Two tile shapes (thin and thick rhombi). They tile the plane — but never periodically. The pattern has 5-fold symmetry, forbidden in any periodic crystal. Every finite patch you see appears infinitely many times across the plane, yet no translation of the plane maps the whole tiling to itself. Roger Penrose, 1974. Shechtman's quasicrystals (Nobel 2011) are this pattern made physical in matter.
how aperiodic order is possible
Penrose's construction uses two rhombus shapes:
• Thick rhombus (the "kite"): interior angles 72° and 108°
• Thin rhombus (the "dart"): interior angles 36° and 144°
With the right edge-matching rules (here implemented as a substitution / deflation), the tiles fit perfectly but never form a period. The substitution rule: each tile in generation n is replaced by a cluster of smaller tiles in generation n+1, with a scale factor of 1/φ (the golden ratio). After many iterations, what looks like a regular pattern at one scale is revealed to have no translational periodicity.
Why 5-fold is forbidden in periodic tilings: the only rotational symmetries compatible with a 2D lattice are 1, 2, 3, 4, and 6 (the crystallographic restriction theorem). 5, 7, 8, 9, … are impossible for any tiling that repeats. Penrose's tiles get around this by giving up on repetition entirely while keeping 5-fold rotational symmetry.
Shechtman's discovery: in 1982, while studying an aluminum-manganese alloy, Shechtman saw an electron diffraction pattern with 10-fold symmetry. This was supposed to be impossible — diffraction comes from periodic structure, and 5/10-fold periodicity is forbidden. He took 2 years to convince the community he wasn't making a mistake; his Nobel arrived in 2011 for "the discovery of quasicrystals." The atoms in the alloy were arranged in a 3D Penrose-style pattern.