Figurate Numbers

figurate_numbers is the most comprehensive and specialized gem for figurate numbers written in Ruby to date. It implements 241 infinite number sequences based on the remarkable book

Figurate Numbers by Elena Deza and Michel Deza, published in 2012.

This implementation uses the Enumerator class to handle infinite sequences. It is intended for use in mathematical projects and with Sonic Pi.

Following the order of the book, the methods are divided into 3 types according to the spatial dimension (see complete list below):

1. Plane figurate numbers implemented = 79
2. Space figurate numbers implemented = 86
3. Multidimensional figurate numbers implemented = 70
4. Zoo of figurate-related numbers implemented = 6

• TOTAL = 241 infinite sequences of figurate numbers implemented

Installation and use

• gem install figurate_numbers

How to use in Ruby

If the sequence is defined with lazy, to make the numbers explicit we must include the converter method to_a at the end.

require 'figurate_numbers'

## Using take(integer)
FigurateNumbers.pronic_numbers.take(10).to_a

## Storing and iterating
f = FigurateNumbers.centered_octagonal_pyramid_numbers
f.next
f.next
f.next

How to use in Sonic Pi

1. Locate or download the file in the path lib/figurate_numbers.rb
2. Drag the file to a buffer in Sonic Pi (this generates the <PATH>)

run_file "<PATH>"

pol_num = FigurateNumbers.polygonal_numbers(8)
80.times do
  play pol_num.next % 12 * 7  # Some mathematical function or transformation
  sleep 0.25
end

List of implemented sequences

• Note that = means that you can call the same sequence with different names.

1. Plane Figurate Numbers

1. polygonal_numbers(m)
2. triangular_numbers
3. square_numbers
4. pentagonal_numbers
5. hexagonal_numbers
6. heptagonal_numbers
7. octagonal_numbers
8. nonagonal_numbers
9. decagonal_numbers
10. hendecagonal_numbers
11. dodecagonal_numbers
12. tridecagonal_numbers
13. tetradecagonal_numbers
14. pentadecagonal_numbers
15. hexadecagonal_numbers
16. heptadecagonal_numbers
17. octadecagonal_numbers
18. nonadecagonal_numbers
19. icosagonal_numbers
20. icosihenagonal_numbers
21. icosidigonal_numbers
22. icositrigonal_numbers
23. icositetragonal_numbers
24. icosipentagonal_numbers
25. icosihexagonal_numbers
26. icosiheptagonal_numbers
27. icosioctagonal_numbers
28. icosinonagonal_numbers
29. triacontagonal_numbers
30. centered_triangular_numbers
31. centered_square_numbers = diamond_numbers (equality only by quantity)
32. centered_pentagonal_numbers
33. centered_hexagonal_numbers
34. centered_heptagonal_numbers
35. centered_octagonal_numbers
36. centered_nonagonal_numbers
37. centered_decagonal_numbers
38. centered_hendecagonal_numbers
39. centered_dodecagonal_numbers = star_numbers (equality only by quantity)
40. centered_tridecagonal_numbers
41. centered_tetradecagonal_numbers
42. centered_pentadecagonal_numbers
43. centered_hexadecagonal_numbers
44. centered_heptadecagonal_numbers
45. centered_octadecagonal_numbers
46. centered_nonadecagonal_numbers
47. centered_icosagonal_numbers
48. centered_icosihenagonal_numbers
49. centered_icosidigonal_numbers
50. centered_icositrigonal_numbers
51. centered_icositetragonal_numbers
52. centered_icosipentagonal_numbers
53. centered_icosihexagonal_numbers
54. centered_icosiheptagonal_numbers
55. centered_icosioctagonal_numbers
56. centered_icosinonagonal_numbers
57. centered_triacontagonal_numbers
58. centered_mgonal_numbers(m)
59. pronic_numbers = heteromecic_numbers = oblong_numbers
60. polite_numbers
61. impolite_numbers
62. cross_numbers
63. aztec_diamond_numbers
64. polygram_numbers(m) = centered_star_polygonal_numbers(m)
65. pentagram_numbers
66. gnomic_numbers
67. truncated_triangular_numbers
68. truncated_square_numbers
69. truncated_pronic_numbers
70. truncated_centered_pol_numbers(m) = truncated_centered_mgonal_numbers(m)
71. truncated_centered_triangular_numbers
72. truncated_centered_square_numbers
73. truncated_centered_pentagonal_numbers
74. truncated_centered_hexagonal_numbers = truncated_hex_numbers
75. generalized_mgonal_numbers(m, left_index = 0)
76. generalized_pentagonal_numbers(left_index = 0)
77. generalized_hexagonal_numbers(left_index = 0)
78. generalized_centered_pol_numbers(m, left_index = 0)
79. generalized_pronic_numbers(left_index = 0)

2. Space Figurate Numbers

1. r_pyramidal_numbers(r)
2. triangular_pyramidal_numbers = tetrahedral_numbers
3. square_pyramidal_numbers = pyramidal_numbers
4. pentagonal_pyramidal_numbers
5. hexagonal_pyramidal_numbers
6. heptagonal_pyramidal_numbers
7. octagonal_pyramidal_numbers
8. nonagonal_pyramidal_numbers
9. decagonal_pyramidal_numbers
10. hendecagonal_pyramidal_numbers
11. dodecagonal_pyramidal_numbers
12. tridecagonal_pyramidal_numbers
13. tetradecagonal_pyramidal_numbers
14. pentadecagonal_pyramidal_numbers
15. hexadecagonal_pyramidal_numbers
16. heptadecagonal_pyramidal_numbers
17. octadecagonal_pyramidal_numbers
18. nonadecagonal_pyramidal_numbers
19. icosagonal_pyramidal_numbers
20. icosihenagonal_pyramidal_numbers
21. icosidigonal_pyramidal_numbers
22. icositrigonal_pyramidal_numbers
23. icositetragonal_pyramidal_numbers
24. icosipentagonal_pyramidal_numbers
25. icosihexagonal_pyramidal_numbers
26. icosiheptagonal_pyramidal_numbers
27. icosioctagonal_pyramidal_numbers
28. icosinonagonal_pyramidal_numbers
29. triacontagonal_pyramidal_numbers
30. triangular_tetrahedral_numbers [finite]
31. triangular_square_pyramidal_numbers [finite]
32. square_tetrahedral_numbers [finite]
33. square_square_pyramidal_numbers [finite]
34. tetrahedral_square_pyramidal_number [finite]
35. cubic_numbers = perfect_cube_numbers != hex_pyramidal_numbers (equality only by quantity)
36. tetrahedral_numbers
37. octahedral_numbers
38. dodecahedral_numbers
39. icosahedral_numbers
40. truncated_tetrahedral_numbers
41. truncated_cubic_numbers
42. truncated_octahedral_numbers
43. stella_octangula_numbers
44. centered_cube_numbers
45. rhombic_dodecahedral_numbers
46. hauy_rhombic_dodecahedral_numbers
47. centered_tetrahedron_numbers = centered_tetrahedral_numbers
48. centered_square_pyramid_numbers = centered_pyramid_numbers
49. centered_mgonal_pyramid_numbers(m)
50. centered_pentagonal_pyramid_numbers != centered_octahedron_numbers (equality only in quantity)
51. centered_hexagonal_pyramid_numbers
52. centered_heptagonal_pyramid_numbers
53. centered_octagonal_pyramid_numbers
54. centered_octahedron_numbers
55. centered_icosahedron_numbers = centered_cuboctahedron_numbers
56. centered_dodecahedron_numbers
57. centered_truncated_tetrahedron_numbers
58. centered_truncated_cube_numbers
59. centered_truncated_octahedron_numbers
60. centered_mgonal_pyramidal_numbers(m)
61. centered_triangular_pyramidal_numbers
62. centered_square_pyramidal_numbers
63. centered_pentagonal_pyramidal_numbers
64. centered_hexagonal_pyramidal_numbers = hex_pyramidal_numbers
65. centered_heptagonal_pyramidal_numbers
66. centered_octagonal_pyramidal_numbers
67. centered_nonagonal_pyramidal_numbers
68. centered_decagonal_pyramidal_numbers
69. centered_hendecagonal_pyramidal_numbers
70. centered_dodecagonal_pyramidal_numbers
71. hexagonal_prism_numbers
72. mgonal_prism_numbers(m)
73. generalized_mgonal_pyramidal_numbers(m, left_index = 0)
74. generalized_pentagonal_pyramidal_numbers(left_index = 0)
75. generalized_hexagonal_pyramidal_numbers(left_index = 0)
76. generalized_cubic_numbers(left_index = 0)
77. generalized_octahedral_numbers(left_index = 0)
78. generalized_icosahedral_numbers(left_index = 0)
79. generalized_dodecahedral_numbers(left_index = 0)
80. generalized_centered_cube_numbers(left_index = 0)
81. generalized_centered_tetrahedron_numbers(left_index = 0)
82. generalized_centered_square_pyramid_numbers(left_index = 0)
83. generalized_rhombic_dodecahedral_numbers(left_index = 0)
84. generalized_centered_mgonal_pyramidal_numbers(m, left_index = 0)
85. generalized_mgonal_prism_numbers(m, left_index = 0)
86. generalized_hexagonal_prism_numbers(left_index = 0)

3. Multidimensional figurate numbers

1. pentatope_numbers = hypertetrahedral_numbers = triangulotriangular_numbers
2. k_dimensional_hypertetrahedron_numbers(k) = k_hypertetrahedron_numbers(k) = regular_k_polytopic_numbers(k) = figurate_numbers_of_order_k(k)
3. five_dimensional_hypertetrahedron_numbers
4. six_dimensional_hypertetrahedron_numbers
5. biquadratic_numbers
6. k_dimensional_hypercube_numbers(k) = k_hypercube_numbers(k)
7. five_dimensional_hypercube_numbers
8. six_dimensional_hypercube_numbers
9. hyperoctahedral_numbers = hexadecachoron_numbers = four_cross_polytope_numbers = four_orthoplex_numbers
10. hypericosahedral_numbers = tetraplex_numbers = polytetrahedron_numbers = hexacosichoron_numbers
11. hyperdodecahedral_numbers = hecatonicosachoron_numbers = dodecaplex_numbers = polydodecahedron_numbers
12. polyoctahedral_numbers = icositetrachoron_numbers = octaplex_numbers = hyperdiamond_numbers
13. four_dimensional_hyperoctahedron_numbers
14. five_dimensional_hyperoctahedron_numbers
15. six_dimensional_hyperoctahedron_numbers
16. seven_dimensional_hyperoctahedron_numbers
17. eight_dimensional_hyperoctahedron_numbers
18. nine_dimensional_hyperoctahedron_numbers
19. ten_dimensional_hyperoctahedron_numbers
20. k_dimensional_hyperoctahedron_numbers(k) = k_cross_polytope_numbers(k)
21. four_dimensional_mgonal_pyramidal_numbers(m) = mgonal_pyramidal_numbers_of_the_second_order(m)
22. four_dimensional_square_pyramidal_numbers
23. four_dimensional_pentagonal_pyramidal_numbers
24. four_dimensional_hexagonal_pyramidal_numbers
25. four_dimensional_heptagonal_pyramidal_numbers
26. four_dimensional_octagonal_pyramidal_numbers
27. four_dimensional_nonagonal_pyramidal_numbers
28. four_dimensional_decagonal_pyramidal_numbers
29. four_dimensional_hendecagonal_pyramidal_numbers
30. four_dimensional_dodecagonal_pyramidal_numbers
31. k_dimensional_mgonal_pyramidal_numbers(k, m) = mgonal_pyramidal_numbers_of_the_k_2_th_order(k, m)
32. five_dimensional_mgonal_pyramidal_numbers(m)
33. five_dimensional_square_pyramidal_numbers
34. five_dimensional_pentagonal_pyramidal_numbers
35. five_dimensional_hexagonal_pyramidal_numbers
36. five_dimensional_heptagonal_pyramidal_numbers
37. five_dimensional_octagonal_pyramidal_numbers
38. six_dimensional_mgonal_pyramidal_numbers(m)
39. six_dimensional_square_pyramidal_numbers
40. six_dimensional_pentagonal_pyramidal_numbers
41. six_dimensional_hexagonal_pyramidal_numbers
42. six_dimensional_heptagonal_pyramidal_numbers
43. six_dimensional_octagonal_pyramidal_numbers
44. centered_biquadratic_numbers
45. k_dimensional_centered_hypercube_numbers(k)
46. five_dimensional_centered_hypercube_numbers
47. six_dimensional_centered_hypercube_numbers
48. centered_polytope_numbers
49. k_dimensional_centered_hypertetrahedron_numbers(k)
50. five_dimensional_centered_hypertetrahedron_numbers(k)
51. six_dimensional_centered_hypertetrahedron_numbers(k)
52. centered_hyperoctahedral_numbers = orthoplex_numbers
53. nexus_numbers(k)
54. k_dimensional_centered_hyperoctahedron_numbers(k)
55. five_dimensional_centered_hyperoctahedron_numbers
56. six_dimensional_centered_hyperoctahedron_numbers
57. generalized_pentatope_numbers(left_index = 0)
58. generalized_k_dimensional_hypertetrahedron_numbers(k = 5, left_index = 0)
59. generalized_biquadratic_numbers(left_index = 0)
60. generalized_k_dimensional_hypercube_numbers(k = 5, left_index = 0)
61. generalized_hyperoctahedral_numbers(left_index = 0)
62. generalized_k_dimensional_hyperoctahedron_numbers(k = 5, left_index = 0) [even or odd dimension only changes sign]
63. generalized_hyperdodecahedral_numbers(left_index = 0)
64. generalized_hypericosahedral_numbers(left_index = 0)
65. generalized_polyoctahedral_numbers(left_index = 0)
66. generalized_k_dimensional_mgonal_pyramidal_numbers(k, m, left_index = 0)
67. generalized_k_dimensional_centered_hypercube_numbers(k, left_index = 0)
68. generalized_k_dimensional_centered_hypertetrahedron_numbers(k, left_index = 0)[provisional symmetry]
69. generalized_k_dimensional_centered_hyperoctahedron_numbers(k, left_index = 0)[provisional symmetry]
70. generalized_nexus_numbers(k, left_index = 0) [even or odd dimension only changes sign]

6. Zoo of figurate-related numbers

1. cuban_numbers = cuban_prime_numbers
2. quartan_numbers [Needs to improve the algorithmic complexity for n > 70]
3. pell_numbers
4. carmichael_numbers [Needs to improve the algorithmic complexity for n > 20]
5. stern_prime_numbers(infty = false) [Quick calculations up to 8 terms]
6. apocalyptic_numbers

Errata

• 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