From: veritasium
Quasicrystals are a class of materials that exhibit a highly ordered, yet non-periodic, atomic structure. Their discovery challenged long-held scientific assumptions about the fundamental symmetries of crystals. The story of quasicrystals begins with mathematical patterns thought to be impossible [00:00:06] and materials that weren’t supposed to exist [00:00:11].
Historical Precursors and Geometric Explorations
The journey towards understanding quasiperiodic structures dates back over 400 years to Johannes Kepler in Prague [00:00:11]. Kepler, renowned for identifying elliptical planetary orbits [00:00:33], also explored geometric regularities in the universe:
Platonic Solids and Planetary Orbits
Before discovering elliptical orbits, Kepler developed a model of the solar system where planets were on nested spheres separated by the Platonic solids [00:00:44]. Platonic solids are objects with identical faces and vertices, allowing them to look the same when rotated through certain angles [00:00:54]. There are only five such solids: the cube, tetrahedron, octahedron, dodecahedron (12 pentagonal sides), and icosahedron (20 sides) [00:01:09]. Kepler used these five solids as spacers between the six known planetary spheres, aligning their order to match astronomical observations as closely as possible [00:01:26]. He held a deep belief in geometric regularity in the universe [00:01:48].
Sphere Packing and Kepler’s Conjecture
Kepler’s interest in geometry extended to practical questions, such as how to stack cannonballs most efficiently [00:02:02]. By 1611, he proposed that hexagonal close packing and face-centered cubic arrangements were optimally efficient, filling about 74% of the volume [00:02:06]. This became known as Kepler’s Conjecture, which, though seemingly obvious, was only formally proven in 2017, nearly 400 years later [00:02:28].
The Mystery of Snowflakes
Kepler published his conjecture in a pamphlet titled Deniva Sexangula (“On the Six-Cornered Snowflake”) [00:02:46]. In it, he pondered why snowflakes invariably display a six-cornered shape, questioning why they don’t form with five or seven corners [00:02:54]. While theories of atoms and molecules didn’t exist in his time, Kepler speculated about the “smallest natural unit” of water and how these units might stack mechanically to form hexagonal crystals, similar to his hexagonal close-packed cannonballs [00:03:12]. His musings bordered on an understanding of crystal structure [00:03:17].
Tiling the Plane and Forbidden Symmetries
Kepler understood that regular hexagons can perfectly cover a flat surface with no gaps, a concept known as tiling the plane periodically [00:03:42]. Periodic tilings allow for duplication of a pattern through translation without rotation or reflection [00:03:52]. They can also exhibit rotational symmetries:
- Rhombus patterns: two-fold symmetry [00:04:03]
- Equilateral triangles: three-fold symmetry [00:04:11]
- Squares: four-fold symmetry [00:04:15]
- Hexagons: six-fold symmetry [00:04:19]
However, only two, three, four, and six-fold symmetries are possible for periodic tilings [00:04:22]. Regular pentagons, for example, cannot tile the plane [00:04:29]. Despite this, Kepler attempted to create a pattern with five-fold symmetry, publishing it in his book Harmonics Mundi (“Harmony of the World”) [00:04:36]. While it had a certain five-fold symmetry, it was not entirely clear how it would continue to tile the whole plane [00:04:46].
The Rise of Aperiodic Tilings
This background set the stage for a key question: Can some tiles only tile the plane non-periodically [00:05:32]?
Early Aperiodic Sets
In 1961, Hao Wang studied multi-colored square tiles with rules about matching edge colors and disallowing rotation or reflection [00:05:38]. Wang conjectured that if a set of these tiles could tile the plane, they could do so periodically [00:05:56]. His student, Robert Berger, disproved this conjecture by finding a set of 20,426 tiles that could tile the plane only non-periodically [00:06:04]. Such a set of tiles is called an aperiodic tiling [00:06:35]. Mathematicians then sought smaller sets:
- Robert Berger reduced the set to 104 tiles [00:06:48].
- Donald Knuth got the number down to 92 [00:06:51].
- In 1969, Raphael Robinson found a set of just six tiles [00:06:56].
Penrose Tilings: Two Tiles to Infinity
Roger Penrose eventually reduced the number of tiles needed for an aperiodic tiling to just two [00:07:07]. His process involved starting with a pentagon, adding others, and then subdividing them, revealing gaps that eventually resolved into specific shapes [00:07:14]. This yielded an aperiodic tiling with an “almost five-fold symmetry” [00:08:04]. Notably, Penrose’s pattern perfectly matched an earlier pentagon pattern published by Kepler [00:08:12].
Penrose distilled the geometry down to two specific tiles: a thick rhombus and a thin rhombus [00:08:27], or alternatively, two shapes called “kites and darts” [00:10:08]. Rules for matching these tiles (e.g., bumps/notches or continuous curves) ensure that they can only tile the plane non-periodically, extending to infinity without ever repeating the same pattern [00:08:40], [00:10:23].
Characteristics of Penrose Tilings:
- Infinite Variations: There are an uncountably infinite number of different Penrose patterns that tile the plane [00:11:25].
- Local Indistinguishability: Despite infinite variations, any finite region of one tiling appears infinitely many times in all other versions, making it impossible to tell them apart by local observation alone [00:11:52].
- Golden Ratio: The ratio of kites to darts in a Penrose tiling approaches the golden ratio (approximately 1.618) [00:12:31]. This irrational ratio provides evidence that the pattern cannot be periodic, as a periodic pattern would result in a ratio of two whole numbers [00:13:35]. The golden ratio is heavily associated with pentagons, and the kite and dart pieces themselves incorporate the golden ratio in their construction [00:13:15].
- Fibonacci Sequence: When specific straight lines are drawn on the Penrose tiles, they connect to form five sets of parallel lines [00:13:54]. Each set contains two different spacings, “long” and “short,” and the ratio of long to short segments in any section corresponds to consecutive numbers in the Fibonacci sequence (e.g., 13 shorts and 21 longs), whose ratio also approaches the golden ratio [00:14:18].
The Physical Analog: Quasicrystals
The existence of Penrose tilings raised a profound question: Could such patterns have a physical analog, perhaps in crystal structures [00:15:04]? At the time, this seemed unlikely, as crystals were understood to be made of repeating units following one of 14 established basic unit cells, none of which allowed for five-fold symmetry [00:15:15]. Furthermore, Penrose tilings seemed to require “long-range coordination,” meaning a local tile placement could lead to an impossible configuration far away [00:15:43].
Theoretical Prediction
In the early 1980s, Paul Steinhardt and his students used computers to model how atoms come together in condensed matter [00:16:47]. They found that locally, atoms liked to form icosahedrons, a shape considered “forbidden” due to its five-fold symmetries [00:16:58]. Inspired by Penrose tilings, they designed a new kind of structure – a 3D analog of Penrose tilings, later known as a quasi-crystal [00:17:15]. Their simulations of x-ray diffraction from such a structure showed a pattern with rings of 10 points, reflecting the five-fold symmetry [00:17:24].
Experimental Discovery
Unaware of Steinhardt’s theoretical work, Dan Shechtman, a few hundred kilometers away, created a flaky material from aluminum and manganese [00:17:35]. When he scattered electrons off this material, the diffraction pattern he obtained almost perfectly matched Steinhardt’s simulated pattern, exhibiting clear five-fold symmetry [00:17:41]. This was the first experimental evidence of a quasicrystal.
The reconciliation of long-range order with local rules was explained by Steinhardt: while matching rules for edges are insufficient for long-term tiling, rules for how vertices connect are strong enough locally to prevent mistakes, allowing the pattern to extend infinitely [00:18:03].
Recognition and Impact
One of the seminal papers on quasicrystals was titled Deniva Quinquangula (“On the Pentagonal Snowflake”), a direct shout-out to Kepler’s earlier work [00:18:33].
Not everyone accepted the existence of quasicrystals initially. Linus Pauling, a double Nobel Prize winner, famously declared, “There are no quasicrystals, only quasi scientists” [00:18:41]. However, Shechtman’s discovery was validated when he was awarded the Nobel Prize in Chemistry in 2011 [00:19:01].
Quasicrystals have since been grown with beautiful dodecahedral shapes [00:19:07] and are being explored for various applications, including:
- Non-stick electrical insulation [00:19:12]
- Cookware [00:19:17]
- Ultra-durable steel [00:19:18]
The existence of quasicrystals highlights that patterns and materials thought to be impossible can exist, expanding our understanding of structure and order in the universe [00:19:23].