There are a number of regular shapes which "tile the plane", just by using large numbers of the same tile. Squares and rectangles, for example, can be laid down in a neat matching pattern that will go on forever. Hexagons, the sort that we see on bathroom floors, are another example of a tile which will cover the whole of an infinite plane, ignoring the raggedy bits around the edges (because infinite planes don't have edges!). NOTE: these diagrams are incomplete, so use the ones below, which have circle segments in them.
Other shapes do the same thing: right-angled triangles can be put together to make rectangles and squares, while equilateral triangles can be assembled into hexagons, and so on. Then there are sets of tiles that work together. Octagons cannot tile the plane by themselves, but if you add in some small squares, you have a complementary set of two shapes which will tile the plane.
All of these tiles and tile sets produce a periodic tiling pattern, where if you travel long enough across the plane, you will find the same pattern repeating itself. The main thing about Penrose tiles is that they seem to be able to tile the plane in an aperiodic way. That is, you can entirely cover an infinite surface, always moving outwards, but without producing any repeating patterns.
Penrose tiles are named after their inventor, Sir Roger Penrose, and they typically show a sort of five-fold symmetry, which derives from the fact that the angles are all multiples of 36 degrees. In the tile sets shown here, the angles are 216, 36, 36 and 72 degrees in the top tile, and 144, 72, 72 and 72 degrees in the lower tile.
In another tile set, the angles are 144, 144, 36 and 36 degrees for one tile, and 108, 108, 72 and 72 degrees for the other. Your task is to make a tile set in large numbers: at least 30 or 40 of each of the two tiles in that set, and to see if you can fit them all together.
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