Figurate Numbers

figurate_numbers is a Ruby module that implements 151 infinite number sequences based on the formulas from the wonderful book

Figurate Numbers (2012) by Elena Deza and Michel Deza.

This implementation uses the Enumerator class to deal with INFINITE SEQUENCES.

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 = 74
2. Space figurate numbers implemented = 42
3. Multidimensional figurate numbers implemented = 29

Additionally we have the sequences mentioned in chapter 6:

4. Zoo of figurate-related numbers implemented = 6

• TOTAL = 151 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

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
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. star_numbers
41. centered_tridecagonal_numbers
42. centered_tetradecagonal_numbers
43. centered_pentadecagonal_numbers
44. centered_hexadecagonal_numbers
45. centered_heptadecagonal_numbers
46. centered_octadecagonal_numbers
47. centered_nonadecagonal_numbers
48. centered_icosagonal_numbers
49. centered_icosihenagonal_numbers
50. centered_icosidigonal_numbers
51. centered_icositrigonal_numbers
52. centered_icositetragonal_numbers
53. centered_icosipentagonal_numbers
54. centered_icosihexagonal_numbers
55. centered_icosiheptagonal_numbers
56. centered_icosioctagonal_numbers
57. centered_icosinonagonal_numbers
58. centered_triacontagonal_numbers
59. centered_mgonal_numbers(m)
60. pronic_numbers
61. cross_numbers
62. aztec_diamond_numbers
63. polygram_numbers(m) = centered star polygonal numbers
64. centered_star_polygonal_numbers(m)
65. gnomic_numbers
66. truncated_triangular_numbers
67. truncated_square_numbers
68. truncated_pronic_numbers
69. truncated_center_pol_numbers(k)
70. truncated_centered_triangular_numbers
71. truncated_centered_square_numbers
72. truncated_centered_pentagonal_numbers
73. truncated_centered_hexagonal_numbers
74. generalized_mgonal_numbers(m, left_index = 0)
75. generalized_centered_pol_numbers(m, left_index = 0)
76. generalized_pronic_numbers(left_index = 0)

2. Space Figurate Numbers

1. r_pyramidal_numbers(r)
2. cubic_numbers = hex pyramidal numbers (equality only by quantity)
3. tetrahedral_numbers
4. octahedral_numbers
5. dodecahedral_numbers
6. icosahedral_numbers
7. truncated_tetrahedral_numbers
8. truncated_cubic_numbers
9. truncated_octahedral_numbers
10. stella_octangula_numbers
11. centered_cube_numbers
12. rhombic_dodecahedral_numbers
13. hauy_rhombic_dodecahedral_numbers
14. centered_tetrahedral_numbers
15. centered_square_pyramid_numbers
16. centered_pentagonal_pyramid_numbers = centered octahedron numbers (equality only in quantity)
17. centered_hexagonal_pyramid_numbers
18. centered_heptagonal_pyramid_numbers
19. centered_octagonal_pyramid_numbers
20. centered_octahedron_numbers
21. centered_icosahedron_numbers = centered cuboctahedron numbers
22. centered_cuboctahedron_numbers
23. centered_dodecahedron_numbers
24. centered_truncated_tetrahedron_numbers
25. centered_truncated_cube_numbers
26. centered_truncated_octahedron_numbers
27. centered_mgonal_pyramid_numbers(m)
28. centered_triangular_pyramidal_numbers
29. centered_square_pyramidal_numbers
30. centered_pentagonal_pyramidal_numbers
31. centered_hexagonal_pyramidal_numbers = hex_pyramidal_numbers
32. hex_pyramidal_numbers
33. centered_mgonal_pyramidal_numbers(m)
34. hexagonal_prism_numbers
35. mgonal_prism_numbers(m)
36. generalized_mgonal_pyramidal_numbers(m, left_index = 0)
37. generalized_cubic_numbers(left_index = 0)
38. generalized_octahedral_numbers(left_index = 0)
39. generalized_icosahedral_numbers(left_index = 0)
40. generalized_dodecahedral_numbers(left_index = 0)
41. generalized_centered_cube_numbers(left_index = 0)
42. generalized_centered_tetrahedron_numbers(left_index = 0)
43. generalized_centered_square_pyramid_numbers(left_index = 0)
44. generalized_rhombic_dodecahedral_numbers(left_index = 0)
45. generalized_centered_mgonal_pyramidal_numbers(m, left_index = 0)
46. generalized_hexagonal_prism_numbers(left_index = 0)

3. Multidimensional figurate numbers

1. pentatope_numbers = hypertetrahedral_number = triangulotriangular_number
2. hypertetrahedral_number
3. triangulotriangular_number
4. k_dimensional_hypertetrahedron_numbers(k) = k hypertetrahedron numbers = regular k-polytopic number = figurate number of order k = k-simplex numbers
5. k_hypertetrahedron_numbers(k)
6. regular_k_polytopic_numbers(k)
7. figurate_number_of_order_k(k)
8. five_dimensional_hypertetrahedron_numbers
9. six_dimensional_hypertetrahedron_numbers
10. biquadratic_numbers
11. k_dimensional_hypercube_numbers(k) = k-measure polytope numbers
12. five_dimensional_hypercube_numbers
13. six_dimensional_hypercube_numbers
14. hyperoctahedral_numbers = 4D hyperoctahedron numbers = hexadecachoron_numbers = four_cross_polytope_numbers = four_orthoplex_numbers
15. hexadecachoron_numbers
16. four_cross_polytope_numbers
17. four_orthoplex_numbers
18. hypericosahedral_numbers = tetraplex numbers = polytetrahedron numbers
19. tetraplex_numbers
20. polytetrahedron_numbers
21. hexacosichoron_numbers
22. hyperdodecahedral_numbers = hecatonicosachoron_numbers = dodecaplex numbers = polydodecahedron numbers
23. hecatonicosachoron_numbers
24. dodecaplex_numbers
25. polydodecahedron_numbers,
26. polyoctahedral_numbers = icositetrachoron numbers = octaplex numbers = hyperdiamond numbers
27. icositetrachoron_numbers
28. octaplex_numbers
29. hyperdiamond_numbers
30. four_dimensional_hyperoctahedron_numbers
31. five_dimensional_hyperoctahedron_numbers
32. six_dimensional_hyperoctahedron_numbers
33. seven_dimensional_hyperoctahedron_numbers
34. eight_dimensional_hyperoctahedron_numbers
35. nine_dimensional_hyperoctahedron_numbers
36. ten_dimensional_hyperoctahedron_numbers
37. k_dimensional_hyperoctahedron_numbers(k) = k-cross polytope numbers
38. k_cross_polytope_numbers(k)
39. four_dimensional_mgonal_pyramidal_numbers(m) = mgonal pyramidal number of the second order
40. mgonal_pyramidal_number_of_the_second_order(m)
41. four_dimensional_square_pyramidal_numbers
42. four_dimensional_pentagonal_pyramidal_numbers
43. four_dimensional_hexagonal_pyramidal_numbers
44. four_dimensional_heptagonal_pyramidal_numbers
45. four_dimensional_octagonal_pyramidal_numbers
46. four_dimensional_nonagonal_pyramidal_numbers
47. four_dimensional_decagonal_pyramidal_numbers
48. four_dimensional_hendecagonal_pyramidal_numbers
49. four_dimensional_dodecagonal_pyramidal_numbers
50. five_dimensional_mgonal_pyramidal_numbers(m)
51. six_dimensional_mgonal_pyramidal_numbers(m)
52. k_dimensional_mgonal_pyramidal_numbers(k, m) = mgonal pyramidal of the (k-2)-th order
53. mgonal_pyramidal_number_of_the_k_2_th_order(k, m)
54. centered_biquadratic_numbers
55. k_dimensional_centered_hypercube_numbers(k)
56. five_dimensional_centered_hypercube_numbers
57. six_dimensional_centered_hypercube_numbers
58. centered_polytope_numbers
59. k_dimensional_centered_hypertetrahedron_numbers(k)
60. five_dimensional_centered_hypertetrahedron_numbers(k)
61. six_dimensional_centered_hypertetrahedron_numbers(k)
62. centered_hyperotahedral_numbers = orthoplex numbers
63. orthoplex numbers
64. nexus_numbers(k)
65. k_dimensional_centered_hyperoctahedron_numbers(k)
66. five_dimensional_centered_hyperoctahedron_numbers(k)
67. six_dimensional_centered_hyperoctahedron_numbers(k)
68. generalized_pentatope_numbers(left_index = 0)
69. generalized_k_dimensional_hypertetrahedron_numbers(k = 5, left_index = 0)
70. generalized_k_dimensional_hypercube_numbers(k = 5, left_index = 0)
71. generalized_k_dimensional_hyperoctahedron_numbers(k = 5, left_index = 0) = even or odd dimension only changes sign
72. 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, formula (truncated centered pentagonal numbers) of page 72 says:

  TCSS_5(n) = (35n^2 - 55n) / 2 + 3

  It should be:

  TCSS_5(n) = (35n^2 - 55n) / 2 + 11

• 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