Snowflake Symmetry

About Snowflake Symmetry

On New Year's Day in 1611, Johannes Kepler — court mathematician in Prague, broke, and looking for a gift — handed his patron Johannes Matthäus Wackher von Wackenfels a small pamphlet titled Strena seu de Nive Sexangula, "A New Year's Gift, On the Six-Cornered Snowflake." Twenty-four pages, in Latin, the first scientific paper ever written about why snowflakes have the shape they do. Kepler had crossed the Vltava Bridge in a snowstorm, noticed the flakes on his sleeve, and asked the question that no one before him had bothered to write down: why six? Not five, not seven, not random — always six. He worked through several candidate explanations: was it the cold air that imposed the shape? was it some property of water vapor? was it a mathematical necessity of how spheres pack together? He could not answer the question, but in the process of asking it he wrote down a conjecture about the densest packing of equal spheres in space — what we now call Kepler's conjecture, finally proved by Thomas Hales in 1998. The snowflake essay was the seed.

The answer to Kepler's question turned out to take three hundred and twenty years to assemble. The chain of discoveries went roughly: in 1885, the Vermont farmer Wilson A. Bentley took the first photomicrograph of a snow crystal (January 15, 1885, on a glass plate exposed for ninety seconds against a black velvet backdrop in the woodshed of his family's farm in Jericho, Vermont). Over the next forty-six years he would photograph more than 5,000 individual snow crystals, the largest single-author record in the history of crystallography. His 1931 book Snow Crystals, co-authored with William J. Humphreys and containing more than 2,400 photomicrographs, remains a reference work. In the 1930s and 1940s, the Japanese physicist Ukichiro Nakaya at Hokkaido University grew snow crystals artificially in a refrigerated cloud chamber, varying temperature and humidity independently. His 1954 Harvard University Press monograph Snow Crystals: Natural and Artificial — 510 pages, 514 figures, 188 plates — established the Nakaya diagram: a chart of crystal morphology as a function of temperature and water-vapor supersaturation. The diagram is canonical. At temperatures just below 0°C, plates and stellars form. Near −5°C, columns and needles. Around −15°C, the large dendritic stars that dominate popular images. Below about −25°C, plates and columns again, depending on humidity. The Nakaya diagram answered Kepler's question in a form not available to him: the six-fold symmetry is fixed by the underlying ice lattice; what varies with weather is the elaboration on the six arms.

The third piece — why the lattice itself is hexagonal — came from twentieth-century structural chemistry. The water molecule H₂O has an H–O–H bond angle of approximately 104.45° in the gas phase (Benedict, Gailar, and Plyler, Journal of Chemical Physics, 1956). When water freezes into ordinary ice — what crystallographers call ice Ih, the form that makes up essentially all the ice on Earth — each oxygen atom hydrogen-bonds to four neighbors arranged at the corners of a tetrahedron. The tetrahedral coordination opens the bond angle slightly, to about 109.5° in the crystal, and produces a lattice in which the oxygen atoms sit at the vertices of puckered hexagonal rings stacked in layers. The lattice is unambiguously hexagonal — space group P6₃/mmc — and the six-fold rotational symmetry of an entire snow crystal is the visible expression of the lattice's six-fold microscopic symmetry. Kenneth Libbrecht's review article (Reports on Progress in Physics, 2005) gives the modern synthesis: the lattice fixes the symmetry, the temperature fixes which faces grow fastest, and the supersaturation fixes how much elaboration the arms can support before they branch or smooth.

This is where the popular literature on snowflakes tends to take a wrong turn, so the correction has to be made early. The most-repeated claim about snowflakes is that no two are alike. This is presented as a measurement, but it is not. It is a probabilistic argument: a typical snow crystal contains roughly 10^18 water molecules, arranged in a complex history of growth steps each of which can vary slightly, so the number of microscopically distinguishable configurations is astronomically large, much larger than the number of snow crystals that have ever existed. The probability that any two crystals are microscopically identical is essentially zero. But this is a calculation about a vast configuration space, not an observation of every snowflake. And there is at least one published exception: in 1988, Nancy Knight, a scientist at the National Center for Atmospheric Research in Boulder, photographed two snow crystals collected from cirrus cloud over Wausau, Wisconsin, that appeared identical under her microscope — both thick hollow columns, both with the same internal hollow geometry, both the same dimensions to the resolution she could measure. Guinness World Records logged them. Kenneth Libbrecht, the Caltech physicist whose laboratory has produced more controlled snow-crystal growth experiments than anyone else's, has been emphatic in interviews: at the visible-microscope level you can occasionally find crystals that look identical; at the molecular level, with 10^18 atoms each, true identity is essentially impossible. The familiar claim is correct as a probabilistic statement and wrong as an absolute one. The two framings are not interchangeable.

The second-most-repeated claim is that snowflakes are perfectly symmetric. They are not. They are statistically six-fold symmetric — the six arms of any single crystal grew under essentially identical conditions (because they grew on the same crystal at the same time), so they elaborated in nearly identical ways. But the symmetry departs from perfect six-fold at the micrometer scale because of random collisions with cloud droplets, brief excursions into different temperature zones, and tiny defects in the underlying lattice. Bentley's 5,000+ photographs were taken on specimens selected for their visual perfection — he discarded the broken, fragmented, and asymmetric ones, which constitute the great majority of real snow. Libbrecht's catalogue at snowcrystals.com documents the full range: most natural snow is a tangle of irregular plates and broken arms, with the iconic six-pointed star forms making up perhaps 10–20% of typical winter snowfall in conditions suitable for their formation.

Mathematical Properties

The symmetry group of an ideal snow crystal in plan view is D₆, the dihedral group of order 12, which combines a six-fold rotational symmetry (rotations by 0°, 60°, 120°, 180°, 240°, 300°) with six reflection axes passing through the center. In three dimensions the symmetry is D₆h, which adds a horizontal mirror plane perpendicular to the six-fold axis. The point group D₆h is the same point group that describes a regular hexagonal prism, which is what a snow crystal essentially is — a hexagonal prism that has elaborated branches off its six side faces or smoothed itself into a thin plate, depending on the growth conditions.

The underlying space group of ordinary ice (ice Ih) is P6₃/mmc. The 6₃ is a six-fold screw axis: a rotation by 60° combined with a translation by half the c-lattice parameter. The mmc indicates the mirror planes and glide planes. The lattice parameters at 0°C and 1 atm are a = 4.51 Å and c = 7.35 Å (Hobbs, Ice Physics, Oxford 1974). Each oxygen atom hydrogen-bonds to four others; the four bonds point toward the vertices of a regular tetrahedron with tetrahedral angle 109.47°. This is slightly different from the gas-phase bond angle in an isolated H₂O molecule (104.45°), which is one of the unusual structural features of water and the reason the ice lattice has the open hexagonal structure that makes ice less dense than liquid water.

The six-fold symmetry of the macroscopic crystal is the visible expression of the six-fold symmetry of the underlying lattice — but there is a subtlety. The basal plane of ice Ih (the plane perpendicular to the c-axis) is hexagonal, with growth rates that are equal in the six equivalent <1120> directions. This six-fold equality is what produces six identical primary arms. Each arm then grows as a column or plate depending on the ratio of growth rates along the c-axis versus the a-axes — and that ratio is exactly what depends sensitively on temperature, which is the underlying mechanism of the Nakaya diagram. Near −2°C and again near −15°C, the a-axis grows faster than the c-axis, producing plate-like and stellar crystals. Near −5°C and below −30°C, the c-axis grows faster, producing columns and needles. The transitions between these regimes occur at temperatures where surface-energy considerations (the Wulff construction applied to ice) flip the relative growth velocities.

The mathematical description of dendritic growth — the branching of crystal arms into smaller side-arms — was developed by James Langer and colleagues at Carnegie Mellon and UC Santa Barbara in the 1980s (Langer, Reviews of Modern Physics, 1980). The Mullins–Sekerka instability shows that a growing flat interface is unstable to perturbations: any small bump on the interface gets amplified because it sticks out further into the supersaturated vapor and grows faster. The result is dendritic branching, with side-arms that themselves bud sub-arms, producing the tree-like elaboration on each of the six primary arms. The fractal dimension of a well-developed dendritic snow crystal in plan view, measured by box-counting, is approximately 1.7 — between a smooth curve (dimension 1) and a filled plane (dimension 2).

Occurrences in Nature

Snow crystals form in cold clouds when water vapor deposits directly onto an ice nucleus (a small foreign particle — often a clay mineral, soot grain, or fragment of biological debris). Growth proceeds by molecular addition at supersaturation pressures above the equilibrium vapor pressure of ice. The crystal's morphology is set by where in the Nakaya diagram (temperature × supersaturation) the growth occurs.

The Nakaya diagram identifies the major morphologies: at temperatures from 0°C to about −3°C, thin hexagonal plates and small stellar crystals. From −3°C to −10°C, needles (very thin columns), then hollow columns. From −10°C to about −22°C, plates again at low supersaturations and the large dendritic stars at high supersaturations — this is the regime that produces the iconic, much-photographed snowflakes. Below −22°C, the morphology becomes more columnar again, with mixed plates and columns at very cold temperatures. Each transition corresponds to a change in the dominant growth face of the ice lattice.

In any given storm, the crystals at the ground are a mixture: the snow has fallen through several thousand meters of atmosphere, passing through cloud layers at different temperatures and humidities. A single crystal can have a column at its center (formed at −5°C in upper cloud) and dendritic plates radiating from its ends (formed at −15°C as it fell into warmer, more humid air below). These mixed-morphology crystals are called capped columns or scrolls and are catalogued extensively in Libbrecht's snowcrystals.com archive and in his 2016 book Ken Libbrecht's Field Guide to Snowflakes.

Outside Earth's atmosphere, hexagonal ice crystals form in the polar caps of Mars (where seasonal CO₂ ice deposits on top of permanent H₂O ice produce visible hexagonal facets imaged by the Mars Reconnaissance Orbiter HiRISE camera), on comet nuclei (where ice Ih sublimates and re-deposits as the comet rotates), on the surfaces of icy moons in the outer solar system (Europa, Enceladus, Triton — though in these environments other ice phases like ice II or ice III may also form at depth), and probably on planets and exoplanets at the right temperature range. The hexagonal symmetry follows from the molecular geometry of water, which is the same throughout the universe, so any snow that ever forms on any world made of the same chemistry will have the same six-fold structure.

The form is also reproduced in laboratory crystal growth from many other substances with similar tetrahedral coordination — silicon, germanium, ammonia ice, certain carbohydrates — whenever the underlying lattice is hexagonal and the growth is dendritic. The six-fold symmetry of snow is a special case of a more general crystallographic principle, not a unique property of water.

Architectural Use

The hexagonal-prism geometry that underlies snow crystals is itself widely used in architecture, though usually as a free-standing geometric choice rather than a direct snowflake reference. Hexagonal column geology — the basalt columns at Giant's Causeway in Northern Ireland (formed ~60 million years ago), the Devil's Postpile in California, Fingal's Cave in Scotland — has been studied since the eighteenth century as a parallel example of hexagonal close-packing emerging from a different physical process (thermal contraction in cooling basalt, modeled by Eldon Spotorno and colleagues in Physics Today, 2010). These natural hexagonal-column formations have inspired architectural references in many buildings, including Gaudí's Park Güell in Barcelona (1900–1914) and Frank Lloyd Wright's Hexagonal House for Robert Berger in San Anselmo, California (1950–1958).

Direct snowflake-form architecture is rarer and is mostly modern. The Snowflake Pavilion at the Expo 2010 Shanghai world exposition, designed by Norman Foster's office, used a six-fold radial roof structure modeled directly on a snowflake's plan view. The Princess Diana Memorial Fountain in Hyde Park (designed by Kathryn Gustafson, 2004) uses six-fold radial symmetry in part of its layout. The Las Vegas Bellagio's seasonal winter installation traditionally features a large six-pointed snowflake centerpiece. In the Sapporo Snow Festival (Hokkaido, annually since 1950), six-fold ice and snow sculptures are constructed as direct architectural homages to Nakaya's work at Hokkaido University — Nakaya is a local figure of pride in Sapporo and the festival celebrates the city's role in snow-crystal science.

The six-pointed star geometry — the hexagram, two overlapping triangles — is independent of snowflakes but visually related. It appears in Jewish tradition as the Star of David (the Magen David, documented in synagogue ornament from the medieval period onward, codified as a Jewish symbol in the seventeenth century in Prague), in Hindu and tantric traditions as the shatkona (the union of male and female energies, Shiva and Shakti), and in early Christian and Islamic geometric art. The convergence of these traditions on six-fold symmetry is not coincidence but follows from the same mathematical fact Kepler identified — six is the smallest number that allows a regular tessellation of the plane combined with a clear central point.

Construction Method

A simple regular six-pointed snowflake can be constructed with compass and straightedge in the classical Euclidean tradition. Start with a circle. Without changing the radius, step the compass around the circumference, marking six points where consecutive arcs meet — these are the vertices of an inscribed regular hexagon. Connect alternating vertices to form two overlapping equilateral triangles (the hexagram), or connect adjacent vertices to form the hexagon itself. From each of the six points, draw a line outward; from each line draw smaller side-branches at 60° to the main line; from each side-branch draw still smaller sub-branches at 60°. This recursive construction produces a stylized dendritic snowflake.

A more rigorous physical construction is the one Nakaya pioneered in 1936: grow real snow crystals in a controlled laboratory. The setup is a refrigerated cloud chamber, a thin nucleating filament (Nakaya used rabbit hair; modern setups use very fine glass fibers), and tightly controlled temperature and humidity. With practice the crystal grows visibly over several minutes, and by varying the temperature one can step through the morphologies of the Nakaya diagram in a single experiment. Libbrecht's lab at Caltech maintains a setup of this kind and has produced thousands of photographs of artificially grown crystals, several of which are documented at snowcrystals.com.

A modern variant uses a fine glass capillary or carbon fiber as the nucleating filament, suspended in a temperature-gradient diffusion chamber where the bottom plate is held a few degrees below freezing at the warm end and much colder at the other, with the air space held at supersaturation. By moving the filament along the gradient, one can step through plate, column, needle, and dendritic-star morphologies in a single growth run — directly tracing the Nakaya diagram in real time.

Spiritual Meaning

The snowflake has carried symbolic weight in several traditions, almost always tied to purity, impermanence, or the transient beauty of the visible world. The cross-tradition convergence is striking: the same set of meanings recurs across Japanese, Christian, and Tibetan Buddhist contexts, though the underlying theology differs.

In Japanese Shintō and the literary tradition that developed alongside it, snow (yuki) is one of the four classical poetic seasonal markers (with cherry blossoms, the moon, and autumn leaves) and is associated with purity (misogi, the purification ritual at Shinto shrines). The Sapporo Snow Festival, the haiku tradition of snow imagery (Bashō's furu yuki ya verses are canonical), and the Hokkaido folk tradition of snowflake-pattern textiles all draw on the same complex of meanings: snow as a brief visitation of perfection that will pass, a reminder that purity is real but not permanent. Nakaya's own life — Hokkaido University physicist who chose to publish his major work in English at Harvard rather than Japanese in Tokyo because he wanted the science to be accessible to the world — sat at the intersection of Japanese poetic tradition and modern crystallography. His 1954 book opens with the observation that "a snow crystal is a letter from the sky," and the phrase has been quoted in Japanese science writing ever since.

In Christian symbolism, snowfall is associated with the cleansing of sin — Psalm 51:7, in the King James translation: "Wash me, and I shall be whiter than snow." The image of a snow-white soul, a clean conscience, a forgiven debt recurs throughout the Western liturgical tradition. Snowflake imagery in Christmas iconography is more recent (mostly nineteenth and twentieth century) and ties to the Northern European seasonal association of the Nativity with winter weather.

In Tibetan and Himalayan Buddhist iconography, snow is associated with the holy mountains (Kangrinboqe / Mount Kailash, the Himalayan peaks where the Buddhist saint Milarepa meditated), with the purification of the mind (in Dzogchen literature, snow-imagery sometimes accompanies descriptions of rigpa, the naked primordial awareness that is the practice's central recognition), and with the bodhisattva ideal (the white lion of Tibet, whose roar echoes through the snows, as an image of the awakened mind). The Tibetan word for the Himalayas, gangs ri, literally means "snow mountains," and the entire region's Buddhist culture is suffused with snow imagery as a contemplative image.

The modern New Age and "sacred geometry" literature has added a layer to all of this by treating the snowflake's six-fold symmetry as a manifestation of cosmic order — sometimes specifically linked to the Star of David, the hexagram, or the Merkaba (the chariot-throne in Jewish mystical literature, sometimes drawn as a three-dimensional hexagram). The mathematical reading is more austere: the six-fold symmetry is the consequence of the 104.5° bond angle of the water molecule (or rather its tetrahedral expansion in the ice lattice), which is in turn the consequence of the quantum mechanics of oxygen's two unbonded electron pairs. The wonder, in the rigorous version, is that a microscopic quantum-mechanical fact about a three-atom molecule expresses itself macroscopically as a six-pointed star you can see with the naked eye when one lands on your sleeve. This kind of scale-jump — from quantum mechanics to visible symmetry — is rare in everyday experience. The snowflake provides one of the few accessible examples.