The tiles are called Penrose Tiles and they have many
unusual properties. When joined together in a mosaic the
tiles make beautiful patterns related to the medieval
Islamic building decoration called "Girih", which predates
the western discovery of the tiles by 500 years.
The geometric patterns in the tiles are also found in nature
where ever the Fibonacci series occurs, for instance in
sunflowers and artichokes and some kinds of crystals and in
the ancestry of bees. Penrose tiles are mathematically
related to the irrational number phi, also know as the
"golden ratio", and this explains at some level why the
patterns are aperioidic.
The tiles cover the infinite plane without making any regularly recurring pattern and there are countless possible tilings. Any finite tiling within an infinite tiling occurs an infinite number of times in that tiling, and in all of the other possible tilings.