## Building a Toratope by Rotations from a Circle

Discussion of shapes with curves and holes in various dimensions.

### Building a Toratope by Rotations from a Circle

I recently put this nice list together, organizing toratopes by a similar attribute. It's a culmination of several things we've discovered.

Starting from a circle, here are all possible ways to bisecting rotate and non-intersecting sweep a toratope into n+1 dimensions. The implicit definition is shown with a fiber bundle sequence and toratopic notation, which defines the equation. By squeezing all of the (),√,+,-,²,R,x,y,z math symbols out, we get an abstract representation of the diameter structure. And, by simplifying a complex mathematical idea into an easy to read format, we can peer into deeper truths of the geometry. In studying the unique differences between the hypertoric rings, we only care about four things:

1. How many dimensions?
2. How many diameters?
3. How are the dimensions distributed among the diameters?
4. What is the combinatoric nesting of the diameters?

In this case, dimensions are defined by a capitol ‘I’ , and diameters are defined by a pair of parentheses ‘()’, with at least two dimensions inside ‘(II)’. The n-spheres have all available dimensions contained within just one diameter, and take on the form (II), (III), (IIII), (IIIII), etc. Nested diameters (one inside the other), ‘((II)I)’ defines toroidal shapes, the donut ring-like objects with one or more holes. More nested circles means more diameters, and more holes: ((II)I) , ((II)(II)) , (((II)I)(II)) , (((II)I)(II)I)), etc.

Toratopic notation follows the discrete combinatoric integer sequence A000669, rooted trees with nested leaves. Number of possible shapes per dimension is:

1D - 1
2D - 1
3D - 2
4D - 5
5D - 12
6D - 33
7D - 90
8D - 261
9D - 766
10D - 2312

How to rotate into N+1
------------------------

In notation form, for a circle with a dimension marked as ‘x’ : (xI), replace x with,

• Bisecting Rotate : x = II

(xI) ---> (III)

• Non-Intersecting Sweep : x = (II)

(xI) ---> ((II)I)

In mathematical form, for a circle defined as x²+y² - r², replace x² or y² with,

• Bisecting Rotate Around x : y² = y²+z²

x²+ -r² ---> x²+y²+z² -r²

• Non-Intersecting Sweep Around y : x² = (√(x²+z²)-R)²

+y² -r² ---> (√(x²+z²)-R)² +y² -r²

Terms of the Fiber Bundles:
----------------------------
Sn = n-sphere

Tn = n-torus

Cn = Clifford flat n-torus, Tn embedded into R2n

• C2=[S1*S1] = T2 embedded in R4
• C3=[S1*S1*S1 = T3 embedded in R6
• C4=[S1*S1*S1*S1] = T4 embedded in R8
• C5=[S1*S1*S1*S1]*S1 = T5 embedded in R10

[Sm*Sn] = Clifford flat (m+n)-manifold, made by embedding SmxSn or SnxSm into R(m+n+2)

• [S2*S1] = S2xS1 or S1xS2 embedded in R5
• [S2*S2] = S2xS2 embedded in R6
• [S3*S1] = S3xS1 or S1xS3 embedded in R6
• [S3*S2] = S3xS2 or S2xS3 embedded in R7

[(SmxSn)*Sk] = Clifford flat (m+n+k)-manifold, made by surface product of SmxSn and Sk (helps define some surfaces)

1 Compact 1-manifold in R2 ; xy

Degree-2 surface of genus-0
-----------------------------
(II) - S1 : x²+y² = r²

2 Compact 2-manifolds in R3 ; xyz

Degree-2 surface of genus-0
----------------------------
(III) - S2 : x²+y²+z² = r²

Degree-4 of genus-1
-------------------
((II)I) - T2 : (√(x²+y²)-R)² +z² = r²

5 Compact 3-manifolds in R4 ; xyzw

Degree-2 surface of genus-0
-----------------------------
(IIII) - S3 : x²+y²+z²+w² = r²

Degree-4 of genus-1
-------------------
((II)II) - S2xS1 : (√(x²+y²)-R)² +z²+w² = r²
((III)I) - S1xS2 : (√(x²+y²+z²)-R)² +w² = r²

Degree-8 of genus-2
-------------------
((II)(II)) - S1xC2 : (√(x²+y²)-R1a)² + (√(z²+w²)-R1b)² = Rmin²
(((II)I)I) - T3 : (√((√(x²+y²)-R1)² +z²)-R2)² +w² = Rmin²

12 Compact 4-manifolds in R5 ; xyzwv

Degree-2 surface of genus-0
----------------------------
(IIIII) - S4 : x²+y²+z²+w²+v² = r²

Degree-4 of genus-1
--------------------
((II)III) - S3xS1 : (√(x²+y²)-R)² +z²+w²+v² = r²
((III)II) - S2xS2 : (√(x²+y²+z²)-R)² +w²+v² = r²
((IIII)I) - S1xS3 : (√(x²+y²+z²+w²)-R)² +v² = r²

Degree-8 of genus-2
--------------------
(((II)I)II) - S2xT2 : (√((√(x²+y²)-R1)² +z²)-R2)² +w²+v² = Rmin²
(((II)II)I) - S1xS2xS1 : (√((√(x²+y²)-R1)² +z²+w²)-R2)² +v² = Rmin²
(((III)I)I) - T2xS2 : (√((√(x²+y²+z²)-R1)² +w²)-R2)² +v² = Rmin²
-
((II)(II)I) - S2xC2 : (√(x²+y²)-R1a)² + (√(z²+w²)-R1b)² +v² = Rmin²
((III)(II)) - S1x[S2*S1] : (√(x²+y²+z²)-R1a)² + (√(w²+v²)-R1b)² = Rmin²

Degree-16 of genus-3
----------------------
(((II)I)(II)) - S1xC2xS1 : (√((√(x²+y²)-R1a)²+z²)-R2)² + (√(w²+v²)-R1b)² = Rmin²
(((II)(II))I) - T2xC2 : (√((√(x²+y²)-R1a)² + (√(z²+w²)-R1b)²)-R2)² +v² = Rmin²
((((II)I)I)I) - T4 : (√((√((√(x²+y²)-R1)² +z²)-R2)² +w²)-R3)² +v² = Rmin²

33 Compact 5-manifolds in R6 ; xyzwvu

Degree-2 surface of genus-0
----------------------------
(IIIIII) - S5 : x²+y²+z²+w²+v²+u² = r²

Degree-4 of genus-1
--------------------
((II)IIII) - S4xS1 : (√(x²+y²)-R)² +z²+w²+v²+u² = r²
((III)III) - S3xS2 : (√(x²+y²+z²)-R)² +w²+v²+u² = r²
((IIII)II) - S2xS3 : (√(x²+y²+z²+w²)-R)² +v²+u² = r²
((IIIII)I) - S1xS4 : (√(x²+y²+z²+w²+v²)-R)² +u² = r²

Degree-8 of genus-2
---------------------
(((II)I)III) - S3xT2 : (√((√(x²+y²)-R1)² +z²)-R2)² +w²+v²+u² = Rmin²
(((II)III)I) - S1xS3xS1 : (√((√(x²+y²)-R1)² +z²+w²+v²)-R2)² +u² = Rmin²
(((IIII)I)I) - T2xS3 : (√((√(x²+y²+z²+w²)-R1)² +v²)-R2)² +u² = Rmin²
-
(((II)II)II) - S2xS2xS1 : (√((√(x²+y²)-R1)² +z²+w²)-R2)² +v²+u² = Rmin²
(((III)I)II) - S2xS1xS2 : (√((√(x²+y²+z²)-R1)² +w²)-R2)² +v²+u² = Rmin²
(((III)II)I) - S1xS2xS2 : (√((√(x²+y²+z²)-R1)² +w²+v²)-R2)² +u² = Rmin²
-
((II)(II)II) - S3xC2 : (√(x²+y²)-R1a)² + (√(z²+w²)-R1b)² +v²+u² = Rmin²
((III)(II)I) - S2x[S2*S1] : (√(x²+y²+z²)-R1a)² + (√(w²+v²)-R1b)² +u² = Rmin²
((III)(III)) - S1x[S2*S2] : (√(x²+y²+z²)-R1a)² + (√(w²+v²+u²)-R1b)² = Rmin²
((IIII)(II)) - S1x[S3*S1] : (√(x²+y²z²+w²)-R1a)² + (√(v²+u²)-R1b)² = Rmin²

Degree-16 of genus-3
----------------------
((((II)I)I)II) - S2xT3 : (√((√((√(x²+y²)-R1)² +z²)-R2)² +w²)-R3)² +v²+u² = Rmin²
((((II)I)II)I) - S1xS2xT2 : (√((√((√(x²+y²)-R1)² +z²)-R2)² +w²+v²)-R3)² +u² = Rmin²
((((II)II)I)I) - T2xS2xS1 : (√((√((√(x²+y²)-R1)² +z²+w²)-R2)² +v²)-R3)² +u² = Rmin²
((((III)I)I)I) - T3xS2 : (√((√((√(x²+y²+z²)-R1)² +w²)-R2)² +v²)-R3)² +u² = Rmin²
-
((II)(II)(II)) - S2xC3 : (√(x²+y²)-R1a)² + (√(z²+w²)-R1b)² + (√(v²+u²)-R1c)² = Rmin²
(((II)I)(II)I) - S2xC2xS1 : (√((√(x²+y²)-R1a)²+z²)-R2)² + (√(w²+v²)-R1b)² +u² = Rmin²
(((II)(II)I)I) - S1xS2xC2 : (√((√(x²+y²)-R1a)² + (√(z²+w²)-R1b)² +v²)-R2)² +u² = Rmin²
(((II)(II))II) - S2xS1xC2 : (√((√(x²+y²)-R1a)² + (√(z²+w²)-R1b)²)-R2)² +v²+u² = Rmin²
(((III)(II))I) - T2x[S2*S1] : (√((√(x²+y²+z²)-R1a)² + (√(w²+v²)-R1b)²)-R2)² +u² = Rmin²
-
(((II)I)(III)) - S1x[T2*S2] : (√((√(x²+y²)-R1a)²+z²)-R2)² + (√(w²+v²+u²)-R1b)² = Rmin²
(((II)II)(II)) - S1x[S2*S1]xS1 : (√((√(x²+y²)-R1a)²+z²+w²)-R2)² + (√(v²+u²)-R1b)² = Rmin²
(((III)I)(II)) - S1xC2xS2 : (√((√(x²+y²+z²)-R1a)²+w²)-R2)² + (√(v²+u²)-R1b)² = Rmin²

Degree-32 of genus-4
----------------------
(((II)(II))(II)) - T2xC3 : (√((√(x²+y²)-R1a)² + (√(z²+w²)-R1b)²)-R2)² + (√(v²+u²)-R1c)² = Rmin²
(((II)I)((II)I)) - S1xC2xC2 : (√((√(x²+y²)-R1a)²+z²)-R2a)² + (√((√(w²+v²)-R1b)²+u²)-R2b)² = Rmin²
-
((((II)I)I)(II)) - S1xC2xT2 : (√((√((√(x²+y²)-R1a)²+z²)-R2)²+w²)-R3)² + (√(v²+u²)-R1b)² = Rmin²
((((II)I)(II))I) - T2xC2xS1 : (√((√((√(x²+y²)-R1a)²+z²)-R2)² + (√(w²+v²)-R1b)²)-R3)² +u² = Rmin²
((((II)(II))I)I) - T3xC2 : (√((√((√(x²+y²)-R1a)² + (√(z²+w²)-R1b)²)-R2)² +v²)-R3)² +u² = Rmin²
(((((II)I)I)I)I) - T5 : (√((√((√((√(x²+y²)-R1)² +z²)-R2)² +w²)-R3)² +v²)-R4)² +u² = Rmin²

90 Compact 6-manifolds in R7 ; xyzwvut

Degree-2 surface of genus-0
----------------------------
(IIIIIII) - S6 : x²+y²+z²+w²+v²+u²+t² = r²

Degree-4 of genus-1
--------------------
((II)IIIII) - S5xS1 : (√(x²+y²)-R)² +z²+w²+v²+u²+t² = r²
((III)IIII) - S4xS2 : (√(x²+y²+z²)-R)² +w²+v²+u²+t² = r²
((IIII)III) - S3xS3 : (√(x²+y²+z²+w²)-R)² +v²+u²+t² = r²
((IIIII)II) - S2xS4 : (√(x²+y²+z²+w²+v²)-R)² +u²+t² = r²
((IIIIII)I) - S1xS5 : (√(x²+y²+z²+w²+v²+u²)-R)² +t² = r²

Degree-8 of genus-2
---------------------
(((II)I)IIII) - S4xS1xS1 : (√((√(x²+y²)-R1)² +z²)-R2)² +w²+v²+u²+t² = Rmin²
(((II)II)III) - S3xS2xS1 : (√((√(x²+y²)-R1)² +z²+w²)-R2)² +v²+u²+t² = Rmin²
(((II)III)II) - S2xS3xS1 : (√((√(x²+y²)-R1)² +z²+w²+v²)-R2)² +u²+t² = Rmin²
(((II)IIII)I) - S1xS4xS1 : (√((√(x²+y²)-R1)² +z²+w²+v²+u²)-R2)² +t² = Rmin²
(((III)III)I) - S1xS3xS2 : (√((√(x²+y²+z²)-R1)² +w²+v²+u²)-R2)² +t² = Rmin²
(((IIII)II)I) - S1xS2xS3 : (√((√(x²+y²+z²+w²)-R1)² +v²+u²)-R2)² +t² = Rmin²
(((IIIII)I)I) - S1xS1xS4 : (√((√(x²+y²+z²+w²+v²)-R1)² +u²)-R2)² +t² = Rmin²
-
(((III)I)III) - S3xS1xS2 : (√((√(x²+y²+z²)-R1)² +w²)-R2)² +v²+u²+t² = Rmin²
(((III)II)II) - S2xS2xS2 : (√((√(x²+y²+z²)-R1)² +w²+v²)-R2)² +u²+t² = Rmin²
(((IIII)I)II) - S2xS1xS3 : (√((√(x²+y²+z²+w²)-R1)² +v²)-R2)² +u²+t² = Rmin²
-------------
((II)(II)III) - S4x[S1*S1] : (√(x²+y²)-R1a)² + (√(z²+w²)-R1b)² +v²+u²+t² = Rmin²
((III)(II)II) - S3x[S2*S1] : (√(x²+y²+z²)-R1a)² + (√(w²+v²)-R1b)² +u²+t² = Rmin²
((III)(III)I) - S2x[S2*S2] : (√(x²+y²+z²)-R1a)² + (√(w²+v²+u²)-R1b)² +t² = Rmin²
((IIII)(II)I) - S2x[S3*S1] : (√(x²+y²+z²+w²)-R1a)² + (√(v²+u²)-R1b)² +t² = Rmin²
((IIII)(III)) - S1x[S3*S2] : (√(x²+y²+z²+w²)-R1a)² + (√(v²+u²+t²)-R1b)² = Rmin²
((IIIII)(II)) - S1x[S4*S1] : (√(x²+y²+z²+w²+v²)-R1a)² + (√(u²+t²)-R1b)² = Rmin²

Degree-16 of genus-3
----------------------
((((II)I)I)III) - S3xT3 : (√((√((√(x²+y²)-R1)² +z²)-R2)² +w²)-R3)² +v²+u²+t² = Rmin²
((((II)I)III)I) - S1xS3xT2 : (√((√((√(x²+y²)-R1)² +z²)-R2)² +w²+v²+u²)-R3)² +t² = Rmin²
((((II)III)I)I) - T2xS3xS1 : (√((√((√(x²+y²)-R1)² +z²+w²+v²)-R2)² +u²)-R3)² +t² = Rmin²
((((IIII)I)I)I) - T3xS3 : (√((√((√(x²+y²+z²+w²)-R1)² +v²)-R2)² +u²)-R3)² +t² = Rmin²
-
((((II)II)II)I) - S1xS2xS2xS1 : (√((√((√(x²+y²)-R1)² +z²+w²)-R2)² +v²+u²)-R3)² +t² = Rmin²
((((II)II)I)II) - S2xS1xS2xS1 : (√((√((√(x²+y²)-R1)² +z²+w²)-R2)² +v²)-R3)² +u²+t² = Rmin²
((((II)I)II)II) - S2xS2xT2 : (√((√((√(x²+y²)-R1)² +z²)-R2)² +w²+v²)-R3)² +u²+t² = Rmin²
-
((((III)I)I)II) - S2xT2xS2 : (√((√((√(x²+y²+z²)-R1)² +w²)-R2)² +v²)-R3)² +u²+t² = Rmin²
((((III)I)II)I) - S1xS2xS1xS2 : (√((√((√(x²+y²+z²)-R1)² +w²)-R2)² +v²+u²)-R3)² +t² = Rmin²
((((III)II)I)I) - T2xS2xS2 : (√((√((√(x²+y²+z²)-R1)² +w²+v²)-R2)² +u²)-R3)² +t² = Rmin²
---------------
((II)(II)(II)I) - S3xC3 : (√(x²+y²)-R1a)² + (√(z²+w²)-R1b)² + (√(v²+u²)-R1c)² +t² = Rmin²
((III)(II)(II)) - S2x[S2*S1*S1] : (√(x²+y²+z²)-R1a)² + (√(w²+v²)-R1b)² + (√(u²+t²)-R1c)² = Rmin²
-
(((II)I)(II)II) - S3xC2xS1 : (√((√(x²+y²)-R1a)²+z²)-R2)² + (√(w²+v²)-R1b)² +u²+t² = Rmin²
(((II)I)(III)I) - S2x[T2*S2] : (√((√(x²+y²)-R1a)²+z²)-R2)² + (√(w²+v²+u²)-R1b)² +t² = Rmin²
(((II)II)(II)I) - S2x[S2*S1]xS1 : (√((√(x²+y²)-R1a)²+z²+w²)-R2)² + (√(v²+u²)-R1b)² +t² = Rmin²
(((III)I)(II)I) - S2xC2xS2 : (√((√(x²+y²+z²)-R1a)²+w²)-R2)² + (√(v²+u²)-R1b)² +t² = Rmin²
-
(((III)I)(III)) - S1x[(S1xS2)*S2] : (√((√(x²+y²+z²)-R1a)²+w²)-R2)² + (√(v²+u²+t²)-R1b)² = Rmin²
(((III)II)(II)) - S1x[S2*S1]xS2 : (√((√(x²+y²+z²)-R1a)²+w²+v²)-R2)² + (√(u²+t²)-R1b)² = Rmin²
(((IIII)I)(II)) - S1xC2xS3 : (√((√(x²+y²+z²+w²)-R1a)²+v²)-R2)² + (√(u²+t²)-R1b)² = Rmin²
-
(((II)I)(IIII)) - S1x[T2*S3] : (√((√(x²+y²)-R1a)²+z²)-R2)² + (√(w²+v²+u²+t²)-R1b)² = Rmin²
(((II)II)(III)) - S1x[S2*S2]xS1 : (√((√(x²+y²)-R1a)²+z²+w²)-R2)² + (√(v²+u²+t²)-R1b)² = Rmin²
(((II)III)(II)) - S1x[S3*S1]xS1 : (√((√(x²+y²)-R1a)²+z²+w²+v²)-R2)² + (√(u²+t²)-R1b)² = Rmin²
-
(((II)(II))III) - S3xS1xC2 : (√((√(x²+y²)-R1a)² + (√(z²+w²)-R1b)²)-R2)² +v²+u²+t² = Rmin²
(((III)(II))II) - S2xS1x[S2*S1] : (√((√(x²+y²+z²)-R1a)² + (√(w²+v²)-R1b)²)-R2)² +u²+t² = Rmin²
(((III)(III))I) - T2x[S2*S2] : (√((√(x²+y²+z²)-R1a)² + (√(w²+v²+u²)-R1b)²)-R2)² +t² = Rmin²
(((IIII)(II))I) - T2x[S3*S1] : (√((√(x²+y²+z²+w²)-R1a)² + (√(v²+u²)-R1b)²)-R2)² +t² = Rmin²
-
(((II)(II)I)II) - S2xS2xC2 : (√((√(x²+y²)-R1a)² + (√(z²+w²)-R1b)² +v²)-R2)² +u²+t² = Rmin²
(((II)(II)II)I) - S1xS3xC2 : (√((√(x²+y²)-R1a)² + (√(z²+w²)-R1b)² +v²+u²)-R2)² +t² = Rmin²
(((III)(II)I)I) - S1xS2x[S2*S1] : (√((√(x²+y²+z²)-R1a)² + (√(w²+v²)-R1b)² +u²)-R2)² +t² = Rmin²

Degree-32 of genus-4
----------------------
(((((II)I)I)I)II) - S2xT4 : (√((√((√((√(x²+y²)-R1)² +z²)-R2)² +w²)-R3)² +v²)-R4)² +u²+t² = Rmin²
(((((II)I)I)II)I) - S1xS2xT3 : (√((√((√((√(x²+y²)-R1)² +z²)-R2)² +w²)-R3)² +v²+u²)-R4)² +t² = Rmin²
(((((II)I)II)I)I) - T2xS2xT2 : (√((√((√((√(x²+y²)-R1)² +z²)-R2)² +w²+v²)-R3)² +u²)-R4)² +t² = Rmin²
(((((II)II)I)I)I) - T3xS2xS1 : (√((√((√((√(x²+y²)-R1)² +z²+w²)-R2)² +v²)-R3)² +u²)-R4)² +t² = Rmin²
(((((III)I)I)I)I) - T4xS2 : (√((√((√((√(x²+y²+z²)-R1)² +w²)-R2)² +v²)-R3)² +u²)-R4)² +t² = Rmin²
-----------------
(((II)(II))(II)I) - S2xS1xC3 : (√((√(x²+y²)-R1a)² + (√(z²+w²)-R1b)²)-R2)² + (√(v²+u²)-R1c)² +t² = Rmin²
(((II)(II))(III)) - S1x[(S1xC2)*S2] : (√((√(x²+y²)-R1a)² + (√(z²+w²)-R1b)²)-R2)² + (√(v²+u²+t²)-R1c)² = Rmin²
(((II)(II)I)(II)) - S1x[(S2xC2)*S1] : (√((√(x²+y²)-R1a)² + (√(z²+w²)-R1b)² +v²)-R2)² + (√(u²+t²)-R1c)² = Rmin²
(((III)(II))(II)) - T2x[S2*S1*S1] : (√((√(x²+y²+z²)-R1a)² + (√(w²+v²)-R1b)²)-R2)² + (√(u²+t²)-R1c)² = Rmin²
-
(((II)I)((II)I)I) - S2xC2xC2 : (√((√(x²+y²)-R1a)²+z²)-R2a)² + (√((√(w²+v²)-R1b)²+u²)-R2b)² +t² = Rmin²
(((II)II)((II)I)) - S1x[S2*S1]xC2 : (√((√(x²+y²)-R1a)²+z²+w²)-R2a)² + (√((√(v²+u²)-R1b)²+t²)-R2b)² = Rmin²
(((III)I)((II)I)) - S1xC2x[S2*S1] : (√((√(x²+y²+z²)-R1a)²+w²)-R2a)² + (√((√(v²+u²)-R1b)²+t²)-R2b)² = Rmin²
-
((((II)I)I)(II)I) - S2xC2xT2 : (√((√((√(x²+y²)-R1a)²+z²)-R2)²+w²)-R3)² + (√(v²+u²)-R1b)² +t² = Rmin²
((((II)I)I)(III)) - S1x[T3*S2] : (√((√((√(x²+y²)-R1a)²+z²)-R2)²+w²)-R3)² + (√(v²+u²+t²)-R1b)² = Rmin²
((((II)I)II)(II)) - S1x[S2*S1]xT2 : (√((√((√(x²+y²)-R1a)²+z²)-R2)²+w²+v²)-R3)² + (√(u²+t²)-R1b)² = Rmin²
((((II)II)I)(II)) - S1xC2xS2xS1 : (√((√((√(x²+y²)-R1a)²+z²+w²)-R2)²+v²)-R3)² + (√(u²+t²)-R1b)² = Rmin²
((((III)I)I)(II)) - S1xC2xS1xS2 : (√((√((√(x²+y²+z²)-R1a)²+w²)-R2)²+v²)-R3)² + (√(u²+t²)-R1b)² = Rmin²
-
((((II)(II))I)II) - S2xT2xC2 : (√((√((√(x²+y²)-R1a)² + (√(z²+w²)-R1b)²)-R2)² +v²)-R3)² +u²+t² = Rmin²
((((II)(II))II)I) - S1xS2xS1xC2 : (√((√((√(x²+y²)-R1a)² + (√(z²+w²)-R1b)²)-R2)² +v²+u²)-R3)² +t² = Rmin²
((((II)(II)I)I)I) - T2xS2xC2 : (√((√((√(x²+y²)-R1a)² + (√(z²+w²)-R1b)² +v²)-R2)² +u²)-R3)² +t² = Rmin²
((((III)(II))I)I) - T3x[S2*S1] : (√((√((√(x²+y²+z²)-R1a)² + (√(w²+v²)-R1b)²)-R2)² +u²)-R3)² +t² = Rmin²
-
((((II)I)(II))II) - S2xS1xC2xS1 : (√((√((√(x²+y²)-R1a)²+z²)-R2)² + (√(w²+v²)-R1b)²)-R3)² +u²+t² = Rmin²
((((II)I)(II)I)I) - S1xS2xC2xS1 : (√((√((√(x²+y²)-R1a)²+z²)-R2)² + (√(w²+v²)-R1b)² +u²)-R3)² +t² = Rmin²
((((II)I)(III))I) - T2x[T2*S2] : (√((√((√(x²+y²)-R1a)²+z²)-R2)² + (√(w²+v²+u²)-R1b)²)-R3)² +t² = Rmin²
((((II)II)(II))I) - T2x[S2*S1]xS1 : (√((√((√(x²+y²)-R1a)²+z²+w²)-R2)² + (√(v²+u²)-R1b)²)-R3)² +t² = Rmin²
((((III)I)(II))I) - T2xC2xS2 : (√((√((√(x²+y²+z²)-R1a)²+w²)-R2)² + (√(v²+u²)-R1b)²)-R3)² +t² = Rmin²
-
(((II)I)(II)(II)) - S2xC3xS1 : (√((√(x²+y²)-R1a)²+z²)-R2)² + (√(w²+v²)-R1b)² + (√(u²+t²)-R1c)² = Rmin²
(((II)(II)(II))I) - S1xS2xC3 : (√((√(x²+y²)-R1a)² + (√(z²+w²)-R1b)² + (√(v²+u²)-R1c)²)-R2)² +t² = Rmin²

Degree-64 of genus-5
----------------------
((((II)(II))(II))I) - T3xC3 : (√((√((√(x²+y²)-R1a)² + (√(z²+w²)-R1b)²)-R2)² + (√(v²+u²)-R1c)²)-R3)² +t² = Rmin²
(((II)(II))((II)I)) - S1xC2xC3 : (√((√(x²+y²)-R1a)² + (√(z²+w²)-R1b)²)-R3)² + (√((√(v²+u²)-R1c)²+t²)-R2)² = Rmin²
((((II)(II))I)(II)) - S1xC2xS1xC2 : (√((√((√(x²+y²)-R1a)² + (√(z²+w²)-R1b)²)-R2)² +v²)-R3)² + (√(u²+t²)-R1c)² = Rmin²
((((II)I)(II))(II)) - T2xC3xS1 : (√((√((√(x²+y²)-R1a)²+z²)-R2)² + (√(w²+v²)-R1b)²)-R3)² + (√(u²+t²)-R1c)² = Rmin²
-
((((II)I)((II)I))I) - T2xC2xC2 : (√((√((√(x²+y²)-R1a)²+z²)-R2a)² + (√((√(w²+v²)-R1b)²+u²)-R2b)²)-R3)² +t² = Rmin²
((((II)I)I)((II)I)) - S1xC2xC2xS1 : (√((√((√(x²+y²)-R1a)²+z²)-R2a)²+w²)-R3)² + (√((√(v²+u²)-R1b)²+t²)-R2b)² = Rmin²
-
(((((II)I)I)I)(II)) - S1xC2xT3 : (√((√((√((√(x²+y²)-R1a)²+z²)-R2)²+w²)-R3)²+v²)-R4)² + (√(u²+t²)-R1b)² = Rmin²
(((((II)I)I)(II))I) - T2xC2xT2 : (√((√((√((√(x²+y²)-R1a)²+z²)-R2)²+w²)-R3)² + (√(v²+u²)-R1b)²)-R4)² +t² = Rmin²
(((((II)I)(II))I)I) - T3xC2xS1 : (√((√((√((√(x²+y²)-R1a)²+z²)-R2)² + (√(w²+v²)-R1b)²)-R3)² +u²)-R4)² +t² = Rmin²
(((((II)(II))I)I)I) - T4xC2 : (√((√((√((√(x²+y²)-R1a)² + (√(z²+w²)-R1b)²)-R2)² +v²)-R3)² +u²)-R4)² +t² = Rmin²
((((((II)I)I)I)I)I) - T6 : (√((√((√((√((√(x²+y²)-R1)² +z²)-R2)² +w²)-R3)² +v²)-R4)² +u²)-R5)² +t² = Rmin²
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### Re: Building a Toratope by Rotations from a Circle

This looks great!

I'm not sure "genus" is the right word here, I thought we were calling it "order"?

PWrong
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### Re: Building a Toratope by Rotations from a Circle

Oh, that's right. I suppose they mean the same thing. I came across this wikipedia definition of a torus as being a compact 2-manifold of genus-1, and liking the way it sounded, used a similar term for the others. And, in that respect, degree is related to "depth" of the toratope.

So, the fiber bundle terms are clearly defined, and all that? I recently visualized how the 3-frame of a (sphere,circle) prism is also a spheritorus OR torisphere embedded in R5. Which was kind of interesting: both the spheritorus and torisphere cells (of the cylspherinder) are completely bound to each other by their surface, like they're reaching out to a common surface, the [S2*S1].
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### Re: Building a Toratope by Rotations from a Circle

Genus only really applies in 3D and is about the number of holes, so a genus 2 surface is just two torii glued together. There is a more sophisticated definition of "hole" involving homology groups, which I've worked on here before.

Your notation is different from what I'm used to, and some things are in the reverse order of what I'd expect, but everything looks correct.

You have some of the radii notated as R1a, R1b, e.t.c. There is a more consistent and straightforward notation for the radii which I've talked about before. The same pattern works for variables. Here's a large example (which will also be a test to see if I understand your notation right):

(((III)(III)II)(((II)I)(II))) - (S3x[S2*S2]) * (S1x[T2*S1])

√{
(√{ (√{x111²+x112²+x113²}-R11)² + (√{x121²+x122²+x123²}-R12)² + x13² + x14² } - R1)²
+ (√{ (√{ (√{x2111²+x2112²}-R211)² + x212²} - R21)^2 + (√{x221²+x222²}-R22)²} - R2)² )
} = R

Essentially, closed toratopes correspond to trees, the variables are the leaves of the tree and the radii are the nodes. The notation is just the coordinates of your position on the tree.

PWrong
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### Re: Building a Toratope by Rotations from a Circle

Genus only really applies in 3D and is about the number of holes, so a genus 2 surface is just two torii glued together. There is a more sophisticated definition of "hole" involving homology groups, which I've worked on here before.

Oh, cool, I'm always down to learn some new terms! Especially if there's a better, more clear way. It's probably in one of the ancient posts, so I'll have a look. That would be a nice update to the list, and it might be good to revive the old thread.

Your notation is different from what I'm used to, and some things are in the reverse order of what I'd expect, but everything looks correct.

Yeah, I noticed you use a more condensed version of what I learned from Marek. You know how it goes, you spend so much time getting used to certain terms, it becomes a force of habit. But, which part is backwards? The equation?

You have some of the radii notated as R1a, R1b, e.t.c. There is a more consistent and straightforward notation for the radii which I've talked about before. The same pattern works for variables.

That's a better notation, since it gets around the problem of so many dimension variables, that may be mistaken as a diameter, as in r =/= R.

[/√{
(√{ (√{x111²+x112²+x113²}-R11)² + (√{x121²+x122²+x123²}-R12)² + x13² + x14² } - R1)²
+ (√{ (√{ (√{x2111²+x2112²}-R211)² + x212²} - R21)^2 + (√{x221²+x222²}-R22)²} - R2)² )
} = R

That's exactly the format I would have written. I see how the radii work, too, with the deeper nested ones taking the form 2 -> 21 -> 211.
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### Re: Building a Toratope by Rotations from a Circle

The homology group idea is that holes come in different dimensions. Homology groups intuitively have something to do with the number of ways you can wrap a q-surface around the object without it falling off. The 0'th homology group is a bit less intuitive, but it has to do with how many disconnected parts the object has. You can wrap a circle around another circle in as many ways as there are integers, so the 1st homology group of the circle is the group of integers, Z. So we say that the circle has a single 1-hole. The 0th homology group is also Z, so we say that the circle has a single 0-hole, because it has one connected component.

There are some simple examples on the Wikipedia article.

In a quicker notation:
S^0 is two points. It has two 0-holes and no other holes.

S^1 is a circle. It has a 0-hole and a 1-hole.

S^2 is a sphere, it has a 0-hole and a 2-hole.

S^1 x S^1 is a torus, it has a 0-hole, two 1-holes and a 2-hole.

I came up with the following conjecture, which I'm pretty confident about.
viewtopic.php?f=24&t=1469

Consider a min-frame rotatope A = S^{a_1} x S^{a_2} x ... x S^{a_k}. Then the number of q-holes of A is the number of subsets of {a_1, a_2, ..., a_k} that sum to give q.

Example:
A = S^3 x S^2 x S^2
We're looking for subsets of {3, 2, 2} that sum to q.
0-holes: 1 (since the empty set sums to zero)
1-holes: 0
2-holes: 2
3-holes: 1
4-holes: 1
5-holes: 2
6-holes: 0
7-holes: 1

Now for toratopes, we just convert into the corresponding rotatope. If two shapes are homeomorphic (can be deformed into each other) then they have the same homology groups. The tiger ((II)(II)) is homeomorphic to (II)(II)(II), so we're looking for subsets of {1,1,1}.
0-holes: 1
1-holes: 3
2-holes: 3
3-holes: 1

PWrong
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### Re: Building a Toratope by Rotations from a Circle

But, which part is backwards? The equation?

The equations are fine. The difference is that I used x for the direct product (e.g. S1 x S1 is the duocylinder (II)(II)), and # for the spheration product, but I also used the spheration product backwards from how you use it.

So where you say
((II)(II)III) - S4x[S1*S1],

I would say
((II)(II)III) - (S1xS1)#S4

One key property of spheration is that it doesn't commute (e.g. S2#S3 and S3#S2 aredifferent shapes), and in fact sometimes it doesn't even exist in both directions. So (S1xS1)#S1 is a real shape, the tiger. But S1#(S1xS1) doesn't even exist. There's no consistent way to perform that action because the shape you're using to blow up the circle isn't isotropic.

PWrong
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### Re: Building a Toratope by Rotations from a Circle

I remember that use, PWrong. I think I also used something similar before going to toratopic notation.

BTW, you say that tiger ((II)(II)) is homeomorphic to (II)(II)(II). So, does that mean that you can "expand" a toratope like this by simply listing the dimensions of its parentheses?

For example triger ((II)(II)(II)) has 3 parentheses with 2 members and 1 with 3 members, so it would be homeomorphic to (III)(II)(II)(II), with subsets of 2,1,1,1?

So triger would have:
1 0-hole (1 shape)
3 1-holes (3 x (1))
4 2-holes of 2 kinds (1 x (2), 3 x (1,1))
4 3-holes of 2 kinds (3 x (2,1), 1 x (1,1,1))
3 4-holes (3 x (2,1,1))
1 5-hole (1 x (2,1,1,1))

This theory also works with disconnected toratopes:

((I)(I)) (a rectangle of four circles) would be homeomorphic to (II)(I)(I). We look for subsets of (1,0,0), which DOES have 4 0-holes and 4 1-holes, like 4 circles should have.

Now, the way this works, the list of holes must be symmetrical (since each combination corresponds to a complementary combination).

If the hypothesis is correct, there should presumably be a connection between specific numbers chosen in a combination and the hole formed by them. To find these, let's skin the tiger.

Imagine you've hunted a tiger. Now you peel off its 3-dimensional skin and stretch it to dry out. You end up with a S1 x S1 x S1, which can be imagined in 3D as a cube with opposite sides joined, a 3D version of Asteroids screens.

Tiger has 3 1-holes -- and the stretched skin has 3 distinct nonvanishing 1D curves; one that goes through front and back side of cube, one that goes through left and right side and one that goes through top and bottom side. Each of these corresponds to one "1" in the product.

And there are also 3 distinct nonvanishing 2D surfaces: one that goes through front/back and left/right, one that goes through front/back and top/bottom and one that goes through left/right and top/bottom. Each is, in a way, dual to one 1-hole. If you introduce a system of 2D surfaces and complementary 1D lines, each pair of these will intersect in one unique point.

The 3-hole then would be just the complete skin. And notice that it, too, intersects in 1 point with every possible choice of dual 0-hole, i.e. a point on the skin.

Now let's skin the torisphere. ((III)I) spreads out to (III)(II), S2 x S1. If we stretch this out, it won't come out nice and Euclidean, but we can imagine it as a cylinder. Top and bottom of the cylinder is joined and all the vertical lines are actually one and the same line.
Then, we get one 1-hole -- a line that extends throughout the cylinder and passes through the top and bottom surfaces that belong to S1. A plane that cuts through all the side vertical lines is the solitary 2-hole.

If we try something a bit more complex, like a 32-torus ((III)II), the skin will pass 4 dimensions as S2 x S2 and stretching will make a solid duocylinder. You have 2 2-holes, each wrapping one of the S2.

So for triger, S2 x S1 x S1 x S1, the stretched skin will span 5 dimensions and can be imagined as circle x cube. So we have:

1 0-hole (point anywhere in the skin)
3 1-holes (3 lines parallel to each of the three "cube" directions)
4 2-holes (plane parallel to the "circle" plane and 3 planes parallel to 2 of the three "cube" directions)
4 3-holes (3 hyperplanes parallel to the "circle" plane and one of the three "cube" directions and hyperplane parallel to all three "cube" directions)
3 4-holes (3 hyperplanes parallel to the "circle" plane and two of the three "cube" directions)
1 5-hole (the whole skin)

You can see that 1-holes and 4-holes, as well as 2-holes and 3-holes form complementary systems.

The remaining question is how those holes look when on a complete animal.
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### Re: Building a Toratope by Rotations from a Circle

Thinking about it a bit more, the rotations actually can help us understand the holes as well.

First, spheres. Spheres are simple -- they have 1 0-hole and 1 n-hole, depending on dimension.

Let's consider a toratope that can be built from a nonbisecting rotation. It has twice as many holes than the toratope it's built from by rotation. Well, those can be understood like this:

Half of them are holes that exist on the original toratope.
The second half are holes that are created by taking the original holes and extending them throughout the rotation.

So, torus. Torus is a nonbisecting rotation of circle. Circle has two holes, a 0-hole (point on the circle) and a 1-hole (the whole circle). Those translate to two holes of the torus (point on torus and circle around its minor diameter). When we rotate them, we get the remaining two holes (circle around major diameter and whole torus).

Spheritorus is nonbisecting rotation of sphere. Direct translation gives us a 0-hole (point on spheritorus) and a 2-hole (sphere around minor diameter). Rotation boosts both by 1, giving us a 1-hole (circle around major diameter) and a 3-hole (whole spheritorus).

Ditorus -- nonbisecting rotation of torus. Direct translation gives us a 0-hole (point on ditorus), two 1-holes (circle around minor diameter and circle around medium diameter) and a 2-hole (torus wrapped around medium and minor diameter). Rotation boosts those to a 1-hole (circle around major diameter), two 2-holes (torus around major and minor diameter and torus around major and medium diameter) and a 3-hole (whole ditorus).

Tiger -- nobisecting rotation of torus. Direct translation gives us a 0-hole (point on tiger), two 1-holes (circle around 1st major diameter and circle around minor diameter) and a 2-hole (torus wrapped around 1st major and minor diameter). Rotation boosts those to a 1-hole (circle around 2nd major diameter), two 2-holes (torus around 2nd major and minor diameter and S1 x S1 around both major diameters) and a 3-hole (whole tiger).

What about toratopes that don't have nonbisecting rotations, like torisphere? Well, we just flip the perspective a bit. To perform a nonbisecting rotation of a toratope is the same thing as to take a circle and replace each of its points with the toratope. This is why, in toratopic notation, you replace one I with (II), the symbol for a circle.

So, for torisphere, analogically, you replace a I in (II) with (III), symbol for a sphere, to get ((III)I). Then you can use the same argument:

The basic circle we "rotate" gives us a 0-hole (point on torisphere) and a 1-hole (circle around minor diameter).
"Rotation" boosts each by 2 (because we use 2-dimensional sphere, instead of 1-dimensional circle), giving us a 2-hole (sphere around major diameter) and a 3-hole (whole torisphere).

So this ties together rotations and homological holes and explains why PWrong's idea works. Could a proof be made on this basis?
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### Re: Building a Toratope by Rotations from a Circle

Awesome information guys. The breakdown of the triger is still blowing my mind. I wasn't able to see how triger unfolded until now. A few questions:

• How do you represent just a Clifford torus and other varieties in homology group notation? My guess is that it's identical to a torus.

• How would you define (((II)I)(II)) , (((II)(II))I) and ((((II)I)I)I) ? It seems like they would unfold to (II)(II)(II)(II) , but I'm not sure of the complete breakdown.

The breakdown of a tiger:

1x 0-hole: point on tiger
3x 1-holes: circle around 1st major diameter, 2nd major diameter, minor diameter
3x 2-holes : 2-plane of torus wrapped around 1st major and minor diameter , torus around 2nd major and minor diameter, and S1 x S1 around both major diameters
1x 3-hole: whole tiger

So, does this mean a tiger has three distinct torus-shaped holes? It seems to be a combination of all available diameters and surfaces, for each dimension. Especially the 2-surfaces. Judging by this, I think the ditorus goes something like this:

(((II)I)I)
1x 0-hole: point on ditorus
3x 1-holes: circle around major, medium, minor diameter
3x 2-holes : 2-plane of torus around major and medium diameter , 2-plane of torus around major and minor, 2-plane of torus around medium and minor
1x 3-hole: whole ditorus

Now if I understand this correctly, perhaps I can extrapolate to (((II)I)(II)) :

(((II)I)(II)) - (((maj1)med)(maj2)min)

1x 0-hole: point on tiger torus
4x 1-holes: circle around major1 , major2 , medium , minor diameter
6x 2-holes : S1 x S1 from maj1-maj2 , torus from: {maj1-med , maj2-med , maj1-min , maj2-min , med-min}
4x 3-holes: tiger from maj1-maj2-med , tiger from maj1-maj2-min , ditorus from maj1-med-min , ditorus from maj2-med-min
1x 4-hole : whole tiger torus

Hmm, looks like binomial expansion. Because tiger groups list out as 1,3,3,1 and torus is 1,2,1. Very interesting. Must investigate.
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### Re: Building a Toratope by Rotations from a Circle

ICN5D wrote:Awesome information guys. The breakdown of the triger is still blowing my mind. I wasn't able to see how triger unfolded until now. A few questions:

• How do you represent just a Clifford torus and other varieties in homology group notation? My guess is that it's identical to a torus.

• How would you define (((II)I)(II)) , (((II)(II))I) and ((((II)I)I)I) ? It seems like they would unfold to (II)(II)(II)(II) , but I'm not sure of the complete breakdown.

Well, the "skinning" works by extracting parentheses from within. So (((II)I)(II)) would go:
(((II)I)(II))
(II)((II)(II))
(II)(II)((II)I)
(II)(II)(II)(II)

In each step, we move one innermost pair of parentheses outside and replace it by "I"

The breakdown of a tiger:

1x 0-hole: point on tiger
3x 1-holes: circle around 1st major diameter, 2nd major diameter, minor diameter
3x 2-holes : 2-plane of torus wrapped around 1st major and minor diameter , torus around 2nd major and minor diameter, and S1 x S1 around both major diameters
1x 3-hole: whole tiger

So, does this mean a tiger has three distinct torus-shaped holes? It seems to be a combination of all available diameters and surfaces, for each dimension. Especially the 2-surfaces. Judging by this, I think the ditorus goes something like this:

(((II)I)I)
1x 0-hole: point on ditorus
3x 1-holes: circle around major, medium, minor diameter
3x 2-holes : 2-plane of torus around major and medium diameter , 2-plane of torus around major and minor, 2-plane of torus around medium and minor
1x 3-hole: whole ditorus

Yes, seems like it

Now if I understand this correctly, perhaps I can extrapolate to (((II)I)(II)) :

(((II)I)(II)) - (((maj1)med)(maj2)min)

1x 0-hole: point on tiger torus
4x 1-holes: circle around major1 , major2 , medium , minor diameter
6x 2-holes : S1 x S1 from maj1-maj2 , torus from: {maj1-med , maj2-med , maj1-min , maj2-min , med-min}
4x 3-holes: tiger from maj1-maj2-med , tiger from maj1-maj2-min , ditorus from maj1-med-min , ditorus from maj2-med-min
1x 4-hole : whole tiger torus

Hmm, looks like binomial expansion. Because tiger groups list out as 1,3,3,1 and torus is 1,2,1. Very interesting. Must investigate.

Yes, it's binomial expansion since one hole is defined for every combination of diameters.
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### Re: Building a Toratope by Rotations from a Circle

Everything I've read here looks correct to me up until the shapes of the tiger holes. I hadn't thought about the shapes of each hole before. In terms of homology groups a "2-hole" isn't something that has a shape, it's much more abstract. This isn't something we can calculate rigorously (partly because I've completely forgotten how to do Mayor-Vietoris sequences anyway). But maybe it works intuitively.

A circle has a 0-hole and a 1-hole shaped like a circle.
A sphere has a 0-hole and a sphere-hole.
A torus has a 0-hole, a torus-hole, and two perpendicular circle-shaped 1-holes. One is in the xy plane, the other is in the rz plane, where r can be any vector in the xy plane.
A clifford torus or duocylinder has a 0-hole, two circle shaped 1-holes (one in xy and the other in zw), and a duocylinder shaped 2-hole.

Tiger is a spherated duocylinder. So there is a circle-hole in xy and a circle-hole in zw, and a third circle-hole where the spheration happens, which we could call the r1r2 plane.
Then you can wrap a duocylinder around the duocylinder (xyzw), or you can wrap a torus in the xyr1 realm, or in the zwr2 realm.

So in response to
So, does this mean a tiger has three distinct torus-shaped holes?

I'll conjecture that a tiger has two distinct torus-shaped holes and a duocylinder-shaped hole, provided that what we're talking about makes sense at all. Note that this new way of talking about holes is NOT invariant under homeomorphism. ((II)(II)), (((II)I)I), ((II)I)(II), and (II)(II)(II) are homeomorphic to each other, so they have the same homology groups (Z, Z^3, Z^3, Z) and the same hole dimensions: (1,3,3,1). But they have different hole shapes.
Last edited by PWrong on Thu May 07, 2015 5:29 am, edited 2 times in total.

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### Re: Building a Toratope by Rotations from a Circle

The homology holes and the naive idea of "holes" are, of course, distinct. Homology holes correspond to nonvanishing shapes on the surface of toratope, i.e. classes of shapes which can't be deformed in a way that reduces their dimension (that explains why the point is there -- its dimension is already 0, therefore can't be reduced).

The other idea of holes stems from this: imagine the shapes are no longer on the surface of the toratope: in fact, they are anywhere BUT. Can they be reduced under these conditions?

For torus, if we detach the circles from the surface, they can be reduced if we put them in wrong direction -- the horizontal circle must be stuck inside the torus to stay nonreducing and the vertical circle must be put outside.

The "naive hole" concept, however, would maintain that torus has only one hole, through the middle. This means that it only accepts nonreducing shapes if they are OUTSIDE the toratope. The naive concept doesn't see the horizontal circle as defining a hole since when you detach it from surface, you can easily pull it to point.

Another thing that the naive concept won't accept is "compound" holes. If there's a hole in one dimension, than only the simplest class of curves/surfaces going through it will be accepted. The 2-hole on torus, torus as such, can't be reduced if you slip the whole skin and stretch it -- it will get stuck in the hole, but the naive concept still sees it as the same hole the vertical circles get stuck on.

Basically, a naive hole is something that allows you to put a rod or a board or something through the toratope in such a way that they become linked and can't be separated without breaking one or the other shape.

So, how does tiger fare here? Well, none of its 3 1-holes actually stays nonreducing when moved outside the tiger itself. We need to use 2-holes. 1st major/minor and 2nd major/minor 2-holes mean there are two directions where you can "thread" a 2D plane, a sphere, or a thin board or torisphere through the tiger (I think I showed some examples when I was researching locks). The 1st major/2nd major 2-hole can then be completely detached from the tiger as "jacket" and reduce or vanish; this is probably the case for any hole made only of major (outermost) dimensions of any toratope.

So this shows that tiger has 2 holes in the traditional sense, both of dimension 2.

This mainly applies to chaining; it can show you why you can chain torisphere and spheritorus together, but you can't make a workable chain of two torispheres or two spheritoruses. It might show a possible way to chain two tigers.
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### Re: Building a Toratope by Rotations from a Circle

Homology holes correspond to nonvanishing shapes on the surface of toratope, i.e. classes of shapes which can't be deformed in a way that reduces their dimension

Kind of. Officially, homology groups can be defined rigorously a few different ways that mostly give the same answers. They do have something to do with wrapping nonvanishing shapes around the surface, but so do homotopy groups, which are completely different and more complicated thanks to things like Hopf fibrations. So we can come up with intuitive ideas about what homology holes "mean", but there's a good chance we'll be wrong.

It's probably time for a table of toratopes and their holes. Homology holes, their "shapes", and the "naive holes". Maybe we can find a pattern that will lead to a rigorous definitions of the hole shapes and naive holes. I suspect that naive holes will either turn out to be an easily calculated subset of the homology holes, or we won't be able to agree on what counts as a naive hole for more complicated toratopes.

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### Re: Building a Toratope by Rotations from a Circle

The "naive hole" concept, however, would maintain that torus has only one hole, through the middle. This means that it only accepts nonreducing shapes if they are OUTSIDE the toratope. The naive concept doesn't see the horizontal circle as defining a hole since when you detach it from surface, you can easily pull it to point.

You could argue that this is only because a naive observer would assume that a torus is like a solid donut, not a 2-dimensional surface. A solid torus does in fact only have one 1-hole, because it's homeomorphic to a circle. So perhaps the "naive hole" concept is actually talking about the homology groups of max-frame toratopes.

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### Re: Building a Toratope by Rotations from a Circle

PWrong wrote:
The "naive hole" concept, however, would maintain that torus has only one hole, through the middle. This means that it only accepts nonreducing shapes if they are OUTSIDE the toratope. The naive concept doesn't see the horizontal circle as defining a hole since when you detach it from surface, you can easily pull it to point.

You could argue that this is only because a naive observer would assume that a torus is like a solid donut, not a 2-dimensional surface. A solid torus does in fact only have one 1-hole, because it's homeomorphic to a circle. So perhaps the "naive hole" concept is actually talking about the homology groups of max-frame toratopes.

Or, it could be talking about homology groups of the whole space MINUS the solid toratope.
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### Re: Building a Toratope by Rotations from a Circle

That would give the same answer but it doesn't sound very "naive"

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### Re: Building a Toratope by Rotations from a Circle

I never thought about the shapes of the holes, either. But, since they have to do with non-vanishing surfaces, which do end up having different shapes, then maybe it's worth investigating. It looks intuitive to me, when thinking about diameters in terms of a hierarchy: major, secondary, trinary, quaternary, minor, etc. Combining certain ones together will make certain surfaces.

I would agree that tiger has a duocylinder-margin shape hole, plus two torus holes. Pairing both major diameters will make the margin, and combos of major/minor lead to a torus.

The ditorus and tiger are straightforward. But, the (((II)I)(II)) has enough of a variety of diameters, leading to new questions. If we're trying to differentiate between margins and non-margin surfaces, then there is one property that stands out. What do we get when combining two majors, and a non-minor diameter? In the case of (((II)I)(II)), we get the 3-surfaces of:

(((II)I)(II)) - (((maj1)med)(maj2)min)

3-frame : maj1-maj2-med --> both majors and a medium which make the duocylinder-torus margin, 3-frame of ((II)I)(II)

tiger : maj1-maj2-min --> both majors + minor make a tiger ((II)(II))

ditorus : maj1-med-min (((II)I)I)

ditorus : maj2-med-min (((II)I)I)

On that topic, PWrong mentions how (S1xS1)#S1 is the tiger, but S1#(S1xS1) is an invalid shape. You could actually use S1#(S1xS1) to define the duocylinder-torus margin. It's a non-spherated surface, made by small duocylinder margin over a big circle, which I would define as [S1*S1]xS1 , where tiger is S1x[S1*S1]. So, you can still use the notation, and expand on it, too! The only aspect that's backwards is the size of S1's, in the chained sequence.

So, how many distinct surfaces are there per dimension, considering margins and non-margins? I'll try out your notation here, with expanded definitions,

0-hole
---------
() - point

1-hole
---------
(II) - circle , S1

2-hole
--------
(III) - sphere , S2

((II)I) - torus , S1#S1
-
(II)(II) - duocylinder margin , (S1xS1)

3-hole
--------
(IIII) - glome , S3

((III)I) - torisphere , S2#S1
((II)II) - spheritorus , S1#S2
-
(III)(II) - cylspherinder margin , (S2xS1)

(((II)I)I) - ditorus , S1#S1#S1
((II)(II)) - tiger , (S1xS1)#S1
-
((II)I)(II) - duocylinder-torus margin , S1#(S1xS1)
(II)(II)(II) - triocylinder margin , (S1xS1xS1)

4-hole , here is where things can get a bit ambiguous
-----------------------------
(IIIII) - pentasphere , S4

((IIII)I) - toriglome , S3#S1
((III)II) - spherisphere , S2#S2
((II)III) - glomitorus , S1#S3
-
(IIII)(II) - glome*circle margin , (S3xS1)
(III)(III) - sphere*sphere margin , (S2xS2)

(((III)I)I) - ditorisphere , S2#S1#S1
(((II)II)I) - torispheritorus , S1#S2#S1
(((II)I)II) - spheriditorus , S1#S1#S2
((II)(II)I) - spheritiger , (S1xS1)#S2
-
((III)I)(II) - torisphere*circle margin , ((S2#S1)xS1) , or S2#(S1xS1) as duocylinder margin over the sphere, the S2 commutes
((II)II)(II) - spheritorus*circle margin , ((S1#S2)xS1), or S1#(S2xS1) as cylspherinder-margin torus, the S1 commutes, type 1
((II)I)(III) - torus*sphere margin , ((S1#S1)xS2), or S1#(S2xS1) the other case of cylspherinder-margin torus, type 2
(III)(II)(II) - sphere*circle*circle margin , (S2xS1xS1)

(((II)I)(II)) - tiger torus , S1#(S1xS1)#S1
(((II)(II))I) - toratiger , (S1xS1)#S1#S1
((((II)I)I)I) - tritorus , S1#S1#S1#S1
-
((II)(II))(II) - tiger*circle margin , (((S1xS1)#S1)xS1), or (S1xS1)#(S1xS1) as duocylinder-margin over duocylinder-margin, type 1
(((II)I)I)(II) - ditorus*circle margin , ((S1#S1#S1)xS1), or S1#S1#(S1xS1) as duocylinder-margin over the torus
((II)I)(II)(II) - torus*circle*circle margin , ((S1#S1)xS1xS1), or S1#(S1xS1xS1) as triocylinder-torus margin
((II)I)((II)I) - torus*torus margin , ((S1#S1)x(S1#S1)), or (S1xS1)#(S1xS1) as duocylinder-margin over duocylinder-margin, type 2
(II)(II)(II)(II) - tetracylinder margin , (S1xS1xS1xS1)

Trying to get ((II)I)((II)I) torus*torus margin by pulling out circles won't work. You have to pull a whole torus out of ((((II)I)I)I), which is a little different than the process I've seen so far. It's a valid 4-surface, but the algorithm changes slightly.

EDIT : added in the ((II)(II)I)
Last edited by ICN5D on Tue May 05, 2015 6:08 am, edited 2 times in total.
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### Re: Building a Toratope by Rotations from a Circle

In addition to the above, it came across my mind that (((II)(II))I) won't hold up to combining maj1-maj2-med as a margin.

(((II)(II))I) - (((maj1)(maj2)med)min)

0-hole : 1x point

1-hole : 4x circles : maj1, maj2, med, min

2-hole : 1x duocyl margin : maj1-maj2 / 5x torus : maj1-med , maj1-min , maj2-med , maj2-min , med-min

3-hole : 2x tigers: maj1-maj2-med , maj1-maj2-min / 2x ditoruses: maj1-med-min , maj2-med-min

4-hole : 1x toratiger

The two distinct non-vanishing tigers have combos that do not agree with the logic behind (((II)I)(II)). A (((II)(II))I) has only one type of non-spherated margin, as a 2-hole. All four of the 3-holes are either tigers or ditoruses. So, maybe there's an even more rigorous approach, considering the different roles the medium and minor diameters play in (((II)I)(II)) and (((II)(II))I).

Plus, I forgot about the ((II)(II)I) as one of the 4-holes:

((II)(II)I) - (S1xS1)#S2
((II)II)(II)
(III)(II)(II)
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### Re: Building a Toratope by Rotations from a Circle

On that topic, PWrong mentions how (S1xS1)#S1 is the tiger, but S1#(S1xS1) is an invalid shape. You could actually use S1#(S1xS1) to define the duocylinder-torus margin. It's a non-spherated surface, made by small duocylinder margin over a big circle, which I would define as [S1*S1]xS1 , where tiger is S1x[S1*S1]. So, you can still use the notation, and expand on it, too!

We've been down that road before. To see why it's ambiguous, look at the shape S1#(S0xS0). It's a circle "spherated" by a square. But how do you arrange the square? Like a square tube or a diamond tube? There are infinitely many ways to describe this thing geometrically. Then there are lots of other ugly shapes that don't feel like they belong, including the shapes that could be described by S1#(S1xS1).

So, how many distinct surfaces are there per dimension, considering margins and non-margins? I'll try out your notation here, with expanded definitions,

I'm not sure what's going in the list that follows this. It looks like you just listed the toratopes by their internal dimension (e.g. where (II)(II) is a 2-manifold).

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### Re: Building a Toratope by Rotations from a Circle

Going back to the shape of holes, I'm not sure about (((II)I)I).

I worked out that ((II)(II)) has the following holes

One 0-hole
Three circle holes: in the xy, zw, and r1r2 planes.
Two torus holes: in the xyr2 and r1zw realms.
One duocylinder hole
One tiger hole

This gives the hole pattern {1,3,3,1} as expected.

But (((II)I)I) should have the same hole dimension numbers, but there's a hole unaccounted for. The 0-hole and 3-hole are trivial.

Three circle holes: in the xy, r11z, r1w planes.

For 2-holes, there's a torus in ((xy)z) and (r1z)w. But I don't know how to describe the middle one. I'm not sure if it's a torus or a duocylinder or what.

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### Re: Building a Toratope by Rotations from a Circle

I think the holes can be found by looking at cross-sections. Write out the implicit equation for the toratope. Set some of the variables to zero. This gives an equation for a manifold that is "stuck" on the shape, because there's a hole. You just have to be careful that you don't count the same hole twice (because some can be moved onto others).

Using this method, the three 3-holes in (((II)I)I) are three toruses: xyz, xyw, xzw. (Note that the xzw torus and the yzw torus are the same).

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### Re: Building a Toratope by Rotations from a Circle

Another way to find them: simply write out the spheration notation for the shape, and cross out some spheres.

Tiger: (S1xS1)#S1
0-holes: {0}
1-holes: S1, S1, S2
2-holes: S1#S1, S1#S1, S1xS1
3-holes: (S1xS1)#S1

Ditorus: (S1#S1)#S1
2-holes: S1#S1, S3

((II)I)(II): (S1#S1)xS1
2-holes: S1#S1, S1xS1, S1xS1

(((II)(II))I): ((S1xS1)#S1)#S1 (hole pattern {1,4,6,4,1})
1-holes: S1, S1, S1, S1
2-holes: S1xS1, S1#S1, S1#S1, S1#S1, S1#S1, S1#S1
3-holes: (S1xS1)#S1, (S1xS1)#S1, (S1#S1)#S1, (S1#S1)#S1
4-holes: ((S1xS1)#S1)#S1

This matches what you said before: the 2-holes include one duocylinder and 5 torii, and the 3-holes include two tigers and two ditorii.

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### Re: Building a Toratope by Rotations from a Circle

PWrong wrote:Going back to the shape of holes, I'm not sure about (((II)I)I).

I worked out that ((II)(II)) has the following holes

One 0-hole
Three circle holes: in the xy, zw, and r1r2 planes.
Two torus holes: in the xyr2 and r1zw realms.
One duocylinder hole
One tiger hole

This gives the hole pattern {1,3,3,1} as expected.

But (((II)I)I) should have the same hole dimension numbers, but there's a hole unaccounted for. The 0-hole and 3-hole are trivial.

Three circle holes: in the xy, r11z, r1w planes.

For 2-holes, there's a torus in ((xy)z) and (r1z)w. But I don't know how to describe the middle one. I'm not sure if it's a torus or a duocylinder or what.

(Noticed you posted new messages while I was writing this.)

Well, one way to work it out is to look for holes on the ditorus mid-cuts.

Any mid-cut has the shape of two toruses (in various configurations) and any hole ends up as either a hole on one of these toruses (which can be moved to the other through the 4th dimension) or a lower-dimensional hole on both toruses (which extends into 4th dimension).

So, if we consider two toruses ((I)I)I) the view A, major pair ((II))I) the view B and minor pair ((II)I)) the view C:

Point hole: displays as a point on one torus in A, B and C

xy hole: displays as a point on both toruses in A and as a major circle on one torus in B and C.

r1z hole: displays as a point on both toruses in B, as a major circle on one torus in A and as a minor circle on one torus in C.

r1w hole: displays as a point on both toruses in C and as a minor circle on one torus in A and B.

xyz hole: displays as a major circle on both toruses in A and B and as one whole torus in C.

xyw hole: displays as a minor circle on both toruses in A, a major circle on both toruses in C and as one whole torus in B.

r1zw hole: displays as a minor circle on both toruses in B and C and as one whole torus in A.

xyzw hole: displays as both whole toruses in A, B and C.

Checking for tiger with mid-cuts A ((II)(I)) and B ((I)(II)):

Point hole - point on one torus in A and B
xy hole - major circle on one torus in A, point on both toruses in B
r1r2 hole - minor circle on one torus in A and B
zw hole - point on both toruses in A, major circle on one torus in B
xyr2 hole - one whole torus in A, minor circle on both toruses in B
r1zw hole - minor circle on both toruses in A, one whole torus in B
r1r2r3 hole - major circle on both toruses in A and B
xyzw hole - both whole toruses in A and B
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### Re: Building a Toratope by Rotations from a Circle

So for the tiger torus (((II)(II))I), we can consider the non-empty 3D cuts and how the holes would display there:

A1: (((II)(I))) (vertical stack of two minor pairs of toruses)
A2: (((I)(II))) (vertical stack of two minor pairs of toruses)
B: (((I)(I))I) (2x2 array of toruses)

Point hole -- A1: point on one torus, A2: point on one torus, B: point on one torus

xy hole -- A1: major circle on one torus, A2: point on two vertically aligned toruses, B: point on two toruses displaced in r1 direction
r1r2 hole -- A1: minor circle on one torus, A2: minor circle on one torus, B: major circle on one torus
zw hole -- A1: point on two vertically aligned toruses, A2: major circle on one torus, B: point on two toruses displaced in r2 direction
(r1/r2)v hole -- A1: point on two toruses in a pair, A2: point on two toruses in a pair, B: minor circle on one torus

xyr2 hole -- A1: one whole torus, A2: minor circle on two vertically aligned toruses, B: major circle on two toruses displaced in r1 direction
r1r2r3 hole -- A1: major circle on two vertically aligned toruses, A2: major circle on two vertically aligned toruses, B: point on all four toruses
xyv hole -- A1: major circle on two toruses in a pair , A2: point on all four toruses, B: minor circle on two toruses displaced in r1 direction
r1zw hole -- A1: minor circle on two vertically aligned toruses, A2: one whole torus, B: major circle on two toruses displaced in r2 direction
r1r2v hole -- A1: minor circle on two toruses in a pair, A2: minor circle on two toruses in a pair, B: one whole torus
zwv hole -- A1: point on all four toruses, A2: major circle on two toruses in a pair, B: minor circle on two toruses displaced in r2 direction

xyzw hole -- A1: two vertically aligned toruses, A2: two vertically aligned toruses, B: major circle on all four toruses
xyr1v hole -- A1: pair of toruses, A2: minor circle on all four toruses, B: two toruses displaced in r1 direction
r1r2r3v hole -- A1: major circle on all four toruses, A2: major circle on all four toruses, B: minor circle on all four toruses
r1zwv hole -- A1: minor circle on all four toruses, A2: pair of toruses, B: two toruses displaced in r2 direction

xyzwv hole -- A1: all four toruses, A2: all four toruses, B: all four toruses
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### Re: Building a Toratope by Rotations from a Circle

And for a torus tiger (((II)I)(II)):
axes (((xy)z)(wv))

A = (((II))(I)) (vertical stack of two major pairs of toruses)
B = (((I)I)(I)) (two vertical stacks of two toruses)
C = (((I))(II)) (vertical stack of four toruses)

Point hole: A: point on one torus, B: point on one torus, C: point on one torus

xy hole: A: major circle on one torus, B: point on two horizontally aligned toruses, C: point on either outer or inner toruses
r1z hole: A: point on a pair of toruses, B: major circle on one torus, C: point on either top or bottom toruses
(r1/z)r3 hole: A: minor circle on one torus, B: minor circle on one torus, C: minor circle on one torus
wv hole: A: point on two vertically aligned toruses, B: point on two vertically aligned toruses, C: major circle on one torus

xyz hole: A: major circle on a pair of toruses, B: major circle on two horizontally aligned toruses, C: point on all four toruses
xyr3 hole: A: one whole torus, B: minor circle on two horizontally aligned toruses, C: minor circle on either outer or inner toruses
r1r3r4 hole: A: major circle on two vertically aligned toruses, B: point on all four toruses, C: major circle on either outer or inner toruses
r1zr3 hole: A: minor circle on a pair of toruses, B: one whole torus, C: minor circle on either top or bottom toruses
zwv hole: A: point on all four toruses, B: major circle on two vertically aligned toruses, C: major circle on either top or bottom toruses
r1wv hole: A: minor circle on two vertically aligned toruses, B: minor circle on two vertically aligned toruses, C: one whole torus

xyzr3 hole: A: a pair of toruses, B: two horizontally aligned toruses, C: minor circle on all four toruses
r1zr3r4 hole: A: major circle on all four toruses, B: major circle on all four toruses, C: major circle on all four toruses
xywv hole: A: two vertically aligned toruses, B: minor circle on all four toruses, C: either two outer or two inner toruses
r1zwv hole: A: minor circle on all four toruses, B: two vertically aligned toruses, C: either two top or two bottom toruses

xyzwv hole: A: all four toruses, B: all four toruses, C: all four toruses
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### Re: Building a Toratope by Rotations from a Circle

We've been down that road before. To see why it's ambiguous, look at the shape S1#(S0xS0). It's a circle "spherated" by a square. But how do you arrange the square? Like a square tube or a diamond tube? There are infinitely many ways to describe this thing geometrically. Then there are lots of other ugly shapes that don't feel like they belong, including the shapes that could be described by S1#(S1xS1).

In a strictly abstract sense of defining the surface, yeah I can see how it's ambiguous. There would be an infinite number of angles that the square could be tilted before spherating the circle. One would have to add in the tegum notation in order to distinguish them. A diamond vs square torus are easily defined apart with an implicit equation.

But, considering coordinate planes of rotation only, a S1#(S0xS0) would always end up as a square torus. Filling in the shape entirely, we'd get S1#(S1xS1) , which isn't really that horrible of a surface, is it? It describes the min-frame of a duocylinder spherating a circle, which is a closed 3-surface of ((II)I)(II). It works really well if trying to separate the closed toratopes from the min-frames of opens.

I'm not sure what's going in the list that follows this. It looks like you just listed the toratopes by their internal dimension (e.g. where (II)(II) is a 2-manifold).

Yes, there I describe closed toratopes as you would, then the min-frames of the opens in a similar fashion. There ends up being two different but equal ways to describe things like ((III)I)(II) or ((II)I)((II)I).

Using this method, the three 3-holes in (((II)I)I) are three toruses: xyz, xyw, xzw. (Note that the xzw torus and the yzw torus are the same).

Yes, all three distinct mid-cuts of ditorus are couples of tori, in 3 different sizes. That's another good way of looking at it.
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### Re: Building a Toratope by Rotations from a Circle

Filling in the shape entirely, we'd get S1#(S1xS1) , which isn't really that horrible of a surface, is it? It describes the min-frame of a duocylinder spherating a circle, which is a closed 3-surface of ((II)I)(II).

So is S1#(S1xS1) actually embedded in 5D? That hadn't occurred to me but it make sense.

So you start with a circle in xy. The duocylinder is 2D embedded in 4D, and the symmetry separates the four into two special planes. We have to pick one of the two planes to be special, and then pick an axis from that plane to correspond to the direction of whatever point in the circle is being spherated. If I'm picturing this right, there are several geometrically distinct shapes that have an equal claim to the name "S1#(S1xS1)". I'm not sure they actually are equivalent to ((II)I)(II). I could be convinced with some algebraic argument, but the algebraic definition of spheration only currently works for spheres.

It works really well if trying to separate the closed toratopes from the min-frames of opens.

Doesn't our frame notation cover that pretty well? ((II)I)(II) has three frames,
F_3((II)I)(II) has one cell with two equations.
F_4((II)I)(II) has two cells, each with an equality and an inequality.
F_5((II)I)(II) has one cell with two inequalities.

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### Re: Building a Toratope by Rotations from a Circle

Going back to your table, you have

((II)II)(II) - spheritorus*circle margin , ((S1#S2)xS1), or S1#(S2xS1) as cylspherinder-margin torus, the S1 commutes, type 1
((II)I)(III) - torus*sphere margin , ((S1#S1)xS2), or S1#(S2xS1) the other case of cylspherinder-margin torus, type 2

So you have S1#(S2xS1) interpreted as two very different shapes. Unless you want to keep associativity but say x is no longer commutative. Then you could say
(S1#S2)xS1 = S1#(S2xS1), and
(S1#S1)xS2 = S1#(S1xS2), but
S2xS1 ≠ S1xS2.

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### Re: Building a Toratope by Rotations from a Circle

The duocylinder is 2D embedded in 4D, and the symmetry separates the four into two special planes. We have to pick one of the two planes to be special, and then pick an axis from that plane to correspond to the direction of whatever point in the circle is being spherated.

Yes, that sounds about right. But overall, wouldn't the axes be ambiguous? No matter which one you pick to be the spherating direction, all will equally make S1#(S1xS1). We get the combinations ((II)I)(II) , (I(II))(II) , (II)((II)I) , (II)(I(II)) of an identical surface.

If I'm picturing this right, there are several geometrically distinct shapes that have an equal claim to the name "S1#(S1xS1)". I'm not sure they actually are equivalent to ((II)I)(II). I could be convinced with some algebraic argument, but the algebraic definition of spheration only currently works for spheres.

Which ones are you thinking? It does in fact describe a tiger or ditorus embedded in 5D, which flattens it out a bit. But, they have different definitions using spherate and product:

(((II)I)I) - S1#S1#S1
((II)(II)) - (S1xS1)#S1
((II)I)(II) - S1#(S1xS1)
(II)(II)(II) - (S1xS1xS1)

For spherating a circle with a duocylinder margin, it's more intuitive to work backwards from your notation. This is why you see mine as the one that's backwards I start with the smallest shape first, then work my way up. It follows Marek's algorithm for rotations of a toratope. Start with circle (II) , mark a dimension as x, making (xI). To bisecting rotate, x = II , (III). To non-intersecting rotate, x = (II) , ((II)I). It's easier to build the shape up in smaller pieces, rather than combining a large, complex term to a simpler one.

(√(x²+y²)-a)² + (√(z²+w²)-b)² = 0

To spherate this around a larger circle, we simply replace x with (√(x²+v²)-c) , which is the same as (xI)(II) , x = (II) , ((II)I)(II):

(√((√(x²+v²)-c)² +y²)-a)² + (√(z²+w²)-b)² = 0

√(x²+y²) = a

it won't be very intuitive or clear as to how to proceed with embedding (√(x²+y²)-a)² + (√(z²+w²)-b)² into it. Stuffing this 2-frame into a single dimension of the circle will make a 3-frame of (((II)(II))I). Where, technically you could start with just the circle, and simply repeat the process above.

Doesn't our frame notation cover that pretty well? ((II)I)(II) has three frames,
F_3((II)I)(II) has one cell with two equations.
F_4((II)I)(II) has two cells, each with an equality and an inequality.
F_5((II)I)(II) has one cell with two inequalities.

Yes, it works perfectly! I was referring to S1#(S1xS1) having the value of defining min-frames apart from closed.

So you have S1#(S2xS1) interpreted as two very different shapes. Unless you want to keep associativity but say x is no longer commutative. Then you could say
(S1#S2)xS1 = S1#(S2xS1), and
(S1#S1)xS2 = S1#(S1xS2), but
S2xS1 ≠ S1xS2.

That's a good idea, actually. I hadn't thought of that. I don't like how it's ambiguous, and this minor addition helps.

((II)II)(II) - (S1#S2)xS1 = S1#(S2xS1)
((II)I)(III) - (S1#S1)xS2 = S1#(S1xS2)

It has a nice relation in describing which parameter of (III)(II), sphere or circle, has the nested circle. The important thing is the partial sequence "S1#(S2" which means the S2 sphere diameter has the nested circle, leading to ((II)II)(II) .

This brings me back to an earlier suggestion of using [] to denote an open toratope. With it, we can define min-frames, open and closed toratopes. All will share the common sequence of the min-frame, where the brackets tell us whether it's a sharp-edged prism or a toroidal ring. The () is rounded, for smooth ring-like objects. The [] is flat with sharp ends, for filled-in prisms with edges. And a lack of brackets is for flat Clifford torus-like objects, such as 3-frames embedded in 5D and the like.

[(II)(II)] - solid duocylinder

|√(x²+y²) - √(z²+w²)| + |√(x²+y²) + √(z²+w²)| = a

(II)(II) - minimum 2-frame of duocylinder

(√(x²+y²)-a)² + (√(z²+w²)-b)² = 0

((II)(II)) - spherated 2-frame makes tiger

(√(x²+y²)-a)² + (√(z²+w²)-b)² = c²
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