FigurateNum

FigurateNum generates and visualizes 235+ plane, space, and multidimensional figurate number sequences

Documentation  ·  API Reference

From polygonal numbers to Platonic solids and higher-dimensional analogues, FigurateNum unifies 235+ sequences and families of figurate numbers, following Deza & Deza (2012).

Gallery

Phase portraits in ℂ and modular patterns in ℤₙ, generated with the code below.

Quickstart

Core sequence generators are stable. Visualization modules remain experimental.

# Stable release (sequence generators only)
pip install figuratenum

# Experimental visualization (ComplexViz and DiscreteViz)
# Requires: numpy, sympy, matplotlib
pip install figuratenum[figurate-viz]

Sequence generators

from figuratenum import FigurateNum

seq1 = FigurateNum().k_dim_centered_hypertetrahedron(4)
print([next(seq1) for _ in range(7)])
# [1, 7, 28, 210, 462, 923, 1709]

seq2 = FigurateNum().k_dim_hyperoctahedron(5)
print([next(seq2) for _ in range(7)])
# [1, 10, 51, 180, 501, 1182, 2471]

Visualization

• Phase portraits of generating functions f(z): ℂ → ℂ in the complex plane, based on Elias Wegert (2012).
• Modular patterns in ℤₙ, inspired by Rogelio Pérez Buendía (2025), generalized here for this family.

from figuratenum.figurate_viz import ComplexViz, DiscreteViz

# Phase portrait of the generating function f(z) in ℂ
c = ComplexViz()
c.visualize_multidim(
    "k_dim_centered_hypertetrahedron", k=4,
    cmap_color="cividis", disk_radius=2.0
)

# Modular pattern in ℤₙ
d = DiscreteViz()
d.visualize_multidim(
    "k_dim_hyperoctahedron", k=5, n_terms=704,
    circ_color="m", bg_color="k", num_text=False,
    num_color="g", ext_circle=True, rotate=-1
)

Mathematical background

Sequences are organized by dimension:

• Plane figurate numbers (2D): polygonal, centered polygonal, pronic, and related forms, including their generalized variants.
• Space figurate numbers (3D): pyramidal, polyhedral (tetrahedral, cubic, octahedral, dodecahedral, icosahedral), and centered space forms, including their generalized variants.
• Multidimensional figurate numbers (4D and beyond): hypertetrahedra, hypercubes, hyperoctahedra, and their centered variants for arbitrary dimension k, including generalized forms.

Documentation

Implementation Notes

FigurateNum uses optimized implementations of figurate number formulas for performance, while preserving mathematically equivalent results. Original expressions from Deza & Deza (2012) are available via *_from_book() methods for reference and validation.

Tests

Core sequence generators are validated against known closed-form values and classical results from number theory and OEIS.

Contributing

FigurateNum is under active development. Contributions are welcome: fork the project and submit a pull request.

• New sequences, with a mathematical reference (OEIS, book, or paper).
• Tests for existing sequences.
• Documentation improvements.
• Errata corrections at docs/errata/errata-figuratenum.tex.

When making commits, use conventional prefixes: feat, refactor, fix, docs, test.

Citation

If you use FigurateNum in your research, thesis, or project, please cite it: