How much detail should be gone into with toratopic notation? Should there be cross section tables with each shape? Or like a term glossary with a specific shape and all of its different arrangements? How about the theory?

There's the various clifford tori as the common surface between open and closed. The edge of one is the wireframe of the other, before spheration. It would seem that the sequence of Cliffords and n-spheres is the actual toratope combinatorics. And, what differentiates open from closed is the set of points from the edge. With using and describing the Cliffords, we can easily describe very high-D toratopes with a simple sequence of them. As in, the generic structure of toratopes comes from the generic structure of their cliffords.

Take the tiger as the most basic example: we have the first carteisan product of surfaces, the Clifford 2-torus, related to (II)(II) and ((II)(II)). We can call the 2-frame of the duocylinder [S^1]^2, if no such notation exists (or, does it?). The implicit equation for the cliffords are easy, as a toratope with its minor diameter removed. By using these basic Cliffords as building blocks, the arrays and structure of base-species toratopes can be easily defined.

So, as a general rule,

(II)(II) - duocylinder , 2-frame = [S^1]^2

(II)(II)(II) - (circle^3) prism , 3-frame = [S^1]^3

(II)(II)(II)(II) - (circle^4) prism , 4-frame = [S^1]^4

(II)(II)(II)(II)(II) - (circle^5) prism , 5-frame = [S^1]^5

(II)(II)(II)(II)(II)(II) - (circle^6) prism , 6-frame = [S^1]^6

(circle^n) has n-frame of [S^1]^n

((II)(II)) - tiger , 1-sphere over the Clifford 2-torus = S^1 x [S^1]^2

((II)(II)(II)) - triger , 2-sphere over the Clifford 3-torus = S^2 x [S^1]^3

((II)(II)(II)(II)) - tetriger , 3-sphere over the Clifford 4-torus = S^3 x [S^1]^4

((II)(II)(II)(II)(II)) - pentatiger , 4-sphere over the Clifford 5-torus = S^4 x [S^1]^5

((II)(II)(II)(II)(II)(II)) - hexatiger , 5-sphere over the Clifford 6-torus = S^5 x [S^1]^6

n-tiger is (n-1)-sphere over clifford n-torus = S^(n-1) x [S^1]^n

All Clifford tori have the lowest cut of 0D points in an n-cube array. This makes them all equal sized diameters:

[S^1]^2 = intercepts 2-plane as 2x2 square array of 4 points

[S^1]^3 = intercepts 3-plane as 2x2x2 cube array of 8 points

[S^1]^4 = intercepts 4-plane as 2x2x2x2 tesseract array of 16 points

[S^1]^5 = intercepts 5-plane as 2x2x2x2x2 5-cube array of 32 points

[S^1]^6 = intercepts 6-plane as 2x2x2x2x2x2 6-cube array of 64 points

[S^1]^7 = intercepts 7-plane as 2x2x2x2x2x2x2 7-cube array of 128 points

[S^1]^8 = intercepts 8-plane as 2x2x2x2x2x2x2x2 8-cube array of 256 points

[S^1]^9 = intercepts 9-plane as 2x2x2x2x2x2x2x2x2 9-cube array of 512 points

[S^1]^n = cuts to n-plane as n-cube array of 2^n points

When you get to something like a (((II)I)(II)), it ends up being a torus with a tiger cross-cut, which I think is called tiger bundle over a circle. This makes the cyltorinder ((II)I)(II) also a torus with a duocylinder cross-cut. The 3-frame is a clifford torus bundle over a circle. And for a (((II)I)I)(II) , we get a duocylindric ditorus , or duocyliner over a torus. The way this nested circle works in higher-D toratopes, are that you end up with nested clifford tori combined with T^n 's.

So, take a duotorus tiger, for instance: (((II)I)((II)I)) , what we get is a circle bundle over medium clifford torus over major clifford torus. There are two differently sized clifford tori, playing the role of equal sized medium and major diameters. If we use the inner-most circle parameters, we can establish the major diameters. I belive Marek helped with that one, by establishing the rule about freely adjustable diameters. These would be the major most, and could be any size without self-intersecting. I feel that the number of these inner-most major diameters would be the major clifford n-torus. Then, the next level down would be the medium/secondary clifford n-torus.

(((II)I)(II)) - tiger over circle = S^1 x [S^1]^2 x S^1

(((II)I)((II)I)) - tiger over clifford torus = S^1 x [S^1]^2 x [S^1]^2

((((II)I)I)((II)I)) - tiger over clifford torus over circle = S^1 x [S^1]^2 x [S^1]^2 x S^1

((((II)I)I)(((II)I)I)) - tiger over clifford torus over clifford torus = S^1 x [S^1]^2 x [S^1]^2 x [S^1]^2

(((II)I)(II)(II)) - 2-sphere over clifford 3-torus over circle = S^1 x [S^1]^3 x S^1

(((II)I)((II)I)(II)) - 2-sphere over clifford 3-torus over clifford torus = S^1 x [S^1]^3 x [S^1]^2

(((II)I)((II)I)((II)I)) - 2-sphere over clifford 3-torus over clifford 3-torus = S^1 x [S^1]^3 x [S^1]^3

((((II)I)I)((II)I)((II)I)) - 2-sphere over clifford 3-torus over clifford 3-torus over circle = S^1 x [S^1]^3 x [S^1]^3 x S^1

((((II)I)I)(((II)I)I)((II)I)) - 2-sphere over clifford 3-torus over clifford 3-torus over clifford torus = S^1 x [S^1]^3 x [S^1]^3 x [S^1]^2

((((II)I)I)(((II)I)I)(((II)I)I)) - 2-sphere over clifford 3-torus over clifford 3-torus over clifford 3-torus = S^1 x [S^1]^3 x [S^1]^3 x [S^1]^3

((II)I) - 2-torus

(((II)(II))I) - 2-torus over clifford 2-torus = T^2 x [S^1]^2

(((II)(II))(II)) - 2-torus over clifford 3-torus = T^2 x [S^1]^3

(((II)I)I) - 3-torus

((((II)(II))I)I) - 3-torus over clifford 2-torus = T^3 x [S^1]^2

((((II)(II))(II))I) - 3-torus over clifford 3-torus = T^3 x [S^1]^3

((((II)(II))(II))(II)) - 3-torus over clifford 4-torus = T^3 x [S^1]^4

((((II)I)I)I) - 4-torus

(((((II)(II))I)I)I) - 4-torus over clifford 2-torus = T^3 x [S^1]^2

(((((II)(II))(II))I)I) - 4-torus over clifford 3-torus = T^3 x [S^1]^3

(((((II)(II))(II))(II))I) - 4-torus over clifford 4-torus = T^3 x [S^1]^4

(((((II)(II))(II))(II))(II)) - 4-torus over clifford 5-torus = T^3 x [S^1]^5

All the large complex arrays come from the clifford combinations, before the small-shape toratope inflates it:

[S^1]^2 x S^1 = intercepts 2-plane as 4x2 square array of 8 points

[S^1]^2 x [S^1]^2 = intercepts 2-plane as 4x4 square array of 16 points

[S^1]^2 x [S^1]^2 x S^1 = intercepts 2-plane as 8x4 square array of 32 points

[S^1]^2 x [S^1]^2 x [S^1]^2 = intercepts 2-plane as 8x8 square array of 64 points

[S^1]^3 x S^1 = intercepts 3-plane as 4x2x2 square array of 16 points

[S^1]^3 x [S^1]^2 = intercepts 3-plane as 4x4x2 square array of 32 points

[S^1]^3 x [S^1]^3 = intercepts 3-plane as 4x4x4 square array of 64 points

[S^1]^3 x [S^1]^3 x S^1 = intercepts 3-plane as 8x4x4 square array of 64 points

[S^1]^3 x [S^1]^3 x [S^1]^2 = intercepts 3-plane as 8x8x4 square array of 128 points

[S^1]^3 x [S^1]^3 x [S^1]^3 = intercepts 3-plane as 8x8x8 square array of 256 points

I also came up with a surtope algorithm

here and

here for the opens, a little while ago. It might be useful in some way for double-checking or deriving surface hypervolumes.

The list so far, using A || B for prisms , and [A + B] for orthogonally bound curved cells

- Code: Select all
`Surtopes of the Open Toratopes`

--------------------------------

2D:

II - [line || line]+[line || line]

3D:

III - [sqr || sqr] + [sqr || sqr] + [sqr || sqr] = [sqr || sqr]^3

(II)I - [circle || circle] + [ line-->circle ]

4D:

IIII - [cube || cube]^4

(II)II - [(II)I || (II)I]^2 + [square-->circle]

(II)(II) - ((II)I)+((II)I)

(III)I - [(III) || (III)] + [line-->sphere]

((II)I)I - cylinder-->circle , [((II)I) || ((II)I)] + [line-->torus]

5D:

IIIII - [tess || tess]^5

(II)III - [[(II)II || (II)II]^3 + [cube-->circle]

(II)(II)I - [(II)(II) || (II)(II)] + [((II)I)I+((II)I)I]

(III)II - [(III)I || (III)I]^2 + [square-->sphere]

((II)I)II - cubinder-->circle , [((II)I)I || ((II)I)I]^2 + [square-->torus]

(III)(II) - ((III)I)+((II)II)

((II)I)(II) - duocylinder-->circle , (((II)I)I)+(((II)I)I)

(IIII)I - [(IIII) || (IIII)] + [line-->glome]

((II)II)I - spherinder-->circle , [((II)II) || ((II)II)] + [line-->spheritorus]

((II)(II))I - cylinder-->duoring , [((II)(II)) || ((II)(II))] + [line-->tiger]

((III)I)I - cylinder-->sphere , [((III)I) || ((III)I)] + [line-->torisphere]

(((II)I)I)I - cylinder-->torus , [(((II)I)I) || (((II)I)I)] + [line-->ditorus]

6D:

IIIIII - [penteract || penteract]^6

(II)IIII - [(II)III || (II)III]^4 + [tesseract-->circle]

(II)(II)II - [(II)(II)I || (II)(II)I]^2 + [((II)I)II+((II)I)II]

(II)(II)(II) - ((II)I)(II)+((II)I)(II)+((II)I)(II)

(III)III - [[(III)II || (III)II]^3 + [cube-->sphere]

((II)I)III - tesserinder-->circle , [((II)I)II || ((II)I)II]^3 + [cube-->torus]

(III)(II)I - [(III)(II) || (III)(II)] + [((III)I)I+((II)II)I]

((II)I)(II)I - [((II)I)(II) || ((II)I)(II)] + [(((II)I)I)I+(((II)I)I)I]

(III)(III) - ((III)II)+((III)II)

((II)I)(III) - (((III)I)I)+(((II)I)II) Type-1 , (((II)I)II)+(((II)II)I) Type-2

((II)I)((II)I) - ((((II)I)I)I)+((((II)I)I)I)

(IIII)II - [(IIII)I || (IIII)I]^2 + [square-->glome]

((II)II)II - cubspherinder-->circle , [((II)II)I || ((II)II)I]^2 + [square-->sphere-->circle]

((II)(II))II - cubinder-->duoring , [((II)(II))I || ((II)(II))I]^2 + [square-->tiger]

((III)I)II - cubinder-->sphere , [((III)I)I || ((III)I)I]^2 + [square-->circle-->sphere]

(((II)I)I)II - cubinder-->torus , [(((II)I)I)I || (((II)I)I)I]^2 + [square-->torus-->circle]

(IIII)(II) - ((II)III)+((IIII)I)

((II)II)(II) - (((II)I)II)+(((II)II)I)

((II)(II))(II) - (((II)I)(II))+(((II)(II))I)

((III)I)(II) - (((II)II)I)+(((III)I)I)

(((II)I)I)(II) - ((((II)I)I)I)+((((II)I)I)I)

(IIIII)I - [(IIIII) || (IIIII)] + [line-->pentasphere]

((II)III)I - glominder-->circle , [((II)III) || ((II)III)] + [line-->glome-->circle]

((II)(II)I)I - spherinder-->duoring , [((II)(II)I) || ((II)(II)I)] + [line-->sphere-->duoring]

((III)II)I - spherinder-->sphere , [((III)II) || ((III)II)] + [line-->sphere-->sphere]

(((II)I)II)I - spherinder-->torus , [(((II)I)II) || (((II)I)II)] + [line-->sphere-->torus]

((III)(II))I - cylinder-->(sphere x circle) , [((III)(II)) || ((III)(II))] + [line-->cylspherintigroid]

(((II)I)(II))I - cylinder-->duoring-->circle , [(((II)I)(II)) || (((II)I)(II))] + [line-->tiger torus]

((IIII)I)I - cylinder-->glome , [((IIII)I) || ((IIII)I)] + [line-->circle-->glome]

(((II)II)I)I - cylinder-->spheritorus , [(((II)II)I) || (((II)II)I)] + [line-->torus-->sphere-->circle]

(((II)(II))I)I - cylinder-->tiger , [(((II)(II))I) || (((II)(II))I)] + [line-->torus-->duoring]

(((III)I)I)I - cylinder-->torisphere , [(((III)I)I) || (((III)I)I)] + [line-->ditorus-->sphere]

((((II)I)I)I)I - cylinder-->ditorus , [((((II)I)I)I) || ((((II)I)I)I)] + [line-->torus-->torus]