Quonset HHH - hhh - 74

This has an interesting consequence. For hexagonal tilings there two ways of tiling a plane with them. You can set hexagonal tiles out one by one carefully matching them up - which is tricky. But once you've done that a bit you will find that there is a rectangular region within the tile that, when copied, place the tiles down much more quickly, even automatically as I did here. Pentagons and rectangles form periodic tilings.

The pentagonal tiles do not make periodic tilings; they make aperiodic tilings. Actually, if a tiling is defined as filling a space with no gaps or overlaps then the geometry of a pentagon always forces gaps between them. Roger Penrose's Penrose tiles are pairs of tiles that do fill a plane with a pentagonal symmetry with no gaps or overlaps and that tiling is also aperiodic. Since it's aperiodic you cannot automate the process of tiling the plane. You actually have to place each tile by hand.