# 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
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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 views: 3 posted: 9/24/2011 language: English pages: 16