## A Notation to describe Toratope Cuts

Discussion of shapes with curves and holes in various dimensions.

### A Notation to describe Toratope Cuts

I've been slowly developing this, for a while. There were some things to figure out with the empty sets, and some refining of symbols. I think it does a good job of putting a visual in the mind.

This notation is a middle-man description between a toratope slice symbol: " (((II))(I)) " , and a completely verbal description: " a product of 4 tori, as a vertical stacking of 2 concentric pairs in the major diameter " . According to my notation, it would read: " 1x1x2 , [r1-2] " . In other words,

(((II))(I)) == 1x1x2 , [r1-2] == product of 4 tori ((II)I), as a vertical stacking of 2 concentric pairs in the major diameter

The basic format is :

(Imaginary Disjoint Groups) , Real Disjoint Groups , [Real Concentric Groups]

which breaks down the 3-tiered hierarchy of what we see in the solution, as 'imaginary-number groups' of real-number arrays of concentric groups. I later realized how the imaginary-numbers are not in groups, but circles. Further down, I'll detail why I haven't changed the symbol yet.

(Imaginary Disjoint Groups) : Describes the number and arrangement of distinct locations where we get points or sets of points in the reals, after canceling the imaginary component. Can be a product of points or surfaces, with concentric groups as well. Use single numbers, or arrays, or arrays of surfaces, (more detail below).

- For no imaginary component, omit the (Im Grp) symbol

- Some examples : (2) , (4) , (2x2) , (4x2) , (2x2-(II)) , (2x2,[r-2]-(II)) , (2x2x2-((II)I)) , (2x2x4x4-((II)(II)))

Real Disjoint Groups : Describes the length x width x height x etc, array of disjoint groups of single objects, or multiple concentric groups, following an " X1 x X2 x X3 x X4 x X5 x ... x Xn ", array of n-dimensions. For any disjoint pairing of minimum 2, describe the full array, based on dimensions of the object.

- For no disjoint groups, i.e. singular array, omit the 1x1x...x1 symbol. This will leave behind the (Im Grp) and [Re Conc Grp]

- For single columns of n vertical stackings, use 1x1x...xn

- For single rows of n sideways stackings, use nx1x...x1

- For nxm vertical square/cube/tess arrays (n-rows of m-columns), use nx1xm , or nx1x...x1xm , etc

- Some examples : 2, 4 , 2x2 , 4x2 , 1x1x2 , 2x1x4 , 1x1x1x8 , 2x1x1x4 , 2x2x2x2 , 4x2x4x2 , 8x4x1x2 , 2^5* , 2^6*

* Use 2^5 in place of 2x2x2x2x2 , to make writing out a 5D array of 32 objects a bit easier on the eyes. You may also use 2^3 and 2^4 as well.

[Real Concentric Groups] : Describes the concentric groups by radius type and number. Follows the format [r1-n]x[r2-m]x[r3-p]x... , which describes n-number of reflections on radius r1 , m reflections of radius r2 , p reflections of radius r3, etc. The symbol [rm-2] is a pair in the minor diameter 'rm' , which can be used for all toratopes, since they will have only one minor diameter, but many more above it.

- For no concentric groupings, i.e. imaginary arrays and real disjoint arrays only, omit the respective radius symbol [r-1]. Only use the symbol for a radius that has a concentric grouping.

- Use toratope notation to label the radii : (r) , ((r1)rm) , (((r1)r2)rm) , ((r1a)(r1b)rm) , (((r1a)r2)(r1b)rm) , (((r1a)(r1b)r2)rm) , ((((r1)r2)r3)rm) , etc

- These concentric radius symbols will also be used to describe the imaginary component as well, when applicable

- Some examples : [r-2] , [r-4] , [r1-2] , [r1-2]x[rm-4] , [r1a-4]x[r1b-2] , [r1a-2]x[r1b-2]x[rm-4] , [r1a-4]x[r2-2]x[rm-8]

Some real solutions (non-empty):

Intersections of 0D points:
(I) - 2
((I)) - 4
(((I))) - 8
((((I)))) - 16
(((((I))))) - 32

Intersections of 2D circles, a circle has just one distinct radius, so we only need [r-n] for conc groups

(II) - a single circle:
---------------------------
Disjoint Arrays (XY) -> X x Y
((I)I) - 2x1
(((I))I) - 4x1
((((I)))I) - 8x1
((I)(I)) - 2x2
(((I))(I)) - 4x2
((((I)))(I)) - 8x2
(((I))((I))) - 4x4
((((I)))((I))) - 8x4
-----------------------------
Concentric Groups (r) -> [r-n]
((II)) - [r-2]
(((II))) - [r-4]
((((II)))) - [r-8]
(((((II))))) - [r-16]
-----------------------------
Disjoint Arrays of Concentric Groups
(((I)I)) - 2x1 , [r-2]
((((I))I)) - 4x1 , [r-2]
(((((I))I))) - 4x1 , [r-4]
(((I)(I))) - 2x2 , [r-2]
((((I))(I))) - 4x2 , [r-2]
(((((I))((I))))) - 4x4 , [r-4]

Intersections of 3D Spheres, same as a circle, a sphere has only one distinct radius, and only needs [r-n] symbol. A sphere will have no visually distinct vertical square array, which will also be equal to a flat square, 2x1x2 looks the same as 2x2x1:

(III) - a single sphere
-------------------------------------
Disjoint Arrays (XYZ) -> X x Y x Z
((I)II) - 2x1x1
(((I))II) - 4x1x1
((((I)))II) - 8x1x1
((I)(I)I) - 2x2x1
(((I))(I)I) - 4x2x1
(((I))((I))I) - 4x4x1
((I)(I)(I)) - 2x2x2
(((I))(I)(I)) - 4x2x2
(((I))((I))((I))) - 4x4x4
------------------------------------
Concentric Groups (r) -> [r-n]
((III)) - [r-2]
(((III))) - [r-4]
((((III)))) - [r-8]
(((((III))))) - [r-16]
-----------------------------------------
Disjoint Arrays of Concentric Groups
(((I)II)) - 2x1x1 , [r-2]
((((I))II)) - 4x1x1 , [r-2]
(((I)(I)I)) - 2x2x1 , [r-2]
((((I)II))) - 2x1x1 , [r-4]
(((I)(I)(I))) - 2x2x2 , [r-2]
(((((I))(I)(I)))) - 4x2x2 , [r-4]

Intersections of 3D tori. A torus has two distinct radii, labeled ((r1)rm) , which allows for visually distinct flat and vertical square arrays in 3-space, where a flat 2x2x1 square does not look the same as a vertical 2x1x2 square:

((II)I) - a single torus
---------------------------
Disjoint Arrays ((XY)Z) -> X x Y x Z
(((I)I)I) - 2x1x1
((((I))I)I) - 4x1x1
(((((I)))I)I) - 8x1x1
(((I)(I))I) - 2x2x1
(((I)I)(I)) - 2x1x2
(((I)(I))(I)) - 2x2x2
((((I))I)((I))) - 4x1x4
((((I))(I))(I)) - 4x2x2
((((I))(I))((I))) - 4x2x4
((((I))((I)))((I))) - 4x4x4
------------------------------
Concentric Groups ((r1)rm) -> [r1-q]x[rm-p]
(((II))I) - [r1-2]
((((II)))I) - [r1-4]
(((((II))))I) - [r1-8]
(((II)I)) - [rm-2]
((((II)I))) - [rm-4]
((((II)I)))) - [rm-8]
((((II))I)) - [r1-2]x[rm-2]
(((((II)))I)) - [r1-4]x[rm-2]
((((((II)))I))) - [r1-4]x[rm-4]
---------------------------------
Disjoint Arrays of Concentric Groups
((((I)I))I) - 2x1x1 , [r1-2]
((((I)(I)))I) - 2x2x1 , [r1-2]
((((I)(I)))(I)) - 2x2x2 , [r1-2]
((((I)I))(I)) - 2x1x2 , [r1-2]
((((I)I)(I))) - 2x1x2 , [rm-2]
((((I)(I))I)) - 2x2x1 , [rm-2]
((((I)(I))(I))) - 2x2x2 , [rm-2]
(((((I)(I)))(I))) - 2x2x2 , [r1-2]x[rm-2]

Intersections of 4D Tigers : A tiger has 3 distinct radii labeled ((r1a)(r1b)rm), and 2 distinct ways to stack in a square array, where 2x1x2x1 is not equal to 2x2x1x1

((II)(II)) - a single tiger
-----------------------------
Disjoint Arrays ((XY)(ZW)) -> X x Y x Z x W
(((I)I)(II)) - 2x1x1x1
(((I)(I))(II)) - 2x2x1x1
(((I)I)((I)I)) - 2x1x2x1
(((I)(I))((I)I)) - 2x2x2x1
(((I)(I))((I)(I))) - 2x2x2x2 -> 2^4
----------------------------------
Concentric Groups ((r1a)(r1b)rm) -> [r1a-n]x[r1b-p]x[rm-q]
(((II))(II)) - [r1a-2]
(((II))((II))) - [r1a-2]x[r1b-2]
(((II)(II))) - [rm-2]
((((II))(II))) - [r1a-2]x[rm-2]
((((II))((II)))) - [r1a-2]x[r1b-2]x[rm-2]
---------------------------------------------
Disjoint Arrays of Concentric Groups
((((I)I))(II)) - 2x1x1x1 , [r1a-2]
(((I)I)((II))) - 2x1x1x1 , [r1b-2]
((((I)(I)))(II)) - 2x2x1x1 , [r1a-2]
(((I)(I))((II))) - 2x2x1x1 , [r1b-2]
((((I)I)(II))) - 2x1x1x1 , [rm-2]
(((((I)I))((I)I))) - 2x1x2x1 , [r1a-2]x[rm-2]
(((((I)(I)))(((I)(I))))) - 2^4 , [r1a-2]x[r1b-2]x[rm-2]

A Closer Look at the Empty Intersections:

An empty slice, or complex solution, i.e. : " ((()I)(II)) " is described verbosely as " an empty intersection, where moving out from origin will cause a torus to appear, divide into a 1x1x2 vertical column of two tori, then merge back together and vanish" . In my new notation, this is : " (2) , 1x1x2 " , which loosely describes the imaginary and real component.

In the beginning, I naively understood this type empty intersection as having only two locations of points, where the Im part cancels to leave behind the real part (a single tiger, the 1x1x2 column). Later, I realized the Im part, in this case, is actually an entire circle that exists on a 2-plane, where we could translate away from the origin to cause the real part of a 1x1x2 column to come into view (into the real numbers). This is obvious, since the empty set has two variables, x and y, that were set to zero to make a 3D slice: (((xy)z)(wv)) --> ((()z)(wv)).

I'm still not sure exactly how I should represent this kind of solution, since I already use a circle for an imaginary component in other types of empty cuts: " (2x2-(II)) " . For the time being, I'll just define the empty slice symbol (2) as two points on a coordinate axis, where a real component exists. We can remember this coordinate axis to be part of a coordinate 2-plane, allowing 2 distinct directions to translate away from origin to bring points into the reals. This will keep things simple when we go to visualize far more complicated intersections like ((((II)())())(()(I))) = (2x2x2) , 1x1x4 , [r1-4] , which is a tri-complex 3D solution of a 10D hypertorus.

This doesn't include enough info for the empty slice of a torisphere ((xyz)w) --> (()w), or spherisphere ((xyz)wv) --> (()wv), which has three coordinate axes as part of the imaginary component. The problem with including too much info like this is the notation gets too complex, and strays away from its intended simplicity. I designed this notation to be a quick way to read and visualize the intersection array, apart from reading the toratope slice symbol itself.

Some Complex and Multi-Complex Solutions:

Complex Solutions of 0D Points 1D slices that are empty in one coordinate axis:
(()I) - (2) , 2
((()I)) - (2) , 4
(((()I))) - (2) , 8
((())I) - (4) , 2
(((())I)) - (4) , 4
(((()))I) - (8) , 2
----------------------
Bi-Complex Solutions of 0D Points, 1D slices empty in 2 coordinate axes:
(()()I) - (2x2) , 2
((()()I)) - (2x2) , 4
(((()()I))) - (2x2) , 8
((())()I) - (4x2) , 2
((())(())I) - (4x4) , 2
(((())(())I)) - (4x4) , 4
((()I)()) - (2x2) , 4
((()(I))()) - (2x2) , 8
((()())I)* - (2x2-(II)) , 2
((()())(I)) - (2x2-(II)) , 4
(((()()))I) - (2x2,[r-2]-(II)) , 2 --> a 2x2 square of concentric pairs of circles, as the Im component

* Here is the interesting case of a bi-complex solution, involving a multi-valued expression as the imaginary component. In the case of (()()I) , we get just the sqrt of 2 negative numbers, which describe a product of 4 points, in a 2x2 square array. Solving for ((()())I) will lead to 3 imaginary numbers, in the equation of a circle. Judging by the algebraic expression, it seems to describe a product of 4 circles in a 2x2 square array. The imaginary component, where the real part exists, is on the surface of 4 circles, instead of just 4 points. It's an empty set of empty sets. But, since imaginary numbers actually lie on a 2-plane, this really describes the surface of a tiger in 4D. To keep things simpler, I'll maintain it as (2x2-(II)) , since we do end up seeing imaginary components of (2^4-((II)(II))) in 9D toratopes.

-------------------------
Tri-Complex Solutions of 0D Points, 1D slices empty in 3 coordinate axes
(()()()I) - (2x2x2) , 2
(((()())I)()) - (2x2x2-(II)) , 4 ---> there can be +3D arrays of these Im circles, too
(((()())())I) - (2x2x2-((II)I)) , 2 ---> a 2x2x2 cube array of Im torus surfaces, having 5 Im numbers in the equation of a torus
-------------------------
Tetra-Complex Solutions of 0D Points, empty in 4 coordinate axes
(((((I)())())())()) - (2^4) , 32
((((()())(I))())()) - (2^4-(II)) , 16
((((()())())(I))()) - (2^4-((II)I)) , 8
((((()())())())(I)) - (2^4-(((II)I)I)) , 4 --> a 2^4 array of 16 ditoruses, as the Im component
(((()())I)(()())) - (2^4-(II)*(II)) , 4 ---> cartesian product of two ortho 2x2 square arrays of Im circles = a 2^4 array of Im Clifford tori
(((()())(()())I) - (2^4-((II)(II))) , 2 ---> a 2^4 array of 16 tigers, as the imaginary component

I think that's all that I need to write to describe it. The 2D, 3D, 4D, etc complex solutions would follow all the same rules as detailed above. This new notation can be easy to write from the toratope slice symbol, if you work backwards by deleting the (Im Grp) , Re Grp , [Re Conc] parts separately. So, starting with the object (((((II)(II))((II)I))I)(II)) , making the 3D slice (((((I)())((II))))()) , we follow these steps:

1) Define the Imaginary component, which is two sets of two points as (2x2) , delete the empty () sets. Leaves over:

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

2) Define the disjoint array dimensions, which is a 1x1x4 column. Delete the extra (...) brackets. Leaves over:

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

3) Define the concentric groups, which is [r1-2] and [rm-4]. Delete the extra (...) brackets. Leaves over:

(I(II)) = ((II)I) , a torus

4) We end up with a torus ((II)I) as the root, with the Real and Imaginary components for the solution :

(((((I)())((II))))()) == (2x2) , 1x1x4 , [r1-2]x[rm-4]

Described as a 2x2 square array of points on a 2D imaginary plane, where we have a vertical column of 4 groups of concentric pairs in the major diameter, of concentric quartets in the minor diameter. Exploring a (2x2) Im part means we have to move along two separate axes (which are part of their own 2-plane), instead of just one, to bring points into the reals. We have to move away from origin and stop at a precise location, then move along another axis, to see a 1x1x4 , [r1-2]x[rm-4] array appear, divide, and merge.

Now, for some 1,2,3D solutions of high dimensional tori, expressed in my notation:

############################################################################

(((((II)(II))((II)I))I)(II)) : 10 variables, 9 coefficients, degree-512

5 Distinct Solutions on a coordinate axis, of 512 Point Intercepts
-------------------------------------------------------------------
(((((I)())(())))()) - (2x4x2) , 32
((((()())((I))))()) - (2x2x2-(II)) , 32
((((()())(()I)))()) - (2x2x2x2-(II)) , 16
((((()())(()))I)()) - (2x2x4x2-((II)I)) , 4
((((()())(())))(I)) - (2x2x4,[rm-2]-((II)I)) , 4

14 Distinct Solutions on a coordinate 2-plane, of 256 Circle Intercepts
------------------------------------------------------------------------
Bi-Complex
(((((I)(I))(())))()) - (4x2) , 2x2 , [r-8]
(((((I)())((I))))()) - (2x2) , 4x4 , [r-4]
(((((I)())(())))(I)) - (2x4) , 16x2
((((()())((I))))(I)) - (2x2-(II)) , 16x2

Tri-Complex
(((((II)())(())))()) - (2x4x2) , [r-16]
(((((I)())(()I)))()) - (2x2x2) , 4x2 , [r-4]
(((((I)())(()))I)()) - (2x4x2) , 8x1 , [r-2]
((((()())((II))))()) - (2x2x2-(II)) , [r-16]
((((()())((I)I)))()) - (2x2x2-(II)) , 2x1 , [r-8]
((((()())((I)))I)()) - (2x2x2-(II)) , 8x1
((((()())(()I)))(I)) - (2x2x2-(II)) , 8x2
((((()())(()))I)(I)) - (2x2x4-(II)) , 2x2
((((()())(())))(II)) - (2x2x4,[rm-2]-((II)I)) , [r-2]

Tetra-Complex
((((()())(()I))I)()) - (2x2x2x2-(II)) , 4x1 , [r-2]

27 Distinct Solutions on a coordinate 3-Plane, of 128 Torus Intercepts
-----------------------------------------------------------------------
3 Complex
(((((I)(I))((I))))()) - (2) , 2x2x4 , [rm-4]
(((((I)(I))(())))(I)) - (4) , 2x2x2 , [r1-4]x[rm-2]
(((((I)())((I))))(I)) - (2) , 4x4x2 , [r1-2]

15 Bi-Complex
(((((II)(I))(())))()) - (4x2) , 1x1x2 , [rm-8]
(((((II)())((I))))()) - (2x2) , 1x1x4 , [r1-2]x[rm-4]
(((((II)())(())))(I)) - (2x4) , 1x1x2 , [r1-8]
(((((I)(I))(()I)))()) - (2x2) , 2x2x2 , [rm-4]
(((((I)(I))(()))I)()) - (4x2) , 2x2x1 , [r1-2]x[rm-2]
(((((I)())((II))))()) - (2x2) , 1x1x4 , [r1-2]x[rm-4]
(((((I)())((I)I)))()) - (2x2) , 2x1x4 , [rm-4]
(((((I)())((I)))I)()) - (2x2) , 4x4x1 , [rm-2]
(((((I)())(()I)))(I)) - (2x2) , 4x2x2 , [r1-2]
(((((I)())(()))I)(I)) - (2x4) , 8x1x2
(((((I)())(())))(II)) - (2x4) , 1x1x16
((((()())((II))))(I)) - (2x2-(II)) , 1x1x2 , [r1-8]
((((()())((I)I)))(I)) - (2x2-(II)) , 2x1x2 , [r1-4]
((((()())((I)))I)(I)) - (2x2-(II)) , 8x1x2
((((()())((I))))(II)) - (2x2-(II)) , 1x1x16

9 Tri-Complex
(((((II)())(()I)))()) - (2x2x2) , 1x1x2 , [r1-2]x[rm-4]
(((((I)())(()I))I)()) - (2x2x2) , 4x2x1 , [rm-2]
(((((II)())(()))I)()) - (2x4x2) , [r1-4]x[rm-2]
((((()())((II)I)))()) - (2x2x2-(II)) , [rm-8]
((((()())((II)))I)()) - (2x2x2-(II)) , [r1-4]x[rm-2]
((((()())((I)I))I)()) - (2x2x2-(II)) , 2x1x1 , [r1-2]x[rm-2]
((((()())(()I))I)(I)) - (2x2x2-(II)) , 4x1x2
((((()())(()I)))(II)) - (2x2x2-(II)) , 1x1x8
((((()())(()))I)(II)) - (2x2x4-((II)I)) , 1x1x2

############################################################################

(((((II)I)((II)I))I)((II)I)) : 10 variables, 9 coefficients, degree-512

1-Plane Intersections of 512 roots of a 0D point
--------------------------------------------------
Bi-Complex
(((((I))(())))(())) - (4x4) , 32
((((())(())))((I))) - (4x4,[r-2]-(II)) , 8

Tri-Complex
((((()I)(())))(())) - (2x4x4) , 16
((((())(()))I)(())) - (4x4x4-(II)) , 4
((((())(())))(()I)) - (4x4x2,[r-2]-(II)) , 4

2-Plane Intersections of 256 roots of a 2D circle
---------------------------------------------------
Complex
(((((I))((I))))(())) - (4) , 4x4 , [r-4]
(((((I))(())))((I))) - (4) , 16x4

Bi-Complex
(((((II))(())))(())) - (4x4) , [r-16]
(((((I)I)(())))(())) - (4x4) , 2x1 , [r-8]
(((((I))(()I)))(())) - (2x4) , 4x2 , [r-4]
(((((I))(()))I)(())) - (4x4) , 8x1 , [r-2]
(((((I))(())))(()I)) - (4x2) , 16x2
((((()I)(())))((I))) - (2x4) , 8x4
((((())(()))I)((I))) - (4x4-(II)) , 4x2
((((())(())))((II))) - (4x4,[r-2]-(II)) , [r-4]
((((())(())))((I)I)) - (4x4,[r-2]-(II)) , 2x1 , [r-2]

Tri-Complex
((((()I)(()I)))(())) - (2x2x4) , 2x2 , [r-4]
((((()I)(()))I)(())) - (2x4x4) , 4x1 , [r-2]
((((()I)(())))(()I)) - (2x4x2) , 8x2
((((())(()))I)(()I)) - (4x4x2-(II)) , 2x2

3-Plane Intersections of 128 roots of a 3D torus
-------------------------------------------------
Real
(((((I))((I))))((I))) - 4x4x4 , [r1-2]

Complex
(((((II))((I))))(())) - (4) , 1x1x4 , [r1-2]x[rm-4]
(((((II))(())))((I))) - (4) , 1x1x4 , [r1-8]
(((((I)I)((I))))(())) - (4) , 2x1x4 , [rm-4]
(((((I)I)(())))((I))) - (4) , 2x1x4 , [r1-4]
(((((I))((I)))I)(())) - (4) , 4x4x1 , [rm-2]
(((((I))((I))))(()I)) - (2) , 4x4x2 , [r1-2]
(((((I))(()I)))((I))) - (2) , 4x2x4 , [r1-2]
(((((I))(()))I)((I))) - (4) , 8x1x4
(((((I))(())))((II))) - (4) , 1x1x16 , [r1-2]
(((((I))(())))((I)I)) - (4) , 2x1x16

Bi-Complex
(((((II)I)(())))(())) - (4x4) , [rm-8]
(((((II))(()I)))(())) - (2x4) , 1x1x2 , [r1-2]x[rm-4]
(((((II))(()))I)(())) - (4x4) , [r1-4]x[rm-2]
(((((II))(())))(()I)) - (4x2) , 1x1x2 , [r1-8]
(((((I)I)(()I)))(())) - (2x4) , 2x1x2 , [rm-4]
(((((I)I)(()))I)(())) - (4x4) , 2x1x1 , [r1-2]x[rm-2]
(((((I)I)(())))(()I)) - (4x2) , 2x1x2 , [r1-4]
(((((I))(()I))I)(())) - (2x4) , 4x2x1 , [rm-2]
(((((I))(()I)))(()I)) - (2x2) , 4x2x2 , [r1-2]
(((((I))(()))I)(()I)) - (4x2) , 8x1x2
((((()I)(()I)))((I))) - (2x2) , 2x2x4 , [r1-2]
((((()I)(()))I)((I))) - (2x4) , 4x1x4
((((()I)(())))((II))) - (2x4) , 1x1x8 , [r1-2]
((((()I)(())))((I)I)) - (2x4) , 2x1x8
((((())(()))I)((II))) - (4x4-(II)) , 1x1x2 , [r1-2]
((((())(()))I)((I)I)) - (4x4-(II)) , 2x1x2
((((())(())))((II)I)) - (4x4,[r-2]-(II)) , [rm-2]

Tri-Complex
((((()I)(()I))I)(())) - (2x2x4) , 2x2x1 , [rm-2]
((((()I)(()I)))(()I)) - (2x2x2) , 2x2x2 , [r1-2]
((((()I)(()))I)(()I)) - (2x4x2) , 4x1x2

############################################################################

(((((II)(II))(II))((II)(II)))(II)) : 12 variables, 11 coefficients , degree-2,048

Solutions of One Variable for 2,048 roots of a point
----------------------------------------------------
Penta-Complex Solutions
(((((I)())())(()()))()) - (2^5-(II)) , 32
((((()())(I))(()()))()) - (2^5-(II)*(II)) , 16
((((()())())((I)()))()) - (2^5-((II)I)) , 16
((((()())())(()()))(I)) - (2^5-(((II)I)(II))) , 4

Solutions of Two Variables for 1,024 roots of a circle
-------------------------------------------------------
Tetra-Complex Solutions
(((((I)())())((I)()))()) - (2^4) , 8x4 , [r-2]
(((((I)(I))())(()()))()) - (2^4-(II)) , 2x2 , [r-8]
(((((I)())(I))(()()))()) - (2^4-(II)) , 4x2 , [r-4]
(((((I)())())(()()))(I)) - (2^4-(II)) , 16x2
((((()())(I))((I)()))()) - (2^4-(II)) , 4x4 , [r-2]
((((()())())((I)(I)))()) - (2^4-((II)I)) , 2x2 , [r-4]
((((()())())((I)()))(I)) - (2^4-((II)I)) , 8x2
((((()())(I))(()()))(I)) - (2^4-(II)*(II)) , 8x2

Penta-Complex Solutions
(((((II)())())(()()))()) - (2^5-(II)) , [r-16]
((((()())())((II)()))()) - (2^5-((II)I)) , [r-8]
((((()())(II))(()()))()) - (2^5-(II)*(II)) , [r-8]
((((()())())(()()))(II)) - (2^5-(((II)I)(II))) , [r-2]

Solutions of Three Variables for 512 roots of a torus
------------------------------------------------------
Tri-Complex Solutions
(((((I)(I))())((I)()))()) - (2x2x2) , 2x2x4 , [r1-2]x[rm-2]
(((((I)())(I))((I)()))()) - (2x2x2) , 4x2x4 , [rm-2]
(((((I)())())((I)(I)))()) - (2x2x2) , 2x2x8 , [rm-2]
(((((I)())())((I)()))(I)) - (2x2x2) , 8x4x2
(((((I)(I))(I))(()()))()) - (2x2x2-(II)) , 2x2x2 , [rm-4]
(((((I)(I))())(()()))(I)) - (2x2x2-(II)) , 2x2x2 , [r1-4]
(((((I)())(I))(()()))(I)) - (2x2x2-(II)) , 4x2x2 , [r1-2]
((((()())(I))((I)(I)))()) - (2x2x2-(II)) , 2x2x4 , [rm-2]
((((()())(I))((I)()))(I)) - (2x2x2-(II)) , 4x4x2
((((()())())((I)(I)))(I)) - (2x2x2-((II)I)) , 2x2x2 , [r1-2]

Tetra-Complex Solutions
(((((I)())())((II)()))()) - (2^4) , 1x1x8 , [r1-2]x[rm-2]
(((((II)())())((I)()))()) - (2^4) , 1x1x4 , [r1-4]x[rm-2]
(((((II)(I))())(()()))()) - (2^4-(II)) , 1x1x2 , [rm-8]
(((((II)())(I))(()()))()) - (2^4-(II)) , 1x1x2 , [r1-2]x[rm-4]
(((((II)())())(()()))(I)) - (2^4-(II)) , 1x1x2 , [r1-8]
(((((I)())(II))(()()))()) - (2^4-(II)) , 1x1x4 , [rm-4]
(((((I)())())(()()))(II)) - (2^4-(II)) , 1x1x16
((((()())(II))((I)()))()) - (2^4-(II)) , 1x1x4 , [r1-2]x[rm-2]
((((()())(I))((II)()))()) - (2^4-(II)) , 1x1x4 , [r1-2]x[rm-2]
((((()())())((II)(I)))()) - (2^4-((II)I)) , 1x1x2 , [rm-4]
((((()())())((II)()))(I)) - (2^4-((II)I)) , 1x1x2 , [r1-4]
((((()())())((I)()))(II)) - (2^4-((II)I)) , 1x1x8
((((()())(II))(()()))(I)) - (2^4-(II)*(II)) , 1x1x2 , [r1-4]
((((()())(I))(()()))(II)) - (2^4-(II)*(II)) , 1x1x8

############################################################################

((((((II)I)((II)I))I)((II)(II)))(II)) : 13 variables, 12 coefficients, degree-4,096

Lowest real solution in 5D:
((((((I))((I))))((I)(I)))(I)) - 4x4x2x2x2 , [r1a-2] of 256 (((II)(II))I)

Multi-complex solutions in 3D of a torus:
((((((II)I)(())))(()()))()) - (2x2x2x4-(II)) , [rm-16]
((((((II))((I))))(()()))()) - (2x2x2-(II)) , 1x1x4 , [r1-2]x[rm-8]
((((((II))(()I)))(()()))()) - (2x2x2x2-(II)) , 1x1x2 , [r1-2]x[rm-8]
((((((II))(()))I)(()()))()) - (2x2x2x4-(II)) , [r1-4]x[rm-4]
((((((II))(())))((I)()))()) - (4x2x2) , 1x1x4 , [r1-8]x[rm-2]
((((((II))(())))(()()))(I)) - (2x2x4-(II)) , 1x1x2 , [r1-16]
((((((I)I)((I))))(()()))()) - (2x2x2-(II)) , 2x1x4 , [rm-8]
(((((()I)(()I)))(()()))(I)) - (2x2x2x2-(II)) , 2x2x2 , [r1-4]
(((((()I)(()))I)((I)()))()) - (2x4x2x2) , 4x1x4 , [rm-2]
(((((()I)(()))I)(()()))(I)) - (2x2x2x4-(II)) , 4x1x2 , [r1-2]
(((((()I)(())))((II)()))()) - (2x4x2x2) , 1x1x8 , [r1-2]x[rm-2]
(((((()I)(())))((I)(I)))()) - (2x4x2) , 2x2x8 , [rm-2]
(((((()I)(())))((I)()))(I)) - (2x4x2) , 8x4x2
(((((()I)(())))(()()))(II)) - (2x2x2x4-(II)) , 1x1x16
(((((())(()))I)((II)()))()) - (4x4x2x2-(II)) , 1x1x2 , [r1-2]x[rm-2]
(((((())(()))I)((I)(I)))()) - (4x4x2-(II)) , 2x2x2 , [rm-2]
(((((())(()))I)((I)()))(I)) - (4x4x2-(II)) , 4x2x2
(((((())(()))I)(()()))(II)) - (4x4x2x2-(II)*(II)) , 1x1x4
(((((())(())))((II)(I)))()) - (4x4x2,[r-2]-(II)) , 1x1x2 , [rm-4]
(((((())(())))((I)(I)))(I)) - (4x4,[r-2]-(II)) , 2x2x2 , [r1-2]
(((((())(())))((I)()))(II)) - (4x4x2,[r-2]-(II)) , 1x1x8
in search of combinatoric objects of finite extent
ICN5D
Tetronian

Posts: 1004
Joined: Mon Jul 28, 2008 4:25 am
Location: Orlando, FL

### Re: A Notation to describe Toratope Cuts

Probably should have started off with the simpler shapes, first:

****
*2D*
****

(II) - Circle / (xy) / (r)
--------------------------
(I) - 2 points

****
*3D*
****

(III) - Sphere / (xyz) / (r)
----------------------------
(II) - 1 circle
---
(I) - 2 points

((II)I) - Torus / ((xy)z) / ((r1)rm)
------------------------------------
((II)) - [r-2] pair of (II) circles
((I)I) - 2x1 row of (II) circles
---
((I)) - 4 points in a row
(()I) - (2) . 2 row of points

****
*4D*
****

(IIII) - Glome / (xyzw) / (r)
------------------------------
(III) - 1 sphere
---
(II) - 1 circle
---
(I) - 2 points in a row

((III)I) - Torisphere / ((xyz)w) / ((r1)rm)
-------------------------------------------
((III)) - [r-2] pair of (III) spheres
((II)I) - 1 torus
---
((II)) - [r-2] pair of (II) circles
((I)I) - 2x1 row of (II) circles
---
((I)) - 4 points in a row
(()I) - (2),2 row of points

((II)II) - Spheritorus / ((xy)zw) / ((r1)rm)
--------------------------------------------
((II)I) - 1 torus
((I)II) - 2x1x1 row of (III) spheres
---
((II)) - [r-2] pair of (II) circles
((I)I) - 2x1 row of (II) circles
(()II) - (2),1x1 single circle
---
((I)) - 4 points in a row
(()I) - (2),2 row of points

(((II)I)I) - 3-Torus / (((xy)z)w) / (((r1)r2)rm)
------------------------------------------------
(((I)I)I) - 2x1x1 row of ((II)I) tori
(((II))I) - [r1-2] concentric pair of ((II)I) tori
(((II)I)) - [rm-2] conc pair of ((II)I) tori
---
(((II))) - [r-4] conc pair of (II) circles
(((I)I)) - 2x1.[r-2] of (II) circles
(((I))I) - 4x1 row of (II) circles
((()I)I) - (2),2x1 row of (II) circles
---
(((I))) - 8 points in a row
((()I)) - (2),4 row of points
((())I) - (4),2 row of points

((II)(II)) - Tiger / ((xy)(zw)) / ((r1a)(r1b)rm)
-------------------------------------------------
((II)(I)) - 1x1x2 column of 2 ((II)I) tori
---
((I)(I)) - 2x2 square array of 4 (II) circles
((II)()) - (2),[r-2] conc pair of (II) circles
---
((I)()) - (2),4 row of points
in search of combinatoric objects of finite extent
ICN5D
Tetronian

Posts: 1004
Joined: Mon Jul 28, 2008 4:25 am
Location: Orlando, FL