Learning Toratopic Notation and Cut Algorithm

Discussion of shapes with curves and holes in various dimensions.

Learning Toratopic Notation and Cut Algorithm

Postby ICN5D » Wed Jul 16, 2014 6:53 pm

This is the place to ask questions, and answer them, in regards to toratopic notation. Rather than perusing the hundreds of posts in " Tiger Explained ", you can just come here. It may be beneficial to split and rearrange those earlier posts into this thread ( to the admins/moderators who can do such a thing! Hint, hint .... )
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Re: Learning Toratopic Notation and Cut Algorithm

Postby wendy » Thu Jul 17, 2014 4:21 am

You would need to give a list of posts to be copied/moved here.
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Re: Learning Toratopic Notation and Cut Algorithm

Postby ICN5D » Thu Jul 17, 2014 5:32 pm

Noted. Thanks, wendy!
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Re: Learning Toratopic Notation and Cut Algorithm

Postby Marek14 » Wed Jul 23, 2014 7:05 am

OK, so here are some files: cuts for 7D and toratope lists for 8, 9 and newly also 10D! (Some corrections in 9D are new.)
Attachments
7d.txt
(279.83 KiB) Downloaded 63 times
8d.txt
(11.3 KiB) Downloaded 52 times
9d.txt
(37.9 KiB) Downloaded 50 times
10d.txt
(129.43 KiB) Downloaded 50 times
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Re: Learning Toratopic Notation and Cut Algorithm

Postby wendy » Wed Jul 23, 2014 7:58 am

I thought Philip Icn5d's aim here was a tutorial of the notation and notions behind toratopes.
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Re: Learning Toratopic Notation and Cut Algorithm

Postby Marek14 » Wed Jul 23, 2014 8:16 am

wendy wrote:I thought Philip In5cd's aim here was a tutorial of the notation and notions behind toratopes.


Yes, and these files can work as exercises :)
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Re: Learning Toratopic Notation and Cut Algorithm

Postby Marek14 » Wed Jul 23, 2014 9:03 am

So, the basics:

The toratopic notation is, in its core, recursive notation based on parentheses. Each pair of parentheses can contain symbols "I" and/or other pairs of parentheses. It can also be empty.

So, all these are valid:

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

We can separate these expressions in three groups:

EMPTY sets are expressions which contain the "eye" -- an empty parentheses string ().
Groups or HERDS of toratopes are expressions that are not empty but that contain a pair of parentheses containing one other object. These depict multiple copies of a toratope at once, the total number is two to the power of the number of these "singleton parentheses pairs".
Single toratopes are expressions where every pair parentheses contains at least two other objects, whether "I" symbols or other parentheses.

Explanations:

The number of "I" symbols is the dimension of the toratope.
(I) represents two points (a 1D circle).
(II...I) represents an n-dimensional sphere.

More complex parentheses are read as follows:
1. Evaluate all inner parentheses.
2. Create a cartesian product of all of those. If there's only 1 pair of inner parentheses, it stays unchanged here.
3. Place this shape into a space with as many extra dimensions as there are extra "I" symbols in the parentheses. If there are none, don't add any extra dimensions.
4. The parentheses evaluate to a set of all points with a given distance from toratope or cartesian product of toratopes obtained in step 2.

Examples:
((II)I) - circle placed in 3D (one extra dimension) and "inflated" into torus.
((II)II) - the same circle, but now placed in 4D. The inflation results in a spheritorus.
((II)(II)) - cartesian product of two circles in 4D. It's inflated just in 4D since no extra dimensions are added. This is what we call a "tiger".
((II)(II)II) - now the same 4D cartesian product of two circles is inflated in a 6D space. This results in a "222-tiger".

((I)(I)) - this is a cartesian product of two (I) (pairs of points), inflated in 2D. The product itself is four points in vertices of rectangle; the inflated products produces four circles. ((I)(I)I) would inflate in 3D, resulting in 2x2 array of spheres.

(((II)I)(II)) - a more complex 5D toratope called "torus tiger". It's a 5D cartesian product of torus ((II)I) and circle (II) inflated while staying in 5D.

((II)(II)(II)) - this is a "triger", 6D toratope based on a cartesian product of 3 circles.

Types of dimensions and types of diameters:
Each "I" symbol marks one dimensions. "I" symbols in same parentheses mark the same type of dimension. Each pair of parentheses marks an unique diameter of the shape.
In torus ((II)I), there are two major dimensions and one minor dimension. Torus also has a major diameter (inner parentheses) and a minor diameter (outer parentheses). The precise nomenclature for dimensions and diameters can be found from the 7D file.

Implicit equations:
A pair of parentheses can be converted to an expression like this:
(XYZII), where X,Y,Z are parentheses expressions: sqrt(expr(X)^2 + expr(Y)^2 + expr(Z)^2 + d1^2 + d2^2) - r.
d1 and d2 are coordinates corresponding to the two "I" symbols in the expression and r is the radius corresponding to this pair of parentheses. When you evaluate the outer parentheses, you can just put "= 0" after it, but it's more traditional to modify the top level like this:
(XYZII) -> sqrt(expr(X)^2 + expr(Y)^2 + expr(Z)^2 + d1^2 + d2^2) - r = 0 ->expr(X)^2 + expr(Y)^2 + expr(Z)^2 + d1^2 + d2^2 = r^2.

Rotations:
Higher-dimensional toratopes can be created by rotation of lower-dimensional ones. There are two possibilities:

1. A bisecting rotation through a particular dimension is achieved by replacing an "I" symbol with string "II".
2. A non-bisecting rotation through a particular dimension (with axis not intersecting with the shape) is achieved by replacing an "I" symbol with string "(II)".

Let's take a torus ((II)I) as an example. There are four possible rotations leading to four distinct 4D toratopes:

Bisecting rotation through major dimension: ((III)I) - torisphere
Bisecting rotation through minor dimension: ((II)II) - spheritorus
Nonbisecting rotation through major dimension: (((II)I)I) - ditorus
Nonbisecting rotation through minor dimension: ((II)(II)) - tiger

Cuts:
A cut of toratope through a particular dimension is achieved simply by erasing a particular "I" symbol.

From ditorus (((II)I)I), possible cuts are:
Major cut: (((I)I)I) - two toruses
Medium cut: (((II))I) - major pair of toruses
Minor cut: (((II)I)) - minor pair of toruses

Trace:
A trace is the lowest-dimensional nonempty cut. The dimension of trace is equal to the number of "innermost" pairs of parentheses.
Torus ((II)I) has trace ((I)) - four points on a line.
Torisphere ((III)I) and spheritorus ((II)II) both have the same trace.
The trace of ditorus (((II)I)I) is (((I))) - eight points on a line.
The trace of tiger ((II)(II)) is ((I)(I)) - 2x2 array of circles.
The trace of torus tiger (((II)I)(II)) is (((I))(I)) - 4x2 array of circles.
The trace of triger ((II)(II)(II)) is ((I)(I)(I)) - 2x2x2 array of spheres.

A set of toratopes with the same trace is called a SPECIES. The species always has one unique lowest-dimensional member which will be separated in two toratopes by any coordinate cut. This is the base member; behaviour of whole species can be usually demonstrated on it.

A set of toratopes whose trace is composed of the same type of figures is a GENUS. Torus and ditorus both belong to the same genus. Tiger belongs to same genus as torus tiger or tiger torus (((II)(II))I).

A set of toratopes whose trace is composed of the same species of figures is an ORDER. That's the highest level we usually work with. Tiger and triger are different genuses as tiger's trace is composed of circles and triger's of spheres, but circle and sphere are same species and so tiger and triger are the same order. Two toratopes of the same order have the same structure of parentheses branching.
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Re: Learning Toratopic Notation and Cut Algorithm

Postby ICN5D » Wed Jul 23, 2014 6:46 pm

Introducing Toratopic Notation to the New Reader


The capitalized " I " is a dimension marker

- The amount of 'I' is how many dimensions the shape has


The parentheses " () " is a distinct diameter, that divides up the dimensions into distinct parts

- A () makes the shape round in some way
- The number of '()' is how many diameters the shape has

• For Open Toratopes:
- n-cubes don't have any at all, because they aren't round, i.e. I , II , III , IIII

- For Open-T rounded prisms:
-- a cylinder (II)I has both with 2 flat sides and 1 rounded side
-- a spherinder (III)I also has 2 flat, 1 round
-- a duocylinder (II)(II) has 2 rounded and no flat
-- a triocylinder (II)(II)(II) has 3 rounded and no flat
-- a cubinder (II)II has 4 flat and 1 round
-- a tesserinder (II)III has 6 flat and 1 round
-- a duocylinder diprism (II)(II)II has 4 flat and 2 round


• For Closed Toratopes:
- N-spheres have just one pair, around all dims, i.e. (II) , (III) , (IIII) , (IIIII)
- Toroids ( donut-shapes): these have two or more, making different size diameters, i.e. ((II)I) , ((III)I) , ((II)II) , (((II)I)I) , ((II)(II))
- Totatopes with Multiple Inner () : these have multiple equal-sized diameters, i.e. ((II)(II)) , (((II)(II))(II)) , ((III)(II)) , (((II)I)(II)) , (((II)I)((II)I))


To make a cross section down to N-1 dimensions, we simply remove only the " I " markers

- 3D ((II)I) cuts to ((I)I) or ((II)) or, 1D ((I))
- 3D (III) cuts to (II) or (I)
- 4D (((II)I)I) cuts to (((I)I)I) or (((II))I) or (((II)I)) in 3D ,


For Base-Species Toratopes, when cut into cross sections, the notation has extra parentheses around the dimension markers, i.e. (((I))) , (((I))I) , ((((I))((I))) , etc

- These extras represent a particular number and arrangement of a lower-D toratope
- If no extras are left, it is from a cut diameter that is higher-D than a 1-sphere/circle, " non-base-species" i.e. ((III)I) , ((II)II) , ((IIII)III) , ((III)(III)) , (((III)II)(III))
- There are as many Lower Dimensional Toratopes from cuts as there are total number in that dimension
-- For 2D : only cuts to circles (II)
-- For 3D : only cuts to spheres (III) and tori ((II)I)
-- For 4D : only cuts to glomes (IIII) , spheritori ((II)II) , torispheres ((III)I) , ditori (((II)I)I) , and tigers ((II)(II))
-- For 5D : only cuts to any of the 12
-- For 6D : only cuts to any of the 33
-- For 7D : only cuts to any of the 90
etc


There are as many ways to arrange a toratope as there are dimensions and diameters of it

• For every extra () pair around a single dimension marker, there are 2n shapes along that dimension
• For every extra () pair around a diameter, there are 2n shapes paired concentrically in that diameter

A circle (II) has TWO dimensions and ONE diameter, orientation (XY)
- It can be stacked in two directions, paired concentrically in one diameter

• Stacking a (II) :
((I)I) is 2 along X
(((I))I) is 4 along X
((((I)))I) is 8 along X
(((((I))))I) is 16 along X
(I(I)) is 2 along Y
(I((I))) is 4 along Y
(I(((I)))) is 8 along Y
(I((((I))))) is 16 along Y
((I)(I)) is a 2x2 array in XY plane
(((I))(I)) is 4x2 array in XY plane
(((I))((I))) is 4x4 array in XY
((((I)))((I))) is 8x4 array in XY

• Concentric Pairing a (II) :
((II)) is 2 pair
(((II))) is 4 concentric , a quartet
((((II)))) is 8 concentric, and octet
(((((II))))) is 16 concentric , a 16-plet
((((((((II)))))))) is 32 concentric, a 32-plet

• Stacking and Pairing a (II)
(((I)I)) is 2 along X, paired by 2, 4 total
((((I)I))) is 2 quartets along X, 8 total
(((I)(I))) is 2x2 array of pairs, 8 total
(((I)(I))) is 2x2 array of quartets, 16 total
((((I))((I)))) is 4x4 array of quartets, 64 total
(((((((I)))((I)))))) is 8x4 array of octets, 256 total

more to come on 3D toratope arrangements




Wow, that 10D list is insane Marek!!! I haven't got around to translating 9D yet, into bundling terms. Looks awesome!
Last edited by ICN5D on Thu Jul 24, 2014 12:45 am, edited 1 time in total.
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Re: Learning Toratopic Notation and Cut Algorithm

Postby Marek14 » Wed Jul 23, 2014 6:58 pm

Thanks :)
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Re: Learning Toratopic Notation and Cut Algorithm

Postby PWrong » Thu Jul 24, 2014 3:16 pm

I've probably missed something, but I'm not sure I follow this notation when the extra brackets are involved. Is ((II)) a new thing? What does it mean exactly?
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Re: Learning Toratopic Notation and Cut Algorithm

Postby ICN5D » Thu Jul 24, 2014 4:08 pm

Yes, I guess it's a new thing. Using the notation, Marek developed a way to interpret cross sections, by removing the markers out of it, and translating what all the extra parentheses mean. It also holds up when deriving the implicit functions for the shapes, by using an adjustable parameter in place of the removed dimension.

So, for a torus ((II)I) , we can make 2D slices of it by removing a single dimension, leaving two behind : ((I)I) and ((II)). Where ((I)I) is the two circles in a row, and ((II)) is two concentric circles, as the two distinct axial midsections of a torus. For a sphere (III) , making any cut to 2D gives a circle (II).

And, when getting into some more high-D toratopes, like (((II)I)I) , there are three ways to slice it into 3D, which can be compiled into cross section tables:

4D Hyperplane
(((II)I)I) - 1x ditorus
---------------------------------
3D Hyperplane
(((I)I)I) - 2x tori along a line
(((II))I) - 2x tori as concentric major pair
(((II)I)) - 2x tori as concentric minor pair
------------------------------------------------
2D Hyperplane
((()I)I) - empty, moving out from center makes sudden appearance of circle divide to 2x circles in a row, then merge and vanish
(((I))I) - 4x circles in a row
(((I)I)) - 4x circles as concentric pairs, 2 in a row
(((II))) - 4x concentric circles
-------------------------------------------------
1D Hyperplane
(((I))) - 8x points in a row


Since there are two distinct diameters for a torus, there are two ways to be concentric. Labelled by its diameter in the hierarchy ((major)minor) , if we see (((II))I) , it's a "major pair" . For (((II)I)) it's a "minor pair".


And, since a torus has three dimensions, there are three directions to stack along a row or array.

((II)I) - torus, ((XY)Z)
--------------------
(((I)I)I) - 2 tori along X
((((I))I)I) - 4 along X
(((((I)))I)I) - 8 along X

((I(I))I) - 2 along Y , which is ambiguous with along X, but just to illustrate the concept
((I((I)))I) - 4 along Y
((I(((I))))I) - 8 along Y

((II)(I)) - 2 along Z , making the vertical column of 2 tori, a 3D cut of tiger ((II)(II))
((II)((I))) - 4 along Z
((II)(((I)))) - 8 along Z

For arrays:

(((I)(I))I) - 2x2 array in XY plane of 4 tori , as 3D cut of (((II)(II))I) , circle bundle over tiger, OR torus bundle over duoring ( Clifford 2-torus ) : T^2 x C^2
((((I))((I)))I) - 4x4 array of 16 tori in XY plane , as 3D cut of (((II)I)((II)I))I) , T^2 x C^2 x C^2
(((I)I)(I)) - 2x1x2 vertical square in XZ plane of 4 tori , 3D cut of (((II)I)(II)) , a torus of a tiger, S^1 x C^2 x S^1
((((I))I)((I))) - 4x1x4 vertical square in XZ of 16 tori , 3D cut of ((((II)I)I)((II)I)) , S^1 x C^2 x C^2 x S^1
(((I)(I))(I)) - 2x2x2 cube array in XYZ of 8 tori , 3D cut of (((II)(II))(II)) , T^2 x C^3 , 2-torus bundle over trioring , C^3 is the edge of (II)(II)(II) , (circle^3)-prism
((((I))(I))(((I)))) - 4x2x8 rectangular array of 64 tori, 3D cut of ((((II)I)(II))(((II)I)I)) , T^2 x C^3 x C^2 x S^1

and such and so forth

The diameter paring can be combined with large arrays which is a 3D cut of some very high-D toratope:

(((((((((I)))((I)))))((I))))) - 2,048 tori as major/minor quartets stacked in an 8x4x4 array , 3D cut of (((((((((II)I)I)((II)I))I)I)((II)I))I)I)

It took me a solid 3-4 months to get it, enough to nail the concept down. I learned this algorithm before making renders and animations. It becomes key when seeing something strange in the plotter, and parsing the cuts to figure out what's happening in 4 or 5D. Otherwise, I wouldn't have a clue when exploring a 7D toratope!
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