FigurateNum

FigurateNum is a collection of 233 figurate number generators based on the book Figurate Numbers by Michel Deza and Elena Deza, published in 2012.

What is the purpose of FigurateNum?

FigurateNum facilitates the discovery of new patterns among sequences and enables various numerical calculations in mathematical projects and related applications. It can be integrated with other software to visualize the geometric objects described. Moreover, it serves as a valuable companion to the book.

How to install?

pip install figuratenum

🚨 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.

Features

FigurateNum generates the following categories of infinite sequences:

• 79 sequences of plane figurate numbers Explore all sequences
• 86 sequences of space figurate numbers Explore all sequences
• 68 sequences of multidimensional figurate numbers Explore all sequences

During the development of this package, errata were identified in Figurate Numbers (2012). The corresponding corrections are available here.

How to use?

1. Import all sequences via the FigurateNum class

from figuratenum import FigurateNum as fgn

>>> seq = fgn()
>>> hyperdodecahedral = seq.hyperdodecahedral()

>>> first = next(hyperdodecahedral)
>>> second = next(hyperdodecahedral)
>>> third = next(hyperdodecahedral)
>>> fourth = next(hyperdodecahedral)

>>> print(first, second, third, fourth)
1 600 4983 19468

2. Import sequences through specialized classes: Plane, Space, and Multidimensional

# from figuratenum import PlaneFigurateNum as pfgn
# from figuratenum import SpacedimensionalFigurateNum as sfgn
from figuratenum import MultidimensionalFigurateNum as mfgn

>>> seq_loop = mfgn()
>>> k_dimensional_centered_hypertetrahedron = seq_loop.k_dimensional_centered_hypertetrahedron(21)

>>> figuratenum_arr = []
>>> for _ in range(1, 15):
>>>     next_num = next(k_dimensional_centered_hypertetrahedron)
>>>     figuratenum_arr.append(next_num)

>>> print(figuratenum_arr)
[1, 23, 276, 2300, 14950, 80730, 376740, 1560780, 5852925, 20160075, 64512240, 193536720, 548354040, 1476337800]

3. Using the NumCollector class for sequence collection

from figuratenum import NumCollector as nc

Importing the NumCollector class allows you to use practical methods to return lists, tuples or arrays with the requested number of elements:

• take(n)
• take_to_list(stop, start, step)
• take_to_array(stop, start, step)
• take_to_tuple(stop, start, step)
• pick(n)

>>> seq = fgn()
>>> pentatope = seq.pentatope()

>>> print(nc.take_to_list(pentatope, 10))
[1, 5, 15, 35, 70, 126, 210, 330, 495, 715]

Plane figurate numbers

1. polygonal
2. triangular
3. square
4. pentagonal
5. hexagonal
6. heptagonal
7. octagonal
8. nonagonal
9. decagonal
10. hendecagonal
11. dodecagonal
12. tridecagonal
13. tetradecagonal
14. pentadecagonal
15. hexadecagonal
16. heptadecagonal
17. octadecagonal
18. nonadecagonal
19. icosagonal
20. icosihenagonal
21. icosidigonal
22. icositrigonal
23. icositetragonal
24. icosipentagonal
25. icosihexagonal
26. icosiheptagonal
27. icosioctagonal
28. icosinonagonal
29. triacontagonal
30. centered_triangular
31. centered_square = diamond numbers
32. centered_pentagonal
33. centered_hexagonal
34. centered_heptagonal
35. centered_octagonal
36. centered_nonagonal
37. centered_decagonal
38. centered_hendecagonal
39. centered_dodecagonal = star
40. centered_tridecagonal
41. centered_tetradecagonal
42. centered_pentadecagonal
43. centered_hexadecagonal
44. centered_heptadecagonal
45. centered_octadecagonal
46. centered_nonadecagonal
47. centered_icosagonal
48. centered_icosihenagonal
49. centered_icosidigonal
50. centered_icositrigonal
51. centered_icositetragonal
52. centered_icosipentagonal
53. centered_icosihexagonal
54. centered_icosiheptagonal
55. centered_icosioctagonal
56. centered_icosinonagonal
57. centered_triacontagonal
58. centered_mgonal(m)
59. pronic = heteromecic = oblong
60. polite
61. impolite
62. cross
63. aztec_diamond
64. polygram(m) = centered_star_polygonal(m)
65. pentagram
66. gnomic
67. truncated_triangular
68. truncated_square
69. truncated_pronic
70. truncated_centered_pol(m) = truncated_centered_mgonal(m)
71. truncated_centered_triangular
72. truncated_centered_square
73. truncated_centered_pentagonal
74. truncated_centered_hexagonal = truncated_hex
75. generalized_mgonal(m, start_numb)
76. generalized_pentagonal(start_numb)
77. generalized_hexagonal(start_numb)
78. generalized_centered_pol(m, start_numb)
79. generalized_pronic(start_numb)

Space figurate numbers

1. m_pyramidal(m)
2. triangular_pyramidal
3. square_pyramidal = pyramidal
4. pentagonal_pyramidal
5. hexagonal_pyramidal
6. heptagonal_pyramidal
7. octagonal_pyramidal
8. nonagonal_pyramidal
9. decagonal_pyramidal
10. hendecagonal_pyramidal
11. dodecagonal_pyramidal
12. tridecagonal_pyramidal
13. tetradecagonal_pyramidal
14. pentadecagonal_pyramidal
15. hexadecagonal_pyramidal
16. heptadecagonal_pyramidal
17. octadecagonal_pyramidal
18. nonadecagonal_pyramidal
19. icosagonal_pyramidal
20. icosihenagonal_pyramidal
21. icosidigonal_pyramidal
22. icositrigonal_pyramidal
23. icositetragonal_pyramidal
24. icosipentagonal_pyramidal
25. icosihexagonal_pyramidal
26. icosiheptagonal_pyramidal
27. icosioctagonal_pyramidal
28. icosinonagonal_pyramidal
29. triacontagonal_pyramidal
30. triangular_tetrahedral[finite]
31. triangular_square_pyramidal[finite]
32. square_tetrahedral[finite]
33. square_square_pyramidal[finite]
34. tetrahedral_square_pyramidal[finite]
35. cubic
36. tetrahedral
37. octahedral
38. dodecahedral
39. icosahedral
40. truncated_tetrahedral
41. truncated_cubic
42. truncated_octahedral
43. stella_octangula
44. centered_cube
45. rhombic_dodecahedral
46. hauy_rhombic_dodecahedral
47. centered_tetrahedron = centered_tetrahedral
48. centered_square_pyramid = centered_pyramid
49. centered_mgonal_pyramid(m)
50. centered_pentagonal_pyramid
51. centered_hexagonal_pyramid
52. centered_heptagonal_pyramid
53. centered_octagonal_pyramid
54. centered_octahedron
55. centered_icosahedron = centered_cuboctahedron
56. centered_dodecahedron
57. centered_truncated_tetrahedron
58. centered_truncated_cube
59. centered_truncated_octahedron
60. centered_mgonal_pyramidal(m)
61. centered_triangular_pyramidal
62. centered_square_pyramidal
63. centered_pentagonal_pyramidal
64. centered_heptagonal_pyramidal
65. centered_octagonal_pyramidal
66. centered_nonagonal_pyramidal
67. centered_decagonal_pyramidal
68. centered_hendecagonal_pyramidal
69. centered_dodecagonal_pyramidal
70. centered_hexagonal_pyramidal = hex_pyramidal
71. hexagonal_prism
72. mgonal_prism(m)
73. generalized_mgonal_pyramidal(m, start_num)
74. generalized_pentagonal_pyramidal(start_num)
75. generalized_hexagonal_pyramidal(start_num)
76. generalized_cubic(start_num)
77. generalized_octahedral(start_num)
78. generalized_icosahedral(start_num)
79. generalized_dodecahedral(start_num)
80. generalized_centered_cube(start_num)
81. generalized_centered_tetrahedron(start_num)
82. generalized_centered_square_pyramid(start_num)
83. generalized_rhombic_dodecahedral(start_num)
84. generalized_centered_mgonal_pyramidal(m, start_num)
85. generalized_mgonal_prism(m, start_num)
86. generalized_hexagonal_prism(start_num)

Multidimensional figurate numbers

1. k_dimensional_hypertetrahedron(k) = k_hypertetrahedron(k) = regular_k_polytopic(k) = figurate_of_order_k(k)
2. five_dimensional_hypertetrahedron
3. six_dimensional_hypertetrahedron
4. k_dimensional_hypercube(k) = k_hypercube(k)
5. five_dimensional_hypercube
6. six_dimensional_hypercube
7. hypertetrahedral = pentachoron = pentatope = triangulotriangular = cell_5
8. hypercube = octachoron = tesseract = biquadratic = cell_8
9. hyperoctahedral = hexadecachoron = four_cross_polytope = four_orthoplex = cell_16
10. hypericosahedral = hexacosichoron = polytetrahedron = tetraplex = cell_600
11. hyperdodecahedral = hecatonicosachoron = dodecaplex = polydodecahedron = cell_120
12. polyoctahedral = icositetrachoron = octaplex = hyperdiamond = cell_24
13. four_dimensional_hyperoctahedron
14. five_dimensional_hyperoctahedron
15. six_dimensional_hyperoctahedron
16. seven_dimensional_hyperoctahedron
17. eight_dimensional_hyperoctahedron
18. nine_dimensional_hyperoctahedron
19. ten_dimensional_hyperoctahedron
20. k_dimensional_hyperoctahedron(k) = k_cross_polytope(k)
21. four_dimensional_mgonal_pyramidal(m) = mgonal_pyramidal_of_the_second_order(m)
22. four_dimensional_square_pyramidal
23. four_dimensional_pentagonal_pyramidal
24. four_dimensional_hexagonal_pyramidal
25. four_dimensional_heptagonal_pyramidal
26. four_dimensional_octagonal_pyramidal
27. four_dimensional_nonagonal_pyramidal
28. four_dimensional_decagonal_pyramidal
29. four_dimensional_hendecagonal_pyramidal
30. four_dimensional_dodecagonal_pyramidal
31. k_dimensional_mgonal_pyramidal(k, m) = mgonal_pyramidal_of_the_k_2_th_order(k, m)
32. five_dimensional_mgonal_pyramidal(m)
33. five_dimensional_square_pyramidal
34. five_dimensional_pentagonal_pyramidal
35. five_dimensional_hexagonal_pyramidal
36. five_dimensional_heptagonal_pyramidal
37. five_dimensional_octagonal_pyramidal
38. six_dimensional_mgonal_pyramidal(m)
39. six_dimensional_square_pyramidal
40. six_dimensional_pentagonal_pyramidal
41. six_dimensional_hexagonal_pyramidal
42. six_dimensional_heptagonal_pyramidal
43. six_dimensional_octagonal_pyramidal
44. centered_biquadratic
45. k_dimensional_centered_hypercube(k)
46. five_dimensional_centered_hypercube
47. six_dimensional_centered_hypercube
48. centered_polytope
49. k_dimensional_centered_hypertetrahedron(k)
50. five_dimensional_centered_hypertetrahedron
51. six_dimensional_centered_hypertetrahedron
52. centered_hyperoctahedral = orthoplex
53. nexus(k)
54. k_dimensional_centered_hyperoctahedron(k)
55. five_dimensional_centered_hyperoctahedron
56. six_dimensional_centered_hyperoctahedron
57. generalized_pentatope(start_num = 0)
58. generalized_k_dimensional_hypertetrahedron(k = 5, start_num = 0)
59. generalized_biquadratic(start_num = 0)
60. generalized_k_dimensional_hypercube(k = 5, start_num = 0)
61. generalized_hyperoctahedral(start_num = 0)
62. generalized_k_dimensional_hyperoctahedron(k = 5, start_num = 0)
63. generalized_hyperdodecahedral(start_num = 0)
64. generalized_hypericosahedral(start_num = 0)
65. generalized_polyoctahedral(start_num = 0)
66. generalized_k_dimensional_mgonal_pyramidal(k, m, start_num = 0)
67. generalized_k_dimensional_centered_hypercube(k, start_num = 0)
68. generalized_nexus(start_num = 0)

Errata for Figurate Numbers (2012)

This 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 - 13 * n + 1	546, 728, 936, 1170

  It should be:

  Name	Formula
  Cent. icosihexagonal	1/3n^2 - 13 * n + 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

Contributing

FigurateNumber is currently under development, and we warmly invite your contributions. Just fork the project and then submit a pull request:

• Sequences from Chapters 1, 2, and 3 of the book
• New sequences not included in the book: If you have new sequences, please provide the source.
• Tests, documentation and errata in the book

When making commits, please use the following conventional prefixes to indicate the nature of the changes: feat, refactor, fix, docs, and test.