The Tiger Explained

[Marek14]

The thing is that torisphere, from its torus cut, never becomes a sphere. The central hole becomes filled, then it becomes a blob with dimples, then a convex blob, and finally shrinks into a point -- but there's never a point where it would be a sphere, mathematically speaking.

[ICN5D]

Okay, that makes sense. I guess it would be two cocircular toruses as the final shape before vanishing. When the torisphere collapses into a point, it would be a cartesian product with a torus and point, with extra brackets around making the cocircular arrangement.

(((II)I)((II)I)) --> (((IIi)I)((II)I))

Moving along axis will collapse spherical cut symmetry to point
(*((II)I)) = (((II)I))

Time to go on another bike ride, it's amazingly beautiful outside. BTW, I read the cut array post of yours in the " Pictures of Toratopes" thread. It all makes sense, and I like how you detailed the cut arrays in general terms.

[Marek14]

I thought about it today and realized that non-empty cuts of a certain species can be described in finite amount of terms. For example any non-empty cut of a tiger (()()) is either a tiger, or two toruses stacked vertically, or four spheres in vertices of rectangle.

Exercise: How would you describe the ((II)(I)(I)) cut of the triger?

[ICN5D]

Exercise: How would you describe the ((II)(I)(I)) cut of the triger?

Hmm. That's a good one . Actually, that's a really good one.

Well, to start off, ((II)(I)) is a vert column of 2 toruses. Then, we add a dimension, and a hollow circle cut as 2 points. I know that ((I)(I)(I)) is a cut as 8 spheres at verts of cube. So, ((II)(I)(I)) is connecting the 8 spheres through a higher dimension, joining them into 4 shapes.


(II) - circle

(II)(I) - mult by hollow circle (2 points) adding dimension, makes 2 circles displaced in vertical column along 3rd dim

(II)(I)(I) - mult by another two points along another dimension, makes 4 circles as a vert column of two vert columns of 2, displaced along the 3rd and 4th dimensions

((II)(I)(I)) - spherating turns all circles into 4 toruses, as a vert column of two vert columns of 2, displaced along the 3rd and 4th dimensions

So, we have 4 toruses in vert of square, but this square is orthogonal, not restricted to 3D. I think the true arrangement is two vertical columns in a vertical column of 2, displaced along 3 and 4D respectively.

[ICN5D]

Well, I double checked my description with one of your previous posts, and you say it's 22-toruses in vertices of a square. But, my build up sequence tells me it's four toruses displaced along 3 and 4D in vertical column arrangements. Hmm, not sure what to make of it.

Though, it does make sense from the standpoint of relating ((II)II) to ((II)(I)(I)), in how it retains the 22-torus and puts them in the vertices of a square (I)(I). But, the hollow circle makes two points, displaced in higher dimensions. I must have missed a step in the spheration process, being too focused on the 4 circles and their displaced nature. It just seems like only the solid circles will be spherated, not the hollow circles.

(I)(I) - four points
((I)(I)) - four circles

(I)(I)(I) - 8 points
((I)(I)(I)) - 8 spheres

((II)II) - 22-torus
((II)I(I)) - vert column of 2 spheritoruses
((II)(I)(I)) - quartet of spheritoruses in vertices of square
(II)(I)(I) - four circles in vert column of 2 vert columns of 2

I guess the spheration adds dimensions to the shapes that are missing from displacement! I think I learned something new, maybe just relearned!

Tracing back my steps, I identified the ((I)(I)(I)) as 8 spheres that get connected by ((II)(I)(I)) through a higher dimension into 4 shapes. It now makes a lot more sense when I use this logic in the connection process. The 8 spheres in vertices of a cube become joined into 4 spheritoruses ((II)II) by undoing the cut. That agrees perfectly with the cut.

[Marek14]

The simple reason that ((II)(I)(I)) can't be toruses is that it has four I's -- so it's a four-dimensional object.

That's a general property of extended toratopic notations -- any object can be only displaced along its own dimension -- if it would be displaced in a dimension it doesn't have, it has to turn into some higher-dimensional object.

Describing ((II)(I)(I)) in terms of stacks is possible, but I'd say it's four spheritoruses in vertices of "vertical rectangle". If we call the major dimension of torus "horizontal" and the minor "vertical" then spheritorus has 2 horizontal and 2 vertical directions. Both (I)'s are equivalent.

So, let's have a look at possible arrangement of toruses (x will mean arbitrary amount of I's, though always enough to make the total number of terms in its parenthesis at least 2 - so in (x) it's at least 2, in (()x) at least 1 and in (()()x) it can be zero).

A single torus ((x)y), appears as cut of toruses

Two toruses:
Horizontally displaced (((I)x)y), cut of ditoruses
Vertically displaced ((x)(I)y), cut of tigers
A pair differing in major diameter (((x))y), cut of ditoruses
A pair differing in minor diameter (((x)y)), cut of ditoruses

Four toruses:
Four in horizontal line ((((I))x)y), cut of tritoruses
Four in horizontal rectangle (((I)(I)x)y), cut of tiger toruses
Four in horizontal/vertical rectangle (((I)x)(I)y), cut of torus tigers
Two horizontally displaced pairs differing in major diameter ((((I)x))y), cut of tritoruses
Two horizontally displaced pairs differing in minor diameter ((((I)x)y)), cut of tritoruses
Four in vertical line ((x)((I))y), cut of torus tigers
Four in vertical rectangle ((x)(I)(I)y), cut of trigers
Two vertically displaced pairs differing in major diameter (((x))(I)y), cut of torus tigers
Two vertically displaced pairs differing in minor diameter (((x)(I)y)), cut of tiger toruses
A quartet differing in major diameter ((((x)))y), cut of tritoruses
A quartet differing in major and minor diameter ((((x))y)), cut of tritoruses
A quartet differing in minor diameter ((((x)y))), cut of tritoruses

Eight toruses:
Eight in horizontal line (((((I)))x)y), cut of tetratoruses
Eight in horizontal 4x2 rectangle ((((I))(I)x)y), cut of torus tiger toruses
Eight in horizontal/vertical 4x2 rectangle ((((I))x)(I)y), cut of ditorus tigers
Four pairs differing in major diameter in horizontal line (((((I))x))y), cut of tetratoruses
Four pairs differing in minor diameter in horizontal line (((((I))x)y)), cut of tetratoruses
Eight in vertices of horizontal cuboid (((I)(I)(I)x)y), cut of triger toruses
Eight in vertices of horizontal/horizontal/vertical cuboid (((I)(I)x)(I)y), cut of double tigers
Four pairs differing in major diameter in vertices of horizontal rectangle ((((I)(I)x))y), cut of tiger ditoruses
Four pairs differing in minor diameter in vertices of horizontal rectangle ((((I)(I)x)y)), cut of tiger ditoruses
Eight in horizontal/vertical 2x4 rectangle (((I)x)((I))y), cut of duotorus tigers
Eight in vertices of horizontal/vertical/vertical cuboid (((I)x)(I)(I)y), cut of torus trigers
Four pairs differing in major diameter in vertices of horizontal/vertical rectangle ((((I)x))(I)y), cut of ditorus tigers
Four pairs differing in minor diameter in vertices of horizontal/vertical rectangle ((((I)x)(I)y)), cut of torus tiger toruses
Two horizontally displaced quartets differing in major diameter (((((I)x)))y), cut of tetratoruses
Two horizontally displaced quartets differing in major and minor diameter (((((I)x))y)), cut of tetratoruses
Two horizontally displaced quartets differing in minor diameter (((((I)x)y))), cut of tetratoruses
Eight in vertical line ((x)(((I)))y), cut of ditorus tigers
Eight in vertical 4x2 rectangle ((x)((I))(I)y), cut of torus trigers
Four pairs differing in major diameter in vertical line (((x))((I))y), cut of duotorus tigers
Four pairs differing in minor diameter in vertical line (((x)((I))y)), cut of torus tiger toruses
Eight in vertices of vertical cuboid ((x)(I)(I)(I)y), cut of tetrigers
Four pairs differing in major diameter in vertices of vertical rectangle (((x))(I)(I)y), cut of torus trigers
Four pairs differing in minor diameter in vertices of vertical rectangle (((x)(I)(I)y)), cut of triger toruses
Two vertically displaced quartets differing in major diameter ((((x)))(I)y), cut of ditorus tigers
Two vertically displaced quartets differing in major and minor diameter ((((x))(I)y)), cut of torus tiger toruses
Two vertically displaced quartets differing in minor diameter (((x)(I)y)), cut of tiger ditoruses
An octet differing in major diameters (((((x))))y), cut of tetratoruses
An octet differing in major and minor diameters (4,2) (((((x)))y)), cut of tetratoruses
An octet differing in major and minor diameters (2,4) (((((x))y))), cut of tetratoruses
An octet differing in minor diameters (((((x)y)))), cut of tetratoruses

Homework:
Find configurations of spheres up to 16 and configurations of ditoruses/tigers up to four.

[ICN5D]

Homework:
Find configurations of spheres up to 16 and configurations of ditoruses/tigers up to four.

(III) - sphere
-----------------------------------------------------------------------
(((((III))))) - 16 concentric spheres
(((((I)II)))) - 2 groups of 8 concentric spheres along line
(((((I))II))) - 4 groups of 4 concentric spheres along line
(((((I)))II)) - 8 groups of 2 concentric spheres along line
(((((I))))II) - 16 spheres along line
------------------------------------------------------------------------
((((I)(I)I))) - 4 groups of 4 concentric spheres in square
((((I))(I)I)) - 4x2 array of 2 concentric spheres, 4 along line stacked 2 high
(((I))((I))I) - 4x4 array of spheres, 4 along line stacked 4 high
((((I)))(I)I) - 8x2 array of spheres, 8 along line stacked 2 high
------------------------------------------------------------------------------
(((I)(I)(I))) - 8 groups of 2 concentric spheres in cube



(((II)I)I) - ditorus
--------------------------------------------------------------------------------------------------
(((((II)I)I))) - 4 cocircular ditoruses
(((((I))I)I)I) - 4 ditoruses displ along line
(((II)((I)))I) - 4 ditoruses stacked along middle diameter
(((II)I)((I))) - 4 ditoruses in vertical stack
(((((II)))I)I) - 4 concentric ditoruses in 2x middle1 and 2x middle2 diameter
(((((II)I)))I) - 4 concentric ditoruses in 2x middle2 and 2x middle3 diameter
-----------------------------------------------------------------------------------------------------
(((((I)I)I)I)) - 2 groups of 2 cocircular ditoruses along line
(((((II)I))I)) - 2 concentric groups of 2 cocircular ditoruses, stacked along middle dimension
((((II)I)(I))) - 2 groups of 2 cocircular ditoruses in vertical stack
((((II)I))(I)) - 2 groups of 2 concentric ditoruses in vertical stack
((((II)(I)))I) - 2 groups of 2 concentric ditoruses stacked in middle dimension
------------------------------------------------------------------------------------------------------
((((I)(I))I)I) - 4 ditoruses in square
((((I)I)(I))I) - 4 ditoruses in square, displ along major and middle diameter
(((II)(I))(I)) - 4 ditoruses in square, stacked along middle and minor dimension
((((I)I)I)(I)) - 4 ditoruses in square, stacked along major and minor dimension

^^^ yeah, those are tough ^^^


((II)(II)) - tiger
--------------------------------------------------------------------------------------------------
((((II)(II)))) - 4 cocircular tigers, displ in minor diameter
((((II))(II))) - 4 tigers, 2 concentric in major1 groups of 2 cocircular
(((II)((II)))) - 4 tigers, 2 concentric in major2 groups of 2 cocircular
(((II))((II))) - 4 tigers, 2 concentric in major1 and 2 concentric in major2 diameter
((((II)))(II)) - 4 tigers concentric in major1 diameter
((II)(((II)))) - 4 tigers concentric in major2 diameter
---------------------------------------------------------------------------------------------------
((((I)I)(II))) - 2 displaced groups of 2 cocircular tigers, identical to (((II)((I)I)))
((((I)I))(II)) - 2 displaced groups of 2 concentric in one major diameter, identical to ((II)(((I)I)))
(((I)I)((II))) - 2 groups of 2 concentric in one major diameter along line, identical to (((II))((I)I))
----------------------------------------------------------------------------------------------------
(((I)I)((I)I)) - 4 tigers in 2x2 array of vertical square
(((I)(I))(II)) - 4 tigers in 2x2 array of flat square, identical to ((II)((I)(I)))

Ta Da! That's all I can think of right now. I'll probably dream up some more tonight.

[ICN5D]

Playing with some naming schemes, and classifying them by tigroid-along-rim-of-tigroid reduction. This is the right step in visualizing these wild things, knowing their embedded-rim tigroid parts. Once you know how to cut the main tigroid frame, the embedded tigroid is reflected in place of its cuts.


(((III)(III))(II)) - 33020-tiger , spheritorispherintigroidal tiger , 330-tiger along rim of 220-tiger

(((II)(II))(III)) - 22030-tiger , tigroidal cylspherintigroid , 220-tiger along rim of 320-tiger

(((III)(II))(III)) - 32030-tiger , cylspherintigroidal cylspherintigroid , 320-tiger along rim of 320-tiger

(((II)(II))((II)I)) - 220210-tiger , tigroidal cyltorintigroid , 220-tiger along rim of 2120-cyltorintigroid

(((II)I)(II))((II)I)) - 2120210-tiger , cyltorintigroidal cyltorintigroid , 2120-tigroid along rim of 2120-tigroid


I guess those are some of the coolest in 7 and 8D, without going into 9D



Well, how about a few 9D ones:

((((II)I)((II)I))((II)I)) - 21210210-tiger , duotorus-tigroidal cyltorintigroid , 21210-tiger along rim of 2120-tiger

(((II)(II)(II))((II)I)) - 2220210-tiger , tritigroidal (torus*sphere)-tigroid , 2220-tiger along rim of 2130-tiger


((((II)I)(II))((II)(II))) - 21202200-tiger , cyltorintigroidal double-tiger tigroid , 2120-tiger along rim of 22020-tiger

^^^this one is amazing^^^ , crazy ridiculous complex and beautiful



I have noticed that the (torus*sphere) is related to the (spheritorus*circle), in that they are made from different bisecting rotations of either starting shape of cyltorinder or spheritorinder. I'm inclined to call (torus*sphere) a cylspheritorinder, but it can also be (spheritorus*circle). Open and closed toratopes seem to have too many equal names when getting into higher dimensions. The numerical toratope system overcomes this, but I LOVE worded names, and always will.

I guess " ICN5D " is a bit of a limiting misnomer now. It's more like " ICN9D ", or at least getting there!

[Marek14]

Not sure whether you missed some of the 4 ditoruses / 4 tigers configurations or not. Depends on whether you consider "ditorus" as a shape or as a whole class of shapes. That's why my torus discussion used x,y, to signify that we are talking about arbitrary dimension.

If you consider more-dimensional shapes, there are 2 more ditorus configurations:
(((x)(I)(I)y)z) - four ditoruses in rectangle in middle dimensions
(((x)y)(I)(I)z) - four ditoruses in rectangle in vertical dimensions

And 5 more tiger configurations:
((((I)x)(y)(I)z) - four tigers in major/minor rectangle
(((x)(y)((I))z) - four tigers in line in minor dimension
(((x)(y)(I)(I)z) - four tigers in rectangle in minor dimensions
((((x))(y)(I)z) - two pairs of concentric tigers differing in a major diameter displaced in minor dimension
((((x)(y)(I)z)) - two pairs of concentric tigers differing in the minor diameter displaced in minor dimension

[ICN5D]

According to the arbitrary shape notation, yes I missed a few. I wasn't sure if that's what you were referring to. But, still a good exercise with cut interpretation. That generalized sequence is a good one, since it reduces all of the " I " down into simpler to read terms. I keep feeling that all of the cuts should be compiled into a library. Using the numerical and bracketed notations, and the neato cut formulas, the algorithm should be easier to digest and figure out what it all means. And, if it's too cryptic, then consult the library list. Going by the number sequence condenses it down, and makes it easier for searching. It's very simple to convert from numerical to bracketed, which is quite useful.

[Marek14]

According to the arbitrary shape notation, yes I missed a few. I wasn't sure if that's what you were referring to. But, still a good exercise with cut interpretation. That generalized sequence is a good one, since it reduces all of the " I " down into simpler to read terms. I keep feeling that all of the cuts should be compiled into a library. Using the numerical and bracketed notations, and the neato cut formulas, the algorithm should be easier to digest and figure out what it all means. And, if it's too cryptic, then consult the library list. Going by the number sequence condenses it down, and makes it easier for searching. It's very simple to convert from numerical to bracketed, which is quite useful.

Basically, the cuts can be enumerated in reverse, based on the species they specify.

For example, a triger is generically ((x)(y)(z)w). It has doublet cuts of form ((x)(y)(I)z) (two tigers displaced in a minor dimension), quartet cuts of form ((x)(I)(I)y) (four toruses in vertices of a vertical rectangle) and an octet cut ((I)(I)(I)x) (eight spheres in vertices of a cube).
Of course, a triger is highly symmetrical. Something like a tritorus ((((x)y)z)w) has much more possible cuts of this time, including a cut ((((I)))) which is just sixteen points.

[ICN5D]

Okay, now I see the species thing you have been referring to. So, any shape that has that generic layout is related in its cuts to that original sequence. Got it.

You know, the more I explore some of the basic properties of those 7, 8, 9D tigroids, the more it's revealing itself to me. Suddenly, the duotorus tiger is not so complicated anymore. At first glance, it's forbiddingly foggy, with no discernible features. But, after re-reading your cut breakdown of it, I understand it all. That's a major conceptual accomplishment for me! I didn't realize it, but it's only been two months of learning your cut algorithm. As you have seen, I'm not afraid of high-D stuff. In fact, I'm ever more curious about them. They are amazing.

What do you think about the tigroid-along-rim breakdown of them? It sure simplifies it down to familiar terms, after exploring 5 and 6D, of course.

This also makes me want to render them. Now, I'll know what I'm seeing, or at least, I'll know if it's correct. I have zero skill with rendering programs, but I'd like to learn it. Closed toratopes are much simpler, so it's easier to go into higher dimensions. Comprehending of the structure of a cylconinder was a little tougher than making cut arrays of 5D tigroids, in my opinion. But, that may be because of my shape building algorithm, and its ability to derive all n-cells of their composition. I'm about to compile the list of 6D shapes, and compute their polynomial equations. But, before I do that, I need to provide a thorough explanation of how it works and how to read an expression. 5D feels elementary, now that I've enumerated all of those shapes.

[ICN5D]

Found out who made that cool tiger youtube video from one of your posts. It was made by username Mrrl in a custom made 4D viewer program.

[Marek14]

Hm, sounds like your tigroid along a rim basically means starting with a x20-tiger and replacing the x sphere with another tigroid shape.

Thinking about it, in 8D you have a fun subclass of shapes whose all I's appear as part of (II):

((II)(II)(II)(II)) - tetriger
(((II)(II))(II)(II)) - tiger triger
(((II)(II)(II))(II)) - triger tiger
((((II)(II))(II))(II)) - tiger tiger tiger, or triple tiger
(((II)(II))((II)(II))) - duotiger tiger

In other words, exactly the 5 4D toratopes with their I's replaced with (II).

EDIT: Or how about this 9D thing I call "torus squared"?
((((II)I)((II)I))((II)I))

[ICN5D]

EDIT: Or how about this 9D thing I call "torus squared"?
((((II)I)((II)I))((II)I))

Oh, wow! I didn't think about that one! I wasn't sure if the torus could have extra () around it, but, you're right, it can! It's in place of every (II) in a triger. I guess one of its 6D cuts is 6 concentric trigers that differ in all major diameters. But, now that I think about it, wouldn't it be torus cubed?

I wanted to explore the tiger along rim features after you asked about the tiger along rim of tiger in the double tiger. So, I guess (((III)(II))(III)) could be called double cylspherintigroid, and (((II)I)(II))((II)I)) can be called the double cyltorintigroid.

EDIT: Well, I didn't notice the additional () around the first two ((II)I). So, it would be torus squared as you said. The first two toruses are in a (II) layout, the last torus is the final " I " in ((II)I), so it starts with ((II)I) , and replace each " I " with a whole torus, making:

((II)I) - 21-torus
to
((((II)I)((II)I))((II)I)) - 21210210-tiger , duotorus tiger along rim of cyltorintigroid , or, 21210 along rim of 2120

Then in this case, the 6D cut ((((II))((II)))((II))) is 6 concentric double tigers differing in all major diameters.

I initially was thinking of (((II)I)((II)I)((II)I)), the triotorus tiger, or 2121210-tiger, which would make the 6D cut of 6 concentric trigers differing in all major diameters as (((II))((II))((II))).

[Marek14]

Yup. Every toratope should have its own range of powers

[ICN5D]

Hmm, I've been thinking about that last shape triotorus tiger (((II)I)((II)I)((II)I)), and what to make of its 3D cut of (((I))((I))((I))). It starts with ((I)(I)(I)), which is a cut of a triger, 8 spheres at vertices of cube. Then, at each dimension where 2 spheres lie, it would be turned into 4 along a line. So, this cut: (((I))((I))((I))) would be 4 points along a line to the third power, then inflated into spheres. So, I think it would be a 4x4x4 array of 64 spheres, filling in a cube. Where 8 spheres are at the vertices, 32 are along the edges, 56 are along the square-faces, leaving eight to fill in the center.

[Marek14]

Hmm, I've been thinking about that last shape triotorus tiger (((II)I)((II)I)((II)I)), and what to make of its 3D cut of (((I))((I))((I))). It starts with ((I)(I)(I)), which is a cut of a triger, 8 spheres at vertices of cube. Then, at each dimension where 2 spheres lie, it would be turned into 4 along a line. So, this cut: (((I))((I))((I))) would be 4 points along a line to the third power, then inflated into spheres. So, I think it would be a 4x4x4 array of 64 spheres, filling in a cube. Where 8 spheres are at the vertices, 32 are along the edges, 56 are along the square-faces, leaving eight to fill in the center.

Yup, correct

[wendy]

Marek14 asked about a proper english word for the toratope ((ii)) "annulus", being the space between two concentric circles.

The word is 'annulus'. It occurs in astronomy as well, as an annular eclipse of the sun is where the disk of the moon covers the middle part of the sun, but there is a ring of sun around the sun.

It's a ladin-word 'little ring'.

When i devised a list of words for Secret, i ended up using a construction that permits eg ((iii)) "spherated glomohedrix" = sphere-shell, etc.

[Marek14]

When i devised a list of words for Secret, i ended up using a construction that permits eg ((iii))) "spherated glomohedrix" = sphere-shell, etc.

I presume you meant ((iii))

[ICN5D]

I got it right? Cool, heck yeah! I've also been cutting torus squared in my head, mostly figuring out its only non-empty 3D cut. It's actually quite similar to triotorus tiger, especially since both have only one non-empty in 3D. It always seems like these type of lowest dimensional cuts are the middle-most cut, through the fattest part of the toratope. Both cuts of which are in 4x4x4 cubic arrays.

Thinking back on the triotorus cut, it's interesting to see how almost all of those 64 spheres are along the surface of the cube. Only 8 of them are actually inside the cube, mimicking the 8 vertex-spheres, deeper in. Very curious....




((((II)I)((II)I))((II)I)) - 21210210-tiger , torus squared , 21210 along rim of 2120

So, this means that 21210210 will have a hybridized crosscut of a 21210 in a 2120 cut arrangement.



(((II)I)((II)I)) - 21210, duotorus tiger

3D cuts:
(((II))((I))) - 20100, 8 toruses in 2 concentric stacked 4 high in column

(((I)I)((I))) - 11100, 8 toruses in 2x4 vertical rectangle ( or square? does the height increase, or are there 2 extra in between original 2? )

2D cut:
(((I))((I))) - 10100, 16 circles in 4x4 flat square array

which unspherated would produce:

((I))((I)) - 1010 , 16 points in 4x4 flat square


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


(((II)I)(II)) - 2120, cyltorintigroid

3D cuts:
(((I)I)(I)) - 1110, 4 parallel toruses in a 2x2 vertical square

(((II))(I)) - 2010 , 4 toruses in 2 concentric stacked 2 high in column

(((I))(II)) - 1020 , four toruses in 1x4 column





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




((((II)I)((II)I))((II)I)) - 21210210 , torus squared

6D cut:
((((II))((II)))((II))) - 20200200 , 6 concentric 22020 double tigers, in 2 concentric per 3 major diameters

3D cut:
((((I))((I)))((I))) - 10100100 , 64 parallel toruses in 4x4x4 cubic array




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




((((I))((I)))((I))) - 10100100 , 64 parallel toruses in 4x4x4 cubic array


Exploring the 3D Subcuts:

((((I))((I)))(I)) - 1010010 , 32 parallel toruses in a 4x4x2 tower

((((I))(I))((I))) - 1010100 , 32 parallel toruses in a 4x2x4 tower

(((I)((I)))((I))) - 1100100 , 32 parallel toruses in a 2x4x4 tower

(((I)(I))((I))) - 110100 , 16 toruses in a 2x2x4 vertical tower

(((I)(I))(I)) - 11010 , 8 toruses at vertices of 2x2x2 cube

(((I)(I))I) - 1101 , 4 toruses at vertices of 2x2 flat square

(((I)I)I) - 111 , 2 toruses along a line

((II)I) - 21 , plain ole' torus



To conclude this exploration, I'd like to compare how both of the cuts from 21210 and 2120 interact with each other:


(((I))((I))) - 10100 , 16 circles in 4x4 flat square

and

(((I))(II)) - 1020 , four toruses in 1x4 vertical column


where the (II) in 1020 had its " I " terms replaced with " ((I)) ", to make


((((I))((I)))((I))) - 10100100 , 64 parallel toruses in 4x4x4 cubic array

[Marek14]

Yup, you get the handle of it

[ICN5D]

I'd have to say, I'm really impressed with your cut algorithm. It's very powerful and simple to use. It only took me about 2 1/2 months of time invested be able to go into 9D. It's one of the neatest things I've learned in a while It seems like the really interesting stuff begins to happen in +7D. The duotorus tiger is so very elementary now, after working with even higher shapes. I never expected to be able to see and cut it so clearly! So, what's next? Detail and analyze 9, 10, 11, etc dimensional tigroids? I haven't worked too much on the cut rotation morphs. But, I did visualize them in the 2D of 4D cuts.

[Marek14]

I'd have to say, I'm really impressed with your cut algorithm. It's very powerful and simple to use. It only took me about 2 1/2 months of time invested be able to go into 9D. It's one of the neatest things I've learned in a while It seems like the really interesting stuff begins to happen in +7D. The duotorus tiger is so very elementary now, after working with even higher shapes. I never expected to be able to see and cut it so clearly! So, what's next? Detail and analyze 9, 10, 11, etc dimensional tigroids? I haven't worked too much on the cut rotation morphs. But, I did visualize them in the 2D of 4D cuts.

Well, I was thinking about rooted trees and their relation to toratopes. There are basically two types.

1. Toratopes themselves map to rooted trees where no node has exactly one descendant. I.e. every node is either terminal (corresponding to I) or has at least two descendants (corresponding to pair of parentheses).

2. Species of toratopes map to rooted trees in general. If the root has one descendant, it's a torus-type species based on its descendant, if it has multiple descendants, it's a tigroid based on all the descendants.

Unfortunately, the number of dimensions where a species appears is not strictly given by the number of parenthesis pairs because tigroids can increase the basic amount (all four-P species are in 5D except for triger which requires 6D).

Next step might be this:

We can do various 3D cuts. It should be possible to create a "pilot" program where you see a 3D cut and can freely fly anywhere in the remaining dimensions to see any slice you want. For example, how would the 4x4x4 array of cubes transform in various oblique directions?

[ICN5D]

Analyzing the 4x4x4 Cubic Lattice array of Torus Squared, 10100100, ((((I))((I)))((I)))


The cut evolution of 10100100 is a combination of both cut evolutions of the 21210 and the 2120. The 4x4x4 cube of toruses is a product of two separate arrays. Taken apart, they are a 4x4 flat square of circles ( which reduced further make 16 points, as the actual product array ), multiplied by a 1x4 column of toruses. These two arrays correspond to certain tiger dances in the cube. Each one of the 6 axes in the cut array will belong to either one of the separate arrays:


• The 21210210 is the 21210 along the tigroid rim of 2120, has a 6D cut axis array

• Four cut axes are from 21210, and two cut axes are from 2120

• The cube lattice itself can be reduced into a 2D x 1D array,
- The 21210 will correspond to the 2D flat array of the vertical columns,
- The 2120 will correspond to the 1D vertical linear array of the 2D sheets.



The Cyltorintigroid , 2120 , (((II)I)(II))
I was thinking about the cuts of 2120 cyltorintigroid, and how its cut of the 1x4 vertical column ((II)((I))) represents the 1D cut of a torus as four points along a line ((I)) . So, this column is made from a vertical hollow torus as the tiger-frame, and will have the same cut evolution: in place of the points are entire toruses. This visual trick helps big time with understanding what the frame of the tigroid is shaped like. It's much easier to visualize how the four points will merge when moving anywhere along the 2D cut array. The 1x4 vertical column of toruses will correspond exactly the same way, just like skewering a line through a hollow torus. From this, we can make the generalization that in 3D:

• cyltorintigroid is torus along the tiger-frame of N-2 torus cuts


(((II)I)(II)) - 2120, cyltorintigroid

cut to 4D:
(((I)I)(II)) - 1120 , two tigers in 2x1 flat line

rotate cut plane around:
(((II)I)(I)) - 1110, 2 ditoruses in 1x2 vertical column

cut to 3D
(((II))(I)) - 2010, 4 toruses, 2 concentric in 1x2 vertical column

rotate cut plane around:
(((I))(II)) - 1020, 4 toruses in 4x1 horizontal column, parallel in main diameters

reorient in 3D:
((II)((I))) - 2100, 4 toruses in 1x4 vertical column ¹

where all three have a 2D cut of:
(((I))(I))) - 1010, 4x2 rectangle of circles


¹ Since the arrangement of ((((II)I)((II)I))((II)I)) places the duotorus tiger in place of the (x) in a (((II)I)(x)), we have to rotate the 21x0 around to reflect this arrangement, as in ((x)((II)I)) - x210 . This orientation will make the 1x4 vertical column, instead of the 4x1 horizontal column. The 2D cut array of ((II)((Ii)i)) corresponds to how the 4 stacks of the 4x4 flat square arrays in the cube evolve.




The Duotorus Tiger , 21210 , (((II)I)((II)I))
Then, we have the (((II)I)((II)I)) - 21210, duotorus tiger, which is very much like the 2120 cyltorintigroid. One of the cuts of 21210 is a 1x4 vertical column of ditoruses : (((II)I)((I))) , 21100 . This is analogous to the 1x4 column of four toruses in ((II)((I))), 2100 . So, there would be three cuts in 3D of these 4 ditoruses in this single vert column arrangement. Simplifying to understandable terms, we can make the generalization that in 3D:

• duotorus tiger is N-1 ditorus cuts along the tiger-frame of N-2 torus cuts

Interchanging any one of the basic slices of these two will make duotorus tiger cuts. Since 21210 has a lot of symmetry, there are several duplicate arrangements with a 90 degree reorientation. But, the most important is the 16 circles in the 4x4 array, (((I))((I))) 10100 :

(((II)I)((II)I)) - 21210 , duotorus tiger

cut to 5D:
(((II)I)((II))) - 21200 , 2 concentric 2120-cyltorintigroids

cut to 4D:
(((II))((II))) - 20200 , 4 concentric 220-tigers diff in 2 per 1x2 major diameters

rotate cut plane around:
(((II)I)((I))) - 21100 , 4 ditoruses in 1x4 vertical column

cut to 3D:
(((II))((I))) - 20100 , 8 toruses, 2 concentric in 1x4 vertical column

reoriented:
(((I))((II))) - 10200 , 8 toruses , 2 concentric in 4x1 horizontal column, parallel to main diameters

further making:
(((I))((I))) - 10100 , 16 circles in 4x4 square ²

unspherated into product form:
((I))((I)) - 1010 , 16 points in 4x4 square


² In this cut we have 16 circles that map to the base of the 16 vertical columns in the cube. When we unspherate, we get 16 points, that when multiplied by the 1x4 vertical column of toruses, makes the 4x4x4 cubic lattice. One could observe the N-2 evolution in the 4 concentric tigers, but the vertical column of 4 ditoruses would be simpler to visualize. Nonetheless, they both share the same 16 circles in a 4x4 square.



Mapping the 4x4x4 Cubic Lattice
If we redefine the 4x4x4 cubic array as 16 vertical columns of 4, in a 4x4 square array, the evolution of 10100 and 1010 corresponds to the dance of the 16 columns. The columns won't merge vertically, it's only a horizontal dance. Visualizing the 10200 arrangement will make it easier to figure out the dance of 10100. As mentioned previously, the cubic lattice can be broken down to a 2D x 1D array mapping to its tigroid along rim of tigroid features.


• (((II)I)(II)) - Cyltorintigroid : 2D array is torus along (N-2 torus) cuts

• (((II)I)((II)I)) - Duotorus tiger: 4D array is circle along (N-2 ditorus) x (N-2 torus) cuts


• ((((II)I)((II)I))((II)I)) - Torus Squared : 6D array is torus along (N-4 duotorus tiger) x (N-2 cyltorintigroid) cuts

- which also equals: (N-2 ditorus) x (N-2 torus) x (N-2 torus) in the cubic lattice of 10100100 ((((I))((I)))((I)))

Rewritten another way: ((((Ii)i)((Ii)i))((Ii)i)) = (((Ii)i)I) x ((Ii)i) x ((Ii)i) cube



The cubic lattice has X by Y by Z axes that share two cut axes each, totaling 6:


X : (((Ii)i)I) : four circles along a line , from 2 concentric/displaced toruses , plane through ditorus

Y : ((Ii)i) : four points along line , from 2 concentric/2 displaced circles, line through torus

Z : ((Ii)i) : four points along line , from 2 concentric/2 displaced circles, line through torus


This is the full mathematical reduction of the tiger dancing that is to occur when moving in the 6D cut array. Each axis of the cube maps to tiger-frame shapes, and their individual N-2 cuts. I guess the next part would be a verbal translation of this 6D array! It's now going to be really easy to do, by simply looking at the XYZ breakdown of the 4x4x4 cubic lattice.



Then, I suppose that any oblique motion in the 6D array will make combined vertical-sheet with lateral-column merges and dances. Eventually, the final shape will be a single torus by itself then vanishes, or four points along a 1x4 vertical line. I'm not sure how to rotate this 6D array around, since only one cut exists in 3D. Any one of the 6 rotations out of this midsection will be into an empty hole. But, one could figure those out I think empty cuts are equally interesting to explore, even if we have to move out from center to reveal structure. There's some actual exploring required, into higher dimensional space. However, if we cut down to 4D, then the resulting 5D cut array could be rotated to reveal other cross sections.

What's really amazing is the fact that I can visualize torus squared now. By breaking it down to its tigroid along tigroid structure, I can build up the main cut array based on lower cut arrays. Knowing how to break down the main array itself into separate arrays from easy shapes is key. Thus making it way easier to trace out the hidden frames in the cubic lattice. This has been a very enlightening voyage through 9D space

[Marek14]

Simply said, the torus squared ((((II)I)((II)I))((II)I)) is the smallest example of (duotorus tiger/torus) tiger. The names are getting cumbersome in higher dimensions...

The 4x4x4 torus cube ((((I))((I)))((I))) is the only non-empty 3D cut. It has six possible evolutions which can be classified into four types:

((((Ii))((I)))((I))) and ((((I))((Ii)))((I))) - 32 ditoruses. They are arranged in 4x4 array in their middle and minor dimensions and each point in the array contains a pair of ditoruses differing in their major diameter. This evolution will have two inner and two outer toruses in each row of 4 merge, which can be done in two directions, corresponding to two evolutions of this type.

((((I)i)((I)))((I))) and ((((I))((I)i))((I))) - 32 ditoruses. This time they are arranged in a 2x4x4 array in their major/middle/minor dimensions. This time two toruses at the end of each row merge during the evolution, and the rows can be once again taken in two possible directions.

((((I))((I)))((Ii))) - 32 tigers. They are arranged in 4x4 array in both dimensions of one of their major diameters and each point contains a pair of tigers differing in the other major diameter. This evolution will have two inner toruses and two outer toruses of each column do the tiger dance.

((((I))((I)))((I)i)) - 32 tigers. This time they are arranged in 4x4x2 array in major1/major1/major2 dimensions. The evolution will have two toruses at each end of each column do the tiger dance.

Just for fun, if you arranged the 32 tigers in 4x2x4 array, you'd get something different: ((((I))(I))(((I))I)), which is a cut of ((((II)I)(II))(((II)I)I)), the (torus tiger/ditorus) tiger. Its only nonempty 3D cut would be ((((I))(I))(((I)))), 64 toruses arranged in 2x4 array of 8-torus columns.

[ICN5D]

Just for fun, if you arranged the 32 tigers in 4x2x4 array, you'd get something different: ((((I))(I))(((I))I)), which is a cut of ((((II)I)(II))(((II)I)I)), the (torus tiger/ditorus) tiger. Its only nonempty 3D cut would be ((((I))(I))(((I)))), 64 toruses arranged in 2x4 array of 8-torus columns.

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

Wow, that one's pretty wild, too. And, yep, it would have a 4x2x8 tower. That tall tower is made from eight 4x2 rectangles, as points on a line through the main circle of a ditorus. Very cool! All this time spent reading the cut notation is mind expanding. It becomes possible to do mentally and quickly. That's how you see it, you're doing so much math mentally, the cut arrays materialize instantly. I know this feeling. The last exploration was a very deep voyage for me.

[ICN5D]

I remember reading somewhere about how a torus can have a bitangent plane sliced through it. The cross section would be two overlapping circles, like a Venn diagram. Are there such things as a tritangent n-plane slicing a ditorus? Or, any other equivalents or n-planes in other toratopes?

[Marek14]

I remember reading somewhere about how a torus can have a bitangent plane sliced through it. The cross section would be two overlapping circles, like a Venn diagram. Are there such things as a tritangent n-plane slicing a ditorus? Or, any other equivalents or n-planes in other toratopes?

Hm, what exactly do you mean? Not sure how to imagine it.

And I'm not too sure whether imagining toratopes is "doing math". Almost the only math I'm doing these days when thinking about polytopes is combinatorics -- to help with the enumeration

[ICN5D]

Probably not imagining them, but cutting them is very mathematical It is still performing calculations using a notation and algorithms. At least that's how I look at it!

As for the bitangent plane:

https://www.youtube.com/watch?v=iYy-26JbIv0

I know you will like this video, if you haven't seen it before.


I'll bet that a torisphere can make Villarso spheres overlapping in the same way.