Example 6: a smooth \(C_3\)-symmetric localized field

PETAL2D is naturally suited to analyzing planar rotational symmetry around a chosen center. Consider

\[ f(x,y)=e^{-\alpha(x^2+y^2)} \left[1+\beta(x^3-3xy^2)\right]. \]

Because

\[ x^3-3xy^2=r^3\cos3\theta, \]

the anisotropic contribution is smooth at the origin and contains only the conjugate angular pair \(m=\pm3\).

Run:

python examples/c3_localized_field.py

=== Smooth C3-symmetric localized field ===
domain_consistency: 0.99977646
selected_pairs: [(0,), (3, -3)]
retained_modes: [0, 3, -3]
reconstruction_error_percent: 2.4606e-14
target_reached: True
     m           power      fraction   r_power_support   r_amplitude_support      r_cutoff  retained
----------------------------------------------------------------------------------------------------
     0    5.554281e-01  9.743327e-01           3.91976               3.91837       3.91976      True
     3    7.315958e-03  1.283366e-02           4.87289               5.04603       5.04603      True
    -3    7.315958e-03  1.283366e-02           4.87289               5.04603       5.04603      True

This is the simplest explicit illustration of the \(C_n\) selection rule: an exactly \(C_3\)-invariant scalar field can contain only harmonics with \(m=0\pmod 3\).

In a crystal calculation, weak \(m=\pm6,\pm9,\ldots\) channels may encode additional \(C_3\)-compatible angular structure rather than symmetry breaking. By contrast, robust converged \(m=\pm1\) or \(\pm2\) content would be incompatible with exact \(C_3\) invariance about the chosen center.

The spectrum contains \(m=0\) and the real conjugate pair \(m=\pm3\).

Original field and retained-mode reconstruction.