So, thinking globally about chains, the current hypothesis is that two toratopes are linked if their cuts are dependent, i.e. if one of them must vanish before the others, even if we translate the cuts.

Let's look at the spheritorus/ditorus link once again.

(((xy)z)w) ditorus, ((xy)zw) spheritorus:

w-cut: minor pair of ((xy)z) toruses, ((xy)z) torus. Independent cuts no matter how you place them. If you place the torus into the toroidal hole, you'll see how you can push a spheritorus through the torus-shaped hole in ditorus.

z-cut: major pair of ((xy)w) toruses, ((xy)w) torus. Cuts could be dependent only if you put the spheritorus between the two toruses in pair.

y-cut: two ((xz)w) toruses displaced in x, two (xzw) spheres displaced in x. Independent.

x-cut: analogical to y-cut

(((xy)z)w) ditorus, ((xz)yw) spheritorus:

w-cut: minor pair of ((xy)z) toruses, ((xz)y) torus. The toruses can be linked, but the cuts are generally independent.

z-cut: major pair of ((xy)w) toruses, two (xyw) spheres displaced in x. Independent.

y-cut: two ((xz)w) toruses displaced in x, ((xz)w) torus. Independent.

x-cut: two ((yz)w) toruses displaced in y, two (yzw) spheres displaced in z. Independent.

(((xy)z)w) ditorus, ((xw)yz) spheritorus:

w-cut: minor pair of ((xy)z) toruses, two (xyz) spheres displaced in x. Independent.

z-cut: major pair of ((xy)w) toruses, ((xw)y) torus. Here, dependence is possible if the torus passes through the gap between two toruses of the pair, i.e. if it's linked with exactly one of them. Then the spheritorus must vanish before the pair merges.

y-cut: two ((xz)w) toruses displaced in x, ((xw)z) torus. The torus can pass through one of the ditorus-cut toruses or through both.

x-cut: two ((yz)w) toruses displaced in y, two (yzw) spheres displaced in w. Independent.

So it seems that it IS possible to link a ditorus and spheritorus in this way, by having the spheritorus pass through the toroidal hole.

(((xy)z)w) ditorus, ((zw)xy) spheritorus:

w-cut: minor pair of ((xy)z) toruses, two (xyz) spheres displaced in z. Independent.

z-cut: major pair of ((xy)w) toruses, two (xyw) spheres displaced in w. Independent.

y-cut: two ((xz)w) toruses displaced in x, ((zw)x) torus. The torus can be only pass through one of the ditorus-cut toruses. Spheritorus must vanish before the ditorus vanishes, though it can move between the two toruses by going far enough in y direction that both toruses merge.

x-cut: analogical to y-cut.

Basically, spheritorus linked to one of displaced ((xz)w) toruses can appear as either ((xw)yz) or ((zw)xy), as it goes around. It always has to disappear before the ditorus does. However, if the spheritorus passes through BOTH toruses, then it's possible to move enough to merge both toruses, and then pull the spheritorus out, so there's no true link in this case.

Tiger/ditorus:

((xy)z)w) ditorus, ((xy)(zw)) tiger:

w-cut: minor pair of ((xy)z) toruses, two ((xy)z) toruses displaced in z. Dependence possible if one of tiger-cut toruses is put in the toroidal hole and the other outside, then ditorus must vanish first.

z-cut: major pair of ((xy)w) toruses, two ((xy)w) toruses displaced in w. Independent.

y-cut: two ((xz)w) toruses displaced in x, two ((zw)x) toruses displaced in x. If only one torus from each pair is linked, sliding can transform the ((zw)x) toruses displaced in x into ((xw)z) toruses displaced in z. This case is impossible, as the cuts have to eventually intersect. If both tiger-cut toruses are linked to same torus of if they are linked one to each, it might be possible to move between these two configurations via ditorus cut merging and they could be slid into ((xw)z) toruses. Tiger has to vanish first in this case.

x-cut: analogical to y-cut

((xy)z)w) ditorus, ((xz)(yw)) tiger:

w-cut: minor pair of ((xy)z) toruses, two ((xz)y) toruses displaced in y. If both tiger-cut toruses are linked, situation is independent. If only one is linked, ditorus has to vanish first.

z-cut: major pair of ((xy)w) toruses, two ((yw)x) toruses displaced in x. If both tiger-cut toruses are linked to both ditorus-cut toruses, situation is independent. If only one is linked to the two cuts, ditorus has to vanish first. If both tiger-cut toruses are linked to one ditorus-cut torus, tiger has to vanish first. If just one tiger-cut torus is linked to just one ditorus-cut torus, situation is impossible.

y-cut: two ((xz)w) toruses displaced in x, two ((xz)w) toruses displaced in w. Independent.

x-cut: two ((yz)w) toruses displaced in y, two ((yw)z) toruses displaced in z. If both tiger-cut toruses are linked to one ditorus-cut torus, tiger has to vanish first. If at least one is linked to both, tiger could slide out.

((xy)z)w) ditorus, ((xw)(yz)) tiger:

w-cut: minor pair of ((xy)z) toruses, two ((yz)x) toruses displaced in x - twin of w-cut in previous case

z-cut: major pair of ((xy)w) toruses, two ((xw)y) toruses displaced in y - twin of z-cut in previous case

y-cut: two ((xz)w) toruses displaced in x, two ((xw)z) toruses displaced in z - twin of y-cut in first case

x-cut: two ((yz)w) toruses displaced in y, two ((yz)w) toruses displaced in w - twin of x-cut in first case

Lost here for now. Seems that tiger and ditorus can be linked, but not sure how many different ways exist.

Let's look at the spheritorus/ditorus link once again.

(((xy)z)w) ditorus, ((xy)zw) spheritorus:

w-cut: minor pair of ((xy)z) toruses, ((xy)z) torus. Independent cuts no matter how you place them. If you place the torus into the toroidal hole, you'll see how you can push a spheritorus through the torus-shaped hole in ditorus.

z-cut: major pair of ((xy)w) toruses, ((xy)w) torus. Cuts could be dependent only if you put the spheritorus between the two toruses in pair.

y-cut: two ((xz)w) toruses displaced in x, two (xzw) spheres displaced in x. Independent.

x-cut: analogical to y-cut

(((xy)z)w) ditorus, ((xz)yw) spheritorus:

w-cut: minor pair of ((xy)z) toruses, ((xz)y) torus. The toruses can be linked, but the cuts are generally independent.

z-cut: major pair of ((xy)w) toruses, two (xyw) spheres displaced in x. Independent.

y-cut: two ((xz)w) toruses displaced in x, ((xz)w) torus. Independent.

x-cut: two ((yz)w) toruses displaced in y, two (yzw) spheres displaced in z. Independent.

(((xy)z)w) ditorus, ((xw)yz) spheritorus:

w-cut: minor pair of ((xy)z) toruses, two (xyz) spheres displaced in x. Independent.

z-cut: major pair of ((xy)w) toruses, ((xw)y) torus. Here, dependence is possible if the torus passes through the gap between two toruses of the pair, i.e. if it's linked with exactly one of them. Then the spheritorus must vanish before the pair merges.

y-cut: two ((xz)w) toruses displaced in x, ((xw)z) torus. The torus can pass through one of the ditorus-cut toruses or through both.

x-cut: two ((yz)w) toruses displaced in y, two (yzw) spheres displaced in w. Independent.

So it seems that it IS possible to link a ditorus and spheritorus in this way, by having the spheritorus pass through the toroidal hole.

(((xy)z)w) ditorus, ((zw)xy) spheritorus:

w-cut: minor pair of ((xy)z) toruses, two (xyz) spheres displaced in z. Independent.

z-cut: major pair of ((xy)w) toruses, two (xyw) spheres displaced in w. Independent.

y-cut: two ((xz)w) toruses displaced in x, ((zw)x) torus. The torus can be only pass through one of the ditorus-cut toruses. Spheritorus must vanish before the ditorus vanishes, though it can move between the two toruses by going far enough in y direction that both toruses merge.

x-cut: analogical to y-cut.

Basically, spheritorus linked to one of displaced ((xz)w) toruses can appear as either ((xw)yz) or ((zw)xy), as it goes around. It always has to disappear before the ditorus does. However, if the spheritorus passes through BOTH toruses, then it's possible to move enough to merge both toruses, and then pull the spheritorus out, so there's no true link in this case.

Tiger/ditorus:

((xy)z)w) ditorus, ((xy)(zw)) tiger:

w-cut: minor pair of ((xy)z) toruses, two ((xy)z) toruses displaced in z. Dependence possible if one of tiger-cut toruses is put in the toroidal hole and the other outside, then ditorus must vanish first.

z-cut: major pair of ((xy)w) toruses, two ((xy)w) toruses displaced in w. Independent.

y-cut: two ((xz)w) toruses displaced in x, two ((zw)x) toruses displaced in x. If only one torus from each pair is linked, sliding can transform the ((zw)x) toruses displaced in x into ((xw)z) toruses displaced in z. This case is impossible, as the cuts have to eventually intersect. If both tiger-cut toruses are linked to same torus of if they are linked one to each, it might be possible to move between these two configurations via ditorus cut merging and they could be slid into ((xw)z) toruses. Tiger has to vanish first in this case.

x-cut: analogical to y-cut

((xy)z)w) ditorus, ((xz)(yw)) tiger:

w-cut: minor pair of ((xy)z) toruses, two ((xz)y) toruses displaced in y. If both tiger-cut toruses are linked, situation is independent. If only one is linked, ditorus has to vanish first.

z-cut: major pair of ((xy)w) toruses, two ((yw)x) toruses displaced in x. If both tiger-cut toruses are linked to both ditorus-cut toruses, situation is independent. If only one is linked to the two cuts, ditorus has to vanish first. If both tiger-cut toruses are linked to one ditorus-cut torus, tiger has to vanish first. If just one tiger-cut torus is linked to just one ditorus-cut torus, situation is impossible.

y-cut: two ((xz)w) toruses displaced in x, two ((xz)w) toruses displaced in w. Independent.

x-cut: two ((yz)w) toruses displaced in y, two ((yw)z) toruses displaced in z. If both tiger-cut toruses are linked to one ditorus-cut torus, tiger has to vanish first. If at least one is linked to both, tiger could slide out.

((xy)z)w) ditorus, ((xw)(yz)) tiger:

w-cut: minor pair of ((xy)z) toruses, two ((yz)x) toruses displaced in x - twin of w-cut in previous case

z-cut: major pair of ((xy)w) toruses, two ((xw)y) toruses displaced in y - twin of z-cut in previous case

y-cut: two ((xz)w) toruses displaced in x, two ((xw)z) toruses displaced in z - twin of y-cut in first case

x-cut: two ((yz)w) toruses displaced in y, two ((yz)w) toruses displaced in w - twin of x-cut in first case

Lost here for now. Seems that tiger and ditorus can be linked, but not sure how many different ways exist.

- Marek14
- Pentonian
**Posts:**1140**Joined:**Sat Jul 16, 2005 6:40 pm

Now that's an interesting idea. The rotations can show how and where they chain together. Interesting! I'll have to try out that one, too. Given four adjustable parameters means this would work well up to 5D, in 6D I'll have to manually change the angles. The more tools I have at my disposal, the cooler the renders will be! Im fascinated by the rotations inside ((((II)I)I)I)(II)), there will most definitely be some animations of those later.

in search of combinatorial objects of finite extent

- ICN5D
- Pentonian
**Posts:**1075**Joined:**Mon Jul 28, 2008 4:25 am**Location:**Orlando, FL

((((II)I)I)I)(II)) is tritorus tiger. The trace is 2x16 array of circles, leading to 5 possible 3D cuts:

1. Vertical stack of 16 toruses

2. 8 vertical stacks of 2 toruses in line

3. 4 vertical stacks of 2 major pairs of toruses

4. 2 vertical stacks of 2 major quartets

5. Vertical stack of 2 major octets

Though from 7D, it might be more fun to animate the ditorus/torus tiger ((((II)I)I)((II)I)) that has a trace of 4x8 array of circles.

1. Vertical stack of 16 toruses

2. 8 vertical stacks of 2 toruses in line

3. 4 vertical stacks of 2 major pairs of toruses

4. 2 vertical stacks of 2 major quartets

5. Vertical stack of 2 major octets

Though from 7D, it might be more fun to animate the ditorus/torus tiger ((((II)I)I)((II)I)) that has a trace of 4x8 array of circles.

- Marek14
- Pentonian
**Posts:**1140**Joined:**Sat Jul 16, 2005 6:40 pm

Whoops, I meant ((((II)I)I)(II)). Thats the one Im imaging now.

in search of combinatorial objects of finite extent

- ICN5D
- Pentonian
**Posts:**1075**Joined:**Mon Jul 28, 2008 4:25 am**Location:**Orlando, FL

Here are some of the amazing Oblique Artifacts of ((((II)I)I)(II)):

((((II)I)I)(II)) : (02)0-Tigroid

The 4x2 wall of 8 toruses axial midcut : (((((I))I)(I))

A simultaneous tiger plus ditorus-cut merging of the 4x2 wall

A fine line of separation

A really wild looking structure

Look at this incredible structure!!! Some of these oblique cuts appear in a very narrow margin between multiple transitions. They require discovery of such emergings. At this point, I learned how to make even finer resolution shapes. When they get this complex, it's required for detail's sake.

This structure appears while rotating through an empty cut. At 0 and 90 degrees, there is nothing but empty cuts. But, when rotating from one empty axial to another, this thing appears as a mirror image torus with the sides trimmed.

Another angle slightly off from the above, illustrating the trimmed down scanning effect. I have never seen this in any shapes yet, but I've only explored about 8 or 9 of them. A ditorus has many different holes by itself, and when manifested in 6D, strange things appear to emerge from it. I think this structure appears because this is the rotation past the minor radius, from one empty to another. It's certainly been unexpected, and really cool to find.

Yet another mid-axial transition making something cool.

So, that's about how many cool things I found so far, with only one version of the rotation formula. I think that these class of tigroids are interesting to explore. That is, those that have a 4D-toratope-times-a-circle tigroid, like:

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

(((III)I)(II))

(((II)II)(II))

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

I'm thinking that the others may have cool undiscovered things to find, especially a rotation equation of double tiger (((II)(II))(II)).

((((II)I)I)(II)) : (02)0-Tigroid

The 4x2 wall of 8 toruses axial midcut : (((((I))I)(I))

A simultaneous tiger plus ditorus-cut merging of the 4x2 wall

A fine line of separation

A really wild looking structure

Look at this incredible structure!!! Some of these oblique cuts appear in a very narrow margin between multiple transitions. They require discovery of such emergings. At this point, I learned how to make even finer resolution shapes. When they get this complex, it's required for detail's sake.

This structure appears while rotating through an empty cut. At 0 and 90 degrees, there is nothing but empty cuts. But, when rotating from one empty axial to another, this thing appears as a mirror image torus with the sides trimmed.

Another angle slightly off from the above, illustrating the trimmed down scanning effect. I have never seen this in any shapes yet, but I've only explored about 8 or 9 of them. A ditorus has many different holes by itself, and when manifested in 6D, strange things appear to emerge from it. I think this structure appears because this is the rotation past the minor radius, from one empty to another. It's certainly been unexpected, and really cool to find.

Yet another mid-axial transition making something cool.

So, that's about how many cool things I found so far, with only one version of the rotation formula. I think that these class of tigroids are interesting to explore. That is, those that have a 4D-toratope-times-a-circle tigroid, like:

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

(((III)I)(II))

(((II)II)(II))

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

I'm thinking that the others may have cool undiscovered things to find, especially a rotation equation of double tiger (((II)(II))(II)).

Last edited by ICN5D on Sat Mar 29, 2014 6:03 am, edited 1 time in total.

in search of combinatorial objects of finite extent

- ICN5D
- Pentonian
**Posts:**1075**Joined:**Mon Jul 28, 2008 4:25 am**Location:**Orlando, FL

Your new naming system is a nice expansion to the high-D shapes. Reducing to lowest possible trace gives us the array, and then the axial cuts. Plus, the numbers and quantity of symbols are much less even with 6D toratopes. Getting into 9 and 10D , the original number string gets huge. I like it. Plus, it further simplifies much higher, like the 16D example you gave. Too bad my computer might explode if I fed it that equation. A supercomputer would be awesome!

That array pushed the limits of my imagination, and it's amazing. There's 2,048 spheres in it! So, basically (321)4-tigroid is equal to [(2111,211,21),0]-tigroid-(4)-toroid ?

The notation sequence is (((((((((II)I)I)I)(((II)I)I)((II)I))I)I)I)I) , or 21112112101111 in the old notation. Which, in comparing the old with new, the 1's directly convert over, and we just omit the 2's and consider them an additional leg of the array.

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.

That array pushed the limits of my imagination, and it's amazing. There's 2,048 spheres in it! So, basically (321)4-tigroid is equal to [(2111,211,21),0]-tigroid-(4)-toroid ?

The notation sequence is (((((((((II)I)I)I)(((II)I)I)((II)I))I)I)I)I) , or 21112112101111 in the old notation. Which, in comparing the old with new, the 1's directly convert over, and we just omit the 2's and consider them an additional leg of the array.

Last edited by ICN5D on Sat Mar 29, 2014 7:18 am, edited 1 time in total.

in search of combinatorial objects of finite extent

- ICN5D
- Pentonian
**Posts:**1075**Joined:**Mon Jul 28, 2008 4:25 am**Location:**Orlando, FL

Yes, that's why I devised it As for the double tiger, rotations will be more tricky since there's only one nonempty axial 3D cut...

Let's ask a question: how many parameters do we actually need to specify a 3D cut?

In 4D, a generic hyperplane needs 4 parameters to fix (roughly, you find its intersections with the axes, giving you four numbers -- this can specify any hyperplane except those that are parallel to one or more axes or pass through the origin).

In 5D, a generic 3D hyperplane would intersect the 10 coordinate planes, but we only need to specify 4 since it's given by four points, so it's 8 numbers

In 6D, analogically, we can specify a 3D hyperplane by 12 numbers.

That's how many degrees of freedom the cut has.

Let's ask a question: how many parameters do we actually need to specify a 3D cut?

In 4D, a generic hyperplane needs 4 parameters to fix (roughly, you find its intersections with the axes, giving you four numbers -- this can specify any hyperplane except those that are parallel to one or more axes or pass through the origin).

In 5D, a generic 3D hyperplane would intersect the 10 coordinate planes, but we only need to specify 4 since it's given by four points, so it's 8 numbers

In 6D, analogically, we can specify a 3D hyperplane by 12 numbers.

That's how many degrees of freedom the cut has.

- Marek14
- Pentonian
**Posts:**1140**Joined:**Sat Jul 16, 2005 6:40 pm

Okay, so take the rotation equation for the ((((II)I)I)(II)):

(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

What other distinct combinations are there that won't produce identical results? There are three cut axes, so how do 12 degrees of freedom fit in there?

(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

What other distinct combinations are there that won't produce identical results? There are three cut axes, so how do 12 degrees of freedom fit in there?

in search of combinatorial objects of finite extent

- ICN5D
- Pentonian
**Posts:**1075**Joined:**Mon Jul 28, 2008 4:25 am**Location:**Orlando, FL

ICN5D wrote:Okay, so take the rotation equation for the ((((II)I)I)(II)):

(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

What other distinct combinations are there that won't produce identical results? There are three cut axes, so how do 12 degrees of freedom fit in there?

Well, first of all, all these rotations are hyperplanes passing through the center, so that's one thing. Three degrees of freedom are from moving the hyperplane in three directions perpendicular to it.

So, if we stay with rotations, using this approach basically means setting three directions for the hyperplane from origin via three angles based on triple-circle coordinate system on surface of 6D sphere.

Here's the question: what happens if you set up those three circles in a different way?

For example what if you put

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

The major cuts might be the same, but will the oblique cuts be the same set?

After all, how many different axial 3D cuts are there in ditorus tiger? There's 20 altogether, but duocylindrical symmetry of the shape reduces it to 10.

((((II)I))()) - empty

((((II))I)()) - empty

2x ((((II)))(I)) - nonempty

2x ((((I)I)I)()) - empty

4x ((((I)I))(I)) - nonempty

4x ((((I))I)(I)) - nonempty

2x ((((I)))(II)) - nonempty

2x (((()I)I)(I)) - empty

(((()I))(II)) - empty

(((())I)(II)) - empty

And now: every rotation can be thought as moving a certain "I" between two positions. Three rotations combine three such operations for 8 positions in total.

Your equation (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 could be written in toratopic notation as follows:

(((xz)x)y)(zy)), with 8 possible results:

(((xz))y)())

(((x))y)(z))

(((xz)))(y))

(((x)))(zy))

(((z)x)y)())

((()x)y)(z))

(((z)x))(y))

((()x))(zy))

This shows all four nonempty cuts, but only 4 of 6 possible distinct empty cuts: cuts ((((II)I))()) and (((())I)(II)) are missing. In fact, since there's 10 possible cuts, no single equation with three parameters can show them all, so some oblique cuts will be always missing.

And there's more: you can imagine three specific oblique cuts that will NEVER exist simultaneously under any axial system. From each such combination, new group of rotations can be constructed.

So you see, there IS more degrees of freedom than you'd think at first

- Marek14
- Pentonian
**Posts:**1140**Joined:**Sat Jul 16, 2005 6:40 pm

I was also thinking about how to classify holes. Simplest would probably be to classify a hole by its cross-cut and by the number of extra dimensions it has.

So, a torus has a hole with circular cross-cut and one extra dimension, so a circle (or another torus) can be threaded through it. In 4D, torisphere has a spherical hole with one extra dimension and spheritorus has a circular hole with two extra dimensions, and so these two can be nicely chained together.

Ditorus is interesting since it has two holes: one is remnant from its torus base, and it's circular with two extra dimensions, and the other is toroidal with one extra dimension.

About holes of tiger there were some talk before, but based on this I guess we'd say that tiger has two (circle +2) holes which cross. (Holes in ditorus don't cross since their cross-cuts, circle and torus, don't cross either.)

So, it looks like a toroid ((X)I)has the same holes as X, plus one more in shape of X itself, and tigroid ((X)(Y)(Z)...) has holes in shape of X, Y, Z, ... plus whatever holes those shapes themselves have.

So duotorus tiger (((II)I)((II)I)) should have two holes (torus +3) and two holes (circle +4)?

So, a torus has a hole with circular cross-cut and one extra dimension, so a circle (or another torus) can be threaded through it. In 4D, torisphere has a spherical hole with one extra dimension and spheritorus has a circular hole with two extra dimensions, and so these two can be nicely chained together.

Ditorus is interesting since it has two holes: one is remnant from its torus base, and it's circular with two extra dimensions, and the other is toroidal with one extra dimension.

About holes of tiger there were some talk before, but based on this I guess we'd say that tiger has two (circle +2) holes which cross. (Holes in ditorus don't cross since their cross-cuts, circle and torus, don't cross either.)

So, it looks like a toroid ((X)I)has the same holes as X, plus one more in shape of X itself, and tigroid ((X)(Y)(Z)...) has holes in shape of X, Y, Z, ... plus whatever holes those shapes themselves have.

So duotorus tiger (((II)I)((II)I)) should have two holes (torus +3) and two holes (circle +4)?

- Marek14
- Pentonian
**Posts:**1140**Joined:**Sat Jul 16, 2005 6:40 pm

I played with a different combination of the rotation equation, and produced an interesting modification to one of the oblique cuts. I'm not sure if it was possible with the other eq, but I think this was a new one. Now, obviously, all of the axial cuts were there. But, there was some subtle crossing between emerging oblique cuts that I don't remember seeing before.

This angle is merging a deformed middle structure of a previous oblique. Looks like a Star Wars land speeder

These amazing single structures best represent the shape's unique identity. No two will be alike, and they appear in a very narrow margin of rotation. At some critical angle where all parts are joined. This one is a combination of two emerging empty cuts, overlapping into one structure.

This angle is merging a deformed middle structure of a previous oblique. Looks like a Star Wars land speeder

These amazing single structures best represent the shape's unique identity. No two will be alike, and they appear in a very narrow margin of rotation. At some critical angle where all parts are joined. This one is a combination of two emerging empty cuts, overlapping into one structure.

in search of combinatorial objects of finite extent

- ICN5D
- Pentonian
**Posts:**1075**Joined:**Mon Jul 28, 2008 4:25 am**Location:**Orlando, FL

Explored another cool shape, one I have been anticipating. It was a challenge finding the right layout of parentheses and radii values, for the equation to work properly. Sometimes, the first try makes blobs, but in the right positions. Then, I have to adjust the diameters to try and make what it should be. It definitely takes some experimenting. And, not to mention all of the different angles to make oblique structures! That takes even more play time. I'm cool with that

(((II)(II))(II)) : Double Tiger

Axial Midsection Equation:

(sqrt((sqrt(x^2 + a^2) - 3)^2 + (sqrt(y^2 + b^2) - 3)^2) - 1.5)^2 + (sqrt(z^2 + c^2) - 2.5)^2 - 0.8^2 = 0

Rotation Equation:

(sqrt((sqrt((x*sin(b))^2 + (z*cos(a))^2) - 3)^2 + (sqrt((y*sin(c))^2 + (x*cos(b))^2) - 3)^2) - 1.5)^2 + (sqrt((z*sin(a))^2 + (y*cos(c))^2) - 2.5)^2 - 0.8^2 = 0

The only non-empty axial midcut of 8 toruses in a 2x2x2 array, symbolized by (((I)(I))(I))

Moving along all three cut axes will merge the 8 into a single structure, before morphing into a complex blob and scale to a point

Tilting our 3D viewplane will deform and merge the 8 torii, into the " Four Diving Masks " oblique angle cut

Tilting a little further will pinch off the masks into goggles, and merge into one structure

This is a neat single structure that appeared in a very narrow margin, with some cool alternating touch points

This was another oblique artifact, appearing as a perforated barrel, at a nearly 45x45x45 degree rotation in all three cut planes

(((II)(II))(II)) : Double Tiger

Axial Midsection Equation:

(sqrt((sqrt(x^2 + a^2) - 3)^2 + (sqrt(y^2 + b^2) - 3)^2) - 1.5)^2 + (sqrt(z^2 + c^2) - 2.5)^2 - 0.8^2 = 0

Rotation Equation:

(sqrt((sqrt((x*sin(b))^2 + (z*cos(a))^2) - 3)^2 + (sqrt((y*sin(c))^2 + (x*cos(b))^2) - 3)^2) - 1.5)^2 + (sqrt((z*sin(a))^2 + (y*cos(c))^2) - 2.5)^2 - 0.8^2 = 0

The only non-empty axial midcut of 8 toruses in a 2x2x2 array, symbolized by (((I)(I))(I))

Moving along all three cut axes will merge the 8 into a single structure, before morphing into a complex blob and scale to a point

Tilting our 3D viewplane will deform and merge the 8 torii, into the " Four Diving Masks " oblique angle cut

Tilting a little further will pinch off the masks into goggles, and merge into one structure

This is a neat single structure that appeared in a very narrow margin, with some cool alternating touch points

This was another oblique artifact, appearing as a perforated barrel, at a nearly 45x45x45 degree rotation in all three cut planes

in search of combinatorial objects of finite extent

- ICN5D
- Pentonian
**Posts:**1075**Joined:**Mon Jul 28, 2008 4:25 am**Location:**Orlando, FL

So if this has a barrel, then one of 7D shapes that cut in two double tigers ((((II)I)(II))(II)) or (((II)(II))((II)I)) could have a "double-barreled" oblique cut?

Double tiger is the simplest example of torus order, toratope whose trace is made up of toruses. These "double tigroids" require two tiger operations to be constructed, which we can call "upper" and "lower".

Generic double tigroid's structure can be described by a set of numbers like this:

(horizontal array size) (vertical array size) major minor

Basic double tiger is (11)(1)11

Number of terms in horizontal array size is the size of lower branching. Double tiger has two dimensions there. Triger tiger (((II)(II)(II))(II)) has trace made of torispheres and its horizontal array has three dimensions.

Similarly, vertical array has as many terms as is size of upper branching, not counting the tigroid. Tiger triger (((II)(II))(II)(II)) has spheritorus trace and two vertical dimensions in array.

Sizes of both arrays are given by exact lengths of branches. (((((II)I)I)((II)I))((((II)I)I)I)), ((ditorus/torus) tiger/tritorus) tiger would have 4x2 horizontal array of toruses with vertical size 8.

Finally, every torus extension of lower branch separates toruses into major pairs and every torus extension of upper branch into minor pairs.

Tiger torus tiger ((((II)(II))I)(II)) has major pairs of toruses in trace, double tiger torus (((((II)(II))(II))I) has minor pairs, Tiger torus tiger torus ((((((II)(II))I)(II))I) has major/minor quartets.

Double tiger is the simplest example of torus order, toratope whose trace is made up of toruses. These "double tigroids" require two tiger operations to be constructed, which we can call "upper" and "lower".

Generic double tigroid's structure can be described by a set of numbers like this:

(horizontal array size) (vertical array size) major minor

Basic double tiger is (11)(1)11

Number of terms in horizontal array size is the size of lower branching. Double tiger has two dimensions there. Triger tiger (((II)(II)(II))(II)) has trace made of torispheres and its horizontal array has three dimensions.

Similarly, vertical array has as many terms as is size of upper branching, not counting the tigroid. Tiger triger (((II)(II))(II)(II)) has spheritorus trace and two vertical dimensions in array.

Sizes of both arrays are given by exact lengths of branches. (((((II)I)I)((II)I))((((II)I)I)I)), ((ditorus/torus) tiger/tritorus) tiger would have 4x2 horizontal array of toruses with vertical size 8.

Finally, every torus extension of lower branch separates toruses into major pairs and every torus extension of upper branch into minor pairs.

Tiger torus tiger ((((II)(II))I)(II)) has major pairs of toruses in trace, double tiger torus (((((II)(II))(II))I) has minor pairs, Tiger torus tiger torus ((((((II)(II))I)(II))I) has major/minor quartets.

- Marek14
- Pentonian
**Posts:**1140**Joined:**Sat Jul 16, 2005 6:40 pm

So if this has a barrel, then one of 7D shapes that cut in two double tigers ((((II)I)(II))(II)) or (((II)(II))((II)I)) could have a "double-barreled" oblique cut?

Quite possibly! Maybe even a merged structure, made from two barrels? Hmm, will have to investigate. I've gotten pretty warmed up to 6D imaging and deriving the right equations. Luckily, I have many premade equations ready for splicing together into higher shapes. I was thinking of doing triger next, I feel it may have some amazing oblique structures, at least one of them. I'm imagining a triple axis tiger deformation that joins all eight spheres into one. I know it exists, just need to find it, and photograph it!

EDIT: Acutally, I made a mistake with the number of spheres in (321)-4 tigroid. It's really 8,192 spheres in a 16 concentric groupings along a 16x8x4 cuboid array.

in search of combinatorial objects of finite extent

- ICN5D
- Pentonian
**Posts:**1075**Joined:**Mon Jul 28, 2008 4:25 am**Location:**Orlando, FL

Another quite educational thing would be ordinary tiger ((II)(II)), but rendered with unequal major diameters.

- Marek14
- Pentonian
**Posts:**1140**Joined:**Sat Jul 16, 2005 6:40 pm

Marek14 wrote:Another quite educational thing would be ordinary tiger ((II)(II)), but rendered with unequal major diameters.

Wouldn't that just be the inflated ridge of a duocylinder of unequal radii (IOW the cartesian product of circles of unequal radii)?

- quickfur
- Pentonian
**Posts:**2565**Joined:**Thu Sep 02, 2004 11:20 pm**Location:**The Great White North

Yep, it's the same thing.

in search of combinatorial objects of finite extent

- ICN5D
- Pentonian
**Posts:**1075**Joined:**Mon Jul 28, 2008 4:25 am**Location:**Orlando, FL

All right, check it out:

((II)(II)(II)) : Triger

Rotation and Translation Animations (This is really awesome!)

Axial midcut of 8 spheres in a 2x2x2 cube, represented by ((I)(I)(I))

The triple 45 degree angle cut, which smoothly merges all 8 spheres into one oblique structure!

A very unique angle that makes 8 triangular pillows, I believe it was at 0.5709 on all three rotation axes, which comes out to^{2*pi}/_{11}

This cut actually appears in the middle of a rotation. It is an oblique cut! When we rotate through here, we see two thin rings appear, inflate to two toruses, then deflate and vanish. They never move towards or away from each other. This was an unexpected cut especially at an oblique angle. And, once again, it's structure that lies between empty cuts! Just like with what we saw in ((((II)I)I)(II)), we scan past structure between empty cuts. I think this is a very cool representation of higher dimensions at work. Rotations and translations will trace out the missing structure, but nothing compares to a sudden emergence of cuts that show up unexpectedly.

Here, I had to cut it open, to see what was going on. Nothing else, no cocircular extras.

((II)(II)(II)) : Triger

Rotation and Translation Animations (This is really awesome!)

Axial midcut of 8 spheres in a 2x2x2 cube, represented by ((I)(I)(I))

The triple 45 degree angle cut, which smoothly merges all 8 spheres into one oblique structure!

A very unique angle that makes 8 triangular pillows, I believe it was at 0.5709 on all three rotation axes, which comes out to

This cut actually appears in the middle of a rotation. It is an oblique cut! When we rotate through here, we see two thin rings appear, inflate to two toruses, then deflate and vanish. They never move towards or away from each other. This was an unexpected cut especially at an oblique angle. And, once again, it's structure that lies between empty cuts! Just like with what we saw in ((((II)I)I)(II)), we scan past structure between empty cuts. I think this is a very cool representation of higher dimensions at work. Rotations and translations will trace out the missing structure, but nothing compares to a sudden emergence of cuts that show up unexpectedly.

Here, I had to cut it open, to see what was going on. Nothing else, no cocircular extras.

in search of combinatorial objects of finite extent

- ICN5D
- Pentonian
**Posts:**1075**Joined:**Mon Jul 28, 2008 4:25 am**Location:**Orlando, FL

Very nice indeed!

Would the triger be the inflated ridge of the tricylinder (Cartesian product of 3 circular disks)?

Would the triger be the inflated ridge of the tricylinder (Cartesian product of 3 circular disks)?

- quickfur
- Pentonian
**Posts:**2565**Joined:**Thu Sep 02, 2004 11:20 pm**Location:**The Great White North

quickfur wrote:Very nice indeed!

Would the triger be the inflated ridge of the tricylinder (Cartesian product of 3 circular disks)?

Yes, it is

- Marek14
- Pentonian
**Posts:**1140**Joined:**Sat Jul 16, 2005 6:40 pm

Thanks Those animations were surprisingly easy to make. Each animation sequence took about 30 min of capturing 115 screenshots. If I render them in smooth contour + 50 cube resolution, it comes out really well. And, on a second note, my computer handled the simple triger equation really well, which makes me think about pushing the limits to some similar 9D ones. It would be awesome to see torus squared, or triotorus tiger! At least the second one, triotorus tiger, would be a reasonable test. 3D Calc Plot likes spheres! I'm dreaming up another amazing looking animation, one I saw in low res one time. It was a double rotation through a cyltorintigroid (((II)I)(II)) , with a 3:5 cycle ratio, and one offset by 45 degrees. Talk about tracing out a hidden structure, this one was like multiple lighthouses in the night!

in search of combinatorial objects of finite extent

- ICN5D
- Pentonian
**Posts:**1075**Joined:**Mon Jul 28, 2008 4:25 am**Location:**Orlando, FL

Triotorus triger with its 4x4x4 array of spheres as a trace would be fun, though maybe you could work up to it through torus triger and duotorus triger (arrays 2x2x4 and 2x4x4)? Don't skip the interlinks

Torus squared with its 4x4x4 array of toruses also has important interlinks to check: 2x2x4 arrays of torus double tiger and tiger/torus tiger and 2x4x4 arrays of duotorus double tiger and (torus tiger)/torus tiger.

EDIT: Have you considered contacting the author of the program? His page, IIRC, mentions that people should let him know if they use the program for educational purposes, but I think he might be interested in our toratope exploration as well.

Torus squared with its 4x4x4 array of toruses also has important interlinks to check: 2x2x4 arrays of torus double tiger and tiger/torus tiger and 2x4x4 arrays of duotorus double tiger and (torus tiger)/torus tiger.

EDIT: Have you considered contacting the author of the program? His page, IIRC, mentions that people should let him know if they use the program for educational purposes, but I think he might be interested in our toratope exploration as well.

- Marek14
- Pentonian
**Posts:**1140**Joined:**Sat Jul 16, 2005 6:40 pm

You know, I did join the discussion board, and post a brief intro to what I wanted to use it for. I haven't followed up with anything I made so far, but now would be a good time. I think this application of his program is something of a treasure trove for him. An unexpected potential use that blows everyone's mind with what it's been able to do. We have seen slices of some of the most incomprehensible shapes that no one ever thought they would see. Like duotorus tiger, for example. Hmm, maybe I could see if he would make some refinements to be better suited for toratope exploration. The only thing I would be interested in would be more than four adjustable parameters. I would like up to 7, for use with 10D shapes.

in search of combinatorial objects of finite extent

- ICN5D
- Pentonian
**Posts:**1075**Joined:**Mon Jul 28, 2008 4:25 am**Location:**Orlando, FL

ICN5D wrote:You know, I did join the discussion board, and post a brief intro to what I wanted to use it for. I haven't followed up with anything I made so far, but now would be a good time. I think this application of his program is something of a treasure trove for him. An unexpected potential use that blows everyone's mind with what it's been able to do. We have seen slices of some of the most incomprehensible shapes that no one ever thought they would see. Like duotorus tiger, for example. Hmm, maybe I could see if he would make some refinements to be better suited for toratope exploration. The only thing I would be interested in would be more than four adjustable parameters. I would like up to 7, for use with 10D shapes.

Well, I don't think it's actually that necessary: every animation has actually only one parameter as the others can be expressed as its function -- in this way you can show arbitrary amount of them.

- Marek14
- Pentonian
**Posts:**1140**Joined:**Sat Jul 16, 2005 6:40 pm

Yes, this is true. And, I'll be forced to reckon with it when I decide to pursue 8D It's more a matter of feel you get with moving the slider, and having a direct feedback with the structure it makes. I feel a greater sense of control with the physical input and graphical output. And, there's also the sense of seeing where the angles are by slider positions. It's another piece of graphical information in place of numbers. These things really help with exploration, and exploration is what finds amazing oblique artifacts. Based on what we've seen so far, I can only begin to imagine what some 8D obliques look like. Consider that one incredible duotorus tiger cut, with the outer and inner cage. Now, I wonder what duoditorus tiger ((((II)I)I)(((II)I)I)) would have in place? It came from a perfect 45 degree angle between the two axial cuts of 2 concentric stacked 4 high toruses. The equivalent in duoditorus tiger would be the two axial cuts of 4 concentric stacked 8 high, ((((II)))(((I)))), which would deform perfectly into an amazing four-cage version. Of course, knowing of where this structure sits helps, other obliques won't be as easy.

in search of combinatorial objects of finite extent

- ICN5D
- Pentonian
**Posts:**1075**Joined:**Mon Jul 28, 2008 4:25 am**Location:**Orlando, FL

Here's one I forgot to document:

(((II)I)(II)) : Cyltorintigroid

Rotation and Translation Movie

( ready yourself for this one ... )

( it's an exploration unlike any other that's come before ... )

Axial Midcut of 4 toruses in a vertical column : (((I))(II))

Axial Midcut of a 2x1x2 vertical square : (((I)I)(I))

Axial Midcut of 2 concentric stacked 2 high : (((II))(I))

Oblique Artifact resembling the Land Speeder midsection of ((((II)I)I)(II)), while omitting the central structure.

A slight deformation of the 2x1x2 vertical square midsection, into the 1x1x4 column

Oblique Artifact existing at some angle close to the (((II))(I)) axial cut

Adjusting our angle will pinch off one and join another touch point

(((II)I)(II)) : Cyltorintigroid

Rotation and Translation Movie

( ready yourself for this one ... )

( it's an exploration unlike any other that's come before ... )

Axial Midcut of 4 toruses in a vertical column : (((I))(II))

Axial Midcut of a 2x1x2 vertical square : (((I)I)(I))

Axial Midcut of 2 concentric stacked 2 high : (((II))(I))

Oblique Artifact resembling the Land Speeder midsection of ((((II)I)I)(II)), while omitting the central structure.

A slight deformation of the 2x1x2 vertical square midsection, into the 1x1x4 column

Oblique Artifact existing at some angle close to the (((II))(I)) axial cut

Adjusting our angle will pinch off one and join another touch point

in search of combinatorial objects of finite extent

- ICN5D
- Pentonian
**Posts:**1075**Joined:**Mon Jul 28, 2008 4:25 am**Location:**Orlando, FL

Check it out! I made a bitangent cut of a torus, and produced the two perfect Villarso circles.

in search of combinatorial objects of finite extent

- ICN5D
- Pentonian
**Posts:**1075**Joined:**Mon Jul 28, 2008 4:25 am**Location:**Orlando, FL

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