Cut-and-project method, forbidden symmetries, and diffraction patterns
For centuries, crystallographers believed that ordered matter must be periodic -- atoms arranged in a repeating lattice. In 1982, Dan Shechtman shattered this dogma by discovering an aluminum-manganese alloy with sharp diffraction peaks and 10-fold symmetry -- impossible for any periodic crystal. He was awarded the 2011 Nobel Prize in Chemistry for this discovery of quasicrystals.
Quasicrystals have long-range order without periodicity. Their mathematical foundation is the cut-and-project method: a quasicrystal is a "slice" of a higher-dimensional periodic lattice, projected at an irrational angle. This explains both the sharp diffraction peaks (from the parent lattice's periodicity) and the forbidden rotational symmetries (from the irrational projection angle).
The simplest quasicrystal is one-dimensional: a Fibonacci sequence of long (L) and short (S) intervals. It arises from a 2D square lattice by selecting points inside a diagonal strip and projecting them onto the strip's direction. When the strip's slope is irrational (here, 1/φ), the projected points form an aperiodic sequence. When rational, the result is periodic.
Slice a 2D square lattice with a strip at an irrational angle. Points inside the strip project onto a 1D quasicrystal — a Fibonacci sequence of Long and Short intervals.
Key insight: The L and S intervals follow the Fibonacci substitution rule: L → LS, S → L. The ratio of L's to S's converges to the golden ratio φ. This connects 1D quasicrystals directly to the Fibonacci sequence and explains why φ appears so prominently in Penrose tilings.
Why can't periodic crystals have 5-fold symmetry? Try tiling with regular pentagons and watch the gaps appear. The crystallographic restriction theorem proves that only 2-, 3-, 4-, and 6-fold rotational symmetries are compatible with translational periodicity in 2D. Quasicrystals bypass this restriction by abandoning periodicity while retaining long-range order.
Only 3-, 4-, and 6-fold symmetries can tile the plane periodically. Pentagons (5-fold) leave unavoidable gaps — their 108° interior angle does not divide 360°.
Key insight: The proof is elegant. A rotation by angle θ maps one lattice point to another. The new point minus the old must also be a lattice point. This forces 2cos(θ) to be an integer, which limits θ to 0°, 60°, 90°, 120°, or 180°. Pentagons (θ = 72°) fail because 2cos(72°) = (√5 - 1) is irrational.
While Penrose tilings have 5-fold/10-fold symmetry, the Ammann-Beenker tiling achieves 8-fold symmetry using squares and 45° rhombi. It arises from projecting a 4D hypercubic lattice onto 2D, and its diffraction pattern shows sharp peaks with 8-fold rotational symmetry -- another "forbidden" symmetry realized by aperiodic order.
An aperiodic tiling with 8-fold rotational symmetry, built from squares and 45° rhombi. Click Inflate to apply substitution rules and reveal finer structure.
Key insight: The Ammann-Beenker tiling comes from a 4D lattice (Z⁴), while Penrose tilings come from 5D (Z⁵). In general, an n-fold quasicrystal requires projecting from a φ(n)-dimensional lattice, where φ is Euler's totient function. 12-fold quasicrystals (recently found in nature) come from 4D as well.