The Tiger Explained

Discussion of shapes with curves and holes in various dimensions.

Re: The Tiger Explained

Postby ICN5D » Sun Mar 23, 2014 5:12 pm

Have you figured out how to move out from center with rotated planes? Also, what rendering program are you using? I remember an old 3D graphing calculator on the macintoshes at my junior high school ( 15 yrs ago ) that would have been great. It allowed parametric and implicit equations, and had great rendering speed. Wonder where I can find it now?
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Re: The Tiger Explained

Postby student91 » Sun Mar 23, 2014 5:16 pm

I've been using G-calc. It's pretty the same as what you're describing, although my version is blocked because it should run on an old java-version. Anyway, The time I did use it, it worked great
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Re: The Tiger Explained

Postby ICN5D » Sun Mar 23, 2014 7:57 pm

G-calc? I'll have to check that one out.


Found a really interesting and beautiful 3D structure from one of the (((II)I)((II)I)) duotorus tiger cuts. It's a single connected object, probably some tri-diagonal oblique cut that lies in between all symmetries.

Use equation: (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

Set:

XYZ min/max = -5 / +5

a=11.652
b=10.522
c=6.836

Here are some quick go to pics:

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Last edited by ICN5D on Sun Mar 23, 2014 8:18 pm, edited 4 times in total.
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Re: The Tiger Explained

Postby Marek14 » Sun Mar 23, 2014 8:00 pm

Hm, I can't see any pics...
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Re: The Tiger Explained

Postby ICN5D » Sun Mar 23, 2014 8:12 pm

resized them, they were too big from FB
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Re: The Tiger Explained

Postby student91 » Sun Mar 23, 2014 8:14 pm

ICN5D wrote:G-calc? I'll have to check that one out.
nvm, I just read you were searching for a 3D-graphing calculator, G-cals is mostly 2D (It has asmall 3D-implication, but that is very primitive)
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Re: The Tiger Explained

Postby Keiji » Sun Mar 23, 2014 8:15 pm

You do know you can take screenshots in Windows 8 using Windows key + Print Screen (or Windows button + Volume down on Surface), right? That's much better quality than taking photographs of your screen. :)

(They'll appear in a Screenshots folder in your My Pictures folder.)
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Re: The Tiger Explained

Postby ICN5D » Sun Mar 23, 2014 8:20 pm

You do know you can take screenshots in Windows 8 using Windows key + Print Screen (or Windows button + Volume down on Surface), right? That's much better quality than taking photographs of your screen. :)

(They'll appear in a Screenshots folder in your My Pictures folder.)


No, I didn't :) That's what I've been looking to do, it's a much better way! I can't stand having to take pictures of my stupid monitor :) Thanks a million!
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Re: The Tiger Explained

Postby ICN5D » Sun Mar 23, 2014 10:35 pm

Okay, forget those last pics. These are what I wanted to show everyone. Ladies and gentleman, introducing the duotorus tiger (((II)I)((II)I)) :

The equation for this shape is:

(sqrt((sqrt((x*sin(11.652))^2 + (z*cos(6.836))^2) - 2)^2 + (y*cos(10.522))^2) -1)^2 + (sqrt((sqrt((y*sin(10.522))^2 + (x*cos(11.652))^2) - 2)^2 + (z*sin(6.836))^2) -1)^2 - 0.4^2 = 0



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And these:

The equation for this shape:

(sqrt((sqrt((x*sin(0.746))^2 - 2)^2 + (y*cos(1.57))^2) -1)^2 + (sqrt((sqrt((y*sin(1.57))^2 + (x*cos(0.746))^2) - 2)^2 - 1)^2 - 0.4^2 = 0




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Re: The Tiger Explained

Postby ICN5D » Mon Mar 24, 2014 5:46 am

Made some more cross section renders of duotorus tiger. Had to do the two axial cuts, of course. Can't forget about them! Along with the 1/4 values of pi angles as well. Interchanging those values between the three axes made some interesting cuts. I've definitely explored this shape thoroughly enough by now.



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Re: The Tiger Explained

Postby Marek14 » Mon Mar 24, 2014 5:58 am

Very nice :) Should be put on wiki...
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Re: The Tiger Explained

Postby ICN5D » Mon Mar 24, 2014 8:30 am

I should probably just be uploading them. It'll be better quality, anyways. How do I get an uploaded image to show up on the post like the [img] tag?
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Re: The Tiger Explained

Postby Marek14 » Mon Mar 24, 2014 9:51 am

Nomenclature update:

Exploring higher-dimensional toratopes showed that the names get increasingly cumbersome. The current system would specify toratopes with long descriptive names and long strings of numbers.

So let's have a look at it.

The two basic terms are torus and tiger. Sphere is also a toratope, but the name only refers to one species of toratopes.

Torus is based on an existing toratope. "Sphere torus" gets the "sphere" part taken out, so we have torus ((II)I), tiger torus (((II)(II))I) etc. "Torus torus" is shortened to "ditorus", "Ditorus torus" to tritorus, etc.

Tiger is based on combination of multiple toratopes. The noun (tiger, triger, tetriger, pentiger, etc.) refers to number of terms in the product. If the number of specified terms is lower, the others are automatically assumed to be spheres.

This basic notation (amended with parenthesis and slashes) grants us the ability to name the various toratope species. To name the individual toratopes, we need to specify the dimensions. I propose modifying the existing system so every word would have its own dimensions appended.

For example, the basic torus tiger (((II)I)(II)) was 2120-torus tiger in my original notation. Now it should be called 21-torus 20-tiger -- the dimensions of torus parts are added to the "torus" term, while the remaining dimensions are appended to "tiger". The same way, the basic tiger torus would be 220-tiger 1-torus.

The "double tiger" nomenclature which replaced multiple consecutive instances of the word "tiger" might have to be abandoned now. With dimensions specified, (((II)(II))(II)) would be 220-tiger 20-tiger and (((II)(II))((II)(II))) would be 220-tiger/220-tiger 0-tiger.

Order nomenclature:

However, I'm developing a new kind of nomenclature for toratopes, so called "order nomenclature", based on their traces.
As I already mentioned (hopefully), a TRACE of toratope is its lowest-dimensional nonempty cut. All toratopes of the same species will have identical trace. Order of toratopes is then a set of all toratopes who share the same species of toratopes in their trace.

Toroids:
The basic term in order nomenclature is "toroid". A toroid is a toratope whose trace is a linear arrangement of points. They are exactly those toratopes that never require a tiger construction.
Sphere is a 0-toroid. Torus is a 1-toroid. Ditorus is a 2-toroid, etc. A trace of n-toroid is a linear arrangement of 2^(n+1) points.
A 2D trace of a n-toroid are various linear arrangements of 2^n circles. A 3D trace can have the circles expand into spheres or join pairwise into toruses, etc.
When exact dimensions are to be specified, they are to be placed in brackets. A 3D sphere is 0-toroid[3], a ditorus is 2-toroid[211], etc. This is because in this nomenclature, the exact dimensions are no longer that important as the "fat" toratopes (the ones with extra dimensions) will no longer be that important and we'll be usually working only with the basic members of species.

Lowest dimension of n-toroid is n+2.

Tigroid:
Tigroid is the basic term for toratopes with single tiger construction. While toroid can be described by a single number, tigroid requires at least three numbers, and possibly an unlimited amount. A trace of a tigroid is an arrangement of spheres.
The basic tiger whose trace is four circles in 2x2 array is (00)0-tigroid. This should be understood as a tigroid with branches of 0-toroid (sphere) and 0-toroid (another sphere), which has a torus transformation then applied 0-times (i.e. not at all). A trace of (ab)c tigroid would be a 2^(a+1) x 2^(b+1) array of groups of 2^c concentric circles.
So, a torus tiger would be (01)0-tigroid (trace 2x4 array of circles), a tiger torus (00)1-tigroid (trace 2x2 array of pairs of concentric circles), duotorus tiger would be (11)0-tigroid and a triger would be (000)0-tigroid.
220-tiger would be specified as (00)0-tigroid[(2,2),0] while 43-torus 21-torus 3-tiger would be (11)0-tigroid[(43,21),3]

Lowest dimension of (abcd...)z-tigroid should be (a+2) + (b+2) + (c+2) + ... + z. So a torus/ditorus/tritorus triger tetratorus, (321)4-tigroid, should exist in 16-dimensional space.

Basically, the order nomenclature takes the previous nomenclature and shifts it one "level" higher.
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Re: The Tiger Explained

Postby ICN5D » Tue Mar 25, 2014 4:20 am

Here's a quick little cool slideshow with a rotation animation in it. In tribute to the amazing shape (((II)I)((II)I)) that has been thoroughly explored over the past few days.
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Re: The Tiger Explained

Postby ICN5D » Tue Mar 25, 2014 4:34 am

So, have you figured out how to make an equation that moves out from center and rotates? That one would be the holy grail of high D exploration :)
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Re: The Tiger Explained

Postby Marek14 » Tue Mar 25, 2014 6:30 am

ICN5D wrote:So, have you figured out how to make an equation that moves out from center and rotates? That one would be the holy grail of high D exploration :)


Isn't it enough to have some parameters linear and some angular?
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Re: The Tiger Explained

Postby ICN5D » Tue Mar 25, 2014 6:52 am

Oh, absolutely! The rotation equations are mindblowing, no doubt. I was just curious if you ever found it. On another note, I investigated the formula for tiger torus (((II)(II))I) , and it's:

(((sqrt(x^2 + y^2) - 2.5)^2 + (sqrt(z^2 + a^2) - 2.5)^2 - 1.2)^2 + b^2) - 0.75^2 = 0

(((sqrt(x^2 + a^2) - 2.5)^2 + (sqrt(y^2 + b^2) - 2.5)^2 - 1.2)^2 + z^2) - 0.75^2 = 0


I think some brackets were missing somewhere. But, anyways, made some cool pics:
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Re: The Tiger Explained

Postby Marek14 » Tue Mar 25, 2014 7:04 am

Well, the rotation + translation movement you want to do is basically equivalent to moving in a helix.

Now, the parametric equations of the helix are:

x(t) = a*cos(t)
y(t) = a*sin(t)
z(t) = b*t

So, you might replace three variables with a*cos(c), a*sin(c) and b*c, then set parameters a and b to particular numbers and animate c.
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Re: The Tiger Explained

Postby Marek14 » Tue Mar 25, 2014 11:57 am

Logically speaking, it seems that the most interesting rotation animations would be produced by toroids and simple tigroids. They are the only toratopes with less than 3-dimensional trace, and thus multiple interesting 3D cuts.

In 6D, there are 3 more simple tigroids apart from duotorus tiger (if we pass by triger for now):

Ditorus tiger ((((II)I)I)(II)): Trace is 2x8 array of circles ((((I)))(I)) leading to 4 possible 3D cuts:
Vertical stack of 2 major quartets of toruses ((((II)))(I))
2 vertical stacks of 2 major pairs of toruses ((((I)I))(I))
4 vertical stacks of 2 toruses ((((I))I)(I))
Vertical stack of 8 toruses ((((I)))(II))

Torus tiger torus ((((II)I)(II))I): Trace is 2x4 array of pairs of concentric circles ((((I))(I))) leading to 4 possible 3D cuts:
Vertical stack of 2 major/minor quartets of toruses ((((II))(I)))
Two vertical stacks of 2 minor pairs of toruses ((((I)I)(I)))
Vertical stack of 4 minor pairs of toruses ((((I))(II)))
2x4 array of toruses ((((I))(I))I)

Tiger ditorus ((((II)(II))I)I): Trace is 2x2 array of quartets of concentric circles ((((I)(I)))) leading to 4 possible 3D cuts:
Vertical stack of 2 minor quartets of toruses ((((II)(I))))
Vertical stack of 2 minor quartets of toruses in different orientation ((((I)(II))))
2x2 array of minor pairs of toruses ((((I)(I))I))
2x2 array of major pairs of toruses ((((I)(I)))I)

In all cases there's also a 5th possible 3D cut where the circles become spheres; but that requires at least 7D since there has to be a minor dimension for tiger for this cut to exist.

Your study of torus tiger and tiger torus could have this cut added if you'd move to 21-torus 21-tiger (((II)I)(II)I) and 221-tiger 1-torus (((II)(II)I)I)
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Re: The Tiger Explained

Postby quickfur » Tue Mar 25, 2014 7:59 pm

ICN5D wrote:I should probably just be uploading them. It'll be better quality, anyways. How do I get an uploaded image to show up on the post like the [img] tag?

1) Sign up for a wiki account (click on the "wiki" link at the top of this page, go to signup / login).
2) Login, and click on "add file". Note that you can add multiple files at a time.
3) Once uploaded, you should see links to pages of the individual uploaded files. On each of the pages, you should see the uploaded image; use your browser's function to copy the image address (or click on "view this file on its own" and copy the address from your browser's address bar).
4) Use these URLs wherever you want to show these images. :)

Note that on the wiki, each image file has a unique hash ID (that long string of letters & digits shown after "file hash:" on the image's page). It doesn't remember the original filenames, so for the purposes of uploading, you don't have to bother with naming any of the images. The hash ID is your key to that specific version of that specific image. (So don't lose it! :P)

The best way to not lose it, is to create a wiki page that references those images. You embed images in the wiki page by writing:
Code: Select all
<[#embed [hash ...]]>

where the ... is the hash ID of the image.

And finally... wow!!! I'm glad you found a program that can render what you want, 'cos it's extremely cool!!! These crazy toratope sections are amazing. Hope to see more of them! ;)
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Re: The Tiger Explained

Postby Keiji » Tue Mar 25, 2014 10:00 pm

quickfur wrote:
ICN5D wrote:I should probably just be uploading them. It'll be better quality, anyways. How do I get an uploaded image to show up on the post like the [img] tag?

1) Sign up for a wiki account (click on the "wiki" link at the top of this page, go to signup / login).


For the umpteenth time, this will not work as wiki accounts are invite-only. If you want to experience this for yourself, why not go to Wiki > My account > Invites, make a new invite code, and PM it to ICN5D :P

Also, a nitpick: it's just a file hash, not a file hash ID. :)
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Re: The Tiger Explained

Postby Marek14 » Tue Mar 25, 2014 10:35 pm

Or, if you want to move the view in a spiral (rotate the plane and move it out of the center), then the Archimedean spiral should have parametric equations
x = a*t*sin(t)
y = a*t*cos(t)

By manipulating the parameter a you get loosely or tightly wound spiral.
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Re: The Tiger Explained

Postby quickfur » Tue Mar 25, 2014 11:21 pm

Keiji wrote:
quickfur wrote:
ICN5D wrote:I should probably just be uploading them. It'll be better quality, anyways. How do I get an uploaded image to show up on the post like the [img] tag?

1) Sign up for a wiki account (click on the "wiki" link at the top of this page, go to signup / login).


For the umpteenth time, this will not work as wiki accounts are invite-only. If you want to experience this for yourself, why not go to Wiki > My account > Invites, make a new invite code, and PM it to ICN5D :P

Also, a nitpick: it's just a file hash, not a file hash ID. :)

Right, I keep forgetting. :( :oops:

Anyway, got an invite code, will pm it now.
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Re: The Tiger Explained

Postby ICN5D » Wed Mar 26, 2014 1:39 am

Awesome, I'm glad everyone likes them :) I thought they were way cool, too. It's a very unique way to express and illustrate high-D concepts. And, just to think, all of those cuts come from one equation!



Or, if you want to move the view in a spiral (rotate the plane and move it out of the center), then the Archimedean spiral should have parametric equations
x = a*t*sin(t)
y = a*t*cos(t)

By manipulating the parameter a you get loosely or tightly wound spiral.


Wow, that sounds amazing! I wonder what it would look like? I'll have to play around with that later. I was thinking more about the application of axial plus angular control, and what I realized is that it would be a great way to explore empty cuts, more to derive their oblique cuts. Once we move out from center, we hit structure, which can then be angle-sliced. Now, something that I visualized happening here is that if this was done to an empty cut of an 8 or 9D toratope, we might see a static row of cut shapes that go through an axial cut sequence. Or, an array of cut shapes that also show an entire axial cut sequence or even two of them. Both cases in where the cut plane is stationary, it's just the nature of cutting through an array of high-D toratopes. We can have an angle slicing at different heights along the sequence. This is of course in the future, I'm not quite yet ready to derive 9D equations. But, I'm the type to push the limits as you can tell, high-D doesn't scare me. I like to experiment with proof of concept before really applying it.
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Re: The Tiger Explained

Postby ICN5D » Thu Mar 27, 2014 1:36 am

Made a new slideshow, trying out some animation skills:


http://www.youtube.com/watch?v=7Io2aiWqgYU
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Re: The Tiger Explained

Postby ICN5D » Thu Mar 27, 2014 3:25 am

Some more cool pics from five dimensional outer space:



(((II)I)II) - 212-Ditorus


Image


The axial midcut of four spheres along a line: (((I))II)

Image





A really neat oblique artifact at an angle between the four spheres and the concentric toruses

Image





Another neat midsection smoothing out the transition between axial cuts

Image





This is a trimming of the cocircular toruses by rotation , not restricting the bounding box! We've rotated a higher plane than 4D, very cool.

Image






Another rotation induced trimming example, this time from top to bottom. This trimming effect is from a sphere symmetry, manifested in a very unique way.

Image






This is another really cool double-trimming midsection angle, with a little bit of both combined, making for another oblique gem.

Image

Image
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Re: The Tiger Explained

Postby Marek14 » Thu Mar 27, 2014 4:03 am

Nice, though this should still be the 212-ditorus. Toruses are the "simplest" shapes so they can be kept as simple number strings. Only when toruses and tigers start to combine in various ways will the numbers start getting confusing.
Also, 21-torus 22-torus is confusing since you have 4 numbers there for some reason, but there are only 3 diameters.
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Re: The Tiger Explained

Postby ICN5D » Thu Mar 27, 2014 4:55 am

ah, fixing......
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Re: The Tiger Explained

Postby ICN5D » Thu Mar 27, 2014 7:31 am

Wow, just made the rotation equation of ((((II)I)I)(II)) , and let me tell you, it's even more complex and amazing than (((II)I)((II)I)). There's some incredible things I'm seeing with the interplay of the rotations, and something I've never seen before. It's a rotation through an empty cut that scans along a structure, but in a narrow trimmed down line of sight. It's quite fascinating. Exploring this one is going to take some time, there's so many incredible looking oblique structures.

The equation I found that works is:

(sqrt((sqrt((sqrt((x*sin(b))^2 + (z*cos(a))^2) - 4.5)^2 + (x*cos(b))^2) - 2.2)^2 + (y*sin(c))^2) - 1.1)^2 + (sqrt((z*sin(a))^2 + (y*cos(c))^2) - 2)^2 - 0.7^2 = 0

Xmin/max= -9,+9
Ymin/max= -9,+9
Zmin/max= -9,+9

0<a<1.57
0<b<1.57
0<c<1.57


Some axials and an oblique, just to give you a taste of what's to come....

Image

Image

Image

Image
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Re: The Tiger Explained

Postby Marek14 » Thu Mar 27, 2014 9:58 am

Here's another suggestion for your explorations: two toratopes chained together.

You can plot two toratopes, either as separate graphs, or with the use of fact that if you have two graphs F(x,y,z) = 0 and G(x,y,z) = 0, then F(x,y,z) * G(x,y,z) = 0 is both at once. And you can translate toratopes by replacing the coordinates by (coordinate - constant). For best results for oblique cuts, you should probably translate both of them, putting the origin halfway between their centers.

I think that the possible chains can be explored by cutting them and studying the cuts.
For example, two torispheres won't fit together: if you'd imagine each torisphere in its torus cut, then both toruses start to inflate and they will certainly and inevitably collide.
But a torisphere and a spheritorus would form a nice chain in this case: while the torisphere inflates, the spheritorus grows thinner and can disappear from the hole of torisphere before the torisphere fills the hole.
Another cut of this situation would see torisphere as two concentric spheres and spheritorus as two spheres -- one inside the hollow sphere and one out. Here, the two concentric spheres would merge and disappear before the spheritorus spheres touch.

How about two spheritoruses? Here, starting from the two chained toruses, both would grow thinner and disappear -- and I think they would be actually never truly linked since some other cuts of the same situation (like four spheres/shperoids or two spheres plus one torus) would have the shapes completely unlinked.

How would we go with tigers and ditoruses here?

Let's have a look at tiger first since its 3D cuts are remarkably simple - the 3D vertical stack of toruses.

Here, we could thread another torus through either one or both holes. If we thread it through just one hole, it can still lead to a valid shape if it's a spheritorus that disappears before both toruses of tiger merge. (Torisphere, of course, is right out.)
Another option is to thread the spheritorus through both holes. But the same can be also done with a torisphere -- all we need is for the tiger to disappear before the torisphere cut touches it, which should be possible for a sufficiently large torisphere and sufficiently small tiger.

How about two tigers chained together? We could have a small tiger whose two toruses hang on a single torus of a large tiger projection -- the small tiger would disappear before the large tiger merges. Having doubly-linked tigers also seems as a possibility -- imagine two very thin toruses close to each other for one tiger. Here, however, both could merge and disappear with no regards to the other, which could mean it's not a true link.

Now, the ditorus. It has three different 3D cuts, both made of two toruses.

Torisphere/ditorus: If we put the ditorus in (((xy)z)w) position, then torisphere has three possible orientations: ((xyz)w), ((xyw)z) and ((xzw)y)

Well, with first orientation we could have a ditorus put entirely inside of torisphere, but that's not really what we're looking for. With orientation 2, the w-cut doesn't seem to allow for any simple chaining since all toruses are parallel and z-cut is also suspicious since two concentric spheres can't be really chained with a minor pair of toruses. With orientation 3, however, the chain works:

x-cut is a torus threaded through one of the separated toruses of the ditorus cut. The ditorus disappears before the torisphere torus gets filled.
y-cut is ditorus cut with two separated toruses and one of them is encased in two concentric spheres. The spheres disappear before the two toruses merge.
z-cut is major pair of toruses and perpendicular torus passing through both of them. The ditorus disappears first.
w-cut is minor pair of toruses and perpendicular torus linked through them. The ditorus disappears first.

Spheritorus/ditorus: Once again, we put the ditorus in (((xy)z)w) position. The spheritorus has four possible orientations: ((xy)zw), ((xz)yw), ((xw)yz) and ((zw)xy).

Orientation 1: Doesn't really work, both shapes have parallel main circles.
Orientation 2: While w-cut has a chain, none of the others do -- seems that this is not a true chain.
Orientation 3: Seems similar to orientation 2, some cuts are not chains.
Orientation 4: Doesn't work either -- the problem is that only one cut of ditorus contains separate "outsides" (minor torus pair), while two cuts of spheritorus contain two separate pieces. For this reason, it seems that spheritorus could be manipulated into any position relatively to a ditorus just by moving. I am not 100% sure of this, I can imagine links where spheritorus and ditorus must disappear in certain order (for example torus cut of spheritorus linked to exactly one torus of major pair or of two separated toruses cut of ditorus.

I'll try to have a look at tiger/ditorus and ditorus/ditorus chains later. Do you have any ideas here?
Marek14
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