What is a tiling?
When confronted with a big flat space, and a bunch of little pieces,
it's just human nature to try to fit the pieces into it. Since
every game has to have rules to make it fun and challenging, it only
natural to want the pieces to fit together nicely. And voilá!
A tiling is born.
Intutively, we all understand what a tiling is. A tiling is what you
get when you fit a collection of individual tiles together
with no gaps or overlaps to fill some flat space like a floor or a
table top. Tilings surround us everyday.
Question: How many tilings can you find in your house? In
your classroom? Walking around outdoors?
Tilings Around Us
Brick walls are tilings. The rectanglular face of each
brick is a tile on the wall.
The game checkers is played on a tiling. Each colored square on the board is a tile. The
checker board is an example of a periodic tiling.
Mother nature is a great producer of tilings.
Bees construct tilings! This cross section of a
beehive is a periodic tiling by hexagons.
Here is a picture of a mud flat that has
dried in the sun. The flat is veined with cracks. This is a tiling
of the flat. Each piece of dried mud is a tile.
Unlike or other examples, the mud flat tiling doesn't have a
regular, repeating pattern. In the mud flat, every tile has a
different shape. By contrast, in our other examples there was just
one shape. Can you imagine a tiling that has just two shapes of
tiles? Three shapes?
What makes a Tiling a Tiling?
As we have seen from our examples, a tiling is just a way of covering a
flat surface with smaller shapes or tiles, subject to a
couple of rules: we want the tiles to fit together nicely, without
gaps or overlaps.
So illustrate what we mean when we say "no gaps or overlaps" imagine
we are given square tiles of four different colors.
In the image on the right, there are no gaps or overlaps and the
squares really do form an tiling of the larger rectangle.
However, the images below are not tilings.
Question: Do you know why these two images are not tilings?
In the picture on the left, notice the yellow tile in the middle
row overlaps the light green tile next to
it. Since gaps and overlaps are not allowed, this is not a tiling.
In the picture on the righ, there is a gap between the
light green and yellow tiles in the middle row. Again,
since gaps and overlaps are not allowed, this is not a tiling.