## Toratope Chains

Discussion of shapes with curves and holes in various dimensions.

### Toratope Chains

I wanted to showcase cool examples of them in action. And, wow! It's almost incomprehensible. How do those things fit together? It's so amazing, you start thinking about higher dimensions in a new way. These are unexpectedly cool, thanks to Marek for providing those equations. I can probably represent this as 2D slices of two torii, which would be a great analogy reference for what you see.

Spheritorus + Torisphere Lock

Big sphere eats a little sphere during rotation. Cool to see how both can be a torus in 3D, and we alternate between both kinds of links.

Tiger + Tiger Chain 180 degree flip

A full 180 flip, showing how two tigers can chain together.

EDIT 6/18: Actually, these tigers are not chained. I'm currently searching for an instance of tiger chaining .....

Tiger + Torisphere Chain

A sphere encapsulates a single torus, being the other intercept of the tiger.

Torisphere + Ditorus Chain

A torisphere locked together with a ditorus. Oh my god ..... that's so crazy. I don't even know what to say!

Last edited by ICN5D on Thu Jun 19, 2014 12:04 am, edited 1 time in total.
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ICN5D
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### Re: Toratope Chains

I think the spheritorus + torisphere should be also a "chain" instead of "lock". I don't have a formal definition at this point, but the lock "fits together" more tightly.

Note that a 4D blacksmith who makes chains would probably use tiger links for them -- otherwise he'd have to alternate spheritoruses and torispheres
Marek14
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### Re: Toratope Chains

Let's have a systematic look at this problem.

There are 5 4D toratopes, but we'll omit glome since it has no hole and cannot form chains.

So we have torisphere, ditorus, tiger and spheritorus.

How to recognize when two shapes are interlocked? Well, when we slice them, their images will never intersect, but they will be interlocked at some point. In addition, you can't separate them from this interlocked state by moving either of the shapes in 4D. For example, two spheritoruses can appear as interlocked toruses, but moving one of them in 4D will shrink the minor diameter into nothingness, after which they are separated.

Alternate condition is that the interlock must be evident in every direction.

I guess that "lock" is when centers of both toratopes can be coincident. In "chain", they have to be displaced.

Round 1: torisphere VS. torisphere
No interlock possible since the torisphere's hole has only one free dimension.

Round 2: torisphere vs. ditorus

Torisphere (sqrt(x^2 + y^2 + z^2) - 9)^2 + w^2 = 1
Ditorus (sqrt((sqrt(x^2 + y^2) - 9)^2 + w^2) - 3)^2 + z^2 = 1
x [-13,13]; y [-13,13]; z [-10,10]; w [-4,4]
This is the lock. In x and y-cut, the torisphere's torus passes through both toruses in ditorus cut.

Torisphere (sqrt(x^2 + y^2 + z^2) - 9)^2 + w^2 = 1
Ditorus (sqrt((sqrt((x-9)^2 + w^2) - 9)^2 + y^2) - 3)^2 + z^2 = 1
x [-10,22]; y [-10,10]; z [-10,10]; w [-13,13]
This is simple chain of torisphere and ditorus.

Round 3: ditorus vs. ditorus

Ditorus (sqrt((sqrt(x^2 + y^2) - 9)^2 + z^2) - 3)^2 + w^2 = 1
Ditorus (sqrt((sqrt(x^2 + y^2) - 6)^2 + w^2) - 3)^2 + z^2 = 1
x [-13,13]; y [-13,13]; z [-4,4]; w [-4,4]
This is the double-ditorus lock.

Round 4: torisphere vs. tiger

Torisphere (sqrt(x^2 + y^2 + z^2) - 9)^2 + w^2 = 1
Tiger (sqrt(x^2 + y^2) - 3)^2 + (sqrt((z-9)^2 + w^2) - 3)^2 = 1
x [-10,10]; y [-10,10]; z [-10,13]; w [-4,4]
This is the tiger/torisphere chain.

Round 5: ditorus vs. tiger

Ditorus (sqrt((sqrt(x^2 + y^2) - 9)^2 + z^2) - 3)^2 + w^2 = 1
Tiger (sqrt(x^2 + y^2) - 9)^2 + (sqrt((z-3)^2 + w^2) - 3)^2 = 1
x [-13,13]; y [-13,13]; z [-4,7]; w [-10,10]
This is the ditorus/tiger chain.

Round 6: tiger vs. tiger

Tiger (sqrt(x^2 + y^2) - 6)^2 + (sqrt(z^2 + w^2) - 3)^2 = 1
Tiger (sqrt((x-6)^2 + z^2) - 6)^2 + (sqrt(y^2 + w^2) - 3)^2 = 1
x [-7,13]; y [-7,7]; z [-7,7]; w [-4,4]
This could be the tiger/tiger chain, couldn't it? I thought so before, but when we explore it more closely, we realize that it's not really a chain -- the two tigers are actually independent and can be separated. If you look at the X-cut, it will become clear.

So, how to chain two tigers? Well, you could do this:
Tiger (sqrt(x^2 + y^2) - 6)^2 + (sqrt(z^2 + w^2) - 3)^2 = 1
Tiger (sqrt((x-6)^2 + z^2) - 6)^2 + (sqrt((y-6)^2 + w^2) - 6)^2 = 1
x [-7,13]; y [-7,13]; z [-7,7]; w [-7,7]
But the y-cut shows that these two tigers aren't chained either.

Let's think about this. A tiger has two holes, both with two free dimensions. What happens if we try to thread a second tiger obliquely, so it passes through both holes at once?

No time to finish this thought now. Let me leave you with my best shot:
(sqrt((y*sqrt(2)/2 + w*sqrt(2)/2)^2 + (x-9)^2) - 9)^2 + (sqrt((y*sqrt(2)/2 - w*sqrt(2)/2)^2 + z^2) - 9)^2 = 0.5
(sqrt((x*sqrt(2)/2 + w*sqrt(2)/2)^2 + y^2) - 9)^2 + (sqrt((x*sqrt(2)/2 - w*sqrt(2)/2)^2 + z^2) - 9)^2 = 0.5

The W-cut intersects, sadly. But perhaps there's some small adjustment that would make it work?
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### Re: Toratope Chains

This is cool stuff. It illustrates the different kinds of holes that are possible in 4D and beyond, and how objects with different kinds of holes can interlock with each other. Fascinating stuff.
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### Re: Toratope Chains

So, that " Tiger + Tiger Chain 180 degree flip " animation isn't a tiger chain? Hmmm. How to link either of the torii together, and have them remain chained after rotation? It would seem that one has to be turned 90 degrees and offset a little.

For example, two spheritoruses can appear as interlocked toruses, but moving one of them in 4D will shrink the minor diameter into nothingness, after which they are separated.

Wouldn't it only appear this way, in 3D? If chained in 4D, we only see a 3D slice of them, where some cuts appear as if they can slip past one another, and unhook. A 2D cut rotation of 2 torii looks the same, one morphs into concentric circles, while the other morphs into two displaced circles. But, they're still linked in 3D.

Let's think about this. A tiger has two holes, both with two free dimensions. What happens if we try to thread a second tiger obliquely, so it passes through both holes at once?

No time to finish this thought now. Let me leave you with my best shot:
(sqrt((y*sqrt(2)/2 + w*sqrt(2)/2)^2 + (x-9)^2) - 9)^2 + (sqrt((y*sqrt(2)/2 - w*sqrt(2)/2)^2 + z^2) - 9)^2 = 0.5
(sqrt((x*sqrt(2)/2 + w*sqrt(2)/2)^2 + y^2) - 9)^2 + (sqrt((x*sqrt(2)/2 - w*sqrt(2)/2)^2 + z^2) - 9)^2 = 0.5

The W-cut intersects, sadly. But perhaps there's some small adjustment that would make it work?

I'm looking into this.....

quickfur wrote:This is cool stuff. It illustrates the different kinds of holes that are possible in 4D and beyond, and how objects with different kinds of holes can interlock with each other. Fascinating stuff.

Yeah it is, isn't it? I thought so, too! I've read about 4D links before, but without visuals, it's tough to see. Now after seeing the concept displayed, I can see many other instances. Then, of course, there's 5D chains!
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### Re: Toratope Chains

If you make a cut of two linked toruses in 2D, you'll see a pair of circles from one, and two circles from the other, one inside, one outside. If you try to move one of the toruses in 3D without moving the other, you find that this isn't possible -- at certain point, they would intersect.

In case of torisphere/spheritorus chain, you have a cut where a spheritorus is two spheres, one surrounded by the two concentric spheres of torisphere. From this configuration, they can't escape.

But two spheritoruses are too "thin" to be locked in this way. Let's have a look at something we haven't seen so far: a 4D animation!

(sqrt(x^2 + y^2) - 3)^2 + z^2 + w^2 = 1
(sqrt((x-3)^2 + z^2) - 3)^2 + y^2 + (w-t)^2 = 1

t is time. When you set parameter b to t, you can watch two spheritoruses moving in 4D between "chained" and "unchained"position without intersecting.

X-cut:
(sqrt(a^2 + y^2) - 3)^2 + z^2 + x^2 = 1
(sqrt((a-3)^2 + z^2) - 3)^2 + y^2 + (x-b)^2 = 1

Y-cut:
(sqrt(x^2 + a^2) - 3)^2 + z^2 + y^2 = 1
(sqrt((x-3)^2 + z^2) - 3)^2 + a^2 + (y-b)^2 = 1

Z-cut:
(sqrt(x^2 + y^2) - 3)^2 + a^2 + z^2 = 1
(sqrt((x-3)^2 + a^2) - 3)^2 + y^2 + (z-b)^2 = 1

W-cut:
(sqrt(x^2 + y^2) - 3)^2 + z^2 + a^2 = 1
(sqrt((x-3)^2 + z^2) - 3)^2 + y^2 + (a-b)^2 = 1
Marek14
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### Re: Toratope Chains

Well now, that's interesting. I guess I was thinking in 3D! It makes sense, now that I see it. At least, I'm trying to wrap my head around it. Meditating on these GIFs will help make the concept more clear. When the cuts are in 3D, we still have an extra direction. Unlike a 2D slice of two linked torii, which still remains linked in 2D. That caught me off-guard.

X-cut:
(sqrt(a^2 + y^2) - 3)^2 + z^2 + x^2 = 1
(sqrt((a-3)^2 + z^2) - 3)^2 + y^2 + (x-b)^2 = 1

Y-cut:
(sqrt(x^2 + a^2) - 3)^2 + z^2 + y^2 = 1
(sqrt((x-3)^2 + z^2) - 3)^2 + a^2 + (y-b)^2 = 1

Z-cut:
(sqrt(x^2 + y^2) - 3)^2 + a^2 + z^2 = 1
(sqrt((x-3)^2 + a^2) - 3)^2 + y^2 + (z-b)^2 = 1

These two are identical. One or the other spheritorus moves sideways, as the only difference.

W-cut:
(sqrt(x^2 + y^2) - 3)^2 + z^2 + a^2 = 1
(sqrt((x-3)^2 + z^2) - 3)^2 + y^2 + (a-b)^2 = 1

I missed a sequence in this one. The left spheritorus deflates and vanishes, the other stays put, to illustrate the lack of linking.

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### Re: Toratope Chains

Okay, explored a little tiger chaining. I don't think it's possible, with its unique topology. Those things are totally off the leash, unchainable and untrainable. I hope I'm wrong, though. I'd love to see that animation . I keep thinking, if I could manage to thread just one torus to another, it'll work. But, in two orientations it doesn't work. I'm trying to find some consistent solid region, that stays put during rotation.
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### Re: Toratope Chains

Well, this is as close as I can get. I think if the vertical stacks could be tilted in 3D, a chain is possible. The columns rotate around four circles in a square, and by overlapping those two planes together and perpendicular, and tilting the tigers according to the 3D cut, both torii will remain linked in both axial slices.

This is the oblique of both, mid-rotation. We can probably call this a pseudochain, or something.

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### Re: Toratope Chains

Arbitrary two-torus chains:

Thinking about this, we can derive behaviour of arbitrary two-torus chains from the 3D model. Or, heck, even from the 2D one.

One torus can be described by two numbers -- basic torus is 21-torus, etc. Two toruses can be described by a 2x2 matrix of four numbers: ((a,b),(c,d)), where we have a (a+b),(c+d)-torus and a (a+c),(b+d)-torus.

Now, all possible 2D or 3D combinations of two toruses can be derived from this matrix:

2D:
((2,0),(0,0)) - 2,0 + 2,0 - pair of circles / pair of circles.
((1,1),(0,0))/((1,0),(1,0)) - 2,0 + 1,1 - pair of circles / two circles. These can be chained if one circle is inside the pair and the other is outside.
((1,0),(0,1)) - 1,1 + 1,1 - two circles / two circles. In this configuration, they form two parallel lines.
((0,2),(0,0))/((0,0),(2,0)) - 2,1 + 0,2 - pair of circles / empty.
((0,1),(1,0)) - 1,1 + 1,1 - two circles / two circles. In this configuration, they form two perpendicular lines.
((0,1),(0,1))/((0,0),(1,1)) - 1,1 + 0,2 - two circles / empty.
((0,0),(0,2)) - 0,2 + 0,2 - empty / empty.

The only cut where we can achieve chaining is therefore ((1,1),(0,0)) or ((1,0),(1,0)). The others are not relevant for this (note that we can modify the cuts by moving the toruses relatively to each other in "third dimension").

How can the ((1,1),(0,0)) cut evolve? There are four options:

((2,1),(0,0)): here, both toruses evolve in major dimension into pair of spheres / torus. But in this evolution, they would necessarily intersect at some point, and therefore it's disallowed.
((1,2),(0,0)): major/minor evolution into pair of spheres / two spheres. This preserves the chain. In 3D version we'll get a sphere enclosed by pair of spheres and second sphere outside.
((2,0),(1,0)): minor/major evolution into torus / torus. This preserves the chain. In 3D version we'll get two chained toruses.
((2,0),(0,1)): minor/minor evolution into torus / two spheres. This breaks the chain. In 3D version we'll get a torus and two spheres which cannot be chained.

So, any chained pair of toruses, in arbitrary dimension, should be of form ((1,x),(y,0)), i.e. (x+1),y-torus and (y+1),x-torus. This means that in 5D we can chain 41-torus with 23-torus or two 32-toruses together, in 6D we can chain 51-torus with 24-torus or 42-torus with 33-torus and in 7D we can chain 61-torus with 25-torus, 52-torus with 34-torus or two 43-toruses.

Let's try an experiment with two 32-toruses, divided as ((1,2),(2,0)):

(sqrt(x^2 + y^2 + z^2) - 3)^2 + w^2 + v^2 = 1
(sqrt(x^2 + w^2 + v^2) - 3)^2 + y^2 + z^2 = 1

If we implement it as
(sqrt(x^2 + y^2 + a^2) - 3)^2 + z^2 + b^2 = 1
(sqrt((x-3)^2 + z^2 + b^2) - 3)^2 + y^2 + a^2 = 1

we can see the chain
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### Re: Toratope Chains

As an aside, we didn't even need to go into 2D -- since torus has 1D trace, we could conceivably resolve the problem in 1D, where the basic chain looks like ((1,0),(0,0)), as two points from torus A - two points from torus B - two points from torus A - two points from torus B, with the allowed chain-preserving evolutions into ((1,1),(0,0) and ((1,0),(1,0)).

So, next we'll try to chain a torus and a ditorus. This time, the matrix will have 6 numbers: ((a,b,c),(d,e,f)) for a (a+b+c),(d+e+f)-torus and (a+d),(b+e),(c+f)-ditorus.

In this situation, we WILL start from 1D case ((1,0,0),(0,0,0)) to see what will happen.

In 1D, we can record the configuration using two T's (representing pair of torus points) and four D's (representing pair of ditorus points). Wven evolving into 2D, torus has major evolution into pair of circles and minor evolution into two circles. Ditorus has major evolution in quartet of circles, medium evolution in two pairs of circles and minor evolution into four circles:

TTDDDD - no chain.

TDTDDD - major/minor, minor/major and minor/medium preserve the chain, leading to a group of chains where torus is threaded through the outer hole of ditorus. This is ((1,0,a),(b,c,0)) configuration, i.e. (a+1),(b+c)-torus and (b+1),c,a-ditorus. In 4D, the only option would be spheritorus/ditorus, in 5D you could have 23-torus/311-ditorus, 23-torus/221-ditorus and 32-torus/212-ditorus.

TDDTDD - major/medium, major/minor and minor/major preserve the chain, leading to a group of chains where torus is threaded through the inner hole of ditorus. This is ((1,a,b),(c,0,0)) configuration, i.e. (a+b+1),c-torus and (c+1),a,b-ditorus. In 4D, the only option would be torisphere/ditorus, in 5D you could have 41-torus/221-ditorus, 41-torus/212-ditorus and 32-torus/311-ditorus.

TDDDTD - major/minor, minor/major and minor/medium preserve the chain. In fact, this is identical to TDTDDD configuration, we just rotate the torus around. Nothing new here.

TDDDDT - major/major, major/medium and major/minor preserve the chain. That means the configuration is ((a+1,b,c),(0,0,0)) - and this is a problem since it means no actual connected torus can appear there. This is only valid for torus degenerated in pair of spheres, similarly to ABBA "chain" of two toruses.

DTTDDD - major/major, major/medium, minor/major and minor/medium preserve the chain. The problem is that no configuration with minor ditorus evolution does, so no connected ditorus will work.

DTDTDD - major/minor, minor/major and minor/medium preserve the chain. Looks like TDTDDD group again, except here the torus is threaded through BOTH holes, so it's more of a lock. Looks like the (1,0,a),(b,c,0)) configuration has this alternate expression as well.

DTDDTD - major/major, major/minor, minor/major and minor/medium preserve the chain. The configuration is ((a+1,0,b),(c,d,0)), with (a+b+1),(c+d)-torus and (a+c+1),d,b-ditorus. In 4D we get the torisphere/ditorus lock, but it looks like a spheritorus/ditorus lock is possible as well. In 5D, we would get 41-torus/311-ditorus, 41-torus/212-ditorus, 32-torus/311-ditorus, 32-torus/221-ditorus, 32-torus/212-ditorus, 23-torus/311-ditorus and 23-torus/221-ditorus.

DDTTDD - major/major and minor/major preserve the chain. This configuration disallows connected ditoruses.

So all in all, it seems that torus and ditorus can form two kinds of chains and two kinds of locks. Let's look at the spheritorus/ditorus combinations the algorithm tells us should exist:
((x,0,y),(z,w,0)):
(sqrt((x-12)^2 + y^2) - 3)^2 + z^2 + w^2 = 1
(sqrt((sqrt(x^2 + z^2) - 9)^2 + w^2) - 3)^2 + y^2 = 1
for chain

and
(sqrt((x-6)^2 + y^2) - 3)^2 + z^2 + w^2 = 1
(sqrt((sqrt(x^2 + z^2) - 9)^2 + w^2) - 3)^2 + y^2 = 1
for lock

Is the chain and lock here really different or can they be transformed into each other? I don't know for now.

So, how about the lock? The configuration should be once again ((x,0,y),(z,w,0)), so
(sqrt(x^2 + y^2) - 9)^2 + z^2 + w^2 = 1
(sqrt((sqrt(x^2 + z^2) - 9)^2 + w^2) - 3)^2 + y^2 = 1

It looks pretty loose, doesn't it? Looks like the spheritorus passes through the outer hole of ditorus twice, on the opposite sides of the inner hole, but it's so big that you can't actually slide it to one side and get it free, so it only "rattles" there. This combination (unlike torisphere/ditorus) uses the minor/major evolution of DTDDTD, which has two circles in the same "region" of quartet of circles. But, weird as it looks, it seems to be a genuine lock.
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### Re: Toratope Chains

Now that I think about it, maybe the spheritorus/ditorus lock could be undone by rotating the spheritorus instead of translating? This might put some additional constraints on numbers, but the system is still good.

Here, we'll have a 3x3 matrix of 9 variables. We'll start from ((1,0,0),(0,0,0),(0,0,0)) and from a chain of four A's and four B's.

[pause for writing a program to write the list/eliminate equivalent entries]
Also, maybe "lock" can be more easily defined as stemming from a chain with same letter on both ends?

I derived 8 basic groups of double-ditorus positionings, 5 of which should be doable in 4D.

Group 1:
AAABABBB - major/minor, medium/minor, minor/major, minor/medium. ((1,0,a),(0,0,b),(c,d,0)). (a+1),b,(c+d)-ditorus and (c+1),d,(a+b)-ditorus. Two ditoruses linked through their outer holes. Note that this configuration isn't possible until 5D where you can have two 212-ditoruses link this way.
AAABBBAB - alternate configuration
ABBBAAAB - another alternate configuration. This is probably impossible since the ditoruses wouldn't fit this way -- each would have to be bigger than the other. Could be done with asymmetrical ditoruses, though.

AABABABB - This time, they are threaded through both outer and inner holes. Looks topologically distinct.
AABBABAB - alternate configuration
ABAABBAB - another alternate configuration
ABABBAAB - yet another?

AABABBBA - Here, one ditorus goes through outer hole of the other, which goes through both of the first torus's holes. A lock.
AABBBABA - alternate configuration

ABABABBA - Another lock.

Group 2:
AAABBABB - major/medium, major/minor, medium/medium, medium/minor, minor/major. ((1,a,b),(0,c,d),(e,0,0)). (a+b+1),(c+d),e-ditorus and (e+1),(a+c),(b+d)-ditorus. Two ditoruses linked through outer hole of one and inner hole of other. If (a,d) or (b,c) is 0, this can be done with two ditoruses in 4D.
AABBBAAB - alternate configuration

AABBABBA - This is a lock where second ditorus goes through both holes of the first.

Group 3:
AABABBAB - major/minor, medium/major, medium/minor, minor/major, minor/medium. ((1,0,a),(b,0,c),(d,e,0)). (a+1),(b+c),(d+e)-ditorus and (b+d+1),e,(a+c)-ditorus. Here, the outer hole of one ditorus is locked in outer hole of the other. If c and d is 0, this can be achieved in 4D.

Group 4:
AABBAABB - major/medium, major/minor, medium/major, minor/major. ((1,a,b),(c,0,0),(d,0,0)). (a+b+1),c,d-ditorus and (c+d+1),a,b-ditorus. The ditoruses are linked through their inner holes. Can't be done until two 311-ditoruses in 5D.

Group 5:
ABAABBBA - major/major, major/medium, major/minor, medium/minor, minor/major, minor/medium. ((a+1,b,c),(0,0,d),(e,f,0)). (a+b+c+1),d,(e+f)-ditorus and (a+e+1),(b+f),(c+d)-ditorus. Outer hole of one is locked through both holes of the other. If (a,c,f) or (b,c,e) is 0, this can be done in 4D.

Group 6:
ABABBABA - major/major, major/minor, medium/minor, minor/major, minor/medium. ((a+1,0,b),(0,0,c),(d,e,0)). (a+b+1),c,(d+e)-ditorus and (a+d+1),e,(b+c)-ditorus. With a = 0, this lock is possible in 5D for two 212-ditoruses and with b = 0 or d = 0 for 212-ditorus and 311-ditorus.

Group 7:
ABBAABBA - major/major, major/medium, major/minor, medium/medium, medium/minor, minor/major. ((a+1,b,c),(0,d,e),(f,0,0)). (a+b+c+1),(d+e),f-ditorus and (a+f+1),(b+d),c-ditorus. If (a,b,e) is 0, this is possible in 4D.

Group 8:
ABBABAAB - major/minor, medium/medium, medium/minor, minor/major, minor/medium. ((1,0,a),(0,b,c),(d,e,0)). (a+1),(b+c),(d+e)-ditorus and (d+1),(b+e),(a+c)-ditorus. If (c,e) is 0, this is possible in 4D.

Degenerates:
AAAABBBB - no chain.
AAABBBBA - A has no functional minor dimensions.
AABBBBAA - A has no functional minor dimensions.

Group 2 model:
AAABBABB, (a,d)=0:
((x,0,y),(0,z,0),(w,0,0))
(sqrt((sqrt(x^2 + y^2) - 8)^2 + z^2) - 6)^2 + w^2 = 1
(sqrt((sqrt((x-14)^2 + w^2) - 6)^2 + z^2) - 2)^2 + y^2 = 1
x [-15,23]; y [-15,15]; z [-7,7]; w [-9,9]

AAABBABB, (b,c)=0:
((x,y,0),(0,0,z),(w,0,0))
(sqrt((sqrt(x^2 + y^2) - 8)^2 + z^2) - 6)^2 + w^2 = 1
(sqrt((sqrt((x-14)^2 + w^2) - 6)^2 + y^2) - 2)^2 + z^2 = 1
x [-15,23]; y [-15,15]; z [-7,7]; w [-9,9]

AABBBAAB (or rather BBAAABBA), (a,d)=0:
((x,0,y),(0,z,0),(w,0,0))
(sqrt((sqrt(x^2 + y^2) - 8)^2 + z^2) - 6)^2 + w^2 = 1
(sqrt((sqrt((x+6)^2 + w^2) - 14)^2 + z^2) - 2)^2 + y^2 = 1
x [-23,15]; y [-15,15]; z [-7,7]; w [-17,17]

AABBBAAB (or rather BBAAABBA), (b,c)=0:
((x,y,0),(0,0,z),(w,0,0))
(sqrt((sqrt(x^2 + y^2) - 8)^2 + z^2) - 6)^2 + w^2 = 1
(sqrt((sqrt((x+6)^2 + w^2) - 14)^2 + y^2) - 2)^2 + z^2 = 1
x [-23,15]; y [-15,15]; z [-7,7]; w [-17,17]

AABBABBA, (a,d)=0:
((x,y,0),(0,0,z),(w,0,0))
(sqrt((sqrt(x^2 + y^2) - 12)^2 + z^2) - 6)^2 + w^2 = 1
(sqrt((sqrt((x-6)^2 + w^2) - 6)^2 + y^2) - 2)^2 + z^2 = 1
x [-19,19]; y [-19,19]; z [-7,7]; w [-9,9]

AABBABBA, (b,c)=0:
((x,0,y),(0,z,0),(w,0,0))
(sqrt((sqrt(x^2 + y^2) - 12)^2 + z^2) - 6)^2 + w^2 = 1
(sqrt((sqrt((x-6)^2 + w^2) - 6)^2 + z^2) - 2)^2 + y^2 = 1
x [-19,19]; y [-19,19]; z [-7,7]; w [-9,9]

Group 3 model:
AABABBAB, (c,d)=0:
((x,0,y),(z,0,0),(0,w,0)
(sqrt((sqrt(x^2 + y^2) - 10)^2 + z^2) - 6)^2 + w^2 = 1
(sqrt((sqrt((x-10)^2 + z^2) - 6)^2 + w^2) - 4)^2 + y^2 = 1
x [-17,21]; y [-17,17]; z [-11,11]; w [-5,5]

Group 5 model:
ABAABBBA, (a,c,f)=0:
((x,y,0),(0,0,z),(w,0,0))
(sqrt((sqrt(x^2 + y^2) - 14)^2 + z^2) - 12)^2 + w^2 = 1
(sqrt((sqrt((x-10)^2 + w^2) - 8)^2 + y^2) - 6)^2 + z^2 = 1
x [-27,27]; y [-27,27]; z [-13,13]; w [-15,15]

To be continued...
Marek14
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Posts: 1191
Joined: Sat Jul 16, 2005 6:40 pm

### Re: Toratope Chains

ABAABBBA, (b,c,e)=0:
((xy,0,0),(0,0,z),(0,w,0))
(sqrt((sqrt(x^2 + y^2) - 14)^2 + z^2) - 12)^2 + w^2 = 1
(sqrt((sqrt((x-10)^2 + y^2) - 8)^2 + w^2) - 6)^2 + z^2 = 1
x [-27,27]; y [-27,27]; z [-13,13]; w [-7,7]

Group 7 model:
ABBAABBA, (a,b,e)=0:
((x,0,y),(0,z,0),(w,0,0))
(sqrt((sqrt(x^2 + y^2) - 8)^2 + z^2) - 6)^2 + w^2 = 1
(sqrt((sqrt(x^2 + w^2) - 8)^2 + z^2) - 2)^2 + y^2 = 1
x [-15,15]; y [-15,15]; z [-7,7]; w [-11,11]

Group 8 model:
ABBABAAB -- thinking about it now, it looks like this isn't actually possible. On the left side, medium diameter of B needs to be smaller then medium diameter of A to fit -- but on the right side, it needs to be the other way around.

Some of these models might not be true links -- those would only come in higher dimensions.

So, how to extend this system to tigers? Well, tigers have 2D trace so we'll have to actually work in 2D. But it's not really possible to create a link with two sets of four circles. Does that mean that tigers can't be linked? I think Wendy claimed they can be linked, a long time ago, but I'm not sure if she ever showed a proof.

For torus + tiger, it turns out that the only feasible 2D combination is to enclose two of tiger's circles into a pair of circles. This corresponds to the torisphere/tiger link we've found before. The matrix is ((a+1,1,0),(0,b,0)) to describe a link between (a+2),b-torus and (a+1),(b+1),0-tiger.

An example could be
(sqrt((x-6)^2 + y^2 + z^2) - 6)^2 + w^2 = 1
(sqrt(x^2 + w^2) - 6)^2 + (sqrt(y^2 + z^2) - 2)^2 = 1
x [-7,13]; y [-7,7]; z [-7,7]; w [-7,7]
Marek14
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Posts: 1191
Joined: Sat Jul 16, 2005 6:40 pm

### Re: Toratope Chains

Some new results for today.

First, the spheritorus/ditorus lock. While this doesn't look like a true lock since the spheritorus can be rotated and slid out, there is a way to prevent it from doing so. We can simply stretch the spheritorus in one dimension so it will no longer fit in the outer hole of the ditorus. It seems there are at least two kinds of links -- topological which work no matter how you deform the shapes, and geometrical that require some precise combination of the shapes.

Next, further thinking about torus/tiger link allowed me to realize the existence of spheritorus/tiger chain:

(sqrt(x^2 + y^2) - 4)^2 + z^2 + w^2 = 1
(sqrt((x-4)^2 + z^2) - 4)^2 + (sqrt((y-4)^2 + w^2) - 4)^2 = 1
x [-5,9]; y [-5,9]; z [-5,5]; w [-5,5]

(sqrt(x^2 + y^2) - 23)^2 + z^2 + w^2 = 1
(sqrt((x-10)^2 + z^2) - 10)^2 + (sqrt((y-10)^2 + w^2) - 10)^2 = 1
x [-24,24]; y [-24,24]; z [-11,11]; w [-11,11]

Also, the torus/tiger chain posted yesterday had the matrix a bt wrong -- it's actually ((a+1,1,b),(0,c,0)), so it's a chain between (a+b+2),c-torus and (a+1),(c+1),b-tiger: the tiger CAN have some minor dimension.
On the other hand, the spheritorus/tiger chain is an example of different kind of link with parameters ((1,1,a),(b,c,0)). This is a chain between (a+2),(b+c)-torus and (b+1),(c+1),a-tiger.
Marek14
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Posts: 1191
Joined: Sat Jul 16, 2005 6:40 pm

### Re: Toratope Chains

Some preliminary wonderings about ditorus/tiger chains.

2D picture that will help here is four circles of tiger combined with either quartet of circles or two pairs of circles of ditorus.

However, it's clear now that there are some chains that can't be formed by "pure" forms of ditorus and tiger, but only by topologically deformed forms.

For quartet of circles/four circles combination, we can classify them by number of tiger's circles in the three areas of quartet -- inner area, the area between two solid "rings" that are inside of ditorus (outer hole), and the outer area. I believe that inner and outer area are topologically identical here, so we can switch them at will, which means that it's sufficient to consider cases where number of circles in inner area is <= number in outer area. (See also my spheritorus/tiger chains in previous post; one of them encompasses a single circle while other encompasses three, but they should be topologically identical.)

So, we have:
0-1-3 circles -- here one of the "arms" of tiger passes through the outer hole of ditorus. This has matrix ((1,1,a),(0,0,b),(c,d,0)) and is link of (a+2),b,(c+d)-ditorus and (c+1),(d+1),(a+b)-tiger. This can't be done in 4D since you need a tiger with at least one minor dimension.
1-0-3 circles -- one of the arms of tiger passes through the inner hole of ditorus. This is analogical to spheritorus/ditorus chain from before. Matrix is ((1,1,a),(b,c,0),(d,e,0)) and we link (a+2),(b+c),(d+e)-ditorus and (b+d+1),(c+e+1),a-tiger. This is possible to do in 4D.
0-2-2 circles -- here we have two possible topologies. In one, the middle region contains two adjacent circles; in the other, two diagonal circles. The diagonal configuration is impossible to do with perfect ditorus and tiger, at least one must be deformed. And the orthogonal configuration has problems of its own -- whether it's possible to change quartet of spheres to minor pair of toruses and simultaneously join the two enclosed circles by torus depends on the exact configuration -- it's impossible if those two circles happen to be on the opposite sides, but possible if they're close to each other.
1-2-1 circles and 0-3-1 circles -- those are left as an exercise for reader for now. And I haven't even tried to check on the two pairs of circles configurations...
Marek14
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