Figurate Numbers

Figurate Numbers is the most comprehensive and specialized library for figurate numbers, developed in Ruby to date. It implements 241 infinite number sequences inspired by the groundbreaking work Figurate Numbers by Elena Deza and Michel Deza, published in 2012.

Installation

Install it from the gem repository:

gem install figurate_numbers

Features

Figurate Numbers implements 241 infinite sequences using the Enumerator class in Ruby, each categorized by its geometric dimension. It is ideal for use in mathematical modeling, algorithmic composition, and integration with tools like Sonic Pi.

The sequences are organized following the structure of the Figurate Numbers book:

• PlaneFigurateNumbers: 79 sequences (2D)
• SpaceFigurateNumbers: 86 sequences (3D)
• MultiDimensionalFigurateNumbers: 70 sequences (4D and beyond)
• Zoo of figurate-related numbers: 6 additional sequences (included in the MultiDimensional module)
• ArithTransform: p-adic transformations and other arithmetic operations

See detailed list of sequences below

How to use in Ruby

Version 1.5.0

require 'figurate_numbers'

## Using take(integer)
FigurateNumbers.pentatope.take(10)

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

Starting from version 1.5.0, figurate_numbers not only supports global access via FigurateNumbers and the use of specific classes for separate access, but also introduces new p-adic transformations and other mathematical operations that can be applied directly to sequences.

PlaneFigurateNumbers.polygonal(19)
SpaceFigurateNumbers.rhombic_dodecahedral
MultiDimensionalFigurateNumbers.six_dimensional_hyperoctahedron
seq = MultiDimensionalFigurateNumbers.five_dimensional_hypercube.take(15)
ArithTransform.ring_padic_val(seq, 3)

How to use in Sonic Pi

Simply copy the entry point path from the lib/figurate_numbers.rb file where the gem is installed.

require "<PATH>"

pol_num = FigurateNumbers.polygonal(8)
350.times do
  play pol_num.next % 12 * 7  # Some mathematical function or transformation
  sleep 0.125
end

amp_seq = SpaceFigurateNumbers.centered_hendecagonal_pyramidal
100.times do
  sample :bd_sone, amp: ArithTransform.padic_norm(amp_seq.next, 3) # p-adic amplitude control
  sleep 0.25
end

Version 1.3.0 (legacy)

The main change introduced after version 1.3.0 is that you must now import the file using require instead of run_file; otherwise, it will not function.

See discussion in the Sonic Pi community thread right here!

List of Sequences and Arithmetic Transformations Methods

1. Plane Figurate Numbers

1. polygonal(m)
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 (equality only by quantity)
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 (equality only by quantity)
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, left_index = 0)
76. generalized_pentagonal(left_index = 0)
77. generalized_hexagonal(left_index = 0)
78. generalized_centered_pol(m, left_index = 0)
79. generalized_pronic(left_index = 0)

2. Space Figurate Numbers

1. r_pyramidal(r)
2. triangular_pyramidal = tetrahedral
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_number [finite]
35. cubic = perfect_cube != hex_pyramidal (equality only by quantity)
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 != centered_octahedron (equality only in quantity)
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_hexagonal_pyramidal = hex_pyramidal
65. centered_heptagonal_pyramidal
66. centered_octagonal_pyramidal
67. centered_nonagonal_pyramidal
68. centered_decagonal_pyramidal
69. centered_hendecagonal_pyramidal
70. centered_dodecagonal_pyramidal
71. hexagonal_prism
72. mgonal_prism(m)
73. generalized_mgonal_pyramidal(m, left_index = 0)
74. generalized_pentagonal_pyramidal(left_index = 0)
75. generalized_hexagonal_pyramidal(left_index = 0)
76. generalized_cubic(left_index = 0)
77. generalized_octahedral(left_index = 0)
78. generalized_icosahedral(left_index = 0)
79. generalized_dodecahedral(left_index = 0)
80. generalized_centered_cube(left_index = 0)
81. generalized_centered_tetrahedron(left_index = 0)
82. generalized_centered_square_pyramid(left_index = 0)
83. generalized_rhombic_dodecahedral(left_index = 0)
84. generalized_centered_mgonal_pyramidal(m, left_index = 0)
85. generalized_mgonal_prism(m, left_index = 0)
86. generalized_hexagonal_prism(left_index = 0)

3. Multidimensional figurate numbers

1. pentatope = hypertetrahedral = triangulotriangular
2. k_dimensional_hypertetrahedron(k) = k_hypertetrahedron(k) = regular_k_polytopic(k) = figurate_numbers_of_order_k(k)
3. five_dimensional_hypertetrahedron
4. six_dimensional_hypertetrahedron
5. biquadratic
6. k_dimensional_hypercube(k) = k_hypercube(k)
7. five_dimensional_hypercube
8. six_dimensional_hypercube
9. hyperoctahedral = hexadecachoron = four_cross_polytope = four_orthoplex
10. hypericosahedral = tetraplex = polytetrahedron = hexacosichoron
11. hyperdodecahedral = hecatonicosachoron = dodecaplex = polydodecahedron
12. polyoctahedral = icositetrachoron = octaplex = hyperdiamond
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_numbers_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(k)
51. six_dimensional_centered_hypertetrahedron(k)
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(left_index = 0)
58. generalized_k_dimensional_hypertetrahedron(k = 5, left_index = 0)
59. generalized_biquadratic(left_index = 0)
60. generalized_k_dimensional_hypercube(k = 5, left_index = 0)
61. generalized_hyperoctahedral(left_index = 0)
62. generalized_k_dimensional_hyperoctahedron(k = 5, left_index = 0) [even or odd dimension only changes sign]
63. generalized_hyperdodecahedral(left_index = 0)
64. generalized_hypericosahedral(left_index = 0)
65. generalized_polyoctahedral(left_index = 0)
66. generalized_k_dimensional_mgonal_pyramidal(k, m, left_index = 0)
67. generalized_k_dimensional_centered_hypercube(k, left_index = 0)
68. generalized_k_dimensional_centered_hypertetrahedron(k, left_index = 0)[provisional symmetry]
69. generalized_k_dimensional_centered_hyperoctahedron(k, left_index = 0)[provisional symmetry]
70. generalized_nexus(k, left_index = 0) [even or odd dimension only changes sign]

4. 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

5. Arithmetic Transformations

1. ArithTransform.n_residue(n, pow, mod)
2. ArithTransform.pc_inversion(n, mod)
3. ArithTransform.padic_val(n, p)
4. ArithTransform.ring_padic_val(seq, p)
5. ArithTransform.padic_norm(n, p)
6. ArithTransform.ring_padic_norm(seq, p)
7. ArithTransform.padic_expansion(n, p, precision = 11, reverse: false)
8. ArithTransform.ring_padic_expansion(seq, p, precision = 11, reverse: false)

Book Errata

• Chapter 1, page 6: The formula for Square in the table is given as: 1/2 (n^2 - 0 * n) It should be corrected to: 1/2 (2n^2 - 0 * n)

• Chapter 1, page 51: The formula for Centered icosihexagonal numbers is listed as: 1/3n^2 - 13n + 1 with values 546, 728, 936, 1170. The correct formula and values are: 13n^2 - 13n + 1 with values 547, 729, 937, 1171.

• Chapter 1, page 51: The value for Centered icosiheptagonal number is given as 972. The correct value is 973.

• Chapter 1, page 51: The value for Centered icosioctagonal number is given as 84. The correct value is 85.

• Chapter 1, page 65: The term polite numbers is misspelled as: inpolite numbers It should read: impolite numbers

• Chapter 1, page 72: The formula for truncated centered pentagonal numbers (TCSS_5) is: TCSS_5(n) = (35n^2 - 55n) / 2 + 3 It should be: TCSS_5(n) = (35n^2 - 55n) / 2 + 11

• Chapter 2, page 92: The formula for octagonal pyramidal numbers is stated as: n(n+1)(6n-1) / 6 The correct formula is: n(n+1)(6n-3) / 6

• Chapter 2, page 140: The sequence for centered square pyramidal numbers is listed as: 1, 6, 19, 44, 85, 111, 146, 231, ... The number 111 should be excluded, resulting in: 1, 6, 19, 44, 85, 146, 231, ...

• Chapter 2, page 155: The formula for generalized centered tetrahedron numbers (S_3^3) is: S_3^3(n) = ((2n - 1)(n^2 + n + 3)) / 3 It should include a negative sign: S_3^3(n) = ((2n - 1)(n^2 - n + 3)) / 3

• Chapter 2, page 156: The formula for generalized centered square pyramid numbers (S_4^3) is: S_4^3(n) = ((2n - 1)(n^2 - n + 2)^2) / 3 The correct formula is: S_4^3(n) = ((2n - 1)(n^2 - n + 2)) / 2

• Chapter 3, page 188: The term hyperoctahedral numbers is incorrectly called: hexadecahoron numbers It should be: hexadecachoron numbers

• Chapter 3, page 190: The term hypericosahedral numbers is incorrectly written as: hexacisihoron numbers It should be: hexacosichoron numbers