figuratenum
Advanced tools
Generate 235 infinite figurate number sequences across multiple dimensions for mathematical research, applications, and exploration in Python.
FigurateNum is a collection of 235 figurate number generators based on the book Figurate Numbers by Michel Deza and Elena Deza, published in 2012.
FigurateNum helps discover patterns in figurate number sequences and supports numerical computation in mathematics-related projects. It integrates with other tools for visualizing geometric structures and serves as a companion to the book.
pip install figuratenum
Enable the FigurateViz class (requires numpy and matplotlib) by installing the optional dependencies:
pip install figuratenum[figurate-viz]
The main class, FigurateNum, provides access to all figurate number sequences across different dimensions, while dedicated classes let you work with each dimension separately:
PlaneFigurateNum class — Explore on GitHubSpaceFigurateNum class — Explore on GitHubMultidimensionalFigurateNum class — Explore on GitHubZooFigurateNum class — Explore on GitHublist[int] or tuple[int, ...]).FigurateNum and related classesfrom figuratenum import FigurateNum, MultidimensionalFigurateNum
# 1. General use: generate any figurate sequence via FigurateNum
seq = FigurateNum()
hyperdodecahedral_gen = seq.hyperdodecahedral()
print([next(hyperdodecahedral_gen) for _ in range(4)])
# Output: [1, 600, 4983, 19468]
# 2. Specialized classes: PlaneFigurateNum, SpaceFigurateNum,
# MultidimensionalFigurateNum, ZooFigurateNum
multi = MultidimensionalFigurateNum()
hypertetrahedron_gen = multi.k_dimensional_centered_hypertetrahedron(21)
print([next(hypertetrahedron_gen) for _ in range(12)])
# Output: [1, 23, 276, 2300, 14950, 80730, 376740,
# 1560780, 5852925, 20160075, 64512240, 193536720]
FigurateViz to visualize and exportfrom figuratenum import FigurateNum as fgn
from figuratenum.figurate_viz.FigurateViz import FigurateViz
# Generate figurate numbers
seq_loop = fgn()
gen = seq_loop.five_dimensional_hyperoctahedron()
figuratenum_seq = [next(gen) for _ in range(704)]
# Create and draw the Gaussian plot
viz = FigurateViz(figuratenum_seq, figsize=(6, 6))
viz.gaussian_plot(
circ_color="m", bg_color="k", num_text=False,
num_color="g", ext_circle=True, rotate=-1
).draw()
# Export plots as .svg, .pdf, .png (matplotlib compatible),
# with options e.g., dpi, transparent, bbox_inches, pad_inches, etc.
viz.export_plot(
"figure1.svg", circ_color="cyan",
dpi=300, transparent=True
)
NumCollectorfrom figuratenum import NumCollector as nc, FigurateNum
gen = FigurateNum().pentatope()
print(nc.take_to_tuple(gen, 10)) # first 10 values as tuple
# Output: (1, 5, 15, 35, 70, 126, 210, 330, 495, 715)
# Available methods:
# - take(n) : first n values as iterator
# - take_to_list(stop, start=0, step=1)
# - take_to_tuple(stop, start=0, step=1)
# - take_to_array(stop, start=0, step=1)
# - pick(n) : nth value
🚨 Version 2.0.0 includes renamed methods and changes in class usage. These changes are incompatible with previous versions. Please review the updated usage instructions below to adapt your code to the new structure.
PlaneFigurateNum classpolygonaltriangularsquarepentagonalhexagonalheptagonaloctagonalnonagonaldecagonalhendecagonaldodecagonaltridecagonaltetradecagonalpentadecagonalhexadecagonalheptadecagonaloctadecagonalnonadecagonalicosagonalicosihenagonalicosidigonalicositrigonalicositetragonalicosipentagonalicosihexagonalicosiheptagonalicosioctagonalicosinonagonaltriacontagonalcentered_triangularcentered_square = diamondcentered_pentagonalcentered_hexagonalcentered_heptagonalcentered_octagonalcentered_nonagonalcentered_decagonalcentered_hendecagonalcentered_dodecagonal = starcentered_tridecagonalcentered_tetradecagonalcentered_pentadecagonalcentered_hexadecagonalcentered_heptadecagonalcentered_octadecagonalcentered_nonadecagonalcentered_icosagonalcentered_icosihenagonalcentered_icosidigonalcentered_icositrigonalcentered_icositetragonalcentered_icosipentagonalcentered_icosihexagonalcentered_icosiheptagonalcentered_icosioctagonalcentered_icosinonagonalcentered_triacontagonalcentered_mgonal(m)pronic = heteromecic = oblongpoliteimpolitecrossaztec_diamondpolygram(m) = centered_star_polygonal(m)pentagramgnomictruncated_triangulartruncated_squaretruncated_pronictruncated_centered_pol(m) = truncated_centered_mgonal(m)truncated_centered_triangulartruncated_centered_squaretruncated_centered_pentagonaltruncated_centered_hexagonal = truncated_hexgeneralized_mgonal(m, start_numb)generalized_pentagonal(start_numb)generalized_hexagonal(start_numb)generalized_centered_pol(m, start_numb)generalized_pronic(start_numb)SpaceFigurateNum classm_pyramidal(m)triangular_pyramidalsquare_pyramidal = pyramidalpentagonal_pyramidalhexagonal_pyramidalheptagonal_pyramidaloctagonal_pyramidalnonagonal_pyramidaldecagonal_pyramidalhendecagonal_pyramidaldodecagonal_pyramidaltridecagonal_pyramidaltetradecagonal_pyramidalpentadecagonal_pyramidalhexadecagonal_pyramidalheptadecagonal_pyramidaloctadecagonal_pyramidalnonadecagonal_pyramidalicosagonal_pyramidalicosihenagonal_pyramidalicosidigonal_pyramidalicositrigonal_pyramidalicositetragonal_pyramidalicosipentagonal_pyramidalicosihexagonal_pyramidalicosiheptagonal_pyramidalicosioctagonal_pyramidalicosinonagonal_pyramidaltriacontagonal_pyramidaltriangular_tetrahedral[finite]triangular_square_pyramidal[finite]square_tetrahedral[finite]square_square_pyramidal[finite]tetrahedral_square_pyramidal[finite]cubictetrahedraloctahedraldodecahedralicosahedraltruncated_tetrahedraltruncated_cubictruncated_octahedralstella_octangulacentered_cuberhombic_dodecahedralhauy_rhombic_dodecahedralcentered_tetrahedron = centered_tetrahedralcentered_square_pyramid = centered_pyramidcentered_mgonal_pyramid(m)centered_pentagonal_pyramidcentered_hexagonal_pyramidcentered_heptagonal_pyramidcentered_octagonal_pyramidcentered_octahedroncentered_icosahedron = centered_cuboctahedroncentered_dodecahedroncentered_truncated_tetrahedroncentered_truncated_cubecentered_truncated_octahedroncentered_mgonal_pyramidal(m)centered_triangular_pyramidalcentered_square_pyramidalcentered_pentagonal_pyramidalcentered_heptagonal_pyramidalcentered_octagonal_pyramidalcentered_nonagonal_pyramidalcentered_decagonal_pyramidalcentered_hendecagonal_pyramidalcentered_dodecagonal_pyramidalcentered_hexagonal_pyramidal = hex_pyramidalhexagonal_prismmgonal_prism(m)generalized_mgonal_pyramidal(m, start_num)generalized_pentagonal_pyramidal(start_num)generalized_hexagonal_pyramidal(start_num)generalized_cubic(start_num)generalized_octahedral(start_num)generalized_icosahedral(start_num)generalized_dodecahedral(start_num)generalized_centered_cube(start_num)generalized_centered_tetrahedron(start_num)generalized_centered_square_pyramid(start_num)generalized_rhombic_dodecahedral(start_num)generalized_centered_mgonal_pyramidal(m, start_num)generalized_mgonal_prism(m, start_num)generalized_hexagonal_prism(start_num)MultidimensionalFigurateNum classk_dimensional_hypertetrahedron(k) = k_hypertetrahedron(k) = regular_k_polytopic(k) = figurate_of_order_k(k)five_dimensional_hypertetrahedronsix_dimensional_hypertetrahedronk_dimensional_hypercube(k) = k_hypercube(k)five_dimensional_hypercubesix_dimensional_hypercubehypertetrahedral = pentachoron = pentatope = triangulotriangular = cell_5hypercube = octachoron = tesseract = biquadratic = cell_8hyperoctahedral = hexadecachoron = four_cross_polytope = four_orthoplex = cell_16hypericosahedral = hexacosichoron = polytetrahedron = tetraplex = cell_600hyperdodecahedral = hecatonicosachoron = dodecaplex = polydodecahedron = cell_120polyoctahedral = icositetrachoron = octaplex = hyperdiamond = cell_24four_dimensional_hyperoctahedronfive_dimensional_hyperoctahedronsix_dimensional_hyperoctahedronseven_dimensional_hyperoctahedroneight_dimensional_hyperoctahedronnine_dimensional_hyperoctahedronten_dimensional_hyperoctahedronk_dimensional_hyperoctahedron(k) = k_cross_polytope(k)four_dimensional_mgonal_pyramidal(m) = mgonal_pyramidal_of_the_second_order(m)four_dimensional_square_pyramidalfour_dimensional_pentagonal_pyramidalfour_dimensional_hexagonal_pyramidalfour_dimensional_heptagonal_pyramidalfour_dimensional_octagonal_pyramidalfour_dimensional_nonagonal_pyramidalfour_dimensional_decagonal_pyramidalfour_dimensional_hendecagonal_pyramidalfour_dimensional_dodecagonal_pyramidalk_dimensional_mgonal_pyramidal(k, m) = mgonal_pyramidal_of_the_k_2_th_order(k, m)five_dimensional_mgonal_pyramidal(m)five_dimensional_square_pyramidalfive_dimensional_pentagonal_pyramidalfive_dimensional_hexagonal_pyramidalfive_dimensional_heptagonal_pyramidalfive_dimensional_octagonal_pyramidalsix_dimensional_mgonal_pyramidal(m)six_dimensional_square_pyramidalsix_dimensional_pentagonal_pyramidalsix_dimensional_hexagonal_pyramidalsix_dimensional_heptagonal_pyramidalsix_dimensional_octagonal_pyramidalcentered_biquadratick_dimensional_centered_hypercube(k)five_dimensional_centered_hypercubesix_dimensional_centered_hypercubecentered_polytopek_dimensional_centered_hypertetrahedron(k)five_dimensional_centered_hypertetrahedronsix_dimensional_centered_hypertetrahedroncentered_hyperoctahedral = orthoplexnexus(k)k_dimensional_centered_hyperoctahedron(k)five_dimensional_centered_hyperoctahedronsix_dimensional_centered_hyperoctahedrongeneralized_pentatope(start_num = 0)generalized_k_dimensional_hypertetrahedron(k = 5, start_num = 0)generalized_biquadratic(start_num = 0)generalized_k_dimensional_hypercube(k = 5, start_num = 0)generalized_hyperoctahedral(start_num = 0)generalized_k_dimensional_hyperoctahedron(k = 5, start_num = 0)generalized_hyperdodecahedral(start_num = 0)generalized_hypericosahedral(start_num = 0)generalized_polyoctahedral(start_num = 0)generalized_k_dimensional_mgonal_pyramidal(k, m, start_num = 0)generalized_k_dimensional_centered_hypercube(k, start_num = 0)generalized_nexus(k, start_num = 0)ZooFigurateNum classcuban_primepellThis section lists the errata and corrections for the book Figurate Numbers (2012) by Michel Deza and Elena Deza. If you find any errors in the content, please feel free to contribute corrections.
Chapter 1, formula in the table on page 6 says:
| Name | Formula | |
|---|---|---|
| Square | 1/2 (n^2 - 0 * n) |
It should be:
| Name | Formula | |
|---|---|---|
| Square | 1/2 (2n^2 - 0 * n) |
Chapter 1, formula in the table on page 51 says:
| Name | Formula | |
|---|---|---|
| Cent. icosihexagonal | 1/3n^2 - 13n + 1 | 546, 728, 936, 1170 |
It should be:
| Name | Formula | |
|---|---|---|
| Cent. icosihexagonal | 13n^2 - 13n + 1 | 547, 729, 937, 1171 |
Chapter 1, formula in the table on page 51 says:
| Name | Formula | |
|---|---|---|
| Cent. icosiheptagonal | 972 |
It should be:
| Name | Formula | |
|---|---|---|
| Cent. icosiheptagonal | 973 |
Chapter 1, formula in the table on page 51 says:
| Name | Formula | |
|---|---|---|
| Cent. icosioctagonal | 84 |
It should be:
| Name | Formula | |
|---|---|---|
| Cent. icosioctagonal | 85 |
Chapter 1, page 65 (polite numbers) says:
inpolite numbers
It should read:
impolite numbers
Chapter 1, formula (truncated centered pentagonal numbers) on page 72 says:
TCSS_5(n) = (35n^2 - 55n) / 2 + 3
It should be:
TCSS_5(n) = (35n^2 - 55n) / 2 + 11
Chapter 2, formula of octagonal pyramidal number on page 92 says:
n(n+1)(6n-1) / 6
It should be:
n(n+1)(6n-3) / 6
Chapter 2, page 140 says:
centered square pyramidal numbers are 1, 6, 19, 44, 85, 111, 146, 231, ...
This sequence must exclude the number 111:
centered square pyramidal numbers are 1, 6, 19, 44, 85,
111, 146, 231, ...
Chapter 2, page 155 (generalized centered tetrahedron numbers) says:
S_3^3(n) = ((2n - 1)(n^2 + n + 3)) / 3
Formula must have a negative sign:
S_3^3(n) = ((2n - 1)(n^2 - n + 3)) / 3
Chapter 2, page 156 (generalized centered square pyramid numbers) says:
S_4^3(n) = ((2n - 1)(n^2 - n + 2)^2) / 3
Formula must write:
S_4^3(n) = ((2n - 1)(n^2 - n + 2)) / 2
Chapter 3, page 188 (hyperoctahedral numbers) says:
hexadecahoron numbers
It should read:
hexadecachoron numbers
Chapter 3, page 190 (hypericosahedral numbers) says:
hexacisihoron numbers
It should read:
hexacosichoron numbers
FigurateNum is currently under development, and we warmly invite your contributions. Just fork the project and then submit a pull request:
When making commits, please use the following conventional prefixes to indicate the nature of the changes: feat, refactor, fix, docs, and test.
FAQs
Generate 235 infinite figurate number sequences across multiple dimensions for mathematical research, applications, and exploration in Python.
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