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Tilings of a plane

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					Tilings of a plane
       Meenal Tayal
Contents:
     Introduction

     Basic terminology

     Wallpaper groups

     The types
    Introduction
   A pattern is something that occurs in a
    systematic manner and if it repeats in a regular
    way, it is called periodic.

   Man has been fascinated by patterns for a long
    time.

   The earliest ones known are the five regular
    solids that were discovered by Pythagoras.

   Patterns are used to decorate things that range
    from fabrics, carpets, baskets, utensils, wall
    cover and even weapons.
     Introduction (continued)
   Man-made tilings range from street tilings (figures
    in first row), to designs (second row, first figure)
    and art-work second row, second figure).
Introduction (continued)
   Tilings also exist in nature, like the honeycomb
    of a bee (shown in the diagram below), froth of
    soap bubbles etc.




   We are only considering plane tilings, which is a way
    of covering a 2-dimensional (Euclidean) plane with
    with tiles, which fit together with no gaps or overlaps.
Testing periodicity
   We construct a lattice, which is a grid
    consisting of two sets of evenly spaced
    parallel lines.

   Clearly, the lattice repeats regularly in
    two directions.

   A tiling is periodic when we can place a
    lattice over the tiling in such a way
    such that each parallelogram contains
    identical pieces of the tiling.
Transformations of the plane
        Isometries are transformations that
         preserve distances.
        Four kinds of isometries:

        Translation, along a vector
        Rotation, by an angle around a point
        Reflection in a line
        Glide reflection in a line, with a
         displacement ‘d’.

        Identity (trivial).
Wallpaper groups
   Some of the interesting tilings are the
    wallpaper tilings, which form a group
    called the wallpaper group or periodic
    group or (plane) crystallographic
    group.

   They are symmetric in a way that we
    can start with one tile and build the
    tilings using that.

   There are exactly 17 of them!
Why exactly 17?
    The idea:
    The wallpaper group has one
     (fundamental) tile with which we can
     build the whole plane.
    Now, there are only a few choices for
     the shapes (rectangles, equilateral
     triangles etc) of the tiles that fit together
     to cover the plane with no gaps.
    So the only rotations that can occur are
     with order 2, 3, 4 or 6.
    Now we have the fundamental shapes.
     From here, we just need to figure out in
     how many ways we can put them
     together.
    This should give a total of 17.
More on Wallpaper groups
       A tiling has the wallpaper symmetry if no
        matter which direction we go, we
        eventually come to a spot that is similar
        to the point we started.

       All types of the wallpaper groups have
        translations.

       http://www.scienceu.com/geometry/articl
        es/tiling/wallpaper.html
Without Rotations
   p1: Two translation axes.
   pm: One translation and two
    reflections. The reflection axes are
    perpendicular to each other.
   pg: Two parallel glide reflections.
   cm: Parallel reflection and glide
    reflection.
With Rotation of 180 degrees
(without rotation of 60 or 90)
        p2: Translation axes.
        pmm: Four reflections along the sides
         of a rectangle.
        pgg: Two perpendicular glide
         reflections.
        cmm: Two perpendicular reflections.
        pmg: One reflection.
With Rotation of 90 degrees
       p4: A half turn and a quarter turn.
       p4m: Four reflection axes inclined
        at 45 deg. to each other, passing
        through the centre of the quarter
        turn.
       p4g: One half turn and two
        perpendicular axes of reflection.
With Rotation of 120 degrees
   (without rotation of 60)
       p3: Two 120 deg. rotations.
       p31m: Three reflections inclined at 60
        deg. to each other, with some centres of
        rotation not on the reflection axes.
       p3m1: Three reflections inclined at 60
        deg. to each other, with all centres of
        rotation on the reflection axes.
With Rotation of 60 degrees
       p6: Rotations of order 2 and 3 as
        well, but no reflections.

       p6m: Rotations of order 2 and 3 as
        well. Also has 6 axes of reflections.
Other symmetries
   Just containing parallel translations.
    For example, the brick tiling.

   Some do not contain translations at all
    (I.e. rotations and reflections only).

   There are also aperiodic patterns like
    the petals of a flower.

   Thank you for listening.

   Any questions?

				
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posted:9/24/2011
language:English
pages:16