The most common tiling is the square grid:
The whole plane is covered with squares.
If you rotate it by 90 degrees (one-quarter turn) it looks the same. If you rotate it by 360 degrees (one turn) you see 4 times the same picture, thus it has 4-fold rotational symmetry.
The grid repeats itself: you can shift it by the length of a square and it remains unchanged, thus it is periodic.
With different colors for different squares, the rotational symmetry and the periodicity can be varied in many ways.
The honeycomb-lattice is another well-known tiling:
The whole plane is covered with regular hexagons.
If you rotate it by 60 degrees (one sixth of a full turn) it looks the same. If you rotate it by 360 degrees you see 6 times the same picture, thus it has 6-fold rotational symmetry.
The grid repeats itself: you can shift it by the distance between the centers of the hexagons and it remains unchanged, thus it is periodic.
If you connect the dots with straigth lines you get a tiling with equal-sided triangles, which is also quite common.
Are there tilings with higher rotational symmetry ?
Can I have a tiling with 8-fold rotational symmetry ? I could rotate it by 45 degrees and it should look the same. Then it would have “twice” the rotational symmetry of the square lattice.
A regular octagon has this symmetry, but it does not make a tiling. You cannot cover the entire plane with only octagons and no overlap or gaps. Obviously, the plane can be covered if squares fill up the gaps between octagons, but this results in a slightly modified square tiling with 4-fold rotational symmetry.
Thus we need another approach. We can put 8 rhombs of an acute angle of 45-degrees together to form a star with 8 points. It has the same symmetry as the octagon. Together with squares we can cover the entire plane such that the tiling repeats itself approximately. We then get a quasiperiodic tiling, which has been a long time ago discovered by Amman and Beenker. In the next posts I will show you how we can make this tiling in a very simple way and I will present same variations.