Modular Patterns

DiscreteViz reduces a sequence modulo $n$ and traces the result as an orbit on a circle. Each term $a_k \bmod n$ lands on one of $n$ positions, and consecutive terms are joined by an edge $a_k \to a_{k+1}$. This was inspired by Rogelio Pérez Buendía (2025).

from figuratenum.figurate_viz import DiscreteViz

d = DiscreteViz()
d.visualize_plane("polygonal", m=17, n_terms=330, circ_color="g", bg_color="k")

Three methods, one per dimension

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

d.visualize_space("icosahedral", n_terms=300)

d.visualize_multidim(
    "k_dim_hyperoctahedron", k=5, n_terms=704,
    circ_color="m", bg_color="k", ext_circle=True, rotate=-1
)

visualize_multidim takes k to select the dimension; all three take m instead, for sequences with an $m$-gonal shape.

Interpretation

Symmetry in the orbit depends on the modulus $n$, not on the dimension of the family: a 2D sequence can produce an irregular pattern under one $n$ and a spirograph-like pattern under another, and the same holds for $k$-dimensional sequences.

Trying different values of n_terms and h_modulo on the same sequence is the way to find where the structure shows up.

Parameters

• figuratenum_name: the sequence, passed positionally.
• m: m-gonal parameter, for sequences with that structure.
• k: dimension parameter (visualize_multidim only).
• n_terms: number of terms generated and plotted.
• h_modulo: the modulus $n$ for the circular mapping. Defaults to the sequence length.
• circ_color, bg_color, num_color: color of orbit edges, background, and position labels.
• num_text: whether numeric labels are drawn at each position.
• ext_circle: whether an outer reference circle is drawn.
• rotate: rotational offset applied to the layout.
• show: display immediately (default True) or return the figure unshown.