A regular pentagon cannot tile the plane. The reason is arithmetic: its interior angle is 108 degrees, and 108 does not divide 360, so pentagons meeting at a point leave a gap of 36 degrees. Only three regular polygons tile alone, triangle, square and hexagon, because only 60, 90 and 120 divide 360 exactly.
Irregular convex pentagons are a different and much harder question, and it stayed open for a century. Karl Reinhardt found five tiling types in 1918. More appeared in the 1960s and 1970s, several of them found by Marjorie Rice, an amateur working from a Scientific American column, who discovered four new types. A fifteenth was found by computer search in 2015 by Casey Mann, Jennifer McLoud-Mann and David Von Derau. In 2017 Michael Rao announced an exhaustive computer search showing there are no others, and part of the proof was independently verified. The current position is that there are fifteen types, resting on a computation rather than on a short argument.
Fivefold symmetry is also the signature of a structure long held to be impossible in crystals. The crystallographic restriction permits two-, three-, four- and sixfold rotational symmetry and no other, because those are the only ones compatible with a repeating lattice. In 1982 Dan Shechtman observed a tenfold diffraction pattern in an aluminium-manganese alloy. He was disbelieved, and Linus Pauling opposed the result publicly for years. Quasicrystals, ordered but non-periodic and showing fivefold symmetry, are now an established class of material, and Shechtman received the Nobel Prize in Chemistry in 2011.
Fivefold form is common in living things: starfish and sea urchins, the flowers of the rose family, an apple cut across the core. It is rare in minerals. The old association of five with life rather than with stone is, for once, roughly borne out by where the symmetry actually occurs.