figurate_numbers

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 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

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. PlaneFigurateNumbers figurate numbers implemented = 79
2. SpaceFigurateNumbers figurate numbers implemented = 86
3. MultiDimensionalFigurateNumbers figurate numbers implemented = 70
4. Zoo of figurate-related numbers implemented = 6 (included in MultiDimensional module)

• TOTAL = 241 infinite sequences of figurate numbers implemented

How to use in Ruby

require 'figurate_numbers'

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

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

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

Since version 1.4.0, you can alternatively call from the classes

PlaneFigurateNumbers.polygonal(3)
SpaceFigurateNumbers.rhombic_dodecahedral
MultiDimensionalFigurateNumbers.six_dimensional_hyperoctahedron

This ensures that you only use the numbers belonging to each geometric dimension.

How to use in Sonic Pi

figurate_numbers - Version 1.4.0

Starting from version 1.4.0, you can use the library globally through FigurateNumbersto access all sequences, or you can use the specific classes mentioned above for separate access. The main change compared to version 1.3.0 is that you now need to import the file using require instead of run_file; otherwise, it will not function.

require "<PATH>"

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

figurate_numbers - Version 1.3.0

You can read and comment in the Sonic Pi community thread right here!

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>"
sleep 1 # see the quote below
pol_num = FigurateNumbers.polygonal_numbers(8)
80.times do
  play pol_num.next % 12 * 7  # Some mathematical function or transformation
  sleep 0.25
end

sleep 1 #to allow figurate_numbers to complete load and setup otherwise first run can give error (Robin Newman - Sonic Pi Core Team)

List of implemented sequences

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

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]

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