## Fiber Bundle Terms of the Toratopes

Discussion of shapes with curves and holes in various dimensions.

### Fiber Bundle Terms of the Toratopes

Here is the growing list of my translation of toratopic notation, into a more common method of description in vector bundles over a surface.

Bn - n-Ball, solid n-sphere
Rn - n-Ring , solid n-torus structure , Rn = B2 x T(n-1) , T1 = S1

Sn - n-Sphere, D-1 surface of hollow n-ball
Tn - n-Torus, D-1 surface of hollow n-ring

Cn - n-Clifford manifold, edge of (B2^n) prism as [S1^n] margin
[(Sn/Tn x Cn x Sn)*Sn] - n-Clifford manifold, edge of (B(n+1)/R(n+1) x Cn x Sn)*B(n+1) open toratope prism

Q x R - 'Q' bundle over the 'R'

The one 1-manifold that curves into 2D:
(II) - S1 , the circle

All 2 distinct 2-Manifolds that curve into 3D:
(III) - S2 , sphere
((II)I) -T2 = S1xS1 , torus

All 5 distinct 3-Manifolds that curve into 4D:
(IIII) - S3
((III)I) - S1xS2
((II)II) - S2xS1
(((II)I)I) - T3 = S1xS1xS1
((II)(II)) - S1xC2 = S1x[S1*S1]

All 12 distinct 4-Manifolds that curve into 5D:
(IIIII) - S4
((II)III) - S3xS1
((II)(II)I) - S2xC2 = S2x[S1*S1]
((III)II) - S2xS2
(((II)I)II) - S2xT2 = S2xS1xS1
((III)(II)) - S1x[S2*S1]
(((II)I)(II)) - S1xC2xS1 = S1x[T2*S1] = S1x[(S1xS1)*S1] = S1x[S1*S1]xS1 , the final S1 from T2 commutes to the outside right
((IIII)I) - S1xS3
(((II)II)I) - S1xS2xS1
(((II)(II))I) - T2xC2 = S1xS1x[S1*S1]
(((III)I)I) - T2xS2 = S1xS1xS2
((((II)I)I)I) - T4 = S1xS1xS1xS1

All 33 distinct 5-manifolds that curve into 6D:
(IIIIII) - S5
((II)IIII) - S4xS1
((II)(II)II) - S3xC2 = S3x[S1*S1]
((II)(II)(II)) - S2xC3 = S2x[S1*S1*S1]
((III)III) - S3xS2
(((II)I)III) - S3xT2 = S3xS1xS1
((III)(II)I) - S2x[S2*S1]
(((II)I)(II)I) - S2xS1xC2 = S2x[T2*S1] = S2x[(S1xS1)*S1] = S2x[S1*S1]xS1 = S2xS1x[S1*S1]
((III)(III)) - S1x[S2*S2]
(((II)I)(III)) - S1x[T2*S2] = S1x[(S1xS1)*S2] = S1x[S2*S1]xS1
(((II)I)((II)I)) - S1xC2xC2 = S1x[T2*T2] = S1x[(S1xS1)*(S1xS1)]
((IIII)II) - S2xS3
(((II)II)II) - S2xS2xS1
(((II)(II))II) - S2xS1xC2 = S2xS1x[S1*S1]
(((III)I)II) - S2xS1sS2
((((II)I)I)II) - S2xT3 = S2xS1xS1xS1
((IIII)(II)) - S1x[S3*S1]
(((II)II)(II)) - S1xS2xC2 = S1xS2x[S1*S1] = S1x[(S2xS1)*S1] = S1x[S2*S1]xS1
(((II)(II))(II)) - T2xC3 = S1xS1x[S1*S1*S1]
(((III)I)(II)) - S1xC2xS2 = S1x[(S1xS2)*S1] = S1x[S1*S1]xS2
((((II)I)I)(II)) - S1xC2xT2 = S1x[T3*S1] = S1x[S1*S1]xT2 = S1x[(S1xS1xS1)*S1] = S1x[S1*S1]xS1xS1
((IIIII)I) - S1xS4
(((II)III)I) - S1xS3xS1
(((II)(II)I)I) - S1xS2xC2 = S1xS2x[S1*S1]
(((III)II)I) - S1xS2xS2
((((II)I)II)I) - S1xS2xT2 = S1xS2xS1xS1
(((III)(II))I) - T2x[S2*S1] = S1xS1x[S2*S1]
((((II)I)(II))I) - T2xC2xS1 = S1xS1x[(S1xS1)*S1] = S1xS1x[S1*S1]xS1
(((IIII)I)I) - T2xS3 = S1xS1xS3
((((II)II)I)I) - T2xS2xS1 = S1xS1xS2xS1
((((II)(II))I)I) - T3xC2 = S1xS1xS1x[S1*S1]
((((III)I)I)I) - T3xS2 = S1xS1xS1xS2
(((((II)I)I)I)I) - T5 = S1xS1xS1xS1xS1

All 90 distinct 6-manifolds that curve into 7D

1. Heptasphere (IIIIIII) - S6
2. 61-torus ((IIIIII)I) - S1xS5
3. 511-ditorus (((IIIII)I)I) - T2xS4
4. 4111-tritorus ((((IIII)I)I)I) - T3xS3
5. 31111-tetratorus (((((III)I)I)I)I) - T4xS2
6. Pentatorus ((((((II)I)I)I)I)I) - T6
7. Tiger tritorus (((((II)(II))I)I)I) - T4xC2
8. 22111-tetratorus (((((II)II)I)I)I) - T3xS2xS1
9. 320-tiger 11-ditorus ((((III)(II))I)I) - T3x[S2*S1]
10. Torus tiger ditorus (((((II)I)(II))I)I) - T3xC2xS1
11. 3211-tritorus ((((III)II)I)I) - T2xS2xS2
12. 21211-tetratorus (((((II)I)II)I)I) - T2xS2xT2
13. 221-tiger 11-ditorus ((((II)(II)I)I)I) - T2xS2xC2
14. 2311-tritorus ((((II)III)I)I) - T2xS3xS1
15. 420-tiger 1-torus (((IIII)(II))I) - T2x[S3*S1]
16. 31-torus 20-tiger 1-torus ((((III)I)(II))I) - T2xC2xS2 = T2x[S1*S1]xS2 = T2x[(S1xS2)*S1]
17. Ditorus tiger torus (((((II)I)I)(II))I) - T2xC2xT2 = T2x[T3*S1] = T2x[S1*S1]xT2
18. Double tiger torus ((((II)(II))(II))I) - T3xC3 = T2xC2xC2 = T2x[(S1xC2)*S1]
19. 22-torus 20-tiger 1-torus ((((II)II)(II))I) - T2x[(S2xS1)*S1] = T2x[S2*S1]xS1
20. 421-ditorus (((IIII)II)I) - S1xS2xS3
21. 3121-tritorus ((((III)I)II)I) - S1xS2xS1xS2
22. 21121-tetratorus (((((II)I)I)II)I) - S1xS2xT3
23. 220-tiger 21-ditorus ((((II)(II))II)I) - S1xS2xS1xC2
24. 2221-tritorus ((((II)II)II)I) - S1xS2xS2xS1
25. 330-tiger 1-torus (((III)(III))I) - T2x[S2*S2]
26. 21-torus 30-tiger 1-torus ((((II)I)(III))I) - T2x[S2*S1]xS1 = T2x[T2*S2]
27. Duotorus tiger torus ((((II)I)((II)I))I) - T2xC2xC2
28. 321-tiger 1-torus (((III)(II)I)I) - S1xS2x[S2*S1]
29. 21-torus 21-tiger 1-torus ((((II)I)(II)I)I) - S1xS2xC2xS1 = S1xS2x[T2*S1]
30. 331-ditorus (((III)III)I) = S1xS3xS2
31. 2131-tritorus ((((II)I)III)I) = S1xS3xT2
32. Triger torus (((II)(II)(II))I) = S1xS2xC3
33. 222-tiger 1-torus (((II)(II)II)I) - S1xS3xC2
34. 241-ditorus (((II)IIII)I) - S1xS4xS1
35. 520-tiger ((IIIII)(II)) - S1x[S4*S1]
36. 41-torus 20-tiger (((IIII)I)(II)) - S1xC2xS3 = S1x[(S1xS3)*S1]
37. 311-ditorus 20-tiger ((((III)I)I)(II)) - S1xC2xS1xS2 = S1x[(T2xS2)*S1] = S1x[T2*S1]xS2
38. Tritorus tiger (((((II)I)I)I)(II)) - S1xC2xT3 = S1x[T4*S1]
39. Tiger torus tiger ((((II)(II))I)(II)) - S1xC2xS1xC2
40. 221-ditorus 20-tiger ((((II)II)I)(II)) - S1xC2xS2xS1 = S1x[(S1xS2xS1)*S1] = S1x[(S1xS2)*S1]xS1
41. 320-tiger 20-tiger (((III)(II))(II)) - S1xC2x[S2*S1]
43. 32-torus 20-tiger (((III)II)(II)) - S1x[S2*S1]xS2 = S1x[(S2xS2)*S1]
44. 212-ditorus 20-tiger ((((II)I)II)(II)) - S1x[S2*S1]xT2 = S1x[(S2xT2)*S1]
45. 221-tiger 20-tiger (((II)(II)I)(II)) - S1x[S2*S1]xC2 = S1x[(S2xC2)*S1]
46. 23-torus 20-tiger (((II)III)(II)) - S1x[S3*S1]xS1 = S1x[(S3xS1)*S1]
47. 52-torus ((IIIII)II) - S2xS4
48. 412-ditorus (((IIII)I)II) - S2xS1xS3
49. 3112-tritorus ((((III)I)I)II) - S2xT2xS2
50. 21112-tetratorus (((((II)I)I)I)II) - S2xT3xS1
51. 220-tiger 12-ditorus ((((II)(II))I)II) - S2xT2xC2
52. 2212-tritorus ((((II)II)I)II) - S2xS1xS2xS1
53. 320-tiger 2-torus (((III)(II))II) - S2xS1x[S2*S1]
54. 21-torus 20-tiger 2-torus ((((II)I)(II))II) - S2xS1xC2xS1 = S2xS1x[T2*S1]
55. 322-ditorus (((III)II)II) - S2xS2xS2
56. 2122-tritorus ((((II)I)II)II) - S2xS2xT2
57. 221-tiger 2-torus (((II)(II)I)II) - S2xS2xC2
58. 232-ditorus (((II)III)II) - S2xS3xS1
59. 430-tiger ((IIII)(III)) - S1x[S3*S2]
60. 21-torus 40-tiger (((II)I)(IIII)) - S1x[T2*S3] = S1x[S3*S1]xS1
61. 31-torus 30-tiger (((III)I)(III)) - S1x[(S1xS2)*S2] = S1x[S2*S1]xS2 = S1x[S2*S2]xS1
62. 31-torus 21-torus 0-tiger (((III)I)((II)I)) - S1xC2x[S2*S1] = S1x[(S1xS2)*T2] = S1x[T2*S1]xS2
63. 211-ditorus 30-tiger ((((II)I)I)(III)) - S1x[T3*S2] = S1x[S2*S1]xT2
64. Ditorus/torus tiger ((((II)I)I)((II)I)) - S1xC2xC2xS1 = S1x[T3*T2]
65. 220-tiger 30-tiger (((II)(II))(III)) - S1x[(S1xC2)*S2] = S1x[S2*S1]xC2
66. Tiger/torus tiger (((II)(II))((II)I)) - S1x[(S1xC2)*T2] = S1x[T2*S1]xC2
67. 22-torus 30-tiger (((II)II)(III)) - S1x[(S2xS1)*S2] = S1x[S2*S2]xS1
68. 22-torus 21-torus 0-tiger (((II)II)((II)I)) - S1x[(S2xS1)*T2] = S1x[S2*S1]xC2 = S1x[T2*S2]xS1
69. 421-tiger ((IIII)(II)I) - S2x[S3*S1]
70. 31-torus 21-tiger (((III)I)(II)I) - S2xC2xS2 = S2x[(S1xS2)*S1]
71. 211-ditorus 21-tiger ((((II)I)I)(II)I) - S2xC2xT2 = S2x[T3*S1]
72. 220-tiger 21-tiger (((II)(II))(II)I) - S2xS1xC3 = S2xC2xC2 = S2x[(S1xC2)*S1]
73. 22-torus 21-tiger (((II)II)(II)I) - S2x[(S2xS1)*S1] = S2x[S2*S1]xS1
74. 43-torus ((IIII)III) - S3xS3
75. 313-ditorus (((III)I)III) - S3xS1xS2
76. 2113-tritorus ((((II)I)I)III) - S3xT3
77. 220-tiger 3-torus (((II)(II))III) - S3xS1xC2
78. 223-ditorus (((II)II)III) - S3xS2xS1
79. 331-tiger ((III)(III)I) - S2x[S2*S2]
80. 21-torus 31-tiger (((II)I)(III)I) - S2x[T2*S2] = S2x[S2*S1]xS1
81. 21-torus 21-torus 1-tiger (((II)I)((II)I)I) = S2xC2xC2 = S2x[T2*T2]
82. 3220-triger ((III)(II)(II)) - S2x[S2*S1*S1]
83. Torus triger (((II)I)(II)(II)) - S2xC3xS1 = S2x[T2*S1*S1]
84. 322-tiger ((III)(II)II) - S3x[S2*S1]
85. 21-torus 22-tiger (((II)I)(II)II) - S3xC2xS1 = S3x[T2*S1]
86. 34-torus ((III)IIII) - S4xS2
87. 214-ditorus (((II)I)IIII) - S4xT2
88. 2221-triger ((II)(II)(II)I) - S3xC3
89. 223-tiger ((II)(II)III) - S4xC2
90. 25-torus ((II)IIIII) - S5xS1

All 261 distinct 7-manifolds that curve into 8D:

1. Octasphere (IIIIIIII) - S7
2. 71-torus ((IIIIIII)I) - S1xS6
3. 611-ditorus (((IIIIII)I)I) - T2xS5
4. 5111-tritorus ((((IIIII)I)I)I) - T3xS4
5. 41111-tetratorus (((((IIII)I)I)I)I) - T4xS3
6. 311111-pentatorus ((((((III)I)I)I)I)I) - T5xS2
7. Hexatorus (((((((II)I)I)I)I)I)I) - T7
8. Tiger tetratorus ((((((II)(II))I)I)I)I) - T5xC2
9. 221111-pentatorus ((((((II)II)I)I)I)I) - T4xS2xS1
10. 320-tiger 111-tritorus ((((((III)(II))I)I)I) - T4x[S2*S1]
11. Torus tiger tritorus ((((((II)I)(II))I)I)I) - T4xC2xS1 = T4x[T2*S1]
12. 32111-tetratorus (((((III)II)I)I)I) - T3xS2xS2
13. 212111-pentatorus ((((((II)I)II)I)I)I) - T3xS2xT2
14. 221-tiger 111-tritorus (((((II)(II)I)I)I)I) - T3xS2xC2
15. 23111-tetratorus (((((II)III)I)I)I) - T3xS4xS1
16. 420-tiger 11-ditorus ((((IIII)(II))I)I) - T3x[S3*S1]
17. 31-torus 20-tiger 11-ditorus ((((III)I)(II))I)I) - T3xC2xS2 = T3x[(S1xS2)*S1]
18. Ditorus tiger ditorus (((((II)I)I)(II))I)I) - T3xC2xT2 = T3x[T3*S1]
19. Double tiger ditorus ((((II)(II))(II))I)I) - T4xC3 = T3xC2xC2 = T3x[(S1xC2)*S1]
20. 22-torus 20-tiger 11-ditorus ((((II)II)(II))I)I) - T3x[S2*S1]xS1 = T3x[(S2xS1)*S1]
21. 4211-tritorus (((IIII)II)I)I) - T2xS2xS3
22. 31211-tetratorus ((((III)I)II)I)I) - T2xS2xS1xS2
23. 211211-pentatorus (((((II)I)I)II)I)I) - T2xS2xT3
24. 220-tiger 211-tritorus ((((II)(II))II)I)I) - T2xS2xS1xC2
25. 22211-tetratorus ((((II)II)II)I)I) - T2xS2xS2xS1
26. 330-tiger 11-ditorus ((((III)(III))I)I) - T3x[S2*S2]
27. 21-torus 30-tiger 11-ditorus (((((II)I)(III))I)I) - T3x[T2*S2] = T3x[S2*S1]xS1
28. Duotorus tiger ditorus (((((II)I)((II)I))I)I) - T3xC2xC2 = T3x[T2*T2]
29. 321-tiger 11-ditorus ((((III)(II)I)I)I) - T2xS2x[S2*S1]
30. 21-torus 21-tiger 11-ditorus (((((II)I)(II)I)I)I) - T2xS2xC2xS1 = T2xS2x[T2*S1]
31. 3311-tritorus ((((III)III)I)I) - T2xS3xS2
32. 21311-tetratorus (((((II)I)III)I)I) - T2xS3xT2
33. Triger ditorus ((((II)(II)(II))I)I) - T2xS2xC3
34. 222-tiger 11-ditorus ((((II)(II)II)I)I) - T2xS4xC2
35. 2411-tritorus ((((II)IIII)I)I) - T2xS4xS2
36. 520-tiger 1-torus (((IIIII)(II))I) - T2x[S4*S2]
37. 41-torus 20-tiger 1-torus ((((IIII)I)(II))I) - T2xC2xS3 = T2x[(S1xS3)*S1]
38. 311-ditorus 20-tiger 1-torus (((((III)I)I)(II))I) - T2xC2xS1xS2 = T2x[(T2xS2)*S1]
39. Tritorus tiger torus ((((((II)I)I)I)(II))I) - T2xC2xT3 = T2x[T4*S1]
40. Tiger torus tiger torus (((((II)(II))I)(II))I) - T2xC2xS1xC2 = T2x[(T2xC2)*S1]
41. 221-ditorus 20-tiger 1-torus (((((II)II)I)(II))I) - T2xC2xS2xS1 = T2x[(S1xS2xS1)*S1]
42. 320-tiger 20-tiger 1-torus ((((III)(II))(II))I) - T2xC2x[S2*S1] = T2x[(S1x[S2*S1])*S1]
43. Torus double tiger torus (((((II)I)(II))(II))I) - T2xC2xC2xS1 = T2x[(S1xC2xS1)*S1] = T2x[(S1x[T2*S1])*S1]
44. 32-torus 20-tiger 1-torus ((((III)II)(II))I) - T2x[S2*S1]xS2 = T2x[(S2xS2)*S1]
45. 212-ditorus 20-tiger 1-torus (((((II)I)II)(II))I) - T2x[S2*S1]xT2 = T2x[(S2xT2)*S1]
46. 221-tiger 20-tiger 1-torus ((((II)(II)I)(II))I) - T2x[S2*S1]xC2 = T2x[(S2xC2)*S1]
47. 23-torus 20-tiger 1-torus ((((II)III)(II))I) - T2x[S3*S1]xS1 = T2x[(S3xS1)*S1]
48. 521-ditorus (((IIIII)II)I) - S1xS2xS4
49. 4121-tritorus ((((IIII)I)II)I) - S1xS2xS1xS3
50. 31121-tetratorus (((((III)I)I)II)I) - S1xS2xT2xS2
51. 211121-pentatorus ((((((II)I)I)I)II)I) - S1xS2xT4
52. 220-tiger 121-tritorus (((((II)(II))I)II)I) - S1xS2xT2xC2
53. 22121-tetratorus (((((II)II)I)II)I) - S1xS2xS1xS2xS1
54. 320-tiger 21-ditorus ((((III)(II))II)I) - S1xS2xS1x[S2*S1]
55. 21-torus 20-tiger 21-ditorus (((((II)I)(II))II)I) - S1xS2xS1xC2xS1 = S1xS2xS1x[T2*S1]
56. 3221-tritorus ((((III)II)II)I) - S1xS2xS2xS2
57. 21221-tetratorus (((((II)I)II)II)I) - S1xS2xS2xT2
58. 221-tiger 21-ditorus ((((II)(II)I)II)I) - S1xS2xS2xC2
59. 2321-tritorus ((((II)III)II)I) - S1xS2xS3xS1
60. 430-tiger 1-torus (((IIII)(III))I) - T2x[S3*S2]
61. 21-torus 40-tiger 1-torus ((((II)I)(IIII))I) - T2x[S3*S1]xS1 = T2x[T2xS3]
62. 31-torus 30-tiger 1-torus ((((III)I)(III))I) - T2x[S2*S1]xS2 = T2x[(S1xS2)*S2]
63. 31-torus 21-torus 0-tiger 1-torus ((((III)I)((II)I))I) - T2xC2x[S2*S1] = T2x[(S1xS2)*T2]
64. 211-ditorus 30-tiger 1-torus (((((II)I)I)(III))I) - T2x[S2*S1]xT2 = T2x[T3*S2]
65. Ditorus/torus tiger torus (((((II)I)I)((II)I))I) - T2xC2xC2xS1 = T2x[T3*T2]
66. 220-tiger 30-tiger 1-torus ((((II)(II))(III))I) - T2x[S2*S1]xC2 = T2x[(S1xC2)*S2]
67. Tiger/torus tiger torus ((((II)(II))((II)I))I) - T2xC2xC2xS1 = T2x[(S1xC2)*T2]
68. 22-torus 30-tiger 1-torus ((((II)II)(III))I) - T2x[S2*S2]xS1 = T2x[(S2xS1)*S2]
69. 22-torus 21-torus 0-tiger 1-torus ((((II)II)((II)I))I) - T2x[S2*S1]xC2 = T2x[(S2xS1)*T2]
70. 421-tiger 1-torus (((IIII)(II)I)I) - S1xS2x[S3*S1]
71. 31-torus 21-tiger 1-torus ((((III)I)(II)I)I) - S1xS2xC2xS2 = S1xS2x[(S1xS2)*S1]
72. 211-ditorus 21-tiger 1-torus (((((II)I)I)(II)I)I) - S1xS2xC2xT2 = S1xS2x[T3*S1]
73. 220-tiger 21-tiger 1-torus ((((II)(II))(II)I)I) - S1xS2xC2xC2 = S1xS2x[(S1xC2)*S1]
74. 22-torus 21-tiger 1-torus ((((II)II)(II)I)I) - S1xS2x[S2*S1]xS1 = S1xS2x[(S2xS1)*S1]
75. 431-ditorus (((IIII)III)I) - S1xS3xS3
76. 3131-tritorus ((((III)I)III)I) - S1xS3xS1xS2
77. 21131-tetratorus (((((II)I)I)III)I) - S1xS3xT3
78. 220-tiger 31-ditorus ((((II)(II))III)I) - S1xS3xS1xC2
79. 2231-tritorus ((((II)II)III)I) - S1xS3xS2xS1
80. 331-tiger 1-torus (((III)(III)I)I) - S1xS2x[S2*S2]
81. 21-torus 31-tiger 1-torus ((((II)I)(III)I)I) - S1xS2x[S2*S1]xS1 = S1xS2x[T2*S2]
82. 21-torus 21-torus 1-tiger 1-torus ((((II)I)((II)I)I)I) - S1xS2xC2xC2 = S1xS2x[T2*T2]
83. 3220-triger 1-torus (((III)(II)(II))I) - S1xS2xC2xS2 = S1xS2xS2xC2 = T2x[S2*S1*S1] = S1xS2x[S2*S1*S1]
84. Torus triger torus ((((II)I)(II)(II))I) - S1xS2xC3xS1 = T2x[T2*S1*S1]
85. 322-tiger 1-torus (((III)(II)II)I) - S1xS3x[S2*S1]
86. 21-torus 22-tiger 1-torus ((((II)I)(II)II)I) - S1xS3xC2xS1 = S1xS3x[T2*S1]
87. 341-ditorus (((III)IIII)I) - S1xS4xS2
88. 2141-tritorus ((((II)I)IIII)I) - S1xS4xT2
89. 2221-triger 1-torus (((II)(II)(II)I)I) - S1xS3xC3
90. 223-tiger 1-torus (((II)(II)III)I) - S1xS4xC2
91. 251-ditorus (((II)IIIII)I) - S1xS5xS1
92. 620-tiger ((IIIIII)(II)) - S1x[S5*S1]
93. 51-torus 20-tiger (((IIIII)I)(II)) - S1xC2xS4 = S1x[(S1xS4]*S1]
94. 411-ditorus 20-tiger ((((IIII)I)I)(II)) - S1xC2xS1xS3 = S1x[(T2xS3)*S1]
95. 3111-tritorus 20-tiger (((((III)I)I)I)(II)) - S1xC2xT2xS2 = S1x[(T3xS2)*S1]
96. Tetratorus tiger ((((((II)I)I)I)I)(II)) - S1xC2xT4 = S1x[T5*S1]
97. Tiger ditorus tiger (((((II)(II))I)I)(II)) - S1xC2xT2xC2 = S1x[(T3xC2)*S1]
98. 2211-tritorus 20-tiger (((((II)II)I)I)(II)) - S1xC2xS1xS2xS1 = S1x[(T2xS2xS1)*S1]
99. 320-tiger 1-torus 20-tiger ((((III)(II))I)(II)) - S1xC2xS1x[S2*S1] = S1x[T2*S1]x[S2*S1] = S1x[(T2x[S2*S1])*S1]
100. Torus tiger torus tiger (((((II)I)(II))I)(II)) - S1xC2xC2xS1 = S1x[(T2xC2xS1)*S1] = S1x[(T2x[T2*S1])*S1]
101. 321-ditorus 20-tiger ((((III)II)I)(II)) - S1xC2xS2xS2 = S1x[(S1xS2xS2)*S1]
102. 2121-tritorus 20-tiger (((((II)I)II)I)(II)) - S1xC2xS2xT2 = S1x[(S1xS2xT2)*S1]
103. 221-tiger 1-torus 20-tiger ((((II)(II)I)I)(II)) - S1xC2xS2xC2 = S1x[(S1xS2xC2)*S1]
104. 231-ditorus 20-tiger ((((II)III)I)(II)) - S1xC2xS3xS1 = S1x[(S1xS3xS1)*S1]
105. 420-tiger 20-tiger (((IIII)(II))(II)) - T2x[S3*S1*S1] = S1xC2x[S3*S1] - S1x[(S1x[S3*S1])*S1]
106. 31-torus20-tiger20-tiger ((((III)I)(II))(II)) - T2xC3xS2 = S1xC2xC2xS3 = S1x[(S1xC2xS2)*S1] = S1x[(S1x[(S1xS2)*S1])*S1]
107. Ditorus double tiger (((((II)I)I)(II))(II)) - T2xC3xT2 = S1x[(S1xC2xT2)*S1] = S1x[(S1x[T3*S1])*S1]
108. Triple tiger ((((II)(II))(II))(II)) - T3xC4 = S1x[(T2xC3)*S1] = S1x[(S1xC2xC2)*S1]
109. 22-torus 20-tiger 20-tiger ((((II)II)(II))(II)) - S1xC2x[S2*S1]xS1 = S1x[((S2xS1)*S1)*S1] = S1x[(S1x[S2*S1]xS1)*S1]
110. 42-torus 20-tiger (((IIII)II)(II)) - S1x[S2*S1]xS3 = S1x[(S2xS3)*S1]
111. 312-ditorus 20-tiger ((((III)I)II)(II)) - S1x[S2*S1]xS1xS2 = S1x[(S2xS1xS2)*S1]
112. 2112-tritorus 20-tiger (((((II)I)I)II)(II)) - S1x[S2*S1]xT3 = S1x[(S2xT3)*S1]
113. 220-tiger 2-torus 20-tiger ((((II)(II))II)(II)) - S1x[S2*S1]xS1xC2 = S1x[(S2xS1xC2)*S1]
114. 222-ditorus 20-tiger ((((II)II)II)(II)) - S1x[S2*S1]xS2xS1 = S1x[(S2xS2xS1)*S1]
115. 330-tiger 20-tiger (((III)(III))(II)) - S1xC2x[S2*S2] = S1x[(S1x[S2*S2])*S1]
116. 21-torus 30-tiger 20-tiger ((((II)I)(III))(II)) - S1xC2x[S2*S1]xS1 = S1x[(S1x[T2*S2])*S1]
117. Duotorus double tiger ((((II)I)((II)I))(II)) - T2xC3xC2 = T2xC2xC3 = S1x[(S1xC2xC2)*S1] = S1x[(S1x[T2*T2])*S1]
118. 321-tiger 20-tiger (((III)(II)I)(II)) - S1xS2xS3xC2 = S1x[S2*S1]x[S2*S1] = S1x[(S2x[S2*S1])*S1]
119. 21-torus21-tiger20-tiger ((((II)I)(II)I)(II)) - S1xS2xS2xC3 = S1x[S2*S1]xC2xS1 = S1x[(S2xC2xS1)*S1] = S1x[(S2x[T2*S1])*S1]
120. 33-torus 20-tiger (((III)III)(II)) - S1x[S3*S1]xS2 = S1x[(S3xS2)*S1]
121. 213-ditorus 20-tiger ((((II)I)III)(II)) - S1x[S3*S1]xT2 = S1x[(S3xT2)*S1]
122. Triger tiger (((II)(II)(II))(II)) - S1xS2xC4 = S1x[(S2xC3)*S1]
123. 222-tiger 20-tiger (((II)(II)II)(II)) - S1xS3xC3 = S1x[S3*S1]xC2 = S1x[(S3xC2)*S1]
124. 24-torus 20-tiger (((II)IIII)(II)) - S1x[S4*S1]xS1 = S1x[(S4xS1)*S1]
125. 62-torus ((IIIIII)II) - S2xS5
126. 512-ditorus (((IIIII)I)II) - S2xS1xS4
127. 4112-tritorus ((((IIII)I)I)II) - S2xT2xS3
128. 31112-tetratorus (((((III)I)I)I)II) - S2xT3xS2
129. 211112-pentatorus ((((((II)I)I)I)I)II) - S2xT5
130. 220-tiger 112-tritorus (((((II)(II))I)I)II) - S2xT3xC2 = S2xT2x[S1xC2]
131. 22112-tetratorus (((((II)II)I)I)II) - S2xT2xS2xS1
132. 320-tiger 12-ditorus ((((III)(II))I)II) - S2xT2x[S2*S1]
133. 21-torus 20-tiger 12-ditorus (((((II)I)(II))I)II) - S2xS1x[S1xC2xS1] = S2xT2x[T2*S1]
134. 3212-tritorus ((((III)II)I)II) - S2xS1xS2xS2
135. 21212-tetratorus (((((II)I)II)I)II) - S2xS1xS2xT2
136. 221-tiger 12-ditorus ((((II)(II)I)I)II) - S2xS1xS2xC2
137. 2312-tritorus ((((II)III)I)II) - S2xS1xS3xS1
138. 420-tiger 2-torus (((IIII)(II))II) - S2xS1x[S3*S1]
139. 31-torus 20-tiger 2-torus ((((III)I)(II))II) - S2xS1xC2xS2 = S2xS1x[(S1xS2)*S1]
140. 211-ditorus 20-tiger 2-torus (((((II)I)I)(II))II) - S2xS1xC2xT2 = S2xS1x[T3*S1]
141. 220-tiger 20-tiger 2-torus ((((II)(II))(II))II) - S2xT2xC3 = S2xS1x[(S1xC2)*S1]
142. 22-torus 20-tiger 2-torus ((((II)II)(II))II) - S2xS1x[S2*S1]xS1 = S2xS1x[(S2xS1)*S1]
143. 422-ditorus (((IIII)II)II) - S2xS2xS3
144. 3122-tritorus ((((III)I)II)II) - S2xS2xS1xS2
145. 21122-tetratorus (((((II)I)I)II)II) - S2xS2xT3
146. 220-tiger 22-ditorus ((((II)(II))II)II) - S2xS2xS1xC2
147. 2222-tritorus ((((II)II)II)II) - S2xS2xS2xS1
148. 330-tiger 2-torus (((III)(III))II) - S2xS1x[S2*S2]
149. 21-torus 30-tiger 2-torus ((((II)I)(III))II) - S2xS1x[S2*S1]xS1 = S2xS1x[T2*S2]
150. 21-torus 21-torus 0-tiger 2-torus ((((II)I)((II)I))II) - S2xS1xC2xC2 = S2xS1x[T2xT2]
151. 321-tiger 2-torus (((III)(II)I)II) - S2xS2x[S2*S1]
152. 21-torus 21-tiger 2-torus ((((II)I)(II)I)II) - S2xS2xC2xS1 = S2xS2x[T2*S1]
153. 332-ditorus (((III)III)II) - S2xS3xS2
154. 2132-tritorus ((((II)I)III)II) - S2xS3xT2
155. 2220-triger 2-torus (((II)(II)(II))II) - S2xS2xC3
156. 222-tiger 2-torus (((II)(II)II)II) - S2xS3xC2
157. 242-ditorus (((II)IIII)II) - S2xS4xS1
158. 530-tiger ((IIIII)(III)) - S1x[S4*S2]
159. 21-torus 50-tiger (((II)I)(IIIII)) - S1x[S4*S1]xS1 = S1x[S4*T2]
160. 41-torus 30-tiger (((IIII)I)(III)) - S1x[S2*S1]xS3 = S1x[(S1xS3)*S2]
161. 41-torus 21-torus 0-tiger (((IIII)I)((II)I)) - S1xC2x[S3*S1] = S1x[(S1xS3)*T2]
162. 311-ditorus 30-tiger ((((III)I)I)(III)) - S1x[S2*S1]xS1xS2 = S1x[S2*T2]xS2 = S1x[(T2xS2)*S2]
163. 311-ditorus 21-torus 0-tiger ((((III)I)I)((II)I)) - S1xC2xC2xS2 = S1x[T2*T2]xS2 = S1x[(T2xS2)*T2]
164. 2111-tritorus 30-tiger (((((II)I)I)I)(III)) - S1x[S2*S1]xT3 = S1x[T4*S2]
165. Tritorus/torus tiger (((((II)I)I)I)((II)I)) - S1xC2xC2xT2 = S1x[T4*T2]
166. 220-tiger 1-torus 30-tiger ((((II)(II))I)(III)) - T3x[S2*S1*S1] = S1x[S2*S1]xS1xC2 = S1x[(T2xC2)*S2]
167. (Tiger torus)/torus tiger ((((II)(II))I)((II)I)) - T3xC3xS1 = S1x[(T2xC2)*T2]
168. 221-ditorus 30-tiger ((((II)II)I)(III)) - S1x[S2*S1]xS2xS1 = S1x[(S1xS2xS1)*S2]
169. 221-ditorus 21-torus 0-tiger ((((II)II)I)((II)I)) - S1xC2x[S2*S1]xS1 = S1x[(S1xS2xS1)*T2]
170. 320-tiger 30-tiger (((III)(II))(III)) - T2x[S2*S2*S1] = T2x[S2*S2]xS1 = S1x[S2*S1]x[S2*S1] = S1x[(S1x[S2*S1])*S2]
171. 320-tiger 21-torus 0-tiger (((III)(II))((II)I)) - T2x[T2*S2*S1] = T2xC2x[S2*S1] = S1x[(S1x[S2*S1])*T2]
172. 21-torus 20-tiger 30-tiger ((((II)I)(II))(III)) - S1x[(S1xC2xS1)*S2] = S1x[(S1x[T2*S1])*S2]
173. (Torus tiger)/torus tiger ((((II)I)(II))((II)I)) - T2xC3xC2 = S1x[(S1x[T2*S1])*T2]
174. 32-torus 30-tiger (((III)II)(III)) - S1x[S2*S2]xS2 = S1x[(S2xS2)*S2]
175. 32-torus 21-torus 0-tiger (((III)II)((II)I)) - S1x[(S2xS2)*T2] = S1x[S2*S1]x[S2*S1]
176. 212-ditorus 30-tiger ((((II)I)II)(III)) - S1x[S2*S2]xT2 = S1x[(S2xT2)*S2]
177. 212-ditorus 21-torus 0-tiger ((((II)I)II)((II)I)) = S1x[S2*S1]xC2xS1 = S1x[(S2xT2)*T2]
178. 221-tiger 30-tiger (((II)(II)I)(III)) - S1xS2xC2xS2 = S1x[S2*S2]xC2 = S1x[(S2xC2)*S2]
179. 221-tiger 21-torus 0-tiger (((II)(II)I)((II)I)) - S1xS2xC3xS1 = S1x[(S2xC2)*T2]
180. 23-torus 30-tiger (((II)III)(III)) - S1x[S3*S2]xS1 = S1x[(S3xS1)*S2]
181. 23-torus 21-torus 0-tiger (((II)III)((II)I)) - S1x[S3*S1]xC2 = S1x[(S3xS1)*T2]
182. 521-tiger ((IIIII)(II)I) - S2x[S4*S1]
183. 41-torus 21-tiger (((IIII)I)(II)I) - S2xC2xS3 = S2x[(S1xS3)*S1]
184. 311-ditorus 21-tiger ((((III)I)I)(II)I) - S1xC2xS1xS2 = S2x[(T2xS2)*S1]
185. 2111-tritorus 21-tiger (((((II)I)I)I)(II)I) - S2xC2xT3 = S2x[T4*S1]
186. 220-tiger 1-torus 21-tiger ((((II)(II))I)(II)I) - S2xT2xC3 = S2x[(T2xC2)*S1]
187. 221-ditorus 21-tiger ((((II)II)I)(II)I) - S2xC2xS2xS1 = S2x[(S1xS2xS1)*S1]
188. 320-tiger 21-tiger (((III)(II))(II)I) - S2xS1xC2xS2 = S2x[(S1x[S2*S1])*S1]
189. 21-torus 20-tiger 21-tiger ((((II)I)(II))(II)I) - S2xT2xC3 = S2x[(S1xC2xS1)*S1] = S2x[(S1x[T2*S1])*S1]
190. 32-torus 21-tiger (((III)II)(II)I) - S2x[S2*S1]xS2 = S2x[(S2xS2)*S1]
191. 212-ditorus 21-tiger ((((II)I)II)(II)I) - S2x[S2*S1]xT2 = S2x[(S2xT2)*S1]
192. 221-tiger 21-tiger (((II)(II)I)(II)I) - S2xS2xC3 = S2x[(S2xC2)*S1]
193. 23-torus 21-tiger (((II)III)(II)I) - S2xS3xC2 = S2x[S3*S1]xS1 = S2x[(S3xS1)*S1]
194. 53-torus ((IIIII)III) - S3xS4
195. 413-ditorus (((IIII)I)III) - S3xS1xS3
196. 3113-tritorus ((((III)I)I)III) - S3xT2xS2
197. 21113-tetratorus (((((II)I)I)I)III) - S3xT4
198. 220-tiger 13-ditorus ((((II)(II))I)III) - S3xT2xC2
199. 2213-tritorus ((((II)II)I)III) - S3xS1xS2xS1
200. 320-tiger 3-torus (((III)(II))III) - S3xS1x[S2*S1]
201. 21-torus 20-tiger 3-torus ((((II)I)(II))III) - S3xT2xC2 = S3xS1xC2xS1 = S3xS1x[T2*S1]
202. 323-ditorus (((III)II)III) - S3xS2xS2
203. 2123-tritorus ((((II)I)II)III) - S3xS2xT2
204. 221-tiger 3-torus (((II)(II)I)III) - S3xS2xC2
205. 233-ditorus (((II)III)III) - S3xS3xS1
206. 440-tiger ((IIII)(IIII)) - S1x[S3*S3]
207. 31-torus 40-tiger (((III)I)(IIII)) - S1x[S3*S1]xS3 = S1x[(S1xS2)*S3]
208. 31-torus 31-torus 0-tiger (((III)I)((III)I)) - S1xC2x[S2*S2] = S1x[(S1xS2)*(S1xS2)]
209. 211-ditorus 40-tiger ((((II)I)I)(IIII)) - S1x[S3*S1]xT2 = S1x[T3*S3]
210. 211-ditorus 31-torus 0-tiger ((((II)I)I)((III)I)) - S1xC2x[T2*S2] = S1x[(S1xS2)*T3]
211. Duoditorus tiger ((((II)I)I)(((II)I)I)) - S1xC2xC2xC2 = S1x[T3*T3]
212. 220-tiger 40-tiger (((II)(II))(IIII)) - T2x[S3*C2] = S1x[(S1xC2)*S3]
213. 220-tiger 31-torus 0-tiger (((II)(II))((III)I)) - T2xC3xS2 = S1x[(S1xC2)*(S1xS2)]
214. Tiger/ditorus tiger (((II)(II))(((II)I)I)) - T2xC3xT2 = S1x[(S1xC2)*T3]
215. Duotiger tiger (((II)(II))((II)(II))) - S1x[(S1xC2)*(S1xC2)]
216. 22-torus 40-tiger (((II)II)(IIII)) - S1x[S3*S2]xS1 = S1x[(S2xS1)*S3]
217. 22-torus 31-torus 0-tiger (((II)II)((III)I)) - S1x[(S2xS1)*(S1xS2)] = S1x[S2*S1]x[S2*S1]
218. 211-ditorus 22-torus 0-tiger ((((II)I)I)((II)II)) - S1x[S2*S1]xC2xS1 = S1x[(S2xS1)*T3]
219. 220-tiger 22-torus 0-tiger (((II)(II))((II)II)) - S1x[S2*S1]xC3 = S1x[(S1xC2)*(S2xS1)]
220. 22-torus 22-torus 0-tiger (((II)II)((II)II)) - S1x[S2*S2]xC2 = S1x[(S2xS1)*(S2xS1)]
221. 431-tiger ((IIII)(III)I) - S2x[S3*S2]
222. 21-torus 41-tiger (((II)I)(IIII)I) - S2x[T2*S3] = S2x[S3*S1]xS1
223. 31-torus 31-tiger (((III)I)(III)I) - S2x[(S1xS2)*S2] = S2x[S2*S1]xS2
224. 31-torus 21-torus 1-tiger (((III)I)((II)I)I) - S2xC2x[S2*S1] = S2x[(S1xS2)*T2]
225. 211-ditorus 31-tiger ((((II)I)I)(III)I) - S2x[T3*S2] = S2x[S2*S1]xT2
226. 211-ditorus 21-torus 1-tiger ((((II)I)I)((II)I)I) - S2xC2xC2xS1 = S2x[T3*T2]
227. 220-tiger 31-tiger (((II)(II))(III)I) - S2x[(S1xC2)*S2] = S2x[S2*S1]xC2
228. 220-tiger 21-torus 1-tiger (((II)(II))((II)I)I) - S2xC2xC2xS1 = S2x[(S1xC2)*T2]
229. 22-torus 31-tiger (((II)II)(III)I) - S2x[S2*S2]xS1 = S2x[(S2xS1)*S2]
230. 22-torus 21-torus 1-tiger (((II)II)((II)I)I) - S2x[S2*S1]xC2 = S2x[(S2xS1)*T2]
231. 4220-triger ((IIII)(II)(II)) - S2x[S3*C2]
232. 31-torus 220-triger (((III)I)(II)(II)) - S2xC3xS2 = S2x[(S1xS2)*C2]
233. Ditorus triger ((((II)I)I)(II)(II)) - S2xC3xT2 = S2x[T3*C2]
234. Tiger triger (((II)(II))(II)(II)) - S2xS1xC4 = S2x[(S1xC2)*C2]
235. 22-torus 220-triger (((II)II)(II)(II)) - S2x[(S2xS1)*C2] = S2x[S2*C2]xS1
236. 422-tiger ((IIII)(II)II) - S3x[S3*S1]
237. 31-torus 22-tiger (((III)I)(II)II) - S3xC2xS2 = S3x[(S1xS2)*S1]
238. 211-ditorus 22-tiger ((((II)I)I)(II)II) - S3xC2xT2 = S3x[T3*S1]
239. 220-tiger 22-tiger (((II)(II))(II)II) - S3xS1xC3 = S3x[(S1xC2)*S1]
240. 22-torus 22-tiger (((II)II)(II)II) - S3xS2xC2 = S3x[(S2xS1)*S1]
241. 44-torus ((IIII)IIII) - S4xS3
242. 314-ditorus (((III)I)IIII) - S4xS1xS2
243. 2114-tritorus ((((II)I)I)IIII) - S4xT3
244. 220-tiger 4-torus (((II)(II))IIII) - S4xS1xC2
245. 224-ditorus (((II)II)IIII) - S4xS2xS1
246. 3320-triger ((III)(III)(II)) - S2x[S2*S2*S1]
247. 21-torus 320-triger (((II)I)(III)(II)) - S2x[S2*C2]xS1 = S2x[T2*S2*S1]
248. Duotorus triger (((II)I)((II)I)(II)) - S2xC3xC2 = S2x[T2*T2xS1]
249. 332-tiger ((III)(III)II) - S3x[S2*S2]
250. 21-torus 32-tiger (((II)I)(III)II) - S3x[S2*S1]xS1 = S3x[T2*S2]
251. 21-torus 21-torus 2-tiger (((II)I)((II)I)II) - S3xC2xC2 = S3x[T2*T2]
252. 3221-triger ((III)(II)(II)I) - S3x[S2*C2]
253. 21-torus 221-triger (((II)I)(II)(II)I) - S3xC3xS1 = S3x[T2*C2]
254. 323-tiger ((III)(II)III) - S4x[S2*S1]
255. 21-torus 23-tiger (((II)I)(II)III) - S4xC2xS1 = S4x[T2*S1]
256. 35-torus ((III)IIIII) - S5xS2
257. 215-ditorus (((II)I)IIIII) - S5xT2
258. Tetriger ((II)(II)(II)(II)) - S3xC4
259. 2222-triger ((II)(II)(II)II) - S4xC3
260. 224-tiger ((II)(II)IIII) - S5xC2
261. 26-torus ((II)IIIIII) - S6xS1
Last edited by ICN5D on Sat Oct 04, 2014 8:31 pm, edited 4 times in total.
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### Re: N-Sphere Bundle over N-Sphere Terminology

You're using x as the spheration or inflation product here, but it seems to me that you're using it backwards compared to how we used to do it? But I'm not sure.

Also, x is usually the standard Cartesian product in mathematics. The spheration product is kind of a special case of the Cartesian product but I think it's a confusing notation. Usually S^1 x S^1 could either mean the Clifford torus (II)(II) or the standard torus ((II)I), and you'd have to work out which one from context. Since they're homeomorphic, topologists often don't care which one you're talking about.

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### Re: N-Sphere Bundle over N-Sphere Terminology

Hmm, that's interesting. So, there's only one kind of ambiguous cartesian product? Well, that won't work. Because when getting into 4D, there are two different products at work, in the same shape: the tiger. I don't think many know about this one. So, there would be a need to differentiate those two apart, in order to continue any further. This puts some merit back into the spheration product, since a tiger is a cartesian product of two circles, then spherated by a circle. So, in that context, the two products work together to build the rest of the toratopes.

I guess this is what I was trying to do with S^1x[S^1 * S^1] , where "x" is spherate and " [ ... * ... ] " is cartesian product. The rest of the toratopes need an expansion to the product-types, as the tigroid family are spherated cartesian products.

For simplicity sake, I use a generalized clifford torus Cn to represent multiple S^1 together, as in Cn = [S^1^n]

For the tigroid families, both forms can be used, to show their relationship. Take (((II)I)(II)) , it can be built through S^1 x C^2 x S^1 , or its pure tigroid form S^1 x [T^2 * S^1].

And for a more complicated one, ((((II)I)(II)I)I) can be built by S^1 x S^2 x[T^2 * S^1] , a torisphere ((III)I) inflating the 3-frame of ((II)I)(II), a (torus*circle)-prism.
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### Re: N-Sphere Bundle over N-Sphere Terminology

At the moment, i see only two distinct products in the torotopes, but this confounds naming no end.

The ordinary single-nested brackets is a straight comb product. It's the sibling-product such as met in the inner brackets of ((ii)(ii)) that confounds everything.
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### Re: N-Sphere Bundle over N-Sphere Terminology

Well, that's a good point. Exactly what part is confounded? I would agree that there are two distinct products at work with closed. Though, my intuition tells me it needs an expansion to differentiate between a spheration, and a cartesian product. Because once we get to a tiger, both are combined. This seems to help with the confounding, since it breaks down toratopes into a chain of spheration and cartesian products.

Where, big enough toratopes have multiple solutions, and thus multiple build sequences. By studying the arrays we get in 3D, there are many different combinations of small-shape toratope inflates large-shape toratope/clifford torus.

Take the 3D intercept array of 9D (((((II)I)(II))I)((II)I)) :

In this array, there are several lower-D closed toratopes that inflate some margin of an open. Where, the margins cut down to make points in an array. This seems to be what makes the arrays, when we cut down a clifford torus, or combination of them. In high-D, the large shapes are independent toratopic combos of the clifford tori and toratopes. The small shapes are any closed toratope, that was embedded into all points of the cliffords.

Construction Tree from a circle (II) to a (((((II)I)(II))I)((II)I))
Code: Select all
`2D    3D         4D           5D               6D                 7D                    8D                        9D                                                                                          (((II)I)((II)I))                                                                   (((((II)I)(II))I)I)                                  (((II)I)(II))  ((((II)I)(II))I)                       (((((II)I)(II))I)(II))              ((II)(II))                                   ((((II)(II))I)(II))                                 (II) ((II)I)              (((II)(II))I)  ((((II)(II))I)I)                       ((((II)(II))I)((II)I))  (((((II)I)(II))I)((II)I))              (((II)I)I)                                   ((((II)I)I)((II)I))                                  ((((II)I)I)I)  ((((II)I)I)(II))                       (((((II)I)I)I)((II)I))                                                           (((((II)I)I)I)(II))                                         (((((II)I)I)I)I)                                                                                                            128i    64         32            16               8                  4                     2                          1`

In the chart above, each toratope has spherated an N-2 margin of open, or N-1 surface of closed, of some kind, which produces a unique array for that shape. All of these make that same array. There are five toratopes that inflate a uniquely oriented margin in 6D, where it has the largest variety of intercepts. So, to break them all down,

General Rules:

• Tm = S1x...m : T1 = S1 , T2 = S1xS1 , T3 = S1xS1xS1 , T4 = S1xS1xS1xS1 , etc

• Cn = n-frame of [B2]^n : 1-frame of B2 , 2-frame of [B2*B2] , 3-frame of [B2*B2*B2] , 4-frame of [B2*B2*B2*B2] , etc

• Cn = [S1]^n : C2=[S1*S1] , C3=[S1*S1*S1] , C4=[S1*S1*S1*S1] , etc

• Cn = intercepts n-plane as n-cube array of 2^n points

• Cn x Tm = intercepts n-plane as n-cuboid array of (2^n)*m points

• Cn x Cm = n>m , intercepts n-plane as n-cuboid array of (2^n)*(2^m) points

• Cn x Cm x Tk = n>m , intercepts n-plane as n-cuboid array of (2^n)*(2^m)*k points

• Cn x Cm x Ck = n>m,k : intercepts n-plane as n-cuboid array (2^n)*(2^m)*(2^k) points

• Cn x Cm x Ck x Tj = n>m,k : intercepts n-plane as n-cuboid array (2^n)*(2^m)*(2^k)*j points

2D
(II) S1 inflates 2-frame of T2, then 5-frame of ((II)I)(II)((II)I) [B2]^3xC2 prismic duotorus as 4x2x4 array of 32 points: T3xC3xC2

3D - 64 intercepts
((II)I) T2 inflates 1-frame of S1, then 5-frame of ((II)I)(II)((II)I) [B2]^3xC2 prismic duotorus, as 4x2x4 array of 32 points: T3xC3xC2

4D - 32 intercepts
((II)(II)) S1xC2 inflates 1-frame of S1, then 3-frame of ((II)I)(II)(II) [B2*B2*B2]xS1 prismic torus as 4x2x2 array of 16 points: S1xC2xS1xC3xS1
(((II)I)I) T3 inflates 5-frame of ((II)I)(II)((II)I) (B2*B2*B2]xC2 prismic duotorus, as 4x2x4 array of 32 points: T3xC3xC2

5D - 16 intercepts
(((II)I)(II)) S1xC2xS1 inflates 4-frame of ((II)I)(II)(II) [B2*B2*B2]xS1 prismic torus as 4x2x2 array of 16 points: S1xC2xS1xC3xS1
(((II)(II))I) T2xC2 inflates 4-frame of (((II)I)(II)) S1xC2xS1 as 2x4 array of 8 circles: T2xC2xS1xC2xS1
((((II)I)I)I) T4 inflates 4-frame of ((II)I)((II)I) [B2*B2]xC2 prismic duotorus as 4x4 array of 16 points: T4xC2xC2

6D - 8 intercepts
(((II)I)((II)I)) S1xC2xC2 inflates 3-frame of (((II)I)(II) [B2*B2]xS1 as 4x2 array of 8 points: S1xC2xC2xC2xS1
((((II)I)(II))I) T2xC2xS1 inflates 4-frame of (((II)I)(II)) S1xC2xS1 as a 4x2 array of 8 circles: T2xC2xT2xC2xS1
((((II)(II))I)I) T3xC2 inflates 3-frame of ((II)I)(II) [B2*B2]xS1 as 2x4 array of 8 points: T3xC2xC2xS1
((((II)I)I)(II)) S1xC2xT2 inflates 3-frame of (II)(II)(II) B2*B2*B2 as 2x2x2 array of 8 points: S1xC2xT2xC3
(((((II)I)I)I)I) T5 inflates 3-frame of ((II)I)(II) [B2*B2]xS1 as 2x4 array of 8 points: T5xC2xS1

7D - 4 intercepts
(((((II)I)(II))I)I) T3xC2xS1 inflates 2-frame of ((II)I) T2 as 4 points along line: T3xC2xT3
((((II)(II))I)(II)) T3xC3 inflates 2-frame of (II)(II) B2*B2, as 2x2 array of 4 points: T3xC3xC2
((((II)I)I)((II)I)) S1xC2xC2xS1 inflates 2-frame of (II)(II) B2*B2 as 2x2 array of 4 points: S1xC2xC2xS1xC2
(((((II)I)I)I)(II)) T4xC2 inflates 2-frame of (II)(II) B2*B2 as 2x2 array of 4 points: T4xC2xC2

8D - 2 intercepts
(((((II)I)(II))I)(II)) T4xC3 inflates S1 as 2 points along line: T4xC3xS1
((((II)(II))I)((II)I)) S1xC2xS1xC3 inflates S1 as 2 points along line: S1xC2xS1xC3xS1
(((((II)I)I)I)((II)I)) T3xC2xC2 inflates S1 as 2 points along line: T3xC2xC2xS1

9D - 1 shape
(((((II)I)(II))I)((II)I)) built by all the above

Using diameter hierarchy would be a good reference for the revolution sequence to take. So, for

(((((II)I)(II))I)((II)I)) becomes (((((maj1)sec1)(maj2)tert)quat)((maj3)sec2)min)

maj1,2,3 : three equal majors from a C3
sec1,2 : two equal secondaries from a C2
tert : one tertiary from S1
quat : one quaternary from S1
min : minor diameter from S1

The sequence of revolution then becomes

min x quat x tert x sec1,2 x maj1,2,3
as
S1 x S1 x S1 x C2 x C3 = T3xC2xC3 = (((((II)I)(II))I)((II)I))

Another way can be to read right to left, in increasing order of nested circles:

(((((II)I)(II))I)((II)I)) = (((((maj)sec1)(sec2)tert)quin1)((quat)quin2)min)

maj : one major from an S1
sec1,2 : 2 eq secondaries from a C2
tert : one tertiary from an S1
quat : one quaternary from an S1
quin1,2 : 2 eq quinternaries from a C2
min : one minor from an S1

min x quin1,2 x quat x tert x sec1,2 x maj
as
S1 x C2 x S1 x S1 x C2 x S1 = S1xC2xT2xC2xS1 , which is a nice symmetrical sequence, equal to all above.

So, in the end, the goal is to derive the build sequence as another way to interpret the notation with common terms. Of course, there are many solutions in this definition, that can come out of just one unique notation sequence. The lowest trace array cut of any toratope can be described in many alternate yet equal surfaces of revolution. The 'N bundle over the M' sequence describes the order of revolutions, beginning with the universal starting toratope of a circle.

Implicit functions have been regarded as being largely unsolvable, so here is a way to derive those solutions. I guess there are as many solutions as there are distinct intercepts in a lower-D hyperplane. I could be getting ahead of myself, but it seems that playing around with implicits this way leads to a very effective workaround for unsolvables. Also the fact that these equations follow an exact combinatorial sequence, of an infinite integer sequence, is quite mindblowing. There's something very special about that, I feel.
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### Re: N-Sphere Bundle over N-Sphere Terminology

Another thing I though about was, the fact that a torus and clifford torus can be made by bending a square into higher dimensions. So, as a generalization, couldn't all toratopes be made by folding an n-cube, various ways into N+1? The first fold would create a hollow tube like structure. Then from there, are several more ways to fold it. As far as I know of, there are 4 ways to fold a square. First step is a hollow tube. A hose link or sock roll of the tube in 3D would make a torus, in 4D a clifford torus.

So, how many more ways are there in 4D to fold?

According to toratopic theory, the combination sequence of A000669 is the total amount of topologically distinct toratopes in n-dimensions. If we have a 3-cube, there should be only 5 ways to fold it to make a 4D toratope. Each will exhaust the sequence list in 4D. Cartesian surface products and spheration products both work together to fold high-D cubes. By gluing the cube's 6 faces in various ways into 4D, we create a smooth 3-manifold that curves into 4D. We can also bend it once into 5D, and once into 6D.

Since we have a 3-manifold cube, there are various 5 and 6D clifford tori we can make, too. Only two, I'm afraid, that come out as a :

• clifford torus over a circle C2xS1 = [ 3-frame of ((II)I)(II) ] , xyz bent into wv: (((xw)y)(zv))
• clifford 3-torus C3 = [3-frame of (II)(II)(II) ] , xyz bent into wvu: (xw)(yv)(zu) : it's a 3D space that barely occupies a 3D plane. It's too busy arcing up into 4, 5, and 6D. Where in 3D, C3 comes out as 8 locations of a 0D point, in the vertices of a cube. Each point is the 3-manifold that curves into another 3D universe, evenly occupying 4, 5, and 6D.

So, this means there are 5 ways in 4D , 1 in 5D , and 1 in 6D, to fold a 3-cube. I'd be interested to know if there's a number sequence for cliffords, as it's another new folding that builds a distinct shape. Toratopic notation seems to rely on a chain of revolutions into N+1, where cliffords are made by N+2 , N+3 , etc.

Come to think of it, I feel the relationship would be based solely on which shape has a clifford torus in it. That is, all open toratopes with at least a product of two circles. This will give it a duocylinder margin of some kind, as the universal beginning to all clifford tori. The first surface product encountered is the clifford torus, where all subsequent cliffords are variations of it, in its own combinatoric sequence.

Perhaps the bundling sequence has some relation to folding?
Last edited by ICN5D on Fri Aug 01, 2014 6:27 am, edited 1 time in total.
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### Re: N-Sphere Bundle over N-Sphere Terminology

Actually, I think I forgot another 3-frame in 5D:

(III)(II) : 3-frame is [S2*S1]
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### Re: Surface Bundle over a Surface Terms

If you're creating shapes by gluing faces of squares and cubes, that's how you get things like Mobius strips and Klein bottles. You can get a sphere this way as well. The square with the gluing is called the fundamental polygon. It doesn't have to be a square or a cube, you can use any polygon or polyhedron. There was a hypothesis that the universe might exist on a Poincaré dodecahedral space. Take a dodecahedron, and for each pair of opposite faces, twist one of them by 1/5 of a turn, and glue them together.

Even if we restrict ourselves to cubes, there are many possible 4D shapes you can make this way. I suspect they've all been studied to some extent, since this is very important in topology.

Also note that even though you can make a (2-frame) torus and a (2-frame) duocylinder by gluing the edges of a square, you can't really distinguish between the two topologically.

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### Re: Surface Bundle over a Surface Terms

The comb product is the repetition of surfaces. It's an actual product. The reason that the tiger and the di-torus come out the same is because they're both over the same three circles.

On the other hand, the surface of ((ii)ii) = (ii)(oii) and ((iii)i) = (iii)(oi) are topologically equal, but you can't distort the shapes into each other, because topologically equal surfaces do not enclose topologically equal volumes.
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