The Tiger Explained

Discussion of shapes with curves and holes in various dimensions.

Re: The Tiger Explained

Postby ICN5D » Mon Mar 17, 2014 2:25 am

Okay, so this is the first full cut analysis of a complex tigroid, totally unaided by any of your previous posts. I chose a nice and tough one, the duotorus tiger. Through the insights gained in the previous posts, I have found a new way to visualize these shapes. I can see the whole cut array now, and how to move along it. In both of the 3D cuts they are (N-1 ditorus) x (N-2 torus). Moving along the 3D array follows an n-1 ditorus cut along one axis and an n-2 torus cut along 2 axes. It's actually a very simple shape, and follows some very basic merge sequences. Probably the first remarkable breakthrough in my understanding so far was being able to see these hollow frames in the notation. Once that was understood, it became my new go to visual source when tracing out the cut array.


- (((xy)z)((wv)u))
- (((II)I)((II)I)) - 21210 , duotorus tiger
----------------------------------------------------------------------------------------------
A (((I)I)((II)I)) - 11210 , two 2120-cyltorintigroids along 2x1 displaced line
B (((II))((II)I)) - 20210 , two concentric 2120-cyltorintigroids
-----------------------------------------------------------------------------------------------
A1 (((I))((II)I)) - 10210 , 4 ditoruses in 1x4 horiz column
A2 (((I)I)((I)I)) - 11110 , 4 tigers in 2x2 vertical square
A3 (((I)I)((II))) - 11200 , 4 tigers in 2 concentric along 2x1 flat line
-----------------------------
B1 (((I))((II)I)) - 10210 , 4 ditoruses in 1x4 horiz column
B2 (((II))((I)I)) - 20110 , 4 tigers in 2 concentric along 1x2 vertical line
B3 (((II))((II))) - 20200 , 4 tigers in 2 concentric per 1x2 main diameters
-------------------------------------------------------------------------------------------------------------
A1a (((I))((I)I)) - 10110 , 8 toruses in 4x2 horizontal column, parallel to main circles
A1b (((I))((II))) - 10200 , 8 toruses in 2 concentric along 1x4 horizontal column
------------------------------
A2a (((I))((I)I)) - 10110 , 8 toruses in 4x2 horizontal column, parallel to main circles
------------------------------
A3a (((I))((II))) - 10200 , 8 toruses in 2 concentric along 1x4 horizontal column
A3b (((I)I)((I))) - 11100 , 8 toruses in 2x4 vertical column
------------------------------
B1a (((I))((I)I)) - 10110 , 8 toruses in 4x2 horizontal column, parallel to main circles
B1b (((I))((II))) - 10200 , 8 toruses in 2 concentric along 1x4 horizontal column
------------------------------
B2a (((I))((I)I)) - 10110 , 8 toruses in 4x2 horizontal column, parallel to main circles
B2b (((II))((I))) - 20100 , 8 toruses in 2 concentric along 1x4 vertical column
------------------------------
B3a (((II))((I))) - 20100 , 8 toruses in 2 concentric along 1x4 vertical column


Two cuts by two orientations:
11100 = 10110
10200 = 20100



(((I)I)((I))) - 11100 - 2x4 tower of 8 toruses : has three cut axes: (((Iy)I)((Iv)u))
----------------------------------------------------------------------------------------------------
• (N-1) Ditorus Cut: 2 displaced circles
• (N-2) Torus Cut: 4 points along vertical line
-------------------------------------------------------------
y : (((Ii)I)((I))) - Axis is 4 ditoruses in 1x4 vertical column
- Moving along main circle of (N-1) ditorus cut
--- Merge 2 towers into a single 1x4 vertical tower of 4 toruses, then collapse main circles and vanish

v : (((I)I)((Ii))) - Axis is 4 tigers in 2 concentric along 2x1 flat line
- Moving along main circle (N-2) torus cut
--- Merge rows 2 with 3 and vanish into 2x2 vertical square, then rows 1 with 4 and vanish

u : (((I)I)((I)i)) - Axis is 4 tigers in 2x2 vertical square
- Moving along minor circle of (N-2) torus cut
--- Simultaneously merge rows 1 with 2 and 3 with 4 into 2x2 vertical square of 4, then collapse main circles and vanish



(((II))((I))) - 20100 - 8 toruses in 2 concentric along 1x4 vertical column : has three cut axes: (((II)z)((Iv)u))
-----------------------------------------------------------------------------------------------------------------------------------------
• (N-1) Ditorus Cut: 2 concentric circles
• (N-2) Torus Cut: 4 points along vertical line
-------------------------------------------------------------
Z : (((II)i)((I))) - Axis is 4 ditoruses in 1x4 vertical column
- Moving along main circle of (N-1) ditorus cut
--- Merge concentric pairings into 1x4 vertical column of 4 toruses, then suddenly vanish

V : (((II))((Ii))) - Axis is 4 tigers in 2 concentric per 1x2 main diameters
- Moving along main circle (N-2) torus cut
--- Merge concentric pair 2 with 3 and vanish into 1x2 vertical column of 4, then concentric pair 1 with 4 and vanish

U : (((II))((I)i)) - Axis is 4 tigers in 2 concentric along 2x1 vertical line
- Moving along minor circle of (N-2) torus cut
--- Simultaneously merge pair 1 with 2 and 3 with 4 into 1x2 vertical column of 4, then collapse main diameters and vanish




Awesome, heck yeah. I like this format for doing cut analyses. I've been thinking of condensing the arrangement language down, in its verbal sense. Like using shorter acronyms for cocircular/concentric/displaced/vert column/ etc .
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Re: The Tiger Explained

Postby ICN5D » Mon Mar 17, 2014 5:39 am

Analysis of ((((II)I)I)(II)) - 21120 - cylditorintigroid , (ditorus*circle)-tiger



((((II)I)I)(II)) - 21120 - cylditorintigroid , (ditorus*circle)-tiger
-----------------------------------------------------------------------------------------------
A ((((I)I)I)(II)) - 11120 - 2 displaced 2120-cyltorintigroids along 2x1 flat line
B ((((II))I)(II)) - 20120 - 2 concentric 2120-cyltorintigroids
C ((((II)I))(II)) - 21020 - 2 cocircular 2120-cyltorintigroids
D ((((II)I)I)(I)) - 21110 - 2 tritoruses in 1x2 vertical column
------------------------------------------------------------------------------------------------
A1 ((((I))I)(II)) - 10120 - 4 displaced tigers along 4x1 flat line
A2 ((((I)I))(II)) - 11020 - 4 tigers in 2 concentric along 2x1 flat line
A3 ((((I)I)I)(I)) - 11110 - 4 ditoruses in 2x2 vertical square
-----------------------------
B1 ((((I))I)(II)) - 10120 - 4 displaced tigers along 4x1 flat line
B2 ((((II)))(II)) - 20020 - 4 concentric tigers on 1 main diameter
B3 ((((II))I)(I)) - 20110 - 4 ditoruses in 2 main-concentric along 1x2 vertical column
-----------------------------
C1 ((((I)I))(II)) - 11020 - 4 tigers in 2 concentric along 2x1 flat line
C2 ((((II)))(II)) - 20020 - 4 concentric tigers on 1 main diameter
C3 ((((II)I))(I)) - 21010 - 4 ditoruses in 2 middle-concentric along 1x2 vertical column
-----------------------------
D1 ((((I)I)I)(I)) - 11110 - 4 ditoruses in 2x2 vertical square
D2 ((((II))I)(I)) - 20110 - 4 ditoruses in 2 main-concentric along 1x2 vertical column
D3 ((((II)I))(I)) - 21010 - 4 ditoruses in 2 middle-concentric along 1x2 vertical column


Equal Cuts:
---------------
A1 = B1
A2 = C1
A3 = D1
---------
B2 = C2
B3 = D2
---------
C3 = D3


A1a ((((I)))(II)) - 10020 - 8 toruses in 1x8 horizontal column
A1b ((((I))I)(I)) - 10110 - 8 toruses in 4x2 vertical tower
-------------------------------
A2a ((((I)))(II)) - 10020 - 8 toruses in 1x8 horizontal column
A2b ((((I)I))(I)) - 11010 - 8 toruses in 2 concentric along 2x2 vertical square
-------------------------------
A3a ((((I))I)(I)) - 10110 - 8 toruses in 4x2 vertical tower
A3b ((((I)I))(I)) - 11010 - 8 toruses in 2 concentric along 2x2 vertical square
-------------------------------
B2a ((((I)))(II)) - 10020 - 8 toruses in 1x8 horizontal column
B2b ((((II)))(I)) - 20010 - 8 toruses in 4 concentric along 1x2 vertical tower
-------------------------------
B3a ((((I))I)(I)) - 10110 - 8 toruses in 4x2 vertical tower
B3b ((((II)))(I)) - 20010 - 8 toruses in 4 concentric along 1x2 vertical tower
-------------------------------
C3a ((((I)I))(I)) - 11010 - 8 toruses in 2 concentric along 2x2 vertical square
C3b ((((II)))(I)) - 20010 - 8 toruses in 4 concentric along 1x2 vertical tower


Equal Cuts:
-------------------
A1a = A2a = B2a
A1b = A3a = B3a
A2b = A3b = C3a
B2b = B3b = C3b



Giving 4 unique 3D cuts:

A1a ((((Ix)y)z)(II)) - 10020 - 8 toruses in 1x8 horizontal column
--------------------------------------------------------------------------------
• Torus along (N-3 Ditorus) cut array
-----------------------------------------------
x : Column collapses towards center between 4 and 5, merging 4+5 , 3+6 , 2+7, and 1+8 , then vanishes
y : Column collapses toward two points merging 1+3 and 2+4 plus 5+7 and 6+8 into 1x4 horiz column, then merges into 1x2 horiz column, then vanishes
z : Column collapses toward four points merging 1+2, 3+4 , 5+6 , and 7+8 into 1x4 horiz column, then vanishes



A1b ((((Ix)y)I)(Iz)) - 10110 - 8 toruses in 4x2 vertical tower
---------------------------------------------------------------------------
• Torus along (N-2 Ditorus) x (n-1 Cirle) cut array
----------------------------------------------------------------
x : Merge towers 2+3 then 1+4 into 1x4 tower, then deflate and vanish
y : Simultaneously merge towers 1+2 and 3+4 into 2x2 vert square, then deflate and vanish
z : Merge into 4x1 flat line of toruses



A2b ((((Ix)I)y)(Iz)) - 11010 - 8 toruses in 2 concentric along 2x2 vertical square
-----------------------------------------------------------------------------------------------------
• Torus along (N-2 Ditorus) x (n-1 Cirle) cut array
----------------------------------------------------------------
x : Merge into 2 concentric along 1x2 vertical tower of 4 toruses
y : Merge concentric pairs into 2x2 vertical square of 4 toruses
z : Merge into 2 concentric along 2x1 flat line of 4 toruses



B2b ((((II)x)y)(Iz)) - 20010 - 8 toruses in 4 concentric along 1x2 vertical tower
---------------------------------------------------------------------------------------------------
• Torus along (N-2 Ditorus) x (n-1 Cirle) cut array
----------------------------------------------------------------
x : Simultaneous merging of concentric pairs 2+3 and 1+4 into 2 concentric along 1x2 vertical column
y : Simultaneous merging of concentric pairs 1+2 and 3+4 into 2 concentric along 1x2 vertical column
z : Merge into 4 concentric toruses



Which sort of answers my previous question about the relationship with (((II)I)((II)I)) and ((((II)I)I)(II)). They are NOT the same tigroids, though have a very similar arrangement with their 8 toruses in tower arrays. This must be a key feature with the cartesian dual relationship, the rotation of the tower arrays of vertical toruses. The toruses stay flat, but the dimensions of the tower is rotated 90 degrees.

(((II)I)((II)I)) : has (((I)I)((I))) - 11100 - 2x4 vertical tower of 8 toruses

((((II)I)I)(II)) : has ((((I))I)(I)) - 10110 - 4x2 vertical tower of 8 toruses


Which also relates to, in what you pointed out earlier, how ((((II)I)((II)I))((II)I)) and ((((II)I)(II))(((II)I)I)) have similar arrays of 64 toruses, but differs in either a 4x4x4 or 4x2x8 tower, respectively.
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Re: The Tiger Explained

Postby Marek14 » Mon Mar 17, 2014 5:44 am

ICN5D wrote: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.


Hm, maybe...

Here's a question: if we cut a toratope with arbitrary 3D hyperplane, how many distinct topological shapes can we get?
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Re: The Tiger Explained

Postby Marek14 » Mon Mar 17, 2014 6:17 am

Seeing your mentions of horizontal columns, I think we might need to standardize our terms there. You call them "horizontal" since they are oriented in a direction of the original shape you chose to call "horizontal" -- but you don't explicitly mention this assumption.

Basically, for a torus, the directions are simple; the "horizontal" directions are the major dimensions and "vertical" are the minor dimensions. Standard torus has 2 horizontal and 1 vertical, but a spheritorus has 2 horizontal and 2 vertical (which is why it can be arranged into a vertical square ((II)(I)(I)) ).
But once you get to higher toratopes, the terms "horizontal" and "vertical" are no longer that useful. A ditorus has three categories of dimensions I call "major", "middle" and "minor". A tiger has two distinct groups of major dimensions plus, optionally, a group of minor dimensions.

That's why I only specify positions using the dimensions of figures in the cut. So, for me, all columns of toruses are automatically vertical -- that's what "column" means.

You'll also notice that I started to use the general term "array" for arrangement of toratopes. We don't need a new word for every dimension and calling the 4x4x4 array of toruses a "cube" is not technically correct -- sure, we will visualize them like that, but you must remember that torus square actually has eight different diameters and the 64 identical toruses only have two. Where does the remaining six go? Well, they go into specific distances between toruses, in each of the three dimensions there are two diameters that control the separation; while the cubic arrangement is possible, it's far from the only one.

Besides, the reduction to 3D, as you might notice, will only work with toratopes that have at most three "legs" (i.e. lowest levels of parenthesis). The ones you analysed actually have two, so you could also mention their 2D cuts:
(((I))((I))) - 16 circles in 4x4 array
((((I)))(I)) - 16 circles in 8x2 array

But of course, starting from 8 dimensions we have quadropod toratopes whose lowest non-empty cuts lie in four dimensions...

Actually, every species of toratope can be uniquely described by its "footprint", i.e. by its lowest-dimensional arrangement cut. For example, the "tiger squared", a 16-dimensional toratope of form

((((II)(II))((II)(II)))(((II)(II))((II)(II))))

has an 8-dimensional lowest cut

((((I)(I))((I)(I)))(((I)(I))((I)(I))))

which is an arrangement of 256 tiger/tiger tigers (tora tora tora!) (((II)(II))((II)(II))) in 2x2x2x2x2x2x2x2 array.
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Re: The Tiger Explained

Postby ICN5D » Tue Mar 18, 2014 12:28 am

Marek14 wrote:Here's a question: if we cut a toratope with arbitrary 3D hyperplane, how many distinct topological shapes can we get?


I think that only a sphere or torus will appear. No matter how many dimensions, there must always be three markers. The only way this can be, is inside of many overlapping brackets. If there is an offset of the quantity of brackets around less than the three, we have a torus of some kind, and thus, concentric circles. If there are no extra enclosing brackets within, we have a sphere. I suppose that the minimum number of distinct shapes are the total amount available in the respective dimension.



Seeing your mentions of horizontal columns, I think we might need to standardize our terms there. You call them "horizontal" since they are oriented in a direction of the original shape you chose to call "horizontal" -- but you don't explicitly mention this assumption.



I'm okay with that. I wan't sure what to name it, but it pretty much is a vertical column laid flat on its side. For visual purposes, though, it's better suited in its vertical arrangement. I was examining how ((((x)))(yz)) would make the eight points along x, then have circle yz oriented perpendicular to stack along it, parallel to their main circles. Rotated around, it would be ((xy)(((z)))), which would be the 1x8 tower. I prefer the vertical column visual anyways :)



You'll also notice that I started to use the general term "array" for arrangement of toratopes. We don't need a new word for every dimension and calling the 4x4x4 array of toruses a "cube" is not technically correct -- sure, we will visualize them like that, but you must remember that torus square actually has eight different diameters and the 64 identical toruses only have two. Where does the remaining six go? Well, they go into specific distances between toruses, in each of the three dimensions there are two diameters that control the separation; while the cubic arrangement is possible, it's far from the only one.




Oh, but that 4x4x4 array is so very powerful when taken literally! I found that by decomposing the cube into three mutual axes, I can place certain smaller cut arrays along them. That's when I noticed the patterns, each axis of the cube has two cut array axes from separate shapes, making 6 in total. The three axes each had n-2 dimensional cuts from either a torus or a ditorus. So, by imagining that along each separate axes of the cube, we have:

X - plane slicing ditorus making 4 circles, in 2D array
Y - line piercing a torus making four points, in 2D array
Z - line piercing a torus making four points, in 2D array

Now, by choosing an axis X, Y, or Z, we have a choice of two more axes to move along. Choosing Y gives us the cut array of a line piercing through a hollow torus, any motion along this 2D array will merge the points. Choosing Z gives us the same identical cut array, but a separate one altogether. Choosing X gives us the four circles. These circles correspond to the locations of the 16 vertical columns in the 4x4x4 array. These circles are from the 2D cut array of a knife through a ditorus. Moving along this array mimics that of the line through torus, but with whole circles in place of the points. The 4x4 points in a vertical square from the two line-through-torus arrays are the positions of the 64 toruses. The 4x4 base of the 4x4x4 is the cut array of the duotorus tiger. The 1x4 column that stacks the arrays is part of the cyltorintigroid. Which agrees very well with the distinction that torus squared is a duotorus tiger along the rim of a cyltorintigroid.

By using this little visual trick, each time I focus on a single axis in the " cubic lattice ", I see an entire cut array of a simpler shape. This makes the whole 9D shape a complete composition of all three 2D cut arrays in one. I can freely move along a ditorus cut and two other torus cuts when traversing the 6D cut array. The 4x4x4 array will of course merge into thinner rectangular towers and shorter columns as they all simultaneously collapse toward different points at different rates. All of which can be moved back and forth to undo and redo the dances in six separate directions. Seeing this pattern in the cube was very mind expanding, and some lights had clicked on. More like blazing carbon-arc lamp torches. That A Ha moment allowed me to see every one of the other tigroid arrays easily, and visualize how they dance and merge. That's how I figure out the merge sequences now, I'm still not sure exactly how you are doing it by looking at what the 4D cuts were. At this point in my learning, I can now see what the cut array looks like, and I pile the simpler arrays up to form more complex ones.


Besides, the reduction to 3D, as you might notice, will only work with toratopes that have at most three "legs" (i.e. lowest levels of parenthesis).



Yes, sadly, there aren't very many 9D shapes that can have 3D non-empty cuts. It gets even worse in higher dimensions. The empty cuts, as you can tell, have been the ones I haven't explored very much. I should, though. There is still some structure to be seen by moving out.



Actually, every species of toratope can be uniquely described by its "footprint", i.e. by its lowest-dimensional arrangement cut. For example, the "tiger squared", a 16-dimensional toratope of form

((((II)(II))((II)(II)))(((II)(II))((II)(II))))

has an 8-dimensional lowest cut

((((I)(I))((I)(I)))(((I)(I))((I)(I))))

which is an arrangement of 256 tiger/tiger tigers (tora tora tora!) (((II)(II))((II)(II))) in 2x2x2x2x2x2x2x2 array.



Whoa, that's pretty crazy. I haven't gone out that far yet. But, the notation says it all, that's the beauty of it. I guess sphere squared would be ((III)(III)(III)) , 3330-tiger. And, I also guess that the good ole' tiger is circle squared! It's pretty crazy how we go from a circle to a tiger, which cut down makes 4 circles, all way to torus squared, which explodes into a block of 64 toruses.
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Re: The Tiger Explained

Postby Marek14 » Tue Mar 18, 2014 5:25 am

ICN5D wrote:
Marek14 wrote:Here's a question: if we cut a toratope with arbitrary 3D hyperplane, how many distinct topological shapes can we get?


I think that only a sphere or torus will appear. No matter how many dimensions, there must always be three markers. The only way this can be, is inside of many overlapping brackets. If there is an offset of the quantity of brackets around less than the three, we have a torus of some kind, and thus, concentric circles. If there are no extra enclosing brackets within, we have a sphere. I suppose that the minimum number of distinct shapes are the total amount available in the respective dimension.


Note that I said "arbitrary 3D hyperplane". So this would, for example, include the diagonal cut of tiger that looks like "cage" with 4 bars.
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Re: The Tiger Explained

Postby ICN5D » Tue Mar 18, 2014 5:41 am

Oh, my bad. Is that all of them, just the three? Would there be any more?
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Re: The Tiger Explained

Postby Marek14 » Tue Mar 18, 2014 7:54 am

ICN5D wrote:Oh, my bad. Is that all of them, just the three? Would there be any more?


Almost certainly not... Logically, there have to be "singular cuts" which separate topologies. In case of tiger, the cut where two original toruses from mid-cut touch before merging into a single one would be an example, but I don't know how many types of such cuts there is.
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Re: The Tiger Explained

Postby Marek14 » Tue Mar 18, 2014 8:55 am

Right, as for the "base cuts". Some thinking led to the following results:

1. The dimension of base cut is equal to number of "legs" of toratope species, i.e. number of innermost pairs of parenthesis.
2. The lowest amount of shapes in the base cut is equal to 2^dimension -- there must be at least two shapes in any applicable dimension since if there was only one, a further non-empty cut could be made.
3. We could define a "torus number" for a toratope. The torus number determines how many concentric toratopes appear in every point of the base cut array. Torus number 0 means there's one, torus number 1 means there are two, torus number 2 means there are four, etc. For example, tiger has torus number 0 since its base cut (four circles in 2x2 array) has only one circle in each point. Tiger torus has torus number 1 since its base cut is four PAIRS of circles in 2x2 array. For torus line itself (sphere, torus, ditorus, tritorus...) whose base cuts are points on a line, the torus number equals to the binary logarithm of the number of points - 1
4. Minimum dimension for a toratope with base cut of dimension n is 2n for the basic array. One additional dimension is added for every doubling of number of toratopes, so the total minimum dimension is binary logarithm of number of toratopes + dimension of base cut (slightly different for torus line).

For example, here are the toratopes with minimum of 8 dimensions:

1D base cut: 128 points on a line: hexatorus (((((((II)I)I)I)I)I)I)

2D base cut: 64 circles:
32x2 array: tetratorus tiger ((((((II)I)I)I)I)(II))
16x4 array: tritorus/torus tiger (((((II)I)I)I)((II)I)))
8x8 array: duoditorus tiger ((((II)I)I)(((II)I)I))
16x2 array of pairs: tritorus tiger torus ((((((II)I)I)I)(II))I)
8x4 array of pairs: ditorus/torus tiger torus (((((II)I)I)((II)I))I)
8x2 array of quartets: ditorus tiger ditorus ((((((II)I)I)(II))I)I)
4x4 array of quartets: duotorus tiger ditorus (((((II)I)((II)I))I)I)
4x2 array of octets: torus tiger tritorus ((((((II)I)(II))I)I)I)
2x2 array of 16-plets: tiger tetratorus ((((((II)(II))I)I)I)I)

3D base cut:
32 spheres:
8x2x2 array: ditorus triger ((((II)I)I)(II)(II))
4x4x2 array: duotorus triger (((II)I)((II)I)(II))
4x2x2 array of pairs: torus triger torus ((((II)I)(II)(II))I)
2x2x2 array of quartets: triger ditorus ((((II)(II)(II))I)I)
32 toruses (last dimension is vertical):
8x2x2 array: ditorus double tiger (((((II)I)I)(II))(II))
4x4x2 array: duotorus double tiger ((((II)I)((II)I)))(II))
4x2x4 array: (torus tiger)/torus tiger ((((II)I)(II))((II)I))
2x2x8 array: tiger/ditorus tiger (((II)(II))(((II)I)I))
4x2x2 array of major pairs: torus tiger torus tiger (((((II)I)(II))I)(II))
4x2x2 array of minor pairs: torus double tiger torus (((((II)I)(II))(II))I)
2x2x4 array of major pairs: (tiger torus)/torus tiger ((((II)(II))I)((II)I))
2x2x4 array of minor pairs: tiger/torus tiger torus ((((II)(II))((II)I))I)
2x2x2 array of major quartets: tiger ditorus tiger (((((II)(II))I)I)(II))
2x2x2 array of major/minor quartets: tiger torus tiger torus (((((II)(II))I)(II))I)
2x2x2 array of minor quartets: double tiger ditorus (((((II)(II))(II))I)I)

4D base cut:
16 spheres: tetriger ((II)(II)(II)(II))
16 torispheres: triger tiger (((II)(II)(II))(II))
16 spheritoruses: tiger triger (((II)(II))(II)(II))
16 ditoruses: triple tiger ((((II)(II))(II))(II))
16 tigers: tiger/tiger tiger (((II)(II))((II)(II)))
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Re: The Tiger Explained

Postby Marek14 » Tue Mar 18, 2014 9:07 am

And this allows us to define the COMPLEXITY of a toratope as follows:

Sphere and point has complexity 0
Other toratopes have complexity 1 higher than complexity of the toratope in their base cut.
So, for example, double tiger (((II)(II))(II)) is simplest toratope of complexity 2. Simplest toratope of complexity 3 would have 64 double tigers as its base cut, and so it's the 12-dimensional toratope ((((II)(II))((II)(II)))((II)(II))) [(tiger/tiger tiger)/tiger tiger]

This also allows us to organize the species of toratopes into GENERA and ORDERS:

A GENUS of toratopes is formed by all toratopes whose base cuts are formed from a specific toratope. For example, tiger, torus tiger, tiger torus, etc. all belong to the tiger genus (or circle-based genus) which is characterized by having its base cut formed from circles. Toratope species can be reduced to its base genus toratope by omitting all parentheses pairs which only have 1 nested parentheses. The toratope PATRIARCHS (basic genus toratopes) always have torus number 0 (though not all toratopes with torus number 0 are patriarchs, for example a torus tiger whose base cut is 4x2 array of circles).

An ORDER of toratopes is formed by all toratopes whose base cuts belong to a specific species. All toratopes from a given order have the same complexity. For example, sphere, torus, tiger, triger etc. all belong in the sphere-based order since their base cuts are formed from spheres of any dimension (including points).

And of course, this hierarchy can be extended upwards, a class of toratopes having a certain genus as its base cuts, etc.
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Re: The Tiger Explained

Postby ICN5D » Wed Mar 19, 2014 3:15 am

Those are some cool shapes! I'm seeing some new ones in there to explore later on.




Played around a little with deriving cut arrays, and exploring a new shape:


((((II)I)(II))((II)(II))) - 2120220 | (torus/circle/tiger )- tiger | (torus*circle)*(circle2)-tiger | 2120-cyltorintigroid along rim of 22020-double-tiger
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
((((I))(I))((I)(I))) - 10101100 - 32 tigers in 4x2x2x2 brick



• 5D cut array follows 4x2x2x2 in (torus-2) x (circle-1) x (circle-1) x (circle-1) tesseractoid brick of tigers



Axes of 5D Cut Array : ((((Ix)y)(Iz))((Iw)(Iv)))
------------------------------------------------------------
x: 4 points from line through 2 concentric circles, moving out merges 2+3 and vanish, then 1+4
y: 4 points from line through 2 displaced circles, moving out merges 1+2 and 3+4, into 2 points
z: 2 points from line through circle, moving out merges into one point
w: 2 points from line through circle, moving out merges into one point
v: 2 points from line through circle, moving out merges into one point



Applied Merges in 10101100 : ((((Ix)y)(Iz))((Iw)(Iv))) : 32 tigers in 4x2x2x2 brick
------------------------------------------------------------------------------------------------------------
x: merge rows 2+3 and vanish leaving 2x2x2x2, then merge 1+4 leaving 1x2x2x2 column
y: merge rows 1+2 and 3+4, into 2x2x2x2 vertices of tesseract
z: merge column levels into 4x1x2x2 vertical wall
w: merge column levels into 4x2x1x2 vertical brick-column
v: merge column levels into 4x2x2 brick



Thinking a little deeper, I realized that this 4x2x2x2 array can be visualized as a projection of a tesseract, where there is a smaller 4x2x2 cuboid nested within the closer and larger 4x2x2 cuboid brick. The two 4x2x2 bricks in this arrangement represent the final stacking of 2 into 4D, in which we are seeing as a projection. It's a far leap, but the visual is correct. We have to imagine a 3D projection of a 4D brick array that contains a 5D cut array of a 9D shape. Not really all that difficult, upon learning how to combine multiple visual tricks into one image.

It can be drawn, but the physical image will be ugly. The mind can process several visuals simultaneously, and tends to be the cleanest way to see it. This is pretty much how one learns to see high-D shapes, at least it is for me. I see layers of transparencies of simpler things going on, in each array axis. When imaging them, we maintain that the NxNxNxN brick array can have many separate lower-D cut arrays within. Movement along a cut array makes some n-cuboid lattices obey a multiple dimensions per axis rule. In the 4x2x2x2 brick, we have the row of 4 corresponding to a line piercing through a torus in a whole separate 2D cut array. This row of 4 is only one axis of the tesseractoid brick, but contains two dimensions within it. Like I said, this recent breakthrough was a mind expander like no other. Well, other than my new development on the polynomial formulas. That was cool, too. Makes me want to go back to school. This would be the first time I discovered the wisdom in doing so, myself. Really, it was the moment I was waiting for :)
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Re: The Tiger Explained

Postby Marek14 » Wed Mar 19, 2014 5:56 am

Yes, I think you're correct with the visualization :)
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Re: The Tiger Explained

Postby ICN5D » Thu Mar 20, 2014 7:41 am

Hey Marek, I found a really awesome rendering program that can do toratope cuts and render their merge animations in real time. I found that the tiger doesn't do a cassini oval deformation to the vertical stack of toruses, they just merge together into one, remaining perfectly circular. It's also quite amazing to see the 4D toratopes in action for the first time. I'm pretty sure you're going to like this:


3D Calc renderer:

http://web.monroecc.edu/manila/webfiles ... Plot3D.htm


Go to :
- Graph
- Add Implicit Surface

Copy-paste into the equation field any one of the equations below

Go to:
- Parameters
- Adjust parameters
- Range tab
-- change " A " to whatever range

-In the parameters box, you have to check the box at the right of the slider for " A "

-Click Animate Parameter to see the amazing animations you have been waiting for!


Toripshere ((III)I) = (sqrt(x^2 + y^2 + z^2) - R)^2 + w^2 = r^2
-----------------------------------------------------------------
• ((IIi)I) - torus
(sqrt(x^2 + y^2 + a^2) - 2.5)^2 + z^2 = -0.5^2
-3 > a > 3

• ((III)i) - concentric spheres
(sqrt(x^2 + y^2 + z^2) - 2.5)^2 + a^2 = -0.5^2
-0.7 > a > 0.7
NEED TO CUT OPEN FOR CONCENTRIC ARRANGEMENT




Spheritorus ((II)II) = (sqrt(x^2 + y^2) - R)^2 + z^2 + w^2 = r2
-----------------------------------------------------------------
• ((Ii)II) - displaced spheres
(sqrt(x^2 + a^2) - 2.5)^2 + y^2 + z^2 = -0.5^2
-3 > a > 3

• ((II)Ii) - torus
(sqrt(x^2 + y^2) - 2.5)^2 + z^2 + a^2 = -0.5^2
-0.7 > a > 0.7




Ditorus (((II)I)I) = (sqrt((sqrt(x^2 + y^2) - ρ)^2 + z^2) - r)^2 + w^2 = R^2
------------------------------------------------------------------------------
• (((Ii)I)I) - displaced toruses
(sqrt((sqrt(x^2 + a^2) - 2.5)^2 + y^2) - 1)^2 + z^2 = -0.3^2
-4.2 > a > 4.2

• (((II)i)I) - concentric toruses
(sqrt((sqrt(x^2 + y^2) - 2.5)^2 + a^2) - 1)^2 + z^2 = -0.3^2
-1.4 > a > 1.4

• (((II)I)i) - cocircular toruses
(sqrt((sqrt(x^2 + y^2) - 2.5)^2 + z^2) - 1)^2 + a^2 = -0.3^2
NEED TO CUT OPEN FOR COCIRCULAR ARRANGEMENT





Tiger ((II)(II)) = (sqrt(x^2 + y^2) - Ra)^2 + (sqrt(z^2 + w^2) - Rb)^2 = r^2
--------------------------------------------------------------------------------
• ((II)(Ii)) - vertical stack of torii
(sqrt(x^2 + y^2) - 2.5)^2 + (sqrt(z^2 + a^2) - 2.5)^2 = -0.5^2
-2.5 > a > 2.5




I'm going to start deriving the 3D equations for the 5D toratopes so I can the copy-paste tables tomorrow. I should have a few by the evening.
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Re: The Tiger Explained

Postby Marek14 » Thu Mar 20, 2014 8:05 am

Hm, the previous cuts of tiger on this site DID show Cassini ovals, though :)

EDIT: I tried it. Nope, the Cassini ovals are clearly visible in tiger animation. The horizontal cuts are, of course, perfectly circular -- Cassini ovals are in the vertical cuts. Try looking at the thing cut so you'd only see half of the toruses.
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Re: The Tiger Explained

Postby ICN5D » Thu Mar 20, 2014 4:36 pm

Yes, the minor radius of the toruses do the cassini ovals, I knew about that one. I remember in some previous post how you said the major radius also would do a cassini oval deforming, but I guess it doesn't. I want to show the merge evolution of the concentric spheres / cocircular toruses next. It's as easy as restricting the bounding box so it cuts the shape in half. I'm taking a stab at 5D tonight.
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Re: The Tiger Explained

Postby Marek14 » Thu Mar 20, 2014 5:22 pm

ICN5D wrote:Yes, the minor radius of the toruses do the cassini ovals, I knew about that one. I remember in some previous post how you said the major radius also would do a cassini oval deforming, but I guess it doesn't. I want to show the merge evolution of the concentric spheres / cocircular toruses next. It's as easy as restricting the bounding box so it cuts the shape in half. I'm taking a stab at 5D tonight.


No, I don't think I said anything about major radius there :) As for concentric spheres/cocircular toruses, I'm afraid there is no big revelation there. These shapes start being interesting in pairs. For example, two pairs of concentric spheres that merge.
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Re: The Tiger Explained

Postby ICN5D » Thu Mar 20, 2014 7:39 pm

I'll be exploring some of those later. I suppose (((II)II)I) would be a good go to example first. This is an exciting new frontier!
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Re: The Tiger Explained

Postby Marek14 » Fri Mar 21, 2014 5:41 am

Here's a nice thing you can try:

(sqrt(x^2 + (y*sin(a))^2) - 2.5)^2 + (sqrt(z^2 + (y*cos(a))^2) - 2.5)^2 = -0.5^2

This is a rotation animation, a tiger cut by hyperplane that rotates between y=0 and w=0. You can see how two toruses smoothly change into two toruses with perpendicular orientation. Since a is an angle here, I set its limits to -3.15 -- 3.15.

Similarly for torisphere:

(sqrt(x^2 + y^2 + (z*sin(a))^2) - 2.5)^2 + (z*cos(a))^2 = -0.5^2

For spheritorus:

(sqrt(x^2 + (y*sin(a))^2) - 2.5)^2 + z^2 + (y*cos(a))^2 = -0.5^2

And for ditorus, where there are three different rotations:

(sqrt((sqrt(x^2 + (y*sin(a))^2) - 2.5)^2 + (y*cos(a))^2) - 1)^2 + z^2 = -0.3^2
(sqrt((sqrt(x^2 + (y*sin(a))^2) - 2.5)^2 + z^2) - 1)^2 + (y*cos(a))^2 = -0.3^2
(sqrt((sqrt(x^2 + y^2) - 2.5)^2 + (z*sin(a))^2) - 1)^2 + (z*cos(a))^2 = -0.3^2
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Re: The Tiger Explained

Postby Marek14 » Fri Mar 21, 2014 9:57 am

OK, now, thinking of 5D toratopes...

Pentasphere (IIIII), x^2 + y^2 + z^2 + w^2 + v^2 = R
Only 3D cut (IIIii), sphere -> animation is identical to glome animation (IIIi)

41-torus ((IIII)I), (sqrt(x^2 + y^2 + z^2 + w^2) - R)^2 + v^2 = r^2
((IIii)I), torus -> animation is identical to torisphere animation ((IIi)I)
((IIIi)i), pair of concentric spheres, (sqrt(x^2 + y^2 + z^2 + a^2) - R)^2 + b^2 = r^2

32-torus ((III)II), (sqrt(x^2 + y^2 + z^2) - R)^2 + w^2 + v^2 = r^2
((Iii)II), two separated spheres -> animation is identical to spheritorus animation ((Ii)II)
((IIi)Ii), torus -> (sqrt(x^2 + y^2 + a^2) - R)^2 + z^2 + b^2 = r^2
((III)ii), pair of concentric spheres -> animation is identical to torisphere animation ((III)i)

Double rotation between two separated spheres and pair of concentric spheres:
(sqrt(x^2 + (y*sin(a))^2 + (z*sin(b))^2) - R)^2 + (y*cos(a))^2 + (z*cos(b))^2 = r^2

23-torus ((II)III), (sqrt(x^2 + y^2) - R)^2 + z^2 + w^2 + v^2 = r^2
((ii)III), empty cut with a glome on each side -> animation is double of glome animation (IIIi)
((Ii)IIi), two separated spheres -> (sqrt(x^2 + a^2) - R)^2 + y^2 + z^2 + b^2 = r^2
((II)Iii), torus -> animation is identical to spheritorus animation ((II)Ii)

Rotation between empty cut and two separated spheres:
(sqrt((x*sin(a))^2 + b^2) - R)^2 +(x*cos(a))^2 + y^2 + z^2 = r^2

Double rotation between empty cut and torus:
(sqrt((x*sin(a))^2 + (y*sin(b))^2) - R)^2 + (x*cos(a))^2 + (y*cos(b))^2 + z^2 = r^2
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Re: The Tiger Explained

Postby Marek14 » Fri Mar 21, 2014 2:43 pm

311-ditorus (((III)I)I), (sqrt((sqrt(x^2 + y^2 + z^2) - r1)^2 + w^2) - r2)^2 + v^2 = r3^2
(((Iii)I)I), two toruses in horizontal line -> animation is identical to ditorus animation (((Ii)I)I)
(((IIi)i)I), a pair of toruses differing in major diameter -> animation is identical to ditorus animation (((II)i)I)
(((IIi)I)i), a pair of toruses differing in minor diamater -> animation is identical to ditorus animation (((II)I)i)
(((III)i)i), a quartet of spheres -> (sqrt((sqrt(x^2 + y^2 + z^2) - r1)^2 + a^2) - r2)^2 + b^2 = r3^2

Double rotation between two toruses in horizontal line and quartet of spheres:
(sqrt((sqrt(x^2 + (y*sin(a))^2 + (z*sin(b))^2) - r1)^2 + (y*cos(a))^2) - r2)^2 + (z*cos(b))^2 = r3^2

Rotation between pair of toruses differing in major diameter and quartet of spheres:
(sqrt((sqrt(x^2 + y^2 + (z*sin(a))^2) - r1)^2 + b^2) - r2)^2 + (z*cos(a))^2 = r3^2

Rotation between pair of toruses differing in minor diameter and quartet of spheres:
(sqrt((sqrt(x^2 + y^2 + (z*sin(a))^2) - r1)^2 + (z*cos(a))^2) - r2)^2 + b^2 = r3^2

Further interesting rotations are homework.

221-ditorus (((II)II)I), (sqrt((sqrt(x^2 + y^2) - r1)^2 + z^2 + w^2) - r2)^2 + v^2 = r3^2
(((ii)II)I), empty with horizontally approached torisphere on each side -> animation is double of torisphere animation ((IIi)I)
(((Ii)Ii)I), two toruses in a horizontal line -> (sqrt((sqrt(x^2 + a^2) - r1)^2 + y^2 + b^2) - r2)^2 + z^2 = r3^2
(((Ii)II)i), two pairs of concentric spheres -> (sqrt((sqrt(x^2 + a^2) - r1)^2 + y^2 + z^2) - r2)^2 + b^2 = r3^2
(((II)ii)I), a pair of toruses differing in major diameter -> animation is identical to ditorus animation (((II)i)I)
(((II)Ii)i), a pair of toruses differing in minor diameter -> (sqrt((sqrt(x^2 + y^2) - r1)^2 + z^2 + a^2) - r2)^2 + b^2 = r3^2

212-ditorus (((II)I)II), (sqrt((sqrt(x^2 + y^2) - r1)^2 + z^2 ) - r2)^2 + w^2 + v^2 = r3^2
(((ii)I)II), empty with horizontally approached spheritorus on each side -> animation is double of spheritorus animation ((Ii)II)
(((Ii)i)II), four spheres in a line -> (sqrt((sqrt(x^2 + a^2) - r1)^2 + b^2 ) - r2)^2 + y^2 + z^2 = r3^2
(((Ii)I)Ii), two toruses in a horizontal line -> (sqrt((sqrt(x^2 + a^2) - r1)^2 + y^2 ) - r2)^2 + z^2 + b^2 = r3^2
(((II)i)Ii), a pair of toruses differing in major diameter -> (sqrt((sqrt(x^2 + y^2) - r1)^2 + a^2 ) - r2)^2 + z^2 + b^2 = r3^2
(((II)I)ii), a pair of toruses differing in minor diameter -> animation is identical to ditorus animation (((II)I)i)

320-tiger ((III)(II)), (sqrt(x^2 + y^2 + z^2) - Ra)^2 + (sqrt(w^2 + v^2) - Rb)^2 = r^2
((Iii)(II)), two toruses in a vertical line -> animation is identical to tiger animation ((Ii)(II))
((IIi)(Ii)), two toruses in a vertical line -> (sqrt(x^2 + y^2 + a^2) - Ra)^2 + (sqrt(z^2 + b^2) - Rb)^2 = r^2
((III)(ii)), empty with vertically approached torisphere on each side -> animation is double of torisphere animation ((III)i)

221-tiger ((II)(II)I), (sqrt(x^2 + y^2) - Ra)^2 + (sqrt(z^2 + w^2) - Rb)^2 + v^2 = r^2
((ii)(II)I), empty with vertically approached spheritorus on each side -> animation is double of spheritorus animation ((II)Ii)
((Ii)(Ii)I), four spheres in 2x2 array -> (sqrt(x^2 + a^2) - Ra)^2 + (sqrt(y^2 + b^2) - Rb)^2 + z^2 = r^2
((Ii)(II)i), two toruses in a vertical line -> (sqrt(x^2 + a^2) - Ra)^2 + (sqrt(y^2 + z^2) - Rb)^2 + b^2 = r^2

Tritorus ((((II)I)I)I), (sqrt((sqrt((sqrt(x^2 + y^2) - r1)^2 + z^2 ) - r2)^2 + w^2) - r3)^2 + v^2 = r4^2
((((ii)I)I)I), empty with ditorus approached from major dimension on each side -> animation is double of ditorus animation (((Ii)I)I)
((((Ii)i)I)I), four toruses in a horizontal line -> (sqrt((sqrt((sqrt(x^2 + a^2) - r1)^2 + b^2 ) - r2)^2 + y^2) - r3)^2 + z^2 = r4^2
((((Ii)I)i)I), two pairs of toruses differing in major diameter in a horizontal line -> (sqrt((sqrt((sqrt(x^2 + a^2) - r1)^2 + y^2 ) - r2)^2 + b^2) - r3)^2 + z^2 = r4^2
((((Ii)I)I)i), two pairs of toruses differing in minor diameter in a horizontal line -> (sqrt((sqrt((sqrt(x^2 + a^2) - r1)^2 + y^2 ) - r2)^2 + z^2) - r3)^2 + b^2 = r4^2
((((II)i)i)I), a major quartet of toruses -> (sqrt((sqrt((sqrt(x^2 + y^2) - r1)^2 + a^2 ) - r2)^2 + b^2) - r3)^2 + z^2 = r4^2
((((II)i)I)i), a major/minor quartet of toruses -> (sqrt((sqrt((sqrt(x^2 + y^2) - r1)^2 + a^2 ) - r2)^2 + z^2) - r3)^2 + b^2 = r4^2
((((II)I)i)i), a minor quartet of toruses -> (sqrt((sqrt((sqrt(x^2 + y^2) - r1)^2 + z^2 ) - r2)^2 + a^2) - r3)^2 + b^2 = r4^2

Tiger torus (((II)(II))I), sqrt((sqrt(x^2 + y^2) - r1a)^2 + (sqrt(z^2 + w^2) - r1b)^2 - r2)^2 + v^2 = r3^2
(((ii)(II))I), empty with ditorus approached from middle dimension on each side -> animation is double of ditorus animation (((II)i)I)
(((Ii)(Ii))I), four toruses in 2x2 horizontal array -> sqrt((sqrt(x^2 + a^2) - r1a)^2 + (sqrt(y^2 + b^2) - r1b)^2 - r2)^2 + z^2 = r3^2
(((Ii)(II))i), two pairs of toruses differing in minor diameter in a vertical line -> sqrt((sqrt(x^2 + a^2) - r1a)^2 + (sqrt(y^2 + z^2) - r1b)^2 - r2)^2 + b^2 = r3^2
This doesn't seem to display right... did I make a mistake in the equation somewhere?

Torus tiger (((II)I)(II)), (sqrt((sqrt(x^2 + y^2) - r1a)^2 + z^2) - r2)^2 + (sqrt(w^2 + v^2) - r1b)^2 = r3^2
(((ii)I)(II)), empty with tiger on each side -> animation is double of tiger animation ((Ii)(II))
(((Ii)i)(II)), four toruses in a vertical line -> (sqrt((sqrt(x^2 + a^2) - r1a)^2 + b^2) - r2)^2 + (sqrt(y^2 + z^2) - r1b)^2 = r3^2
(((Ii)I)(Ii)), four toruses in a 2x2 horizontal/vertical array -> (sqrt((sqrt(x^2 + a^2) - r1a)^2 + y^2) - r2)^2 + (sqrt(z^2 + b^2) - r1b)^2 = r3^2
(((II)i)(Ii)), two pairs of toruses differing in major diameter in a vertical line -> (sqrt((sqrt(x^2 + y^2) - r1a)^2 + a^2) - r2)^2 + (sqrt(z^2 + b^2) - r1b)^2 = r3^2
(((II)I)(ii)), empty with ditorus approached from minor dimension on each side -> animation is double of ditorus animation (((II)I)i)
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Re: The Tiger Explained

Postby ICN5D » Sat Mar 22, 2014 12:21 am

Oh AWESOME Marek!!! Thanks for those !!! I was trying to derive the equation for the (((II)I)(II)) last night, by judging the difference between other equations. It is a little tougher than I anticipated. Luckily, I found an old post of yours ( in 2006 ), where you had listed them, and the conversion method. But, they were all midsections, no obliques or rotations. Those new ones are really interesting, I'll have to check them out. I did render the (((II)I)(II)) and all mid-cuts. Watching the diagonal flythrough along the cut array was really cool, as the blob simultaneously split into four toruses. I made a list of some notable topological features during transitions. I found that at a given depth of one axis, animating the other makes a three torus dance from the column of four. Plus, many others, I thoroughly explored cyltorintigroid last night. Found some really bizarre things going on when I cut the shapes open. Here is the list I made:




Cyltorintigroid (((II)I)(II)) = ((Sqrt((Sqrt(x^2+y^2)-Ra)^2+z^2)-Rb)^2+(Sqrt(w^2+v^2)-Rc)^2)-Rd = 0
------------------------------------------------------------------------------------------------------------------------

Set graph box to X, Y, Z min/max all -5 / 5 , unless otherwise noted to set Ymin to 0 or -0.5 for cutting open.

• (((Ii)i)(II)) - 4 torii in 1x1x4 column
((Sqrt((Sqrt(x^2+a^2)-2.5)^2+b^2)-1.2)^2+(Sqrt(y^2+z^2)-1.5)^2)-0.4 = 0
-4 < a < 4
-2.5 < b < 2.5
-----------------------------------
Notable Topological Features:
-----------------------------------
-- Set a=1.5, run b : Tiger dance of 3 torii
-- Set a=1/3 , b= -0.3 : stack of 4 tires
------------------------------------------------------------------------------------



• (((II)i)(Ii)) - 4 torii in 2 concentric along 1x1x2 column
((Sqrt((Sqrt(x^2+y^2)-2.5)^2+a^2)-1.2)^2+(Sqrt(z^2+b^2)-1.5)^2)-0.4 = 0
-1.7 > a > 1.7
-3 > b > 3
------------------------------------
Notable Topological Features:
------------------------------------
-- Set 0.2<a<0.6 , -1.45<b<-1.31 , Ymin=0 : Parameter Field of simultaneous merge instances with cassini torus hole
-- Set a=0.26 , b= -1.38 , Moment of merge of all four torii
-----------------------------------------------------------------------------



• (((Ii)I)(Ii)) - 4 torii in 2x1x2 vertical square
((Sqrt((Sqrt(x^2+a^2)-2.5)^2+y^2)-1.2)^2+(Sqrt(z^2+b^2)-1.5)^2)-.4 = 0
-5<a<5
-2.5<b<2.5
-----------------------------------
Notable Topological Features:
------------------------------------
-- Run a and b , nice 2D oblique flythrough
-- Set a= -1 , b=2 : figure-8 shape
-- Set a= -0.6 , b=0.9 : cassini deformed 2x1x2 with three holes
-- Set a=3 , b=0.9 , Yimn=-0.5 : Cassini clam-shaped void, plus many others with complex tunnel shapes



In the (((II))(I)) cut, there was a moment where the blob developed an inside-out torus shaped hole inside, made from the cassini deformation of the four toruses when they all split apart. In the (((I)I)(I)) cut, there were some really wild looking tunnel morphings inside, during division.


But, now you have me really interested in those cut-plane rotation equations. I have no idea how to get those. Have you ever thought about investigating a possible algorithm using the toratope notation? I'm REALLY interested in some 5D rotation renders :) This is going to keep me busy for a while. I made a little movie of the diagonal flythrough of (((II)I)(II)) to show people, and it's awesome. Mind bending, for them!


EDIT: Well, now that I've been looking at them, I see a simple pattern to the rotation equation. The syntax " (y*sin(a))^2 " is taking place of y^2 . And, there are as many distinct rotations as there are distinct diameters. Wow, a major breakthrough for me. ( says in a robotic voice ) Need MORE INPUT!!!!
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Re: The Tiger Explained

Postby ICN5D » Sat Mar 22, 2014 5:40 am

Oh my gosh, you have to check out cyltorintigroid (((II)I)(II)) :D :

Set 0 < a,b < 1.57

((Sqrt((Sqrt(x^2 + (y*sin(a))^2) - 2.5)^2 + (z*cos(b))^2) - 1.2)^2 + (Sqrt((z*sin(b))^2 + (y*cos(a))^2) - 1.5)^2) - 0.4 = 0

This is a good one, because I put the two different "start to finish" planes on opposing symmetries. This allows you to rotate the planes around and interchange the midsection cuts. Exactly like what you were looking for! By setting a and b from 0 to 1.57, you can move the whole slider over to the other side and end up at another axial cross cut. This allows you to interchange the two rotated planes into all available midcuts ( even the empty one! ), and see their amazing transformations. Cyltorintigroid is a true gem, and has an amazing rotation morphing.

Especially at this one really amazing looking structure when you set a=0.95 and b=0.72. It's a room with pillars and chambers, at some unique interaxial cross section between the circle and the torus products. This is the most amazing exploration I have ever gone on with these shapes. This program can do many more things that I cannot envision, and it illuminates the shapes in an incredible light. Being able to go from one component symmetry to another, and morph the two independently is really amazing!! :D Rotation equations ROCK!!

Another good one: set b=0.65 , run " a " , you get an amazing transformation between interaxial symmetries


This makes me think: can I also plug in a modifying parameter that moves out from center? As in, have the rotation ability AND move out from center. This would allow full control of the angle and depth of the cross section planes, like some sort of inter-dimensional spaceship. I guess it could be as easy as going from:

(y*sin(a))^2
(y*cos(a))^2

to

((y-c)*sin(a))^2
((y-c)*cos(a))^2


I might experiment on this, but it won't be that interesting until getting out to 6 and 7D. You need a decent sized array to do some flying around, without losing too much of the shape. This would also allow for oblique cuts, by rotating and moving out from center. Hmm, oblique cuts for 5 and 6D toratopes.....
Last edited by ICN5D on Sun Mar 23, 2014 11:43 pm, edited 1 time in total.
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Re: The Tiger Explained

Postby Marek14 » Sat Mar 22, 2014 6:49 am

Well, you could put a translation parameter in one dimension and a rotation parameter in the other two...

So, what could be done with double tiger?

(((II)(II))(II)) would have the equation sqrt((sqrt(x^2 + y^2) - Ra)^2 + (sqrt(z^2 + w^2) - Rb)^2 - Rc)^2 + (sqrt(v^2 + u^2) - Rd)^2 = Re^2

The basic sliding cut should be sqrt((sqrt(x^2 + a^2) - Ra)^2 + (sqrt(y^2 + b^2) - Rb)^2 - Rc)^2 + (sqrt(z^2 + c^2) - Rd)^2 = Re^2... but it still doesn't show the toruses, just blobs, no matter how I set the radii... Even Mathematica doesn't render them as toruses... there might be something different here.
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Re: The Tiger Explained

Postby ICN5D » Sat Mar 22, 2014 7:30 am

Yeah, that's what I was wondering. I tried my idea in the previous post, but it only translated the whole array around, no cutting. But, I'll tell you what I did find, is that the sine and cosine make a cyclic repetition when run in animation. If you make the parameter mins and maxes in multiples of pi, then offset both with different multiples, the resulting animation is amazing. It basically goes through many different oblique cuts, with one rotating faster than the other, and a few "harmonic" cuts emerging for a moment, while both cut planes are in the same place. It's a really awesome run through of many of the distinct topological shapes from oblique cuts.

Try setting 0 < a < 9.42 and -1.57 < b < 1.57 , and hit animate. One of the convergent cuts is in the empty zone, and looks really neat when rotating in and out of it.

And, check out 0 < a < 9.42 and 0 < b < 3.14, this one has completely different sequences. Kind of an unanticipated delight, these cyclic animations of rotating
planes.



I HAVE to add this one:

set :
-1.57 < a < 7.85
0 < b < 15.7

This one is very unique. The two planes seem to follow each other around the shape while tracing out both symmetry structures at the same time. As you can tell, the cycles are 3 by five, where 3 is turned 180 to the 5. Something is going on here with this specific frequency of rates and offset. The trace-out of the frame is awesome, and very smooth with nice transitions and pause points at the midsections.
Last edited by ICN5D on Sat Mar 22, 2014 8:45 am, edited 1 time in total.
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Re: The Tiger Explained

Postby Marek14 » Sat Mar 22, 2014 7:54 am

Yes, should be similar principle to Lissajous curves.
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Re: The Tiger Explained

Postby Marek14 » Sat Mar 22, 2014 4:04 pm

Thought a bit about the order notion. Basically, looking at toratopes through their order is the same as ignoring their torus parts and be only interested in the tiger operations.

So:

The point order is made of sphere, torus, ditorus, tritorus etc... toratopes that don't use the tiger operation at all. Their structural graph is single line with no branches.

The sphere order is a tree with a single branching. The trace shows the structure:
Dimension of the sphere corresponds to the size of the branching - circles for tigers, spheres for trigers, glomes for tetrigers etc.
Size of the array corresponds to length of the branches - 2x2 array is tiger, 4x2 is torus tiger, 8x2 ditorus tiger, 4x4 duotorus tiger, 4x4x2 duotorus triger, etc.
Number of concentric spheres at one spot corresnponds to the length of branch above the branching - 2x2 with 1 sphere is tiger, with 2 spheres tiger torus, with 3 spheres tiger ditorus etc.

The torus order is a tree with two branchings. The horizontal/vertical dimensions of the torus correspond to size of lower/upper branching, size of the array corresponds to various terminal branches and major/minor diameter pairs correspond to length of branch between the branchings/the top branch.

This way, the understanding of toratopes can be shifted one level higher.
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Re: The Tiger Explained

Postby ICN5D » Sun Mar 23, 2014 1:33 am

Well, unfortunately this poor program can't do tritorus very well . Strange, because it can do 4 vertical torii, but can't do a horizontal line. I also looked into (((II)(II))I) tiger torus, and I can't seem to find what the problem is. I changed the radii many times and messed with brackets. I did discover some really strange 5D shapes that were hyperbolic, though.


EDIT: I found a good one for tritorus:

(sqrt((sqrt((sqrt(x^2 + a^2) - 9.0)^2 + y^2 ) - 5)^2 + b^2) - 2)^2 + z^2 = 1.2^2

Set all XYZ min/max to -17 / 17


This is a neat rotation equation, check it out:
- You can go through all sequences from cut to cut by setting 0 < a,b < 1.57, and alternate the sliders at their endpoints.

(sqrt((sqrt((sqrt(x^2 + (y*cos(a))^2) - 9.0)^2 + (y*sin(a))^2 ) - 5)^2 + (z*cos(b))^2) - 2)^2 + (z*sin(b))^2 = 1.2^2
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Re: The Tiger Explained

Postby Marek14 » Sun Mar 23, 2014 5:35 am

Yes, looks nice :)
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Re: The Tiger Explained

Postby ICN5D » Sun Mar 23, 2014 5:44 am

(((II)I)((II)I)) rotation
(sqrt((sqrt((x*sin(a))^2 + (z*cos(c))^2) - 2)^2 + (y*cos(b))^2) -1)^2 + (sqrt((sqrt((y*sin(b))^2 + (x*cos(a))^2) - 2)^2 + (z*sin(c))^2) -1)^2 = 0.4^2

xyz min/max :-5 / +5

0.746 < a 2.316
1.57 < b < 3.14
0 < c < 1.57

All sliders left make amazing structure at perfect cross between both hollow torus products. Probably the best cross section to represent as pic! And the first 6D oblique cross section I've ever explored, plus the others.
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Re: The Tiger Explained

Postby Marek14 » Sun Mar 23, 2014 6:30 am

Yes, the transformations seem to show all possible 3D cross-sections that can stem from the duotiger torus's trace, a 4x4 array of circles -- 2x4 array of toruses or four major pairs of toruses in vertical line, both in two possible orientations. (The final 3D cut would be 16 spheres in 4x4 array, but that won't arise with this shape, you'd need a 7D duotorus tiger (((II)I)((II)I)I).)
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