Phase Portraits

Instead of the sequence itself, ComplexViz looks at its generating function $f(z)$ over the complex plane $\mathbb{C}$, exposing global structure that individual terms don’t reveal on their own. The technique is domain coloring, producing a phase portrait following Wegert (2012).

from figuratenum.figurate_viz import ComplexViz

c = ComplexViz()
c.visualize_plane("triangular", cmap_color="viridis", disk_radius=2.0)

Three methods, one per dimension

visualize_plane handles 2D sequences, visualize_space handles 3D, and visualize_multidim handles k-dimensional and generalized families.

c.visualize_space(
    "tetrahedral",
    plot_type="modulus_contours",
    xlim=(-0.5, 0.5), ylim=(-0.5, 0.5), disk_radius=0.5
)

c.visualize_multidim(
    "k_dim_centered_hypertetrahedron", k=4,
    cmap_color="cividis", disk_radius=2.0
)

visualize_multidim takes k to select the dimension; the other two take m instead, for sequences with an $m$-gonal geometry.

Interpretation

The portrait encodes phase and magnitude through color, following Wegert (2012). Certain regions stand out immediately, where $f(z)$ changes rapidly. For generating functions of figurate numbers, this behavior is typically observed near $z = 1$. See Theory for a more detailed explanation.

Main controls

name_seq selects the sequence; m and k select which member of a parametric or k-dimensional family, depending on the method.

Beyond that, cmap_color and disk_radius are the two parameters that most change what a portrait looks like: the colormap shapes how phase is rendered, and the plot domain itself is a rectangle set by domain_grid_radius in the constructor, with disk_radius cropping it to a circular mask.

Parameters

• name_seq: the sequence, passed positionally.
• m: $m$-gonal parameter (visualize_plane, visualize_space, visualize_multidim).
• k: dimension parameter (visualize_multidim only).
• plot_type: "pure_phase_portrait", "phase_contours", "modulus_contours", or "enhanced_phase_portrait" (default).
• cmap_color: colormap, e.g. "hsv" (default), "cividis", "twilight".
• xlim, ylim: explicit rectangular limits, overriding domain_grid_radius.
• disk_radius: radius of the circular mask over rectangular domain.
• show_axes: whether to display the $\Re$ or $\Im$ axes.
• show: display immediately (default True) or return the figure unshown.