Introduction
In 1611, Johannes Kepler wrote a short New Year’s gift called “On the Six-Cornered Snowflake” (Strena seu de nive sexangula), his curiosity piqued by a simple question: why do snowflakes so reliably arrive with six arms—why not five or seven—and why are they so often flat, star-like plates? He imagined a “formative faculty” at work in nature, an idea that sat comfortably inside his theological picture of an ordered cosmos. Today’s scientists may retain the wonder, but they usually encapsulate it in the language of structure, constraints, and (sometimes) “natural laws”. [1]
Even small physical systems teach this. A frosting line on a gas canister is not a “law”—it is a visible signature of heat flow under phase change. You can say what would change it (draw rate, humidity, ambient temperature, insulation), and the pattern shifts in predictable ways.
In the same way, a snowflakes habit is a record of transport limits plus surface kinetics: a visible trace of what the growth process allowed under those conditions. Here “transport limits” means how quickly water vapour (and the latent heat released on freezing) can be moved to and from the crystal, while “surface kinetics” means how readily arriving molecules actually attach to the ice lattice on a given facet.
In a previous piece I framed explanation in law-shaped terms (law + conditions → outcome); here I test that ideal against snowflake morphology.[2]
Thesis
The formation of a snowflake is a powerful example of scientific explanation without a single universal, law-shaped statement. What does the explanatory work is a web of constraints and sensitivities: temperature and supersaturation shape vapour diffusion and heat removal; small instabilities amplify differences at edges and tips; and surface attachment kinetics—how readily the ice surface accepts molecules—depends on temperature, supersaturation, impurities, and the microstate of the surface.
By “lawless” I do not mean that physics lacks laws; I mean that at the level of snowflake morphology no single compact, law-shaped statement dictates the outcome.
Argument
Setup and Background
Terminology note. A “snow crystal” is a single crystal of ice—what scientists grow in the lab —whereas “snowflake” is the broader term for anything that falls from the sky. A snowflake can be a single crystal or a cluster of crystals stuck together. In this essay I’ll follow common usage and treat the two terms interchangeably unless the distinction matters.
Snowflakes are a good test case for that view, because they look like a domain where a tidy law ought to exist. Set the temperature and supersaturation, and nature will hand you plates here, needles there, dendrites at higher supersaturations. The famous morphology diagram (often called the Nakaya diagram) seems to support this.[3] It has the visual authority of a periodic table: neat regions, repeatable outcomes, and a suggestion that the right rule is waiting to be written down.
But things are not quite as they seem. Libbrecht’s 2005 review complicates the picture.[4] Yes, many morphologies correlate reliably with temperature and supersaturation, but the reason those correlations hold is not a clean, law-like story. Diffusion-limited growth provides a strong constraint on how branching happens, yet the decisive “chooser” of plates versus columns lives in microscopic surface processes—how readily water molecules attach to basal and prism facets—and that part is hard to derive from first principles.
Think of this essay as tackling a quality assurance problem. You don’t certify a claim because it carries the right explanation; you certify it because you can state what should be present, what would count as failure, and what trace a real mechanism ought to leave. That is, a good explanation makes its dependencies and failure modes explicit. You say what variables matter, how a change affects the outcome, and under what conditions the explanation would break.
Worked Mini Example
• Claim: near −15°C, increasing supersaturation tends to push growth toward thin plates and dendritic forms.
• Applies when: growth occurs under controlled temperature and known supersaturation in a defined size regime (because diffusion-limited effects strengthen as crystals get larger), with chamber conditions that keep contamination low.
• Depends on: (i) vapour diffusion and heat removal (transport limits), and (ii) facet-specific attachment kinetics (basal vs prism) which can be modified by surface state, impurities, and edge geometry.
• If you change X: raise supersaturation and you should see faster growth and more branching; shift temperature across the habit boundary and plate/column preference should flip; change surface state/impurity conditions and you should see shifts in relative facet growth even at “the same” temperature and supersaturation.
• Would count as failure: repeated runs at matched temperature, supersaturation, and size regime produce systematically different habits without any identifiable change in transport, surface state, or measurement setup — forcing you to revisit what you treated as “controlled.”
• How we check: measure facet growth rates and morphologies under controlled conditions, then test whether diffusion-plus-kinetics models reproduce both the observed shapes and the growth dynamics, and whether deviations track a named channel (impurities, premelting regime, facet width, instrumentation).
Core analysis: lawless explanation (Libbrecht 2005)
One striking feature is the broad sequence of crystal habit changes with temperature: plates at −2˚C, columns at −5˚C, and plates again at −15˚C. Within each region there are variations of form based on supersaturation. But Libbrecht also emphasises that while there is dependence on temperature and supersaturation as observed by Nakaya, this is not the whole story. The other key factors noted are particle diffusion, heat diffusion, and attachment kinetics. So temperature and supersaturation organise the broad pattern, but they do not by themselves explain the morphology; the full explanation also depends on transport (vapour/heat diffusion) and on attachment kinetics at the crystal surface.
By 2005 the two transport effects were well understood and could be used to model how they limit growth in controlled settings. The difficult part was attachment kinetics. This is where molecular-scale processes matter: how easily molecules join a facet, how nucleation proceeds on a surface, how dislocations and other step sources affect growth rates, and how impurities and imperfections can alter what the surface “allows”.[5]
Diffusion also helps explain why branching appears so readily. As a crystal grows, diffusion-limited conditions amplify small perturbations, so branches and side-branches can emerge even when the overall conditions are stable. Because growth is sensitive to local environment, abrupt changes—like movement through slightly different air— a change in the local constraints (temperature, supersaturation, airflow) can also trigger sudden shifts in form, such as arms sprouting from a plate-like crystal.[6]
Further support since 2005
A second development after 2005 is the increasing use of computer simulations to test whether constraint-and-kinetics stories can reproduce “hard” morphologies. By “hard” I mean stringent test cases: shapes that are difficult to generate from the stated constraints because small changes in conditions can flip the outcome.
Demange et al. (2016) present a modified 3D phase-field model intended to capture features such as anisotropic attachment, surface diffusion, and anisotropic surface tension, to reproduce faceting and dendritic growth. They make a strong claim that is directly relevant to the “law-like” temptation of the Nakaya diagram: “We demonstrate that this model reproduces the growth dynamics of the most challenging morphologies of snowflakes from the Nakaya diagram.”[7]
Alongside modelling, there is also a methodological push—especially visible in Libbrecht’s later work—to turn “habit” (shape categories) into numbers that can be fitted, compared, and checked. The aim is not just to say “plates tend to form near −15°C,” but to infer how basal and prism faces behave under specific conditions by growing crystals in controlled settings and fitting diffusion-plus-kinetics models to measured growth rates and shapes. The story becomes more conditional and more accountable: under these constraints, the basal surface behaves as if it has one attachment behaviour, while the prism surface behaves as if it has another.[8] In other words, the field is moving toward explanations that are more quantitative and predictive—but only within carefully specified regimes, not via a single universal law.
One standard way to frame facet growth is in terms of nucleation barriers and Classical Nucleation Theory (CNT) as a baseline idealisation, with the usual cautions about how far those idealisations travel.[9][10]
In the snow-crystal context, the relevant idea is often closer to layer (2D) nucleation on a facet—a way of describing why faceted growth can be slow and strongly temperature-dependent.
This is the background for what Libbrecht calls Structure-Dependent Attachment Kinetics (SDAK): the proposal that attachment is not only a function of temperature and supersaturation, but also of the mesoscale geometry of the facet itself.[8] In plain terms: a wide, flat facet behaves differently from a very narrow facet at an edge, because the local structure can change the effective barrier to adding new molecular layers. SDAK does not introduce a universal new law. It adds a further condition: edge structure can change the kinetics, and the same nucleation picture may not hold in every regime.
Metaphor (SDAK): Starting a new molecular layer is like trying to light a fire with newspaper: it’s easiest when you have a clean, exposed edge to catch, and harder when the paper is packed so there’s no stable edge to take. SDAK is the crystal version of that: a narrow facet near an edge can lower the “start-up cost” for adding a new layer, so attachment can speed up there even when temperature and supersaturation are unchanged.
Libbrecht’s 2019 quantitative physical model is an attempt to assemble these pieces into a semi-empirical account of the morphology diagram:[8] attachment kinetics depend on temperature, premelting behaviour, and facet width, and those dependencies can reproduce the broad features of the Nakaya diagram. This matters because it turns an old, puzzling observation—extremely thin plates near about −15°C, hollow columns in other regions—into a difference-maker story with explicit conditions: if edges reduce the nucleation barrier, then growth accelerates at edges, sharpening plates or hollowing columns.
Objection and Reply
Objection: Isn’t the “lawless” label misleading? Even if morphology looks contingent and regime-dependent, the physics is still lawful underneath. So perhaps the explanation is still law-shaped—we just haven’t yet written it in the right form at the scale we’re describing.
Reply: Read carefully: this is not a return to “laws.” It is a different kind of warrant. The point is not that a model is trustworthy only if it is derived from a law, nor that explanation runs only “bottom-up” from data. What matters is whether the model is answerable to the world: whether it survives confrontation with measurement in the ways that matter for the claim being made. In this case, a model earns trust when changing supersaturation changes growth in the predicted way; when it reproduces not only a visually plausible dendrite but the dynamics of growth; and when it has identifiable regimes where it stops working. That is expectation management: an explanation that travels by conditions, sensitivities, and failure modes, rather than by wearing a law label.
So What
The snowflake story is a reminder that science can move forward without discovering a neat law. It progresses by making claims more accountable: taking a qualitative morphology diagram and turning it into something you can measure, fit, compare, and—crucially—break. That is what gives the explanation its force.
Handoff
But the same success creates a pressure of its own. The more the models work, the more they seem to ask us to believe in things we never directly observe—attachment behavior, nucleation barriers, premelting effects, and the rest. So the next essay takes the natural step: should success make us realists? And if so, realists about what—about the unobservables themselves, or about the difference making constraints the model is reliably tracking?
Notes
- Johannes Kepler, The Six-Cornered Snowflake: A New Year’s Gift, trans. Jacques Bromberg (Philadelphia, PA: Paul Dry Books, 2010), accessed January 23, 2026, Internet Archive, https://archive.org/details/sixcorneredsnowf0000kepl_r8s6.
- Ross Anderson, “Why Induction Is a Problem – and What Scientists Do Anyway,” AfterCertainty, https://aftercertainty.net/index.php/why-induction-is-a-problem/.
- Kenneth G. Libbrecht, “The Physics of Snow Crystals,” Reports on Progress in Physics 68, no. 4 (2005): 860, fig. 2, doi:10.1088/0034-4885/68/4/R03.
- Kenneth G. Libbrecht, “The Physics of Snow Crystals,” Reports on Progress in Physics 68, no. 4 (2005): 855–859, doi:10.1088/0034-4885/68/4/R03.
- Libbrecht, “Physics of Snow Crystals,” 864–871.
- Libbrecht, “Physics of Snow Crystals,” 870–874; 878–881.
- G. Demange, H. Zapolsky, R. Patte, and M. Brunel, “Snowflake Growth in Three Dimensions Using Phase Field Modelling,” arXiv (November 10, 2016), arXiv:1611.03394, PDF, https://arxiv.org/pdf/1611.03394, 1.
- Kenneth G. Libbrecht, “A Quantitative Physical Model of the Snow Crystal Morphology Diagram,” arXiv (October 20, 2019), arXiv:1910.09067, PDF, https://arxiv.org/pdf/1910.09067.pdf.
- Ian Ford, Thermodynamics and Kinetics of Nucleation (lecture notes, Thomas Young Centre / University College London, March 2023), sec. 6.4, https://thomasyoungcentre.org/wp-content/uploads/2023/03/Nucleation_notes-Ian-Ford-March-2023.pdf.
- Aaron R. Finney and Matteo Salvalaglio, “Theoretical and Computational Approaches to Study Crystal Nucleation from Solution,” ChemRxiv (April 25, 2023), version 1, PDF, https://chemrxiv.org/engage/api-gateway/chemrxiv/assets/orp/resource/item/64472df2e4bbbe4bbf338572/original/theoretical-and-computational-approaches-to-study-crystal-nucleation-from-solution.pdf, 7.
Sources cited: Kepler (1611/2010), Libbrecht (2005; 2019), Demange et al. (2016), Ford (2023), Finney & Salvalaglio (2023).