The Tiger Explained

[ICN5D]

Had a quick look at Tiger Ditorus. If you thought the tiger was a beast, this one is a real beast! It took some time in refining the radii values, in addition to the extra parentheses that need to be in this kind of equation. I did the axial cut of 2 concentric along a 2x2 flat square of 8 toruses, and one oblique so far. I don't know about you all, but I never once expected to get a chance to see images of 6D toratopes like this!


((((II)(II))I)I) - (00)-2 Tigroid - Tiger Ditorus





Standard Equation:
(sqrt(((sqrt(x^2 + y^2) - R1)^2 + (sqrt(z^2 + w^2) - R2)^2 - R3)^2 + v^2) - R4)^2 + u^2 - R5^2 = 0


((((I)(I)))I) Cut Equation:
(sqrt(((sqrt(x^2 + a^2) - 3)^2 + (sqrt(y^2 + b^2) - 3)^2 - 3)^2 + c^2) - 1.6)^2 + z^2 - 1^2 = 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 - 3)^2 + (y*cos(c))^2) - 1.6)^2 + (z*sin(a))^2 - 1^2 = 0





Axial Midsection of 2 concentric along a 2x2 flat square of 8 toruses : ((((I)(I)))I)






Slight merging of quartet along both cut axes of the tiger component. Produces a neat square with hole in the center






Further bi-axial translation along tiger component, at mid-merge that produces something topologically equivalent to a sphere, in the center






A slight merging of the concentric pairing will combine into a smoothed out table of holes and valleys






Further translation in tiger component ...






Tiger Ditorus has some really wild oblique midsections!







I'll post more obliques later. They are highly complex and unlike any others I've seen before. It's related to the 4 cocircular groupings of toruses, and how they deform into other arrangements. A lot of rotation trimming happens, and when 4 cocircular toruses are opened up, we see quite a bit going on inside.

[Marek14]

The outer toruses in the first picture look more like flat bands... why is that?

[ICN5D]

I'm, not sure Marek. I played with the radii values a lot, trying to round them out. The process is a matter of separating the pieces, away from being internally nested. No matter what I did, the outer toruses come out as ribbons, or joined the inner quartet. Maybe, it's a matter of parentheses in the function, perhaps they have a peculiar arrangement making elongated minor radii?

[Marek14]

Hm, the function is too complex for me to find whether the parentheses are correct at this time of night

[ICN5D]

The first test formula only made four in a square. The extra concentric four showed up right after adding new parentheses, so I'm thinking it could be right.Its probably some special combination of radii values that will round out the ribbons. That's usually the case I see.

[ICN5D]

Marek, I've been referring back to your new toratope nomenclature, and I noticed something. How do you differentiate between (((III)I)(II)) and (((II)II)(II)), if both have the trace of (10)0-Tigroid?

[Marek14]

Marek, I've been referring back to your new toratope nomenclature, and I noticed something. How do you differentiate between (((III)I)(II)) and (((II)II)(II)), if both have the trace of (10)0-Tigroid?

Well, it can be done by secondary numbers (it's in my post), but mostly, this notation is to classify species of toratopes, not the individual shapes. The two toratopes you mention belong to the same species.

[ICN5D]

Oh, that's right. That's when you use something like 1-toroid (10) 0- tigroid? But maybe more like 31-toroid (10) 0-tigroid for (((III)I)(II))?

[Marek14]

Something like that, yes. Basically, once you get high enough to need this type of notation, you don't really care about the "fat" toratopes, only about the basic member of the species.

[anderscolingustafson]

I used the Parametric Equation for the Tiger to help myself understand it. The Parametric Equation for the Tiger is

x=acos(f)+ccos(f)cos(h)
y=asin(f)+csin(f)cos(h)
z=bcos(g)+ccos(g)sin(h)
w=bsin(g)+csin(g)sin(h)

I found used the Parametric Equation to find some of the equations for circles on the Tiger and found that the Tiger is has Duocylinders on it's surface. In fact every point on the Tiger is part of a Duocylinder.

I was wondering if the Tiger can link up with other Tigers to form chains can they form a chain that is in the shape of a Duocylinder?

[Marek14]

Good question. I started pondering about two-link chains some time ago, so not there yet.

[wendy]

tigers make a chain, but because they have different kinds of hole. One kind of hole is represented by a single duo-cylinder-face, while the other kind of hole is in one duocylinder, and out the other. The parities of these holes match what is needed to make a chain in four dimensions.

[ICN5D]

...

I found used the Parametric Equation to find some of the equations for circles on the Tiger and found that the Tiger is has Duocylinders on it's surface. In fact every point on the Tiger is part of a Duocylinder.

Yep, that's what it means when you see " inflated duocylinder margin". The duocylinder has two curved faces only, and both meet each other at a single 90 degree sharp edge that snakes its way around the surface. If we cut out the faces, leaving behind only the ridge as a wireframe, then inflated this structure like an innertube, we end up with the tiger. It doesn't actually have duocylinders on its surface, but is made from part of one: the ridge.

[quickfur]

Technically speaking, since the tiger is an inflated duocylinder ridge, and (1) the duocylinder ridge is 2D, which doesn't divide space, whereas (2) the tiger not only divides space (it has a 3D surface), it also fills space, its surface must consist of an infinitude of duocylinder ridges, stacked on each other in a circular way. So in that sense, you can cut up the tiger's surface into an infinitude of duocylinder ridges, each one of which can be made into a duocylinder by closing it up with torus-shaped surface patches.

[ICN5D]

Now that's an interesting thinking process. By analogy, one could say the same about a torus having an infinitude of circles stacked in a circular way, being the inflated edge of a circle. One can then fill in the infinite circle cuts with a line torus to make a cylinder, the open toratope cousin of a torus. The tiger frame will always have a memory of its duocylinder heritage, and can be transformed back into one by the same process as you described. Cool stuff

[Marek14]

Now that's an interesting thinking process. By analogy, one could say the same about a torus having an infinitude of circles stacked in a circular way, being the inflated edge of a circle. One can then fill in the infinite circle cuts with a line torus to make a cylinder, the open toratope cousin of a torus. The tiger frame will always have a memory of its duocylinder heritage, and can be transformed back into one by the same process as you described. Cool stuff

Well, torus is made of an infinitude of circles in two different ways (which can be used to give it a coordinate system). Tiger could, analogically, have one coordinate specifying on which duocylinder margin we are, and then two coordinates fixing our position there.

[ICN5D]

Yeah that's the feeling I got, since a torus has two distinct axial cuts. The tiger has only one in both 2 and 3D.





On another note, I revisited the rotation algorithm and saw new things with the ditorus. I wanted to show what happens to the cuts that have no concentric/cocircular arrangements.

(((II)I)I) - ditorus - 2x1x1 line of toruses

(((xy)z)w) - hyperplane arrangement


Non-bisecting around main diameter hyperplane XY
-----------------------------------------------------------------
((((II)I)I)I) - tritorus - 4x1x1 line of toruses


Non-bisecting around middle diameter hyperplane XZ
-------------------------------------------------------------------
(((II)(II))I) - tiger torus - 2x2x1 square of toruses


Non-bisecting around minor diameter hyperplane XW
----------------------------------------------------------------------
(((II)I)(II)) - cyltorintigroid - 2x1x2 vert square of toruses


And so I found that interesting, that a tiger torus can be made by a rotation of a ditorus. I feel it's related to how the middle diameter plays its role in the ditorus. An identical rotation of a torus around its minor diameter in XZ will produce a tiger. The middle diameter is playing the role of the minor in 3D torus. So, when this rotation happens, another tiger is made, naturally, which now resides within the shape. Starting with a torus of a torus, we rotate and make a torus of a tiger! Makes an awesome amount of sense, don't it? I think so, and was wholly impressed with the logic in the scenario.

Even furthermore, we see what happens to the initial cut array of the 2x1x1 line of two toruses. Each non-bisecting rotation doubled the quantity on the respective leg.

[Marek14]

Yes, a nonbisecting rotation doubles the cut in some way.

[ICN5D]

Well, I corrected myself on the 1x1x4 vertical column. They are all laid flat in all arrangements, even better!

[Marek14]

BTW, ICN5D, I looked at your youtube videos, but I'm afraid they are a bit too complex for the average viewer. Maybe you should make some introductory videos for 3D and 4D toratopes?

For 4D toratopes, the basic animations are:

Glome slice (sphere)
Torisphere slice (pair of spheres)
Torisphere slice (torus)
Torisphere rotation (pair of spheres <-> torus)
Ditorus slice (two separate toruses)
Ditorus slice (major pair of toruses)
Ditorus slice (minor pair of toruses)
Ditorus rotation (two separate toruses <-> major pair of toruses)
Ditorus rotation (two separate toruses <-> minor pair of toruses)
Ditorus rotation (major pair of toruses <-> minor pair of toruses)
Tiger slice (stack of 2 toruses)
Tiger rotation (stack of 2 toruses <-> stack of 2 toruses)
Spheritorus slice (torus)
Spheritorus slice (2 spheres)
Spheritorus rotation (torus <-> 2 spheres)

So 15 short animations that could easily fit into a single video.

Since, for example, the torus tiger animation shows clearly the rotation animation of tiger, but people might not know it if they never saw the simpler example.

[Marek14]

5D notable animations:

1. Pentasphere (IIIII)
1 4D slice: glome (IIII)
1 3D slice: sphere (III)
Sphere cut can only evolve into glome slice.
No rotations.

2. 41-torus ((IIII)I)
2 4D slices: pair od glomes ((IIII)), torisphere ((III)I)
2 3D cuts: pair of spheres ((III)), torus ((II)I).
Pair of spheres cut can evolve into torisphere slice or pair of glomes slice.
Torus cut can only evolve into torisphere slice.
1 rotation is animation of torisphere.

3. 311-ditorus (((III)I)I)
3 4D slices: minor pair of torispheres (((III)I)), major pair of torispheres (((III))I), ditorus (((II)I)I)
4 3D cuts: quartet of spheres (((III))), minor pair of toruses (((II)I)), major pair of toruses (((II))I), two toruses (((I)I)I).
Quartet of spheres cut can evolve into minor pair of torispheres slice or major pair of torispheres slice.
Minor pair of toruses cut can evolve into minor pair of torispheres slice or ditorus slice.
Major pair of toruses cut can evolve into major pair of torispheres slice or ditorus slice.
Two toruses cut can only evolve into ditorus slice.
3 rotations are animations of ditorus.
2 new rotations: quartet of spheres - minor pair of toruses (((IIx)x)) and quartet of spheres - major pair of toruses (((IIx))x).
1 double rotation: quartet of spheres - two toruses (((Ixy)x)y)

4. Tritorus ((((II)I)I)I)
4 4D slices: minor pair of ditoruses ((((II)I)I)), medium pair of ditoruses ((((II)I))I), major pair of ditoruses ((((II))I)I) and two ditoruses ((((I)I)I)I)
7 3D cuts: minor quartet of toruses ((((II)I))), major/minor quartet of toruses ((((II))I)), two minor pairs of toruses ((((I)I)I)), major quartet of toruses ((((II)))I), two major pairs of toruses ((((I)I))I), four toruses ((((I))I)I), empty cut (((()I)I)I)
Minor quartet of toruses cut can evolve into minor pair of ditoruses slice or medium pair of ditoruses slice.
Major/minor quartet of toruses cut can evolve into minor pair of ditoruses slice or major pair of ditoruses slice.
Two minor pairs of toruses cut can evolve into minor pair of ditoruses slice or two ditoruses slice.
Major quartet of toruses cut can evolve into medium pair of ditoruses slice or major pair of ditoruses slice.
Two major pairs of toruses cut can evolve into medium pair of ditoruses slice or two ditoruses slice.
Four toruses cut can evolve into major pair of ditoruses slice or two ditoruses slice.
Empty cut can only evolve into two ditoruses slice.
15 Rotations: minor quartet of toruses - major/minor quartet of toruses ((((II)x)x)), minor quartet of toruses - two minor pairs of toruses ((((Ix)I)x)), minor quartet of toruses - major quartet of toruses ((((II)x))x), minor quartet of toruses - two major pairs of toruses ((((Ix)I))x), major/minor quartet of toruses - two minor pairs of toruses ((((Ix)x)I)), major/minor quartet of toruses - major quartet of toruses ((((II))x)x), major/minor quartet of toruses - four toruses ((((Ix))I)x), two minor pairs of toruses - two major pairs of toruses ((((I)I)x)x), two minor pairs of toruses - four toruses ((((I)x)I)x), two minor pairs of toruses - empty cut ((((x)I)I)x), major quartet of toruses - two major pairs of toruses ((((Ix)x))I), major quartet of toruses - four toruses ((((Ix))x)I), two major pairs of toruses - four toruses ((((I)x)y)I), two major pairs of toruses - empty cut ((((x)I)x)I), four toruses - empty cut ((((x)x)I)I)
6 Double rotations: minor quartet of toruses - four toruses ((((Ix)y)x)y) and ((((Ix)y)y)x), minor quartet of toruses - empty cut ((((xy)I)x)y), major/minor quartet of toruses - two major pairs of toruses ((((Ix)x)y)y) and ((((Ix)y)y)x), major/minor quartet of toruses - empty cut ((((xy)x)I)y), two minor pairs of toruses - major quartet of toruses ((((Ix)x)y)y) and ((((Ix)y)x)y), major quartet of toruses - empty cut ((((xy)x)y)I)
Note: Specific double rotations can be shared, for example ((((Ix)y)x)y) can be between minor quartet of toruses and four toruses or between two minor pairs of toruses and major quartet of toruses, based on how the initial values for parameters are chosen.

5. Tiger torus (((II)(II))I)
2 4D slices: minor pair of tigers slice (((II)(II))), two ditoruses stacked in medium dimension (((II)(I))I) and (((I)(II))I)
3 3D cuts: vertical stack of two minor pairs of toruses (((II)(I))) and (((I)(II))), empty cut (((II)())I) and ((()(II))I), 2x2 array of toruses (((I)(I))I)
Vertical stack of two minor pairs of toruses cut can evolve into minor pair of tigers slice or two ditoruses stacked in medium dimension
Empty cut can evolve only into two ditoruses stacked in medium dimension.
2x2 array of toruses cut can evolve into two ditoruses stacked in medium dimension, but in two different orientations.
4 rotations: vertical stack of two minor pairs of toruses - alternate vertical stack of two minor pairs of toruses ((((Ix)(Ix))), vertical stack of two minor pairs of toruses - empty cut ((((II)(x))x), vertical stack of two minor pairs of toruses - 2x2 array of toruses cut (((Ix)(I))x), empty cut - 2x2 array of toruses cut (((Ix)(x))I)
2 double rotations: vertical stack of two minor pairs of toruses - alternate empty cut (((xy)(Ix))y), empty cut - alternate empty cut (((xy)(xy))I)

6. 221-ditorus (((II)II)I)
3 4D slices: minor pair of spheritoruses slice (((II)II)), ditorus slice (((II)I)I), two torispheres slice (((I)II)I)
5 3D cuts: minor pair of toruses (((II)I)), two pairs of spheres (((I)II)), major pair of toruses (((II))I), two toruses (((I)I)I), empty cut ((()II)I)
Minor pair of toruses cut can evolve into minor pair of spheritoruses slice or ditorus slice.
Two pairs of spheres cut can evolve into minor pair of spheritoruses slice or two torispheres slice.
Major pair of toruses cut can evolve only into ditorus slice.
Two toruses cut can evolve into ditorus slice or two torispheres slice.
Empty cut can evolve only into two torispheres slice.
3 rotations are animations of ditorus.
4 new rotations: minor pair of toruses - two pairs of spheres (((Ix)Ix)), two pairs of spheres - two toruses (((I)Ix)x), two pairs of spheres - empty cut (((x)II)x), two toruses - empty cut (((x)Ix)I)
3 double rotations: minor pair of toruses - empty cut (((xy)Ix)y), two pairs of spheres - major pair of toruses (((Ix)xy)y), major pair of toruses - empty cut (((xy)xy)I)

7. 320-tiger ((III)(II))
2 4D slices: vertical stack of two torispheres slice ((III)(I)), tiger slice ((II)(II))
3 3D cuts: empty cut ((III)()), vertical stack of two toruses A ((II)(I)), vertical stack of two toruses B ((I)(II))
Empty cut can evolve only into vertical stack of two torispheres slice.
Vertical stack of two toruses cut A can evolve into vertical stack of two torispheres slice or tiger slice.
Vertical stack of two toruses cut B can evolve only into tiger slice.
1 rotation is animation of tiger.
1 new rotation: empty cut - vertical stack of two toruses A ((IIx)(x))
1 double rotation: empty cut - vertical stack of two toruses B ((Ixy)(xy))

8. Torus tiger (((II)I)(II))
3 4D slices: minor stack of two ditoruses slice (((II)I)(I)), major pair of tigers slice (((II))(II)), two tigers slice (((I)I)(II))
5 3D cuts: empty cut A (((II)I)()), vertical stack of two major pairs of toruses (((II))(I)), two vertical stacks of toruses (((I)I)(I)), vertical stack of four toruses (((I))(II)), empty cut B ((()I)(II))
Empty cut A can evolve only into minor stack of two ditoruses slice.
Vertical stack of two major pairs of toruses cut can evolve into minor stack of two ditoruses slice or major pair of tigers slice.
Two vertical stacks of toruses cut can evolve into minor stack of two ditoruses slice or two tigers slice.
Vertical stack of four toruses cut can evolve into major pair of tigers slice or two tigers slice.
Empty cut B can evolve only into two tigers slice.
7 rotations: empty cut A - vertical stack of two major pairs of toruses (((II)x)(x)), empty cut A - two vertical stacks of toruses (((Ix)I)(x)), vertical stack of two major pairs of toruses - two vertical stacks of toruses (((Ix)x)(I)), vertical stack of two major pairs of toruses - vertical stack of four toruses (((Ix))(Ix)), two vertical stacks of toruses - vertical stack of four toruses (((I)x)(Ix)), two vertical stacks of toruses - empty cut B (((x)I)(Ix)), vertical stack of four toruses - empty cut B (((x)x)(II))
3 double rotations: empty cut A - vertical stack of four toruses (((Ix)y)(xy)), empty cut A - empty cut B (((xy)I)(xy)), vertical stack of two major pairs of toruses - empty cut B (((xy)x)(Iy))

9. 32-torus ((III)II)
2 4D slices: torisphere slice ((III)I), spheritorus slice ((II)II)
3 3D cuts: pair of spheres ((III)), torus ((II)I), two spheres ((I)II)
Pair of spheres cut can evolve only into torisphere slice.
Torus cut can evolve into torisphere slice or spheritorus slice.
Two spheres cut can evolve only into spheritorus slice.
1 rotation is animation of torisphere.
1 rotation is animation of spheritorus.
1 double rotation: pair of spheres - two spheres ((Ixy)xy)

10. 212-ditorus (((II)I)II)
3 4D slices: ditorus slice (((II)I)I), major pair of spheritoruses slice (((II))II), two spheritoruses slice (((I)I)II)
5 3D cuts: minor pair of toruses (((II)I)), major pair of toruses (((II))I), two toruses (((I)I)I), four spheres (((I))II), empty cut ((()I)II)
Minor pair of toruses cut can evolve only into ditorus slice.
Major pair of toruses cut can evolve into ditorus slice or major pair of spheritoruses slice.
Two toruses cut can evolve into ditorus slice or two spheritoruses slice.
Four spheres cut can evolve into major pair of spheritoruses slice or two spheritoruses slice.
Empty cut can evolve only into two spheritoruses slice.
3 rotations are animations of ditorus.
4 new rotations: major pair of toruses - four spheres (((Ix))Ix), two toruses - four spheres (((I)x)Ix), two toruses - empty cut (((x)I)Ix), four spheres - empty cut (((x)x)II)
3 double rotations: minor pair of toruses - four spheres (((Ix)y)xy), minor pair of toruses - empty cut (((xy)I)xy), major pair of toruses - empty cut (((xy)x)Iy)

11. 221-tiger ((II)(II)I)
2 4D slices: tiger slice ((II)(II)), vertical stack of 2 spheritoruses slice ((II)(I)I) and ((I)(II)I)
3 3D cuts: vertical stack of 2 toruses ((II)(I)) and ((I)(II)), empty cut ((II)()I) and (()(II)I), 2x2 array of spheres ((I)(I)I)
Vertical stack of 2 toruses cut can evolve into tiger slice or vertical stack of 2 spheritoruses slice.
Empty cut can evolve only into vertical stack of 2 spheritoruses slice.
2x2 array of spheres cut can evolve only into vertical stack of 2 spheritoruses slice, but in 2 different ways.
1 rotation is animation of tiger.
3 new rotations: vertical stack of 2 toruses - empty cut ((II)(x)x), vertical stack of 2 toruses - 2x2 array of spheres ((Ix)(I)x), empty cut - 2x2 array of spheres ((Ix)(x)I)
2 double rotations: vertical stack of 2 toruses - alternate empty cut ((xy)(Ix)y), empty cut - alternate empty cut ((xy)(xy)I)

12. 23-torus ((II)III)
2 4D slices: spheritorus slice ((II)II), two glomes slice ((I)III)
3 3D cuts: torus ((II)I), two spheres ((I)II), empty cut (()III)
Torus cut can evolve only into spheritorus slice.
Two spheres cut can evolve into spheritorus slice or two glomes slice.
Empty cut can evolve only into two glomes slice.
1 rotation is animation of spheritorus.
1 new rotation: two spheres - empty cut ((x)IIx)
1 double rotation: torus - empty cut ((xy)Ixy)

[ICN5D]

Yeah, you're right I just dove right into 5D with no precursory display of 4D actions. Those are the one's I'll do next, and try to refrain myself from more 6D exotics. It could fit into one animation, I suppose. Good thing is, I made all of the necessary scripts to load into CalcPlot3D, so accessing 3 and 4D will be quick.

But, other than that, the torus tiger rotations are super cool, aren't they?! It's got a solid 3 minutes of random rotations of various kinds. Probably beats out all other high-D youtube videos in display of awesomeness. I liken it to a 3D Lighthouse, rotating in the center of the shape. As the beam sweeps around, it illuminates all of the solid parts and holes in the most incredible way I've ever seen

[Marek14]

Yeah, you're right I just dove right into 5D with no precursory display of 4D actions. Those are the one's I'll do next, and try to refrain myself from more 6D exotics. It could fit into one animation, I suppose. Good thing is, I made all of the necessary scripts to load into CalcPlot3D, so accessing 3 and 4D will be quick.

But, other than that, the torus tiger rotations are super cool, aren't they?! It's got a solid 3 minutes of random rotations of various kinds. Probably beats out all other high-D youtube videos in display of awesomeness. I liken it to a 3D Lighthouse, rotating in the center of the shape. As the beam sweeps around, it illuminates all of the solid parts and holes in the most incredible way I've ever seen

Yeah, it's nice

[ICN5D]

Relating back to the rotations of a ditorus, I just saw a little clearer how the torus-tiger (((II)I)(II)) is built. In the animations, you can clearly see a torus in the images. By noting how a tiger is made by the rotation around the minor diameter of a torus, I'll coin the term " tigering" to describe the transformation. Making a tritorus is pretty straightforward, just another "torusing" of a ditorus. The tiger torus (((II)(II))I) is made by the tigering of the middle diameter, related to a normal torus-->tiger rotation.

Now comes the tigering of the minor diameter, and thus the creation of a torus-tiger (((II)I)(II)). Ditorus can be understood as inflating the surface of a torus with a circle. If we perform a " tigering" transformation of this minor circle, the main torus frame is still left intact. We then have a hollow torus with a skin containing a tiger-symmetry circle. The array of all 3D cuts of a torus-tiger come out as a [ditorus-1 cut] x [circle-1 cut] or [torus-2 cut] x [spherated circle]. Which can also be traced back to one of the 4d cuts of a vertical stack of ditoruses, (((II)I)(I)).




Now, going little further, let's rotate a torus-tiger:

(((II)I)(II)) - (10)0-tigroid , torus-tiger

(((xy)z)(wv)) - hyperplane orientation



Nonbisecting rotation around plane XY
---------------------------------------------------
((((II)I)I)(II)) - (20)0-tigroid , ditorus-tiger


Nonbisecting rotation around plane XZ
---------------------------------------------------
(((II)(II))(II)) - ([(00)0])0-tigroid , double tiger ---->> not sure how you would represent that in the new nomenclature


Nonbisecting rotation around plane WV
---------------------------------------------------
(((II)I)((II)I)) - (11)0-tigroid , duotorus-tiger


Nonbisecting rotation around minor diameter, not sure which hyperplane it would be in
-------------------------------------------------------------------------------------------------------------------
((((II)I)(II))I) - (10)1-tigroid , torus-tiger torus


Which brings me to my next point: the torus-tiger torus ((((II)I)(II))I). It's trace is a 4x2x1 flat rectangle of toruses. If we were to apply another rotation around this last dimensional marker, that is a " tigering " of this main diameter, we get ((((II)I)(II))(II)), which has the trace of a 4x2x2 array of toruses. Which I find interesting, actually, at how this final diameter is transformed. In this nonbisecting rotation, the torus-tiger torus has its main circle held flat, and rotated into a new, tiger-like shape. This embeds the flat ring of torus-tiger torus into a vertical ring, making the duoring of a tiger. That's how we end up with a 4x2x2 array from a 4x2x1. The original array gets a vertical stacking, just like a regular tiger does with the two toruses. Which is how this 7D toratope ((((II)I)(II))(II)) is built: A torus-tiger (((II)I)(II)) embedded along the duoring rim of a tiger. The rotations up to it illuminate the structure fairly well.


I've also been thinking on how many different combinations of rotations each toratope can be made from, and what it might mean, or display. It could be nothing more than a simple permutation of rotations, but it's neat to play with the idea of multiple paths to creation.

Ahh, that felt great! I really needed to talk about shapes for a bit. It's a happy place for me

[Marek14]

Just a note: 4x2x1 array of toruses can't be the trace since it can still be bisected -- trace in this case is 4x2 array of pairs of circles.

And I understand your shapetalk need I'm currently compiling document about slices and rotations of 6D toratopes

[ICN5D]

I'm currently compiling document about slices and rotations of 6D toratopes

Aw, sweet! Any illustrations? I think you mentioned Mathematica before, so probably. Would it be any better for renders, do you think? I fear that treading into 7 and 8D waters will be beyond the capacity of CalcPlot3D. Though, I'm not sure, since I haven't gone there yet. But, if it handled Triger really well, then maybe some higher derivatives of it will be just as easy. As in, those that are a trace of spheres.

I've been thinking of those 6D toratopes that are spheres along a tigroid rim, like spheric tiger ((II)(II)I), spheric torus-tiger (((II)I)(II)I), and the like. It is interesting that the trace of spheres is actually from the lowest circle-array trace, that then had a bisecting rotation of its minor diameter. It's also cool to see how it plays around with the hidden minor diameter of tigroids. It seems to be already defined by the outermost brackets, in the positions of the two markers in (II). Adding another marker, as in ((II)(II)I) has the identity of (III), a sphere. (((II)I)(II)I) ought to have some wild looking obliques, wouldn't you say?

[ICN5D]

I also wanted to put this out there:

Does anyone who reads this thread have any questions about the toratope notation, or cut algorithm? Or, any q's of all this talk about arrangements and rotations into crazy high dimensions? The basic stuff was discussed way back, and it got more advanced really quickly. Now's a perfect time to ask if you want to learn it! We're all ears....

[anderscolingustafson]

I also wanted to put this out there:

Does anyone who reads this thread have any questions about the toratope notation, or cut algorithm? Or, any q's of all this talk about arrangements and rotations into crazy high dimensions? The basic stuff was discussed way back, and it got more advanced really quickly. Now's a perfect time to ask if you want to learn it! We're all ears....

I'm confused about what the roman numerals mean.

[ICN5D]

The roman numerals, made by the capitol letter " i " as " I ", are the dimensions of the shape. The parentheses divide up the dimensions into other diameters, if allowed. Take the n-spheres, for example:

(II) - 2D circle
(III) - 3D sphere
(IIII) - 4D glome
(IIIII) - 5D pentasphere
(IIIIII) - 6D hexasphere

Each n-sphere has the exact matching amount of dimension markers, or " I " in one set of parentheses. One set of parentheses means only one diameter is rounded spherically in all dimensions.

Now, we can also make toroids out of these dimensions, too. Using the same number of dimension markers, we can have a shape with two diameters in 3D, a torus, with two sets of parentheses: ((II)I). First we started with a circle (II), and replaced one marker with an X, making (xI). Then, we take an entire circle (II), and replace it with X, making a circle embedded along the edge of another circle: ((II)I).

(III) - 3D sphere
((II)I) - 3D torus

For 4D, we can play with the markers of a glome, which has one diameter, and thus one pair of parentheses.

(IIII) - 4D sphere, glome

((II)II) - 4D spheritorus, sphere along the edge of a circle

((III)I) - 4D torisphere, circle along edge of sphere: replacing X with a sphere (III) in a circle (xI)

(((II)I)I) - 4D ditorus, circle along circle along circle, or simply torus along circle, has three diameters

((II)(II)) - 4D tiger, circle along duoring edge of duocylinder, has three diameters as two major, one minor


And, then of course, there's 5D. With 5 dimension markers, " I " , we can have a pentasphere (IIIII), with only one diameter, or something like a torus-tiger (((II)I)(II)), with four diameters.

Then, to go even further, there's something like a 12 dimensional (((((II)I)(II))((II)(II)))((II)I)), which has a whopping eleven diameters!! It's lowest dimensional trace is 128 tiger toruses (((II)(II))I) in a 4x2x2x2x4 penteractoid 5-axis array. Starting with a duodecasphere (IIIIIIIIIIII) with only one diameter, we can then turn it into something with a large number of diameters, by dividing up the shape with parentheses.

[anderscolingustafson]

Thank You ICN5D! Now I think I understand the roman numeral notation.