Tilings of a plane
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
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.
Man-made tilings range from street tilings (figures
in first row), to designs (second row, first figure)
and art-work second row, second figure).
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.
We construct a lattice, which is a grid
consisting of two sets of evenly spaced
Clearly, the lattice repeats regularly in
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
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
Some of the interesting tilings are the
wallpaper tilings, which form a group
called the wallpaper group or periodic
group or (plane) crystallographic
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 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
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
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
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
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
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.
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.