1. Tilings of a plane
Meenal Tayal

2. Contents:
Introduction
Basic terminology
Wallpaper groups
The types

3. 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.

4. 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).

5. 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.

6. 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.

7. 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).

8. 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!

9. 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.

10. 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.

11. 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.

12. 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.

13. 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.

14. 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.

15. 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.

16. 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?