1

I. SUPPLEMENTARY MATERIAL 1:COMPARISON OF SIMULATIONS WITH REAL

SNOWFLAKES

(a) Simple star from [1] (b) Simple star

(c) Stellar dendrite from [1] (d) Stellar dendrite

(e) Sectored plate from [1] (f) Sectored plate

FIG. 1. Comparison between real snowflakes photographs(left) taken from [1], and our phase field simulations (right).Visual rendering for our simulations uses the softwareBlender.

SUPPLEMENTARY MATERIAL 2: 3DFACETING ALGORITHM

Faceting requires the anisotropy constants εxy and εzvalues in A(n), to exceed 1/35 for 6-fold horizontal sym-metry, and 1/3 for 2-fold vertical symmetry [2]. Abovesuch values, metastable and unstable crystal orientationsn where the stiffness becomes negative, emerge. A(n)must hence be modified, so that the stiffness remainspositive. A preliminary step is to determine missing ori-

entations. This can be done by detecting sign change inthe stiffness. This causes a local convexity inversion inthe 2D polar plot of 1/A(θ, φ0), for any fixed azimuthalangle φ0, or in the polar plot of 1/A(θ0, φ) for any fixedpolar angle θ0. This reads A+∂2θθA = 0 and A+∂2φφA = 0respectively, providing us with the maximum anisotropyconstants εmxy and εmz before missing orientation:

εmxy =

1 + εz cos(2φ)

35

εmz =1 + εxy cos(6θ)

3.

(1)

Restricting the study to θ ∈ [−π/6, π/6] and φ ∈ [0, π]due to rotation periodicity of A(θ, φ), missing θ orienta-tions lie within a φ-dependant angular sector [−θm(φ) +kπ/3, θm(φ) + kπ/3] (k ∈ Z) for εxy > εmxy. Same goesfor φ orientations lying within [−φm(θ) + kπ, φm(θ) +kπ] for εz > εmz . Following the procedure developedby Eggleston in [3], θm and φm respectively satisfy∂θ[cos(θ)/A(θ, φ)] = 0 and ∂φ[cos(φ)/A(θ, φ)] = 0, lead-ing to:

6εxy sin(6θm) cos(θm)

sin(θm)= (1 + εxy cos(6θm) + εzcos(2φ))

2εz sin(2φm) cos(φm)

sin(φm)= (1 + εxy cos(6θ) + εzcos(2φ

m)) .

(2)The regularization step finally consists in replacingA(θ, φ) by a simple trigonometric function for missingorientation, as suggested by Debierre et al. [4]:

A(n) =

Aθ(θ, φ) = A1(φ) +B1 cos(θ), |θ| < θm, |φ| ≥ φm

Aφ(θ, φ) = A2(θ) +B2 cos(φ), |θ| ≥ θm, |φ| < φm

α(θ, φ)Aθ + (1− α(θ, φ))Aφ, |θ| < θm, |φ| < φm

1 + εxy cos(6θ) + εz cos(2ψ) otherwise.(3)

Coefficients A1(φ), B1, A2(θ), B2 and α(θ, φ) are set toensure continuity of A(n) and its derivatives:

A1(φ) = 1 + εxy cos(6θm) + εz cos(2φ)− 6εxy sin(6θm)

sin(θm)cos(θm)

B1 =6εxy sin(6θm)

sin(θm)

A2(θ) = 1 + εxy cos(6θ) + εz cos(2φm)− 2εz sin(2φm)

sin(φm)cos(φm)

B2 =2εz sin(2φm)

sin(φm)

α(θ, φ) =|θ − θm|√

(θ − θm)2 + (φ− φm)2.

(4)

2

[1] K. G. Libbrecht, “Snowcrystals.com,” http://www.

snowcrystals.com.[2] I. Singer-Loginova and H. Singer, Reports on progress in

physics 71, 106501 (2008).

[3] J. J. Eggleston, G. B. McFadden, and P. W. Voorhees,Physica D: Nonlinear Phenomena 150, 91 (2001).

[4] J.-M. Debierre, A. Karma, F. Celestini, and R. Guerin,Physical Review E 68, 041604 (2003).