## The Tiger Explained

Discussion of shapes with curves and holes in various dimensions.

### Re: The Tiger Explained

Marek14 wrote:Both torisphere and spheritorus have S1xS2 as their surface -- you'd have nonvanishing spheres in two dimensions and nonvanishing circles in the third dimension.

I've seen that description before. What exactly does " non-vanishing sphere/circle " mean? Does it relate to the dimensionality of the curvature, or repeating spaces?
in search of combinatorial objects of finite extent
ICN5D
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### Re: The Tiger Explained

ICN5D wrote:
Marek14 wrote:Both torisphere and spheritorus have S1xS2 as their surface -- you'd have nonvanishing spheres in two dimensions and nonvanishing circles in the third dimension.

I've seen that description before. What exactly does " non-vanishing sphere/circle " mean? Does it relate to the dimensionality of the curvature, or repeating spaces?

It works like this: If you draw a circle on a sphere, you can shrink the circle until it reduces to a point and vanishes. But if you draw a circle on a torus, there are two distinct categories of circles that cannot be shrunk this way: circles that go through the hole and circles parallel to the main plane. These circles also don't divide the surface of torus; you can get from one side to the other by going all the way around, without crossing the circle.

Note that torus has two distinct cuts in shape of two circles. This is related.

Now, let's look at torisphere. Its 3D cuts are pair of spheres and torus. The torus is not nonvanishing since it's single, but the sphere is in pair, and so it is nonvanishing and cannot be shrunk into a point. Similarly, a spheritorus has nonvanishing spheres because of its "two spheres" cut.

If we look at 2D cuts of torisphere, we found that there is pair of circles or two circles. Pair of circles can evolve into either torus or pair of spheres -- since ONE circle can correspond to ONE sphere, it's not nonvanishing (it can be shrunk by moving it on the sphere). But two circles only evolves into one torus, which means it IS nonvanishing. That gives us the total set of one type of spheres and one type of circles.

2D cuts of spheritorus are similar: there's an empty cut (which we don't care about), two circles cut and pair of circles cut. Similar logic leads us to conclusion that two circles cut is not nonvanishing since it can evolve in two spheres, while pair of circles cut shows nonvanishing circles since it only evolves in one torus.

Similarly for ditorus and tiger:

Ditorus (((II)I)I)
3D cuts:
*Minor pair of toruses (((II)I))
*Major pair of toruses (((II))I)
*Two toruses (((I)I)I)
2D cuts:
*Quartet of circles (((II)))
*Two pairs of circles (((I)I))
*Four circles (((I))I)
Empty cut ((()I)I)

Ditorus is easy: it has three distinct types of nonvanishing circles and every combination of two of them leads to one type of nonvanishing torus that cannot be shrunk to a circle.

Tiger ((II)(II))
3D cuts:
*Vertical stack of two toruses A ((II)(I))
*Vertical stack of two toruses B ((I)(II))
2D cuts:
Empty cut A ((II)())
Empty cut B (()(II))
*2x2 array of circles ((I)(I))

Looks like a tiger has one type of nonvanishing circle and two types of nonvanishing toruses. And it seems that I can indeed imagine how a "major" circle on one of its torus cut could be shrunk thanks to topology changes while rotating.
Marek14
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### Re: The Tiger Explained

Time for some totally rad animations. These come from the 7D toroid ((((II)I)((II)I))I) , built by T2xC2xC2 or T2x[T2*T2].

The implicit function I used for the animations is

((((Ia))((A)))I) :

((sqrt((sqrt(x^2+(y*cos(b)-a*sin(b))^2)-5)^2)-2.5)^2+(sqrt((sqrt((y*sin(b)+a*cos(b))^2)-5)^2)-2.5)^2-2)^2+z^2-1.25^2 = 0

This function is exploring the 5D cut of ((((II)I)((I)))I) - 1x1x1x4x1 tertiary column of 4 tritoruses ((((II)I)I)I) with a single translate + rotate parameter.

Each window represents an incremental tilting of our 3D slicing hyperplane by 18° . Starting in the empty cut of ((((II))(()))I), we rotate to ((((I))((I)))I) . For each rotation step, we translate from below the four tritoruses when A = -12.5 up, through the whole column, slicing along the way to A = 12.5. As we rotate, we end up slicing through them at differing angles, until we end up slicing through all of them when A = 0, in the axial midsection of ((((I))((I)))I) , a 4x4x1 array of 16 tori.

Θ = 90° , slicing the four tritoruses top to bottom , evolution of cut ((((II))(()))I)

Θ = 72°

Θ = 54°

Θ = 36°

Θ = 18°

Θ = 0° , slicing the four tritoruses through the middle , evolution of cut ((((I))((I)))I)

I've got another one coming tomorrow, that shows what happens when we go through all four rotations of two dimensions, held at one of the tritoruses.
in search of combinatorial objects of finite extent
ICN5D
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### Re: The Tiger Explained

I think that the 18-degree one is most interesting, really giving you the feeling of oblique cut
Marek14
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### Re: The Tiger Explained

Yeah, 18 degrees is a cool one

Here's the breakdown of all 261 distinct 7-manifolds that curve into 8D:

Code: Select all
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 Thu Nov 27, 2014 3:15 am, edited 1 time in total.
in search of combinatorial objects of finite extent
ICN5D
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### Re: The Tiger Explained

Here's that fascinating morphing sequence of ((((II)I)((II)I))I). In this animation, we have translated to tritorus number 3 when C = 2.5, in the column of 4 in ((((II)I)((I))I). Here, we go through all four 90 degree orientations in the function:

• ((((Ac))((C)a))I) - ((((I))((I)))I)

((sqrt((sqrt((x*sin(b) + a*cos(b))^2+(y*cos(d) - c*sin(d))^2) -5)^2) -2.5)^2 + (sqrt((sqrt((y*sin(d) + c*cos(d))^2) -5)^2 +(x*cos(b) - a*sin(b))^2) -2.5)^2 -2)^2 +z^2 -1.25^2 = 0

A = 0
C = 2.5

Run [B,D] through sequence [0,0] > [1.57,0] > [1.57,1.57] > [0,1.57] > [0,0]

This sequence is similar to the first type you saw in ((((II)I)(II))I)

in search of combinatorial objects of finite extent
ICN5D
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### Re: The Tiger Explained

Here's some cool gifs I made lately. Some are the combo loop of 4 right angle turns, my new favorite. Most of these are exploring an empty cut, by translating out of the hole to an intercept, then do rotations centered on it.

(((II)I)((II)I))

(sqrt((sqrt((x*sin(b) + a*cos(b))^2 + (y*sin(d) + c*cos(d))^2) - 2)^2) -1)^2 + (sqrt((sqrt(z^2 + (y*cos(d) - c*sin(d))^2) - 2)^2 + (x*cos(b) - a*sin(b))^2) -1)^2- 0.3^2 = 0

((((II)I)I)(II))

(sqrt((sqrt((sqrt(x^2 + 0^2) -6)^2 + (y*cos(b) - a*sin(b))^2) -3)^2 + (z*cos(d) - c*sin(d))^2) -1.5)^2 + (sqrt((y*sin(b) + a*cos(b))^2 + (z*sin(d) + c*cos(d))^2) -3)^2 - 0.6^2 = 0

(((II)(II))(II))

(sqrt((sqrt((x*sin(b) + a*cos(b))^2 + 0^2) -3)^2 + (sqrt((y*sin(d) + c*cos(d))^2 + (x*cos(b) - a*sin(b))^2) -3)^2) -1.5)^2 + (sqrt(z^2 + (y*cos(d) - c*sin(d))^2) -3)^2 - 0.5^2 = 0

in search of combinatorial objects of finite extent
ICN5D
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### Re: The Tiger Explained

Looks good I can't spend much time on these right now as I'm on holiday in Australia, but I'll have a proper look once I get back.
Marek14
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### Re: The Tiger Explained

I've been thinking about rotate functions lately. Whenever you have time, Marek or anyone, can you tell me if it's possible to have a single start point with two or more endpoints for rotations? Where, these two cut dimensions ( endpoints ) are independently controllable. Take, for example:

5D (((II)I)(II)) - (sqrt((sqrt(x^2 + a^2) -2)^2 + b^2) -1)^2 + (sqrt(y^2 + z^2) -2)^2 -0.5^2 = 0

One could set up two different start-end rotate parameters with (x*sin(b)) / (x*cos(b)), and (y*sin(a)) / (y*cos(a)), making :

(sqrt((sqrt((x*sin(b))^2 + (y*cos(a))^2) -2)^2 + (x*cos(b))^2) -1)^2 + (sqrt((y*sin(a))^2 + z^2) -2)^2 -0.5^2 = 0

These two are one start dimension of x or y, and a single end dimension of b or a, respectively. Is it possible to have one start dimension of x, y, or z, with two endings, independently adjustable to a and b? This way, it leaves y and z alone, with the ability to go between both cut dimensions a and b. It might even be possible to nest this explore function with rotate + translate, if possible.

EDIT:

Upon further thought, it seems as easy as combining two circles of rotation, into one. Like a binomial expansion to two terms. Maybe the startpoint would be a quadratic function of the sin/cos and the endpoints are single roots of the quadratic?

If (x*sin(b)) / (x*cos(b)) is a single circle of rotation, then we would need a Clifford torus of rotation, combining two ortho circles into one. Let's say the endpoints are (x*cos(a)) and (x*cos(b)) or something like that. Would it work if the startpoint was a multiplication of (x*cos(a)) * (x*cos(b)) with the sin function? Something like (x*sin(a^2 + 2ab + b^2)) ?

EDIT: I got it!!!! With enough experimentation, I found a way to do it! I used the function:

• (((I))(II)) --> (((Ia)b)(YI))

(Sqrt((Sqrt(x^2 + (y*cos(a))^2) -2)^2 + (y*cos(b))^2) -1)^2 + (Sqrt((y*((sin(a))*(sin(b))))^2 + z^2) -2)^2 -0.5^2 = 0

Startpoint Dimension Y = (y*((sin(a))*(sin(b)))
Endpoint a = (y*cos(a))
Endpoint b = (y*cos(b))

Set a,b = 0 ~ 1.57

Adjust A and B to flip between all three cuts. Will produce correct cuts when at least A or B are at 1.57. If both are zero, no familiar midcut.

Cuts of (((II)I)(II))

[A,B]
[1.57,1.57] = (((I))(II))
[1.57,0] = (((I)I)(I))
[0,1.57] = (((II))(I))

How about that? I've been wondering for a while.
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ICN5D
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### Re: The Tiger Explained

Been exploring a really amazing new shape. I'm surprised I've never thought of this one before. Since I've made a mostly complete tour of 6D, I'm on to 7D now. This has got to be one of the coolest ones, for sure. Only one gif, and no pics so far. Also tried out the new rotate function, with nice results.

((((II)I)I)((II)I))

Dimensional Map of ((((II)I)I)((II)I)) Hyperplane Intercepts

7D Hyperplane
((((II)I)I)((II)I)) - 1x Tiger Duotorus Torus
------------------------------------------------------------------------

6D Hyperplane Intercepts : Tiger Duotorus (((II)I)((II)I)) = (((R1a)R2a)((R1b)R2b)R3) // Tiger Ditorus ((((II)I)I)(II)) = ((((R1a)R2)R3)(R1b)R4)

((((I)I)I)((II)I)) - 2x Tiger Duotoruses (((II)I)((II)I)) in 2x1x1x1x1x1 row
((((II))I)((II)I)) - 2x Tiger Duotoruses (((II)I)((II)I)) as R1a concentric pair
((((II)I))((II)I)) - 2x Tiger Duotoruses (((II)I)((II)I)) as R2a conc pair
((((II)I)I)((I)I)) - 2x Tiger Ditoruses ((((II)I)I)(II)) in 1x1x1x1x2x1 column
((((II)I)I)((II))) - 2x Tiger Ditoruses ((((II)I)I)(II)) as R1b conc pair
--------------------------------------------------------------------------

5D Hyperplane Intercepts : Tigritorus (((II)I)(II)) = (((R1a)R2)(R1b)R3) // Tritorus ((((II)I)I)I) = ((((R1)R2)R3)R4)

(((()I)I)((II)I)) - empty, moving out of hole makes row of 2x Tigritori (((II)I)(II)) split/merge
((((I))I)((II)I)) - 4x Tigritori (((II)I)(II)) in 1x1x1x4x1 column
((((I)I))((II)I)) - 4x Tigritori (((II)I)(II)) as R1b conc pair stacked in 1x1x1x2x1 column
((((I)I)I)((I)I)) - 4x Tigritori (((II)I)(II)) in 2x1x1x2x1 vert square array
((((I)I)I)((II))) - 4x Tigritori (((II)I)(II)) as R1b conc pair stacked in 2x1x1x1x1 row
((((II)))((II)I)) - 4x Tigritori (((II)I)(II)) as R1b conc quartet
((((II))I)((I)I)) - 4x Tigritori (((II)I)(II)) as R1a pair stacked in 1x1x1x2x1 column
((((II))I)((II))) - 4x Tigritori (((II)I)(II)) as R1a pair + R1b pair
((((II)I))((I)I)) - 4x Tigritori (((II)I)(II)) as R2 conc pair stacked in 1x1x1x2x1
((((II)I))((II))) - 4x Tigritori (((II)I)(II)) as R1b pair + R2 pair
((((II)I)I)(()I)) - empty, moving out of hole makes column of 2x Tritoruses ((((II)I)I)I) split/merge
((((II)I)I)((I))) - 4x Tritoruses ((((II)I)I)I) in 1x1x1x1x4 column
------------------------------------------------------------------------------

4D Hyperplane Non-Empty Intercepts : Tiger ((II)(II)) = ((R1a)(R1b)R2) // Ditorus (((II)I)I) = (((R1)R2)R3)

((((I)))((II)I)) - 8x Ditoruses (((II)I)I) in 1x1x1x8 column
((((I))I)((I)I)) - 8x Tigers ((II)(II)) in 4x1x2x1 vert rectangle array
((((I))I)((II))) - 8x Tigers ((II)(II)) as R1b conc pair in 4x1x1x1 row
((((I)I))((I)I)) - 8x Tigers ((II)(II)) as R1a conc pair in 2x1x2x1 vert square array
((((I)I))((II))) - 8x Tigers ((II)(II)) as R1a pair + R1b pair in 2x1x1x1 row
((((I)I)I)((I))) - 8x Ditoruses (((II)I)I) in 2x1x1x4 vert rectangle array
((((II)))((I)I)) - 8x Tigers ((II)(II)) as R1a conc quartet in 1x1x2x1 column
((((II)))((II))) - 8x Tigers ((II)(II)) as R1a quartet + R1b pair
((((II))I)((I))) - 8x Ditoruses (((II)I)I) as R1 conc pair in 1x1x1x4 column
((((II)I))((I))) - 8x Ditoruses (((II)I)I) as R2 conc pair in 1x1x1x4 column
---------------------------------------------------------------------------------

3D Hyperplane Non-Empty Intercepts : Torus ((R1)R2)

((((I)))((I)I)) - 16x Tori ((II)I) in 2x1x8 vert rectangle array
((((I)))((II))) - 16x Tori ((II)I) as R1 conc pair in 1x1x8 column
((((I))I)((I))) - 16x Tori ((II)I) in 4x1x4 vert square array+
((((I)I))((I))) - 16x Tori ((II)I) as R1 conc pair in 2x1x4 vert rectangle array
((((II)))((I))) - 16x Tori ((II)I) as R1 conc quartet in 1x1x4 column
----------------------------------------------------------------------

2D Hyperplane Non-Empty Intercepts : Circle : (R)

((((I)))((I))) - 32x Circles (II) in 8x4 rectangle array

Explore functions and basic midcuts:

• ((((II)I)I)((II)I)) --> ((((R1)R2a)R3a)((R2b)R3b)R4) --> R1=4.5 , R2=2 , R3=1 , R4=0.5
(sqrt((sqrt((sqrt(x^2 + y^2) - R1)^2 + z^2) - R2a)^2 + w^2) - R3a)^2 + (sqrt((sqrt(v^2 + u^2) - R2b)^2 + t^2) - R3b)^2 - R4^2 = 0

(sqrt((sqrt((sqrt(x^2 + y^2) - 4.5)^2 + z^2) - 2)^2 + w^2) - 1)^2 + (sqrt((sqrt(v^2 + u^2) - 2.5)^2 + t^2) - 1.25)^2 - 0.4^2 = 0
--- XYZ= -9,+9

• ((((I)))((I)I)) - 16x Tori ((II)I) in 2x1x8 vert rectangle array
(sqrt((sqrt((sqrt(x^2 + 0^2) - 4.5)^2 + 0^2) - 2)^2 + 0^2) - 1)^2 + (sqrt((sqrt(y^2 + 0^2) - 2.5)^2 + z^2) - 1.25)^2 - 0.4^2 = 0

• ((((I)))((II))) - 16x Tori ((II)I) as R1 conc pair in 1x1x8 column
(sqrt((sqrt((sqrt(x^2 + 0^2) - 4.5)^2 + 0^2) - 2)^2 + 0^2) - 1)^2 + (sqrt((sqrt(y^2 + z^2) - 2.5)^2 + 0^2) - 1.25)^2 - 0.4^2 = 0

• ((((I))I)((I))) - 16x Tori ((II)I) in 4x1x4 vert square array
(sqrt((sqrt((sqrt(x^2 + 0^2) - 4.5)^2 + 0^2) - 2)^2 + y^2) - 1)^2 + (sqrt((sqrt(z^2 + 0^2) - 2.5)^2 + 0^2) - 1.25)^2 - 0.4^2 = 0

• ((((I)I))((I))) - 16x Tori ((II)I) as R1 conc pair in 2x1x4 vert rectangle array
(sqrt((sqrt((sqrt(x^2 + 0^2) - 4.5)^2 + y^2) - 2)^2 + 0^2) - 1)^2 + (sqrt((sqrt(z^2 + 0^2) - 2.5)^2 + 0^2) - 1.25)^2 - 0.4^2 = 0

• ((((II)))((I))) - 16x Tori ((II)I) as R1 conc quartet in 1x1x4 column
(sqrt((sqrt((sqrt(x^2 + y^2) - 4.5)^2 + 0^2) - 2)^2 + 0^2) - 1)^2 + (sqrt((sqrt(z^2 + 0^2) - 2.5)^2 + 0^2) - 1.25)^2 - 0.4^2 = 0

---------------------------
Translate Equations
• ((((Ia)b)c)((Id)I))
(sqrt((sqrt((sqrt(x^2 + a^2) - 4)^2 + b^2) - 2)^2 + c^2) - 1)^2 + (sqrt((sqrt(y^2 + d^2) - 2.5)^2 + z^2) - 1.25)^2 - 0.5^2 = 0

• ((((Ia)b)c)((II)d))
(sqrt((sqrt((sqrt(x^2 + a^2) - 4)^2 + b^2) - 2)^2 + c^2) - 1)^2 + (sqrt((sqrt(y^2 + z^2) - 2.5)^2 + d^2) - 1.25)^2 - 0.5^2 = 0

• ((((Ia)b)I)((Ic)d))
(sqrt((sqrt((sqrt(x^2 + a^2) - 4)^2 + b^2) - 2)^2 + y^2) - 1)^2 + (sqrt((sqrt(z^2 + c^2) - 2.5)^2 + d^2) - 1.25)^2 - 0.5^2 = 0

• ((((Ia)I)b)((Ic)d))
(sqrt((sqrt((sqrt(x^2 + a^2) - 4)^2 + y^2) - 2)^2 + b^2) - 1)^2 + (sqrt((sqrt(z^2 + c^2) - 2.5)^2 + d^2) - 1.25)^2 - 0.5^2 = 0

• ((((II)a)b)((Ic)d))
(sqrt((sqrt((sqrt(x^2 + y^2) - 4)^2 + a^2) - 2)^2 + b^2) - 1)^2 + (sqrt((sqrt(z^2 + c^2) - 2.5)^2 + d^2) - 1.25)^2 - 0.5^2 = 0

Translate + Rotate Functions
• ((((IA)a))((C)c))
(sqrt((sqrt((sqrt(x^2 + (y*sin(b) + a*cos(b))^2) - 4)^2 + (y*cos(b) - a*sin(b))^2) - 2)^2 + 0^2) - 1)^2 + (sqrt((sqrt((z*sin(d) + c*cos(d))^2 + 0^2) - 2.5)^2 + (z*cos(d) -c*sin(d))^2) - 1.25)^2 - 0.5^2 = 0

• ((((AC))a)((I)c))
(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + (y*sin(d) + c*cos(d))^2) - 4)^2 + 0^2) - 2)^2 + (x*cos(b) - a*sin(b))^2) - 1)^2 + (sqrt((sqrt(z^2 + 0^2) - 2.5)^2 + (y*cos(d) - c*sin(d))^2) - 1.25)^2 - 0.5^2 = 0

Multi-Position Rotate Functions
• ((((XY)z)a)((Zb)y)) - new multi-position rotate with [X -> a,b][Y->c][Z->d]
(sqrt((sqrt((sqrt((x*((sin(a))*(sin(b))))^2 + (y*sin(c))^2) - 4.5)^2 + (z*cos(d))^2) - 2)^2 + (x*cos(b))^2) - 1)^2 + (sqrt((sqrt((z*sin(d))^2 + (x*cos(a))^2) - 2.5)^2 + (y*cos(c))^2) - 1.25)^2 - 0.4^2 = 0
--- A=1.0635 , B=1.57 , [C,D] => 4x Rotation Cycle , or [C,D] double rotate 0->1.57
--- A=1.0635 , [B,C,D] => 6x Rotation Cycle, has two different cycles, very amazing!
--- A,B,C,D = 1.57 => Rotation sequence to 0 and back to 1.57 : [B,C,B,D,A,C,D,A]

• ((((XY)y)b)((Zx)a)) [Z->a,b][X->c][Y->d]
(sqrt((sqrt((sqrt((x*sin(c))^2 + (y*sin(d))^2) - 4.5)^2 + (y*cos(d))^2) - 2)^2 + (z*cos(b))^2) - 1)^2 + (sqrt((sqrt((z*((sin(a))*(sin(b))))^2 + (x*cos(c))^2) -2.5)^2 + (z*cos(a))^2) - 1.25)^2 - 0.4^2 = 0
--- Very cool with nice alternating midcut sequences

• ((((Xa)z)Y)((Zx)b)) [Y->a,b][X->c][Z->d]
(sqrt((sqrt((sqrt((x*sin(c))^2 + (y*cos(a))^2) - 4.5)^2 + (z*cos(d))^2) - 2)^2 + (y*((sin(a))*(sin(b))))^2) - 1)^2 + (sqrt((sqrt((z*sin(d))^2 + (x*cos(c))^2) -2.5)^2 + (y*cos(b))^2) - 1.25)^2 - 0.4^2 = 0
--- A,B,C,D=1.57 => Rotation Loop Slider Sequence: [A,C,D,A,B,C,D,B] and [A,C,D,C,B,A,D,B]
--- A=0 , B=1.57 => 4x Rotation Loop Sequence [C,D,C,D]=[0->1.57] <ANIMATED>

• ((((Xy)a)b)((YZ)z)) - [X->a,b][Y->c][Z->d]
(sqrt((sqrt((sqrt((x*((sin(a))*(sin(b))))^2 + (y*cos(c))^2) - 4.5)^2 + (x*cos(a))^2) - 2)^2 + (x*cos(b))^2) - 1)^2 + (sqrt((sqrt((y*sin(c))^2 + (z*sin(d))^2) -2.5)^2 + (z*cos(d))^2) - 1.25)^2 - 0.4^2 = 0

And, voi-la, the one animation I made so far:

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### Re: The Tiger Explained

Here are six more animations of ((((II)I)I)((II)I)), showing 5 scans each that make a full 90 degree turn.

What we see when sliding the object through our 3D plane. There are five scan sequences, that together, complete a full 90 degree turn. The two columns of 8 tori are individual hyperdonuts, that are 6D themselves : ((((II)I)I)(II)) . This scan sequence changes the angle to show how they are two objects next to each other. Starting with the two columns of 8, the scans gradually becomes a two-fold appearance of 8 tori split and merge, which is one of the 6D intercepts.

This scan shows how one of the 5D cuts of 4 tritoruses ((((II)I)I)I) changes form. In the first seq, you see all 4 sliced sideways, evenly together. The last seq scans the column from top to bottom, as the four pop in and out of the 3D plane. When you see one of the big fat tori, there's actually three more inside that get skinnier and stack inwards, one of the 3D cuts of tritorus. Another cut is a concentric pair, side by side of four tori, seen in the first sequence.

Here is the second way to slice the four tritoruses ((((II)I)I)I). Just like in [2 of 6] , the first seq is scanning the four in a different column arrangement, making the 4x1x4 array of 16 tori. Just one of the tritoruses cuts down to 4 tori in a row. The last scan seq cuts through the column at 90 degrees to the first. When you see one of the fat concentric pair pop into view, there's another thinner one, inside each.

This is scanning the 4D cut of a column of 8 ditoruses (((II)I)I) . The first sequence shows 8 instances of a ditorus cut (((I)I)I) . The last sequence shows the ditorus cut of (((II)I)), in a vertical column of 8. The five sequences work together to make a full 90 degree turn of a ditorus, defined by rotation (((IY)I)y) .

Here is scanning the 6D cut of two tiger duotoruses (((II)I)((II)I)) , side by side. The first sequence scans across both 6D intercepts evenly. The last scan is 90 degrees to the first, showing the other midcut. In the last one, note how the two pop in and out of the 3D beam, with the gap between them containing a whole 3D universe. The third scan is at 45 degrees, and shows the two 6D shapes in their 45 deg scan sequence, the quad tiger cage.

This is showing the 5D intercept of 4 tigritoruses (((II)I)(II)) , in a vertical column. The first seq scans across the four of them evenly, coming out as a 4x1x4 vertical square of 16 tori. Just one of the (((II)I)(II)) cuts down to four tori in a column. The last scan sequence cuts the column, top to bottom, making 4 tori in a concentric pair stacked two high, appear and vanish.
Last edited by ICN5D on Tue Oct 07, 2014 1:43 am, edited 2 times in total.
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### Re: The Tiger Explained

Do you have a Tumblr, ICN5D? I'd love to reblog all these from you.

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### Re: The Tiger Explained

I do in fact. Haven't posted anything up yet. I'm hypertorusexplorer
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### Re: The Tiger Explained

Cool! I tried to post one of your gifs, but it came out as a static image.

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### Re: The Tiger Explained

Hmm. I've seen this before, it did that to me, too. I can only post an HTML image, or the Imgur link, but no playing gif pic.

EDIT: looked deeper into it, Tumblr plays gifs up to 1MB , all these are 5MB. It took a lot of squeezing down to get 5, so I can't even imagine a pinky nail sized microgif
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### Re: The Tiger Explained

Made a new kind of explore function for toratopes. This one is interesting, in how it modifies a rotate parameter. It actually lets you rotate a rotate function, to see more cuts. The whole idea behind this is to maximize the four adjustable parameters in CalcPlot3D, into as many ways to control the shape as possible. But, this function is clever, in the way it overlaps the endpoints of a rotation.

Normal rotate function for ditorus: (((IY)y)I) --> swaps position from Y to y when adj A

(sqrt((sqrt(x^2 + (y*sin(a))^2) - 2.5)^2 + (y*cos(a))^2) - 1)^2 + z^2 = -0.5^2

-- This is a single position rotate, with one start point Y and one endpoint y.

The multi-dimension rotate for ditorus : (((IY)[yz])Z) --> Y with A , Z with B : two startpoints Y and Z, one endpoint [yz]

(sqrt((sqrt(x^2 + (y*sin(a))^2) - 2.5)^2 + (z*cos(b) - (y*cos(a))*sin(b))^2) - 1)^2 + (z*sin(b) + (y*cos(a))*cos(b))^2 -0.5^2 = 0

-- This allows rotation from two different start dimensions to one endpoint. Using the sliders for A and B together give all possible rotations and cuts, in one single function. One of these rotations will turn the (((I)I)I) intercept 90 degrees, making a non-standard morphing of the intercepts. Treat this as a modifying rotation, not as an actual intercept morph (though, it does look interesting). Using single rotate would require three unique functions to do this. This explore function is a hybrid of the translate+rotate and a single rotate together.

For higher than 4D, one could combine the multiposition rotate with this new one, using four adj parameters to accomplish what a lot of separate functions can do.

In exploring the tritorus ((((II)I)I)I) , I used the explore function: ((((XI)Z)[az])b) - [X->a,b] [Z->z]

(sqrt((sqrt((sqrt((x*((sin(a))*(sin(b))))^2 + y^2) - 9)^2 + (z*sin(c) + (x*cos(a))*cos(c))^2 ) - 4)^2 + (z*cos(c) - (x*cos(a))*sin(c))^2) - 2)^2 + (x*cos(b))^2 - 1= 0

This single function lets you go through just about all possible 3D intercepts for tritorus. It is still possible to link Y with (x*cos(b)) :

((((XY)Z)[az])[yb]) - [X->a,b] [Z->z] [Y->y]

(sqrt((sqrt((sqrt((x*((sin(a))*(sin(b))))^2 + (y*sin(d) + (x*cos(b))*cos(d))^2) - 9)^2 + (z*sin(c) + (x*cos(a))*cos(c))^2 ) - 4)^2 + (z*cos(c) - (x*cos(a))*sin(c))^2) - 2)^2 + (y*cos(d) - (x*cos(b))*sin(d))^2 - 1 = 0
Last edited by ICN5D on Sun Oct 05, 2014 3:54 am, edited 1 time in total.
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### Re: The Tiger Explained

Lately, I put together three new render gifs of 5D toratopes. All are related to a 3-torus, but with a 2-sphere shaped diameter, in one of the coordinate planes.

(((III)I)I) Ditorisphere T2 x S2

Has ditorus (((II)I)I) and torisphere ((III)I) rotations

(((II)II)I) Torispheritorus S1 x S2 x S1

Earlier wireframe render, before using transparent

Has ditorus (((II)I)I) , spheritorus ((II)II) , and torisphere ((III)I) rotations

(((II)I)II) Spheriditorus S2 x T2

Has ditorus (((II)I)I) and spheritorus ((II)II) rotations
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### Re: The Tiger Explained

I managed to render ((III)II) , the spherisphere, S2xS2. My poor computer barely works. It's sooo broken. Waiting for the awesome replacement to arrive on the 17th sucks.

Using the double rotate function:

• ((IYZ)yz)
(sqrt(x^2 + (y*sin(a))^2 + (z*sin(b))^2) -3)^2 + (y*cos(a))^2 + (z*cos(b))^2 -1 = 0

• When you see the 2 spheres in a row ((Iii)II), we have canceled out 2 dimensions of the large sphere, leaving behind 2 points. This will isolate out the small-shape sphere, as the minor diameter. The two spheres join together along a curved 2D plane of the larger sphere, into 4 and 5D.

• When you see the torus ((IIi)Ii), we have canceled out 1 dimension of each sphere, leaving behind circles, as circle over circle : S1xS1. Also note how the 4th vs 5th dimension makes a torus in different orientations. This reflects the property of how 5D is just another perpendicular direction from 4D.

• When you see the 2 concentric spheres ((III)ii), we have canceled out 2 dimensions of the small sphere, leaving 2 points behind again. This will isolate out the large-shape sphere, as the major diameter. Sitting in between the two concentric spheres are an infinite number of smaller spheres, that have been reduced to 2 points. Traversing the distance between spheres will allow a full 2D plane of a sphere, curving into 4 and 5D.
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### Re: The Tiger Explained

Wrote the ten dimensional equation for ((((((II)I)I)((II)I))I)(II)) today. Haven't graphed it yet, but it makes 128 torus intercepts in an 8x4x2 array of major pairs. A few ways to build it are:

Right to Left Diameter Size
S1 x C2 x S1 x C2 x C2 x S1

(II) - S1
((II)(II)) - S1 x C2
(((II)I)(II)) - S1 x C2 x S1
((((II)(II))I)(II)) - S1 x C2 x S1 x C2
(((((II)I)((II)I))I)(II)) - S1 x C2 x S1 x C2 x C2
((((((II)I)I)((II)I))I)(II)) - S1 x C2 x S1 x C2 x C2 x S1

Nested Major Diameter Size
T4 x C2 x C3

(II) - S1
((II)I) - T2
(((II)I)I) - T3
((((II)I)I)I) - T4
(((((II)I)(II))I)I) - T4 x C2
((((((II)I)I)((II)I))I)(II)) - T4 x C2 x C3

Deriving the implicit from the notation:

((((((II)I)I)((II)I))I)(II)) = 0

(((((II)I)I)((II)I))I)(II) = 0

( (( ( (II) I) I) ( (II) I) )I) (II) = 0

( (( ( (xy) z) w) ( (vu) t) )s) (rq) = 0

( (( ( (x+y) +z) +w) + ( (v+u) +t) )+s) + (r+q) = 0

( (( ( (x+y -R1A) +z -R2A) +w -R3) + ( (v+u -R1B) +t -R2B) -R4) +s -R5) + (r+q -R1C) -R6 = 0

( (( ( (x+y -R1A)^2 +z -R2A)^2 +w -R3)^2 + ( (v+u -R1B)^2 +t -R2B)^2 -R4)^2 +s -R5)^2 + (r+q -R1C)^2 -R6^2 = 0

( (( ( (sqrt(x+y) -R1A)^2 +z -R2A)^2 +w -R3)^2 + ( (v+u -R1B)^2 +t -R2B)^2 -R4)^2 +s -R5)^2 + (r+q -R1C)^2 -R6^2 = 0

( (( (sqrt((sqrt(x+y) -R1A)^2 +z) -R2A)^2 +w -R3)^2 + ( (v+u -R1B)^2 +t -R2B)^2 -R4)^2 +s -R5)^2 + (r+q -R1C)^2 -R6^2 = 0

( ((sqrt((sqrt((sqrt(x+y) -R1A)^2 +z) -R2A)^2 +w) -R3)^2 + ( (v+u -R1B)^2 +t -R2B)^2 -R4)^2 +s -R5)^2 + (r+q -R1C)^2 -R6^2 = 0

( ((sqrt((sqrt((sqrt(x+y) -R1A)^2 +z) -R2A)^2 +w) -R3)^2 + ( (sqrt(v+u) -R1B)^2 +t -R2B)^2 -R4)^2 +s -R5)^2 + (r+q -R1C)^2 -R6^2 = 0

( ((sqrt((sqrt((sqrt(x+y) -R1A)^2 +z) -R2A)^2 +w) -R3)^2 + ( sqrt((sqrt(v+u) -R1B)^2 +t) -R2B)^2 -R4)^2 +s -R5)^2 + (r+q -R1C)^2 -R6^2 = 0

(sqrt(((sqrt((sqrt((sqrt(x+y) -R1A)^2 +z) -R2A)^2 +w) -R3)^2 + (sqrt((sqrt(v+u) -R1B)^2 +t) -R2B)^2 -R4)^2 +s) -R5)^2 + (r+q -R1C)^2 -R6^2 = 0

(sqrt(((sqrt((sqrt((sqrt(x+y) -R1A)^2 +z) -R2A)^2 +w) -R3)^2 + (sqrt((sqrt(v+u) -R1B)^2 +t) -R2B)^2 -R4)^2 +s) -R5)^2 + (sqrt(r+q) -R1C)^2 -R6^2 = 0

(sqrt(((sqrt((sqrt((sqrt(x^2+y^2) -R1A)^2+z^2) -R2A)^2+w^2) -R3)^2+(sqrt((sqrt(v^2+u^2) -R1B)^2+t^2) -R2B)^2 -R4)^2+s^2) -R5)^2+(sqrt(r^2+q^2) -R1C)^2 -R6^2 = 0

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### Re: The Tiger Explained

Ah, the 10D number 832 - Ditorus/torus tiger torus tiger
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### Re: The Tiger Explained

I checked out three more toratopes in 7D. They would be (((((II)I)I)I)(II)) , (((II)(II))((II)I)) , and ((((II)I)(II))(II)) . No pics yet of the first, but here's the last two:

Both of these are a double tiger (((II)(II))(II)) bundle over a circle. There are three coordinate planes to do this, where two are distinct. Both 7D shapes have only one 3D real intercept, all others empty ( complex ).

((((II)I)(II))(II))

The 3D intercept ((((I))(I))(I)) , a 4x2x2 rectangle of 16 tori. In fact, there are many more intercepts here, in 4, 5, and 6D. The entire cross section table below can be seen in this one array. This is a property of the lowest dimensional non-empty trace array, being able to link together all higher plane intercepts into one.

7D
((((II)I)(II))(II)) - 1x Double Tiger 1A-Torus ((((R1a)R2)(R1b)R3)(R1c)R4)
-----------------------------------------------------------------------------------------------------------------------------------
6D Intercepts : double tiger (((R1a)(R1b))(R1c)R2) / tiger ditorus ((((R1a)R2)R3)(R1b)R4) / Toritiger Torus ((((R1a)R2)(R1b)R3)R4)

((((I)I)(II))(II)) - 2x Double Tigers (((II)(II))(II)) in 2x1x1x1x1x1 row
((((II))(II))(II)) - 2x Double Tigers (((II)(II))(II)) as R1a concentric pair
((((II)I)(I))(II)) - 2x Tiger Ditoruses ((((II)I)I)(II)) in 1x1x1x2x1x1 row
((((II)I)(II))(I)) - 2x Toritiger Toruses ((((II)I)(II))I) in 1x1x1x1x1x2 vertical column
----------------------------------------------------------------------------------------------------------
5D Intercepts : Tiger Torus (((R1a)R2)(R1b)R3) / Toritiger (((R1a)(R1b)R2)R3) / Tritorus ((((R1)R2)R3)R4)

(((()I)(II))(II)) - Void of R1a in Dbl Tgr Trs, moving out to ring makes 2x Tiger Toruses (((II)I)(II)) split into 1x1x2x1x1 row
((((I))(II))(II)) - 4x Tiger Toruses (((II)I)(II)) in 1x1x4x1x1 row
((((I)I)(I))(II)) - 4x Tiger Toruses (((II)I)(II)) in 2x1x2x1x1 vertical square
((((I)I)(II))(I)) - 4x Toritigers (((II)(II))I) in 2x1x1x1x2 vertical square
((((II))(I))(II)) - 4x Tiger Toruses (((II)I)(II)) in 1x1x2x1x1 row of R1a pairs
((((II))(II))(I)) - 4x Toritigers (((II)(II))I) in 1x1x1x1x2 vertical column of R1a pairs
((((II)I)())(II)) - Void of R1b in Dbl Tgr Trs, moving out to ring makes 2x Tiger Toruses (((II)I)(II)) split into R2 pair
((((II)I)(I))(I)) - 4x Tritoruses ((((II)I)I)I) in 1x1x1x2x2 vertical square
((((II)I)(II))()) - Void of R1c in Dbl Tgr Trs, moving out to ring makes 2x Tiger Toruses (((II)I)(II)) split into R3 pair
-----------------------------------------------------------------------------------------------------------------------------
4D Non-Empty Intercepts : Ditorus (((R1)R2)R3) / Tiger ((R1a)(R1b)R2)

((((I))(I))(II)) - 8x Tigers ((II)(II)) in 4x2x1x1 rectangle array
((((I))(II))(I)) - 8x Ditoruses (((II)I)I) in 1x1x4x2 vertical rectangle array
((((I)I)(I))(I)) - 8x Ditoruses (((II)I)I) in 2x1x2x2 cube array
((((II))(I))(I)) - 8x Ditoruses (((II)I)I) in 1x1x2x2 vertical square of R1 pairs
-------------------------------------------------------------------------------------
3D Non-Empty Intercepts : Torus ((R1)R2)

((((I))(I))(I)) - 16x Tori ((II)I) in 4x2x2 brick array

Deriving the Implicit Function of ((((II)I)(II))(II))
------------------------------------------------------------
((((II)I)(II))(II)) = 0
((((xy)z)(wv))(ut)) = 0
( ( (x y) z) (w v) ) (u t) = 0
( ( (x + y) + z) + (w + v) ) + (u + t) = 0
( ( (x + y -R1a) + z -R2) + (w + v -R1b) -R3) + (u + t -R1c) -R4 = 0
( ( (x + y -R1a)^2 + z -R2)^2 + (w + v -R1b)^2 -R3)^2 + (u + t -R1c)^2 -R4^2 = 0
( ( (sqrt(x + y) -R1a)^2 + z -R2)^2 + (w + v -R1b)^2 -R3)^2 + (u + t -R1c)^2 -R4^2 = 0
( (sqrt((sqrt(x + y) -R1a)^2 + z) -R2)^2 + (w + v -R1b)^2 -R3)^2 + (u + t -R1c)^2 -R4^2 = 0
( (sqrt((sqrt(x + y) -R1a)^2 + z) -R2)^2 + (sqrt(w + v) -R1b)^2 -R3)^2 + (u + t -R1c)^2 -R4^2 = 0
( (sqrt((sqrt(x + y) -R1a)^2 + z) -R2)^2 + (sqrt(w + v) -R1b)^2 -R3)^2 + (sqrt(u + t) -R1c)^2 -R4^2 = 0
(sqrt((sqrt((sqrt(x + y) -R1a)^2 + z) -R2)^2 + (sqrt(w + v) -R1b)^2) -R3)^2 + (sqrt(u + t) -R1c)^2 -R4^2 = 0
(sqrt((sqrt((sqrt(x^2 + y^2) -R1a)^2 + z^2) -R2)^2 + (sqrt(w^2 + v^2) -R1b)^2) -R3)^2 + (sqrt(u^2 + t^2) -R1c)^2 -R4^2 = 0

(((II)(II))((II)I))

The 3D intercept (((I)(I))((I))) , a 2x2x4 tower of 16 tori. Intercepts that can be seen in this array are,

7D
(((II)(II))((II)I)) - 1x Tigritorus 1B-Duotorus : (((R1A)(R1B)R3)((R1C)R2)R4) : S1xC2xS1xC2
---------------------------------------------------------------------------------------------
6D Intercepts : Tiger Duotorus (((R1a)R2a)((R1b)R2b)R3) / Double Tiger (((R1a)(R1b)R2)(R1c)R3)

(((II)(I))((II)I)) - 2x Tiger Duotoruses (((II)I)((II)I)) in 1x1x2x1x1x1 row
(((II)(II))((I)I)) - 2x Double Tigers (((II)(II))(II)) in 1x1x1x1x2x1 row
(((II)(II))((II))) - 2x Double Tigers (((II)(II))(II)) as R1c concentric pair
-------------------------------------------------------------------------------
5D Intercepts : Tigritorus (((R1a)R2)(R1b)R3) / Toritiger (((R1a)(R1b)R2)R3)

(((II)(I))((I)I)) - 4x Tigritoruses (((II)I)(II)) in 1x1x2x2x1 square array
(((II)(II))(()I)) - Void R1C, moving out to ring makes 2x Toritigers (((II)(II))I) split into 1x1x1x1x2 column
(((II)(II))((I))) - 4x Toritigers (((II)(II))I) in 1x1x1x1x4 column
(((II)(I))((II))) - 4x Tigritoruses (((II)I)(II)) in 1x1x2x1x1 row of R1b concentric pairs
(((II)())((II)I)) - Void R1B, moving out to ring makes 2x Tigritoruses (((II)I)(II)) split into R1b pair
---------------------------------------------------------------------------------------------------------
4D Intercepts : Tiger ((R1a)(R1b)R2) / Ditorus (((R1)R2)R3)

(((I)(I))((I)I)) - 8x Tigers ((II)(II)) in 2x2x2x1 cube array
(((II)())((I)I)) - Void R1B, moving out to ring makes 4x Tigers ((II)(II)) in 1x1x2x1 row split into R1a pairs
(((II)(I))(()I)) - Void R1C, moving out to ring makes 4x Ditoruses (((II)I)I) split into 1x1x2x2 vertical square
(((II)(I))((I))) - 8x Ditoruses (((II)I)I) in 1x1x2x4 vertical rectangle
(((II)(II))(())) - Void R1C, moving out to rings make 2x Tigers ((II)(II)) split/merge into R2 pairs, sequenced 2 times
(((I)(I))((II))) - 8x Tigers ((II)(II)) in 2x2x1x1 square of R1b pairs
(((II)())((II))) - Void R1B, moving out to ring makes 4x Tigers ((II)(II)) in R1b pair split into R1a pairs
(((I)())((II)I)) - Void R1B, moving out to ring makes 4x Ditoruses (((II)I)I) split into 1x1x1x4 column
----------------------------------------------------------------------------------------------------------
3D Non-Empty Intercepts : Torus ((R1)R2)

(((I)(I))((I))) - 16x Tori ((II)I) in 2x2x4 vertical tower

Deriving the Implicit Function of (((II)(II))((II)I))
-------------------------------------------------------------
(((II)(II))((II)I)) = 0
(((xy)(zw))((vu)t)) = 0
((xy)(zw))((vu)t) = 0
( (x + y) + (z + w) ) + ( (v + u) + t) = 0
( (x + y -R1a) + (z + w -R1b) -R2) + ( (v + u -R1c) + t -R3) -R4 = 0
( (x + y -R1a)^2 + (z + w -R1b)^2 -R2)^2 + ( (v + u -R1c)^2 + t -R3)^2 -R4^2 = 0
( (sqrt(x + y) -R1a)^2 + (z + w -R1b)^2 -R2)^2 + ( (v + u -R1c)^2 + t -R3)^2 -R4^2 = 0
( (sqrt(x + y) -R1a)^2 + (sqrt(z + w) -R1b)^2 -R2)^2 + ( (v + u -R1c)^2 + t -R3)^2 -R4^2 = 0
( (sqrt(x + y) -R1a)^2 + (sqrt(z + w) -R1b)^2 -R2)^2 + ( (sqrt(v + u) -R1c)^2 + t -R3)^2 -R4^2 = 0
(sqrt((sqrt(x + y) -R1a)^2 + (sqrt(z + w) -R1b)^2) -R2)^2 + ((sqrt(v + u) -R1c)^2 + t -R3)^2 -R4^2 = 0
(sqrt((sqrt(x + y) -R1a)^2 + (sqrt(z + w) -R1b)^2) -R2)^2 + (sqrt((sqrt(v + u) -R1c)^2 + t) -R3)^2 -R4^2 = 0
(sqrt((sqrt(x^2 + y^2) -R1a)^2 + (sqrt(z^2 + w^2) -R1b)^2) -R2)^2 + (sqrt((sqrt(v^2 + u^2) -R1c)^2 + t^2) -R3)^2 -R4^2 = 0
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### Re: The Tiger Explained

I wrote a couple more super-rotate equations, this time for 5D (((II)I)(II)). Each will rotate between all 3D visible intercepts, and one of the empty holes.

(((Xa)Y)(I[xb]))

(sqrt((sqrt((x*sin(c) + (y*cos(b))*cos(c))^2 + (y*cos(a))^2) - 4)^2 + (y*((sin(a))*(sin(b))))^2) - 2)^2 + (sqrt(z^2 + (x*cos(c) - (y*cos(b))*sin(c))^2) - 2)^2 - 1^2

--- a,b,c = 0 ~ 1.57

• set a,b,c to 1.57
• [a,b,c] = [1.57,1.57,1.57] -->> (((I)I)(I))
• [a,b,c] = [0,1.57,1.57] -->> (((II))(I))
• [a,b,c] = [1.57,0,1.57] -->> (((I))(II))
• [a,b,c] = [1.57,1.57,0] -->> ((()I)(II))

(((I[az])Y)(Zb))

(sqrt((sqrt(x^2 + (z*cos(c) - (y*cos(a))*sin(c))^2) - 4)^2 + (y*((sin(a))*(sin(b))))^2) - 2)^2 + (sqrt((z*sin(c) + (y*cos(a))*cos(c))^2 + (y*cos(b))^2) - 2)^2 - 1^2

--- Main difference is rotation into the empty cut, as this is the other 3D hole
--- a,b,c = 0 ~ 1.57

• set a,b,c to 1.57
• [a,b,c] = [1.57,1.57,1.57] -->> (((I)I)(I))
• [a,b,c] = [0,1.57,1.57] -->> (((II))(I))
• [a,b,c] = [1.57,0,1.57] -->> (((I))(II))
• [a,b,c] = [1.57,1.57,0] -->> (((II)I)())
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### Re: The Tiger Explained

Cool photo gallery of the 3D intercepts of (((((II)I)I)I)(II)). I plan on making rotate gifs that accentuate the lower 4,5, and 6D intercepts. Until then, the oblique cages will do the trick.

Building the Tiger Tritorus:

(II) - S1
((II)(II)) - S1 x C2
(((II)I)(II)) - S1 x C2 x S1
((((II)I)I)(II)) - S1 x C2 x T2
(((((II)I)I)I)(II)) - S1 x C2 x T3

or

(II) - S1
((II)I) - T2
(((II)I)I) - T3
((((II)I)I)I) - T4
(((((II)I)I)I)(II)) - T4 x C2

(((((II)I)I)I)(II)) Implicit Surface Equation

I wrote two distinct explore functions for tiger tritorus:

• (((((IY)a)b)c)(Id)) : Five-Position Rotate [Y --> a,b,c,d]
(sqrt((sqrt((sqrt((sqrt(x^2 + (y*((sin(a))*(sin(b))*(sin(c))*(sin(d))))^2) -8)^2 + (y*cos(a))^2) -4)^2 + (y*cos(b))^2) -2)^2 + (y*cos(c))^2) -1)^2 + (sqrt(z^2 + (y*cos(d))^2) -2)^2 -0.5^2 = 0

• (((((Ia)b)c)d)(YI)) : Five-Position Rotate [Y --> a,b,c,d]
(sqrt((sqrt((sqrt((sqrt(x^2 + (y*cos(a))^2) -8)^2 + (y*cos(b))^2) -4)^2 + (y*cos(c))^2) -2)^2 + (y*cos(d))^2) -1)^2 + (sqrt((y*((sin(a))*(sin(b))*(sin(c))*(sin(d))))^2 + z^2) -2)^2 -0.5^2 = 0

Either equation will change the position of the visible Y dimension to the a,b,c, or d positions. Set all a,b,c,d to 1.57, then move any one parameter to 0. Set XYZbox to -16 , +16 , and 45 cubes resolution.

Ring makes 5 distinct, visible intercepts in 3D, and a large number of empty holes:

(((((I))))(II)) - 16x Tori in 1x1x16 vertical column
(((((I)))I)(I)) - 16x Tori in 8x1x2 vertical rectangle array
(((((I))I))(I)) - 16x Tori in 4x1x2 vert rectangle of R1 pairs
(((((I)I)))(I)) - 16x Tori in 2x1x2 vert square of R1 quartets
(((((II))))(I)) - 16x Tori in 1x1x2 column of R1 octets

1x1x16 Column : (((((I))))(II))

8x1x2 Vertical Rectangle : (((((I)))I)(I))

4x1x2x[R1 pair] : (((((I))I))(I))

2x1x2x[R1 quartet] : (((((I)I)))(I))

1x1x2x[R1 octet] : (((((II))))(I))

Row of 8 Tiger Cages
Mid rotation between the 8x1x2 array and the 1x1x16 column.
This angle will accentuate the 4D intercept of (((((I)))I)(II)) , an 8x1x1x1 row of tigers ((II)(II))

Row of 4 Tiger Torus Cages
Mid rotation between the 4x1x2 of R1 pairs and the 1x1x16 column.
This angle will accentuate the 5D intercept of (((((I))I)I)(II)) , a 4x1x1x1x1 row of tiger toruses (((II)I)(II))

Row of 2 Tiger Ditorus Cages
Mid rotation between the 2x1x2 square of R1 quartets and the 1x1x16 column.
This angle will accentuate the 6D intercept of (((((I)I)I)I)(II)) , a 2x1x1x1x1x1 row of tiger ditoruses ((((II)I)I)(II))
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### Re: The Tiger Explained

All right!

Finally got around to graphing that 10D monster (((((II)I)I)((II)I))I)(II))

Defined implicitly:
(sqrt(((sqrt((sqrt((sqrt(x^2 + y^2) -R1A)^2 + z^2) -R2A)^2 + w^2) -R3)^2 + (sqrt((sqrt(v^2 + u^2) -R1B)^2 + t^2) -R2B)^2 -R4)^2 + s^2) -R5)^2 + (sqrt(r^2 + q^2) -R1C)^2 -R6^2 = 0

3D Midcut
(((((I)))((I))))(I)) : 8x4x2x[R1 pair] of 128 tori

(sqrt(((sqrt((sqrt((sqrt(x^2) -R1A)^2) -R2A)^2) -R3)^2 + (sqrt((sqrt(y^2) -R1B)^2) -R2B)^2 -R4)^2) -R5)^2 + (sqrt(z^2) -R1C)^2 -R6^2 = 0

Here's a nifty diameter adjustment equation. By setting the parameters to the ranges listed below, one can build the shape by expansion of diameters. Its starts with the minor diameter hard-set to 1, making a sphere. The remaining diameters are scaled to multiples of the minor, with their own unique ratios. All parameters set to the max range will graph the shape as imaged below.

(sqrt(((sqrt((sqrt((sqrt(x^2) -a)^2) -b)^2) -(b/2))^2 + (sqrt((sqrt(y^2) -(3a/4))^2) -(3b/4))^2 -c)^2) -d)^2 + (sqrt(z^2) -(a/4))^2 -1^2 = 0

Parameter Range
0 < a < 11.25
0 < b < 5.6
0 < c < 3.4
0 < d < 2.1

Grid Size
XYZbox = -22.5 , +22.5
Zscale = 0.5

This is graphing in 300 cubes resolution, with smooth gradient.

Full 10D equation is (((((xy)z)w)((vu)t))s)(rq)) . Solving for plane equation xvr makes 128 solutions of a 3D torus, in an 8x4x2x[R1 pair] array : (((((I)))((I))))(I))

First thing that catches my eye are the 32 tiger tori in an 8x4 array. Just one of these is a

So, we've taken this 5D shape (((II)I)(II)) and stretched it over the edge of a (((II)I)I)((II)I) , a (ditorus*torus)-prism . The n-2 edge is a duocylinder margin nested within a duocyl margin, nested within a circle : C2 x C2 x S1 . This 5D surface is a single structure, and can intercept 2D as an 8x4 array of points. By inflating this 5-surface with a whole tiger torus, we get an 8x4 array of 32 tiger tori : (((((I)))((I)))I)(II)) . Which also means this 5D shape is playing the role of the trace array of 32 circles from ((((II)I)I)((II)I)). Which, furthermore, makes all of those awesome gifs I made earlier on, equal to the wild stuff that happens in 6D, with tiger ditoruses ((((II)I)I)(II)). Just think about that, for a sec. All the 3D torus morphing from ((((II)I)I)((II)I)) is what's going on in 6D, with this 10D shape. But in 6D, it's 16 tiger ditoruses morphing in the same way. Now that would be interesting to show together. A super complex shape, illustrated with simpler animations of higher toratope morphing. That's why I chose this shape, because of the relation.
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### Re: The Tiger Explained

Well, I just realized that some of these equations I wrote have a missing square root! That's why those affected will graph as narrow outer bands in a major pair. The first instance of this was with the 6D ((((II)(II))I)I) , when the 2x2 square of major pairs had distorted tori. The equation I originally wrote was :

(sqrt(((sqrt(x^2 + y^2) - R1a)^2 + (sqrt(z^2 + w^2) - R1b)^2 - R2)^2 + v^2) - R3)^2 + u^2 - R4^2 = 0

when it should have been

(sqrt((sqrt((sqrt(x^2 + y^2) - R1a)^2 + (sqrt(z^2 + w^2) - R1b)^2) - R2)^2 + v^2) - R3)^2 + u^2 - R4^2 = 0

The part I was missing was the square root around the two terms in the tigroid, ((((II)(II))I)I) which are replacing two single dimensions. These two dimensions would have had a square root around them to begin with, and so should the higher toratope.

By adding this key square root, I replotted some earlier 7D and 9D shapes, and found them to resolve as perfect tori. The other good news about this, is that I can easily render 9D shapes with low resolution, which allows them to be explorable. Certain shapes will have too dense of nested diameters to be dynamic enough, case in point is ((((((II)I)(II))I)I)(II)). The 4x2x2 array of 4 concentric tori needs high resolution.

Some retakes and a new one:

6D ((((II)(II))I)I)

9D (((((II)I)(II))I)((II)I)) , a tiger torus bundle over tiger torus

And the second shape made by tiger torus bundle over tiger torus,

9D ((((((II)I)(II))I)I)(II))

Third shape makes a 4D intercept array, and can't fully sit in 3D.
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### Re: The Tiger Explained

Made some new animations of ((((II)I)I)((II)I)). These are double and triple rotations, from one coordinate slice to another.

• ((((Iz))Y)((Zy))) - double rotation YZ -> yz
(sqrt((sqrt((sqrt(x^2 + (z*cos(a))^2) - 5)^2) - 2.5)^2 + (y*sin(a))^2) - 1.25)^2 + (sqrt((sqrt((z*sin(a))^2 + (y*cos(a))^2) - 2.5)^2) - 1.25)^2 = 0.5^2

Going from the 1x1x4x[R1 quartet] column ((((II)))((I))) to the 4x1x4 vertical square ((((I))I)((I))) , of 16 tori.

• ((((Iz)Y))((Zy))) - double rotation YZ -> yz
(sqrt((sqrt((sqrt(x^2 + (z*cos(a))^2) - 5)^2 + (y*sin(a))^2) - 2.5)^2) - 1.25)^2 + (sqrt((sqrt((z*sin(a))^2 + (y*cos(a))^2) - 2.5)^2) - 1.25)^2 = 0.5^2

Going from the 1x1x4x[R1 quartet] column ((((II)))((I))) to the 2x1x4x[R1 pair] vertical rectangle ((((I)I))((I))) , of 16 tori.

• ((((Xy))Y)((Ix))) - double rotation XY -> xy
(sqrt((sqrt((sqrt((x*sin(a))^2 + (y*cos(a))^2) - 5)^2) - 2.5)^2 + (y*sin(a))^2) - 1.25)^2 + (sqrt((sqrt(z^2 + (x*cos(a))^2) - 2.5)^2) - 1.25)^2 = 0.5^2

Going from the 1x1x8x[R1 pair] column ((((I)))((II))) to the 4x1x4 vertical square ((((I))I)((I))), of 16 tori

• ((((Xy))Y)((I)x)) - double rotation XY -> xy
(sqrt((sqrt((sqrt((x*sin(a))^2 + (y*cos(a))^2) - 5)^2) - 2.5)^2 + (y*sin(a))^2) - 1.25)^2 + (sqrt((sqrt(z^2) - 2.5)^2 + (x*cos(a))^2) - 1.25)^2 = 0.5^2

Going from the 2x1x8 vertical rectangle ((((I)))((I)I)) to the 4x1x4 vertical square ((((I))I)((I))), of 16 tori

• ((((Xz))Y)((Zx)y)) - triple rotation XYZ -> xyz
(sqrt((sqrt((sqrt((x*sin(a))^2 + (z*cos(a))^2) - 5)^2) - 2.5)^2 + (y*sin(a))^2) - 1.25)^2 + (sqrt((sqrt((z*sin(a))^2 + (x*cos(a))^2) - 2.5)^2 + (y*cos(a))^2) - 1.25)^2 = 0.5^2
--- unexpectedly awesome central point of a minor pair of tori at a = pi/4

Going from the 2x1x8 vertical rectangle ((((I)))((I)I)) to the 4x1x4 vertical square ((((I))I)((I))), of 16 tori

• ((((Xz)Y))((Zx)y)) - triple rotation XYZ -> xyz
(sqrt((sqrt((sqrt((x*sin(a))^2 + (z*cos(a))^2) - 5)^2 + (y*sin(a))^2) - 2.5)^2) - 1.25)^2 + (sqrt((sqrt((z*sin(a))^2 + (x*cos(a))^2) - 2.5)^2 + (y*cos(a))^2) - 1.25)^2 = 0.5^2

Going from the 2x1x8 vertical rectangle ((((I)))((I)I)) to the 2x1x4x[R1 pair] vertical rectangle ((((I)I))((I))) , of 16 tori.

• ((((Xz)Y)x)((Zy))) - triple rotation XYZ -> xyz
(sqrt((sqrt((sqrt((x*sin(a))^2 + (z*cos(a))^2) - 5)^2 + (y*sin(a))^2) - 2.5)^2 + (x*cos(a))^2) - 1.25)^2 + (sqrt((sqrt((z*sin(a))^2 + (y*cos(a))^2) - 2.5)^2) - 1.25)^2 = 0.5^2

Going from the 4x1x4 vertical square ((((I))I)((I))) to the 2x1x4x[R1 pair] vertical rectangle ((((I)I))((I))) , of 16 tori.

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### Re: The Tiger Explained

Some really cool, new toratope animations. I used a new exploring function that allows for way more control over the shape in a higher dimension. It's the one I used to turn the tesseract so it passes through a 3-plane corner first. This is the 8D ((((II)I)(II))((II)I)) :

(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + ((z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d))*sin(t))^2) -10)^2) -5)^2 + (sqrt((y*sin(c) + (x*cos(b) - a*sin(b))*cos(c))^2) -5)^2) -2.5)^2 + (sqrt((sqrt((z*sin(d) + (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*cos(d))^2) -5)^2 + ((z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d))*cos(t))^2) -2.5)^2 = 1

Set: a=5 , c=pi/4 , d=0 , t=0 , animate 0 < b < 2pi . Shows a momentary (((I)I)((I))) 2x1x4 square in two flip-flopped positions, between a very complex topology change with high level of symmetry.

(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + ((z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d))*sin(t))^2) -10)^2) -5)^2 + (sqrt((y*sin(c) + (x*cos(b) - a*sin(b))*cos(c))^2) -5)^2) -2.5)^2 + (sqrt((sqrt((z*sin(d) + (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*cos(d))^2 + ((z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d))*cos(t))^2) -5)^2) -2.5)^2 = 1

Set at a=5 , c=pi/4 , d=pi/2 , t=0 , animate 0 < b < 2pi . Shows a momentary (((II))((I))) 1x1x4x[R1 pair] column in two flip-flopped positions, between complex topo changes.

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