most polygons (e.g. triangles, parallelograms, rectilinear hexagons,
    ...) are unlocked, but I expect that there is a pair rectifiable
    polyomino (for example) that is locked.

    Some countablegons are locked (e.g. the rectangular spirals) and some
    are unlocked (e.g. the rectangular stepped countablegons).

    Most teragons are locked, but I think that there are exceptions.

    --
    SRH

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  • From Roger Bagula@21:1/5 to All on Wed Apr 27 11:23:29 2016
    Stewart Robert Hinsley
    I'm not sure I understand the question:
    is this question :"to jig saw or not to jig saw?".
    This approach may be totally off from your question...
    In any case is clearly delineates the difference between
    fractal affine edged tiles and straight sided lattice tiles.
    Suppose you have a square lattice tiling
    and you replace one side with an "curve" or "arc",
    then the jig saw effect is that all the symmetrically opposite
    sides have to be made to fit as well with a second matching curve.
    The projective plane-Wang tile set in a square lattice
    makes the minimal square lattice "different" tile set
    out at 13. Here they use 5 colors instead of curves: https://en.wikipedia.org/wiki/Wang_tile
    as five color on sides (or ten jig saw curves).
    The curve option instead of the color one destroys the transformative symmetry options of rotation ( motion) or reflection.
    What your question seems to be asking is if the transformation
    symmetry group of a tile in invariant property of a tiling.
    I have recently been doing work on polyhedral symmetry groups
    which behave much like geodesic tilings of spheres
    and the answer in this kind of case is that many polyhedrons can share the same group number of transforms while have very different polyhedral geometry. If you look at point groups in 3d, many share the same number of transforms, but use very different
    transforms to form the geometry. The 17 basic plane straight line edge tilings and their transforms appear to
    actually be "exceptions" or special cases when modern
    tiling and space filling is taken into account.
    But some polynomial based fractal tilings appear to be as you said "locked" and have very limited transformative ( rotational or reflection) options.
    I know this isn't an answer, but maybe you need to be more clear about your meanings in your definitions.

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