The Tiger Explained

[Marek14]

Note that any rotation between two empty cuts is also empty.

Actually, that's not entirely accurate. In some shapes, structure does appear at an oblique angle between empties. That's what this montage illustrates:

I meant specifically for the ditorus case. Ditorus has only one way to make an empty cut (have the innermost pair of parenthesis empty). To have an interesting transition between two empty cuts, they must be empty in a "different way".

Let's look at the tiger case:

(()())
3D cuts:
Empty cut A1: ((III)()) -> [vertical stack of two toruses 1, empty cut B1]
Vertical stack of two toruses 1: ((II)(I)) -> [empty cut A1, empty cut B1, vertical stack of two toruses 2, 2x2 array of spheres]
Empty cut B1: ((II)()I) -> [empty cut A1, vertical stack of two toruses 1, 2x2 array of spheres, empty cut C1]
Vertical stack of two toruses 2: ((I)(II)) -> [vertical stack of two toruses 1, 2x2 array of spheres, empty cut A2, empty cut B2]
2x2 array of spheres: ((I)(I)I) -> [vertical stack of two toruses 1, empty cut B1, vertical stack of two toruses 2, empty cut C1, empty cut B2, empty cut C2]
Empty cut C1: ((I)()II) -> [empty cut B1, 2x2 array of spheres, empty cut C2, empty cut D]
Empty cut A2: (()(III)) -> [vertical stack of two toruses 2, empty cut B2]
Empty cut B2: (()(II)I) -> [vertical stack of two toruses 2, 2x2 array of spheres, empty cut A2, empty cut C2]
Empty cut C2: (()(I)II) -> [2x2 array of spheres, empty cut C1, empty cut B2, empty cut D]
Empty cut D: (()()III) -> [empty cut C1, empty cut C2]

Now, rotations from one empty cut to another will stay empty whenever you rotate from one "1" cut to another of from one "2" cut to another, but they will have intermediate stages if you rotate between "1" and "2" cut. Empty cut D, however, will stay empty no matter what single rotation you use.

[ICN5D]

oh, whoops! My bad! You did after all just get done talking about a ditorus before writing that. I should of caught that, but oh well

[ICN5D]

Came up with some cool names for the toratopes, in relation to Polyhedron Dude's initial system. The linguistic part of my brain is fully engaged today, which is why I had to do it. Cool and unique names are just as awesome as trace array nomenclature


2D:
(II) - circle

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

4D:
(IIII) - glome
((II)II) - spheritorus
((II)(II)) - tiger
((III)I) - torisphere
(((II)I)I) - ditorus

5D:
(IIIII) - glomoteron , pentasphere?
((II)III) - glomitorus
((II)(II)I) - spheritiger
((III)II) - spheritorisphere
(((II)I)II) - spheriditorus
((III)(II)) - cylspherintigroid
(((II)I)(II)) - cyltorintigroid
((IIII)I) - toriglome
(((II)II)I) - torispheritorus
(((II)(II))I) - toratiger
(((III)I)I) - ditorisphere
((((II)I)I)I) - tritorus

6D:
(IIIIII) - glomoexon
((II)IIII) - glomoteritorus
((II)(II)II) - glomitiger
((II)(II)(II)) - triger
((III)III) - glomitorisphere
(((II)I)III) - glomiditorus
((III)(II)I) - sphericylspherintigroid
(((II)I)(II)I) - sphericyltorintigroid
((III)(III)) - duosphere tiger
(((II)I)(III)) - duocyltorintigroid
(((II)I)((II)I)) - duotorus tiger
((IIII)II) - spheritoriglome
(((II)II)II) - dispheritorus
(((II)(II))II) - spheritoratiger
(((III)I)II) - spheriditorisphere
((((II)I)I)II) - spheritritorus
((IIII)(II)) - cylglomintigroid
(((II)II)(II)) - cylspheritorintigroid
(((II)(II))(II)) - tigritiger, or ditiger?
(((III)I)(II)) - cyltorispherintigroid
((((II)I)I)(II)) - cylditorintigroid
((IIIII)I) - toriglomoteron , toripentasphere?
(((II)III)I) - toriglomitorus
(((II)(II)I)I) - toraspheritiger
(((III)II)I) - torispherisphere
((((II)I)II)I) - torispheriditorus
(((III)(II))I) - toracylspherintigroid
((((II)I)(II))I) - toracyltorintigroid
(((IIII)I)I) - ditoriglome
((((II)II)I)I) - ditorispheritorus
((((II)(II))I)I) - ditoratiger
((((III)I)I)I) - tritorisphere
(((((II)I)I)I)I) - tetratorus

[Polyhedron Dude]

Came up with some cool names for the toratopes, in relation to Polyhedron Dude's initial system. The linguistic part of my brain is fully engaged today, which is why I had to do it. Cool and unique names are just as awesome as trace array nomenclature

5D:
(IIIII) - glomoteron , pentasphere?
((II)III) - glomitorus
((II)(II)I) - spheritiger
((III)II) - spheritorisphere
(((II)I)II) - spheriditorus
((III)(II)) - cylspherintigroid
(((II)I)(II)) - cyltorintigroid
((IIII)I) - toriglome
(((II)II)I) - torispheritorus
(((II)(II))I) - toratiger
(((III)I)I) - ditorisphere
((((II)I)I)I) - tritorus

Interesting, a few days ago I made an attempt to name a few of the 5-D ones and I got similar (or even the same) results, here's what I got:

(IIIII) - pentasphere
((IIII)I) - toriglome
(((III)I)I) - ditorisphere
(((II)(II))I) - toritiger
(((II)II)I) - torispheritorus
((III)II) - spherisphere
(((II)I)II) - spheriditorus
((II)III) - glomitorus

I just thought of some interesting "fused" names for the "tigroids"

((III)(II)) - cysphiger
(((II)I)(II)) - cytoriger

[ICN5D]

Yeah, it's cool, I like it. It's very mathematical, and precise. No two names are alike, and they conform perfectly with the [small shape]-tori/tigri-[large shape] format.

However, I think some can be refined, just as I see you doing, like ((III)II) . I'm now inclined to call it disphere and (((III)II)I) a toridisphere. This would allow (((III)I)II) to be a spheritorisphere in place of spheriditorisphere, since there is only one circle-diameter in between two spherical diameters. The names also help with understanding the build-up of the shapes, if consulting the [small shape]-tori/tigri-[large shape]. Because after all, the name ought to describe the topology!

[ICN5D]

Okay, small update to the list:

(IIIII) - glomoteron
((II)III) - glomitorus
((II)(II)I) - spheritiger
((III)II) - disphere
(((II)I)II) - spheriditorus
((III)(II)) - cylspherintigroid
(((II)I)(II)) - cyltorintigroid
((IIII)I) - toriglome
(((II)II)I) - torispheritorus
(((II)(II))I) - toritiger
(((III)I)I) - ditorisphere
((((II)I)I)I) - tritorus

6D:
(IIIIII) - glomoexon
((II)IIII) - glomoteritorus
((II)(II)II) - glomitiger
((II)(II)(II)) - triger
((III)III) - glomisphere
(((II)I)III) - glomiditorus
((III)(II)I) - sphericylspherintigroid
(((II)I)(II)I) - sphericyltorintigroid
((III)(III)) - duosphere tiger
(((II)I)(III)) - duocyltorintigroid
(((II)I)((II)I)) - duotorus tiger
((IIII)II) - spheriglome
(((II)II)II) - dispheritorus
(((II)(II))II) - spheritoratiger
(((III)I)II) - spheritorisphere
((((II)I)I)II) - spheritritorus
((IIII)(II)) - cylglomintigroid
(((II)II)(II)) - cylspheritorintigroid
(((II)(II))(II)) - tigritiger, or ditiger?
(((III)I)(II)) - cyltorispherintigroid
((((II)I)I)(II)) - cylditorintigroid
((IIIII)I) - toriglomoteron , toripentasphere?
(((II)III)I) - toriglomitorus
(((II)(II)I)I) - toraspheritiger
(((III)II)I) - toridisphere
((((II)I)II)I) - torispheriditorus
(((III)(II))I) - toracylspheritigroid
((((II)I)(II))I) - toracyltorintigroid
(((IIII)I)I) - ditoriglome
((((II)II)I)I) - ditorispheritorus
((((II)(II))I)I) - ditoratiger
((((III)I)I)I) - tritorisphere
(((((II)I)I)I)I) - tetratorus



To further simplify the names without losing any information, it can be just simply [small shape][large shape]

where for different diameters,

tori - circle
ditori - toritori
tritori - toritoritori
spheri - sphere
disphere - spherispheri
trisphere - spherispherispheri
sphere*sphere - duosphere
glomi - glome
pentaspheri/glomoteri - pentasphere
tigri - tiger
n-tigroid - spherated open toratope
ditiger - tigritiger
([...]*circle) - cyl[...]intigroid
([...]*sphere) - duocyl[...]intigroid


Trying to define toritiger (((II)(II))I) was tough at first. I had to realize that the final diameter at the end was actually the minor diameter, and the whole tiger was in place of the major-shape to inflate. Toritiger is the circle-inflation of the 2D skin of a whole tiger, and not just the duocylinder margin. At first, I thought it was a tiger along the rim of a circle, but now I'm thinking it is a circle along the surface of a tiger. I guess it depends on the magnitude of that final diameter, and whether or not it comes out those ways. If (((II)(II))I) is a circle along skin of tiger, then what shape is tiger along edge of circle, a true tiger torus?? I'm not sure here, or even what tiger along sphere would be, if not (((II)(II))II).

[Marek14]

Well, think of it like this:

A circle if (II).
Circle along a circle is torus, ((II)I).
Circle along a sphere is torisphere ((III)I).
In general, circle along (x) is ((x)I).

So, couldn't "tiger along something" be simply a tiger with one I (all are equivalent) being replaced by that something? So tiger along circle would be the torus tiger (((II)I)(II)), and tiger along sphere would be 31-torus 20-tiger (((III)I)(II)).

Would that work?

[ICN5D]

Trace Array Notation for 5D toratopes:

These traces are for 5D arrays of 5D shapes. If there are less than 2 shapes stacked on either of the 5 legs, then the array can be cut further into a 4D trace. Only to be used for sufficiently large enough toratopes in +10 dimensions.


Trace of (IIIII) - pentasphere
---------------------------------------
(abcde)f - tigroid

2^a+1 x 2^b+1 x 2^c+1 x 2^d+1 x 2^e+1 array of
2^f concentric pairs of
2^{(5+a+b+c+d+e+f)} pentaspheres



Trace of ((II)III) - glomitorus
--------------------------------------
((ab)c,def)g - tigroid

2^a+1 x 2^b+1 x 2^d+1 x 2^e+1 x 2^f+1 major/minor array of
2^c major pairs / 2^g minor pairs of
2^{(5+a+b+c+d+e+f+g)} glomitoruses



Trace of ((II)(II)I) - spheritiger
----------------------------------------
((ab)c,(de)f,g)h - tigroid

2^a+1 x 2^b+1 x 2^d+1 x 2^e+1 x 2^g+1 major1/major2/minor array of
2^c major1 pairs / 2^f major2 pairs / 2^h minor pairs of
2^{(5+a+b+c+d+e+f+g+h)} spheritigers


Trace of ((III)II) - disphere
---------------------------------------
((abc)d,ef)g - tigroid

2^a+1 x 2^b+1 x 2^c+1 x 2^e+1 x 2^f+1 major/minor array of
2^d major pairs / 2^g minor pairs of
2^{(5+a+b+c+d+e+f+g+)} dispheres




Trace of (((II)I)II) - spheriditorus
-------------------------------------------
(((ab)c,d)e,fg)h - tigroid

2^a+1 x 2^b+1 x 2^d+1 x 2^f+1 x 2^g+1 major/medium/minor array of
2^c major pairs / 2^e medium pairs / 2^h minor pairs of
2^{(5+a+b+c+d+e+f+g+h)} spheriditoruses


Trace of ((III)(II)) - cylspherintigroid
-----------------------------------------------
((abc)d,(ef)g)h - tigroid

2^a+1 x 2^b+1 x 2^c+1 x 2^e+1 x 2^f+1 major[sphere]/major[circle] array of
2^d major[sphere] pairs / 2^g major[circle] pairs / 2^h minor pairs of
2^{(5+a+b+c+d+e+f+g+h)} cylspherintigroids


Trace of (((II)I)(II)) - cyltorintigroid
----------------------------------------------
(((ab)c,d)e,(fg)h)i - tigroid

2^a+1 x 2^b+1 x 2^d+1 x 2^f+1 x 2^g+1 major[torus] / major[circle] / medium array of
2^c major[torus] pairs / 2^h major[circle] pairs / 2^e medium pairs / 2ⁱ minor pairs of
2^{(5+a+b+c+d+e+f+g+h+i)} cyltorintigroids



Trace of ((IIII)I) - toriglome
---------------------------------------
((abcd)e,f)g

2^a+1 x 2^b+1 x 2^c+1 x 2^d+1 x 2^f+1 major/minor array of
2^e major pairs / 2^g minor pairs of
2^{(5+a+b+c+d+e+f+g)} toriglomes


Trace of (((II)II)I) - torispheritorus
---------------------------------------------
(((ab)c,de)f,g)h - tigroid

2^a+1 x 2^b+1 x 2^d+1 x 2^e+1 x 2^g+1 major/medium/minor array of
2^c major pairs / 2^f medium pairs / 2^h minor pairs of
2^{(5+a+b+c+d+e+f+g+h)} torispheritoruses


Trace of (((II)(II))I) - toritiger
---------------------------------------
(((ab)c,(de)f)g,h)i - tigroid

2^a+1 x 2^b+1 x 2^d+1 x 2^e+1 x 2^h+1 major1/major2/minor array of
2^c major1 pairs / 2^f major2 pairs / 2^g medium pairs / 2ⁱ minor pairs of
2^{(5+a+b+c+d+e+f+g+h+i)} toritigers


Trace of (((III)I)I) - ditorisphere
-----------------------------------------
(((abc)d,e)f,g)h - tigroid

2^a+1 x 2^b+1 x 2^c+1 x 2^e+1 x 2^g+1 major/medium/minor array of
2^d major pairs / 2^f medium pairs / 2^h minor pairs of
2^{(5+a+b+c+d+e+f+g+h)} ditorispheres


Trace of ((((II)I)I)I) - tritorus
---------------------------------------
((((ab)c,d)e,f)g,h)i - tigroid

2^a+1 x 2^b+1 x 2^d+1 x 2^f+1 x 2^h+1 major/secondary/tertiary/minor array of
2^c major pairs / 2^e secondary pairs / 2^g tertiary pairs / 2ⁱ minor pairs of
2^{(5+a+b+c+d+e+f+g+h+i)} tritoruses

[ICN5D]

Well, think of it like this:

A circle if (II).
Circle along a circle is torus, ((II)I).
Circle along a sphere is torisphere ((III)I).
In general, circle along (x) is ((x)I).

So, couldn't "tiger along something" be simply a tiger with one I (all are equivalent) being replaced by that something? So tiger along circle would be the torus tiger (((II)I)(II)), and tiger along sphere would be 31-torus 20-tiger (((III)I)(II)).

Would that work?

I considered torus-tiger, and how the diameters are working together inside the spheration. But, I don't think it is so. I'm starting to believe that (((II)(II))I) is BOTH tiger along circle and circle along skin of tiger. If the trace of four circles becomes four toruses from a non-intersecting rotation, then it was a rotation around the hyperplane of the minor diameter. It's a torusing of the whole tiger, along a 5D ring. Since the frame is a duoring, it evenly occupies a 4-plane. Tiger being the inflated duoring means this non-intersecting rotation around the minor diameter's hyperplane puts it in a 5D circle path, just like it sounds. But, at the same time, we can see the minor diameter itself get an inflation by a circle. This even smaller circle becomes the new minor diameter, and the previous tiger-skin diameter a meduim like a ditorus.

So, that's why I think it's both in one, as strange as that sounds. I feel it's all about the nature of the hidden minor diameter in ((II)(II)), and how it affects 5D toratopes from a rotation of such:

((II)(II)) - ((xy)(zw))
---------------------------
(((II)I)(II)) - non-intersecting rotation around hyperplanes xy , or zw
(((II)(II))I) - non-intersecting rotation around hyperplane xyzw


But, some of the cut rotation evolutions of (((II)I)(II)) do show something of a torus in it! I'm not sure about it, though, since a torus-tiger can also be a non-intersect rotation around the minor hyperplane of a ditorus, which I feel is a better way to grasp torus-tiger. I now understand it to be a tiger-circle along the skin of a torus, however that sounds.

[Marek14]

Well, maybe we should ask: just what is this "A along B"? How exactly is it defined?

We will probably agree that torus is circle along another circle. If we explore the cuts of torus, we find two circles -- that's the first circle -- and pair of circles, which is the second circle.
It should be that A along B is B with every point of its surface removed and replaced by A. The resulting shape will have a+b+1 dimensions where a and b are the dimensions of surfaces, not the dimensions of figures as whole.

The generic ((x)y) torus has, similarly, a cut of two (y+1)-dimensional spheres and a cut of a pair of x-dimensional spheres, which allows us to show it's y+1-sphere along x-sphere.

So maybe A should appear in cuts as separate toratopes and B should appear as pair?

Pure tigroids, of course, have B as cartesian product of lower toratopes' surfaces. For example, tiger is circle along (circle x circle), since circle has surface of dimension 1, circle x circle gets dimension 2 and the total dimension of tiger is 1+2+1 = 4.

So, torus tiger should be tiger along circle -- it has a cut of two separate tigers, it's nonbisecting rotation of tiger so the center of tiger traces a circle and at every point of this circle there's tiger. So there should be also a cut that would show the circle... let's have a look at 4D slices of torus tiger:

Minor stack of two ditoruses (((II)I)(I))
Major pair of tigers (((II))(II))
Two tigers (((I)I)(II))

The "major pair of tigers" is the slice we need. Here, the circle in "tiger along circle" exists as "middle" circle between differing diameters of the pair.

However, from this we can see that torus tiger is also "ditorus along circle" as it's a nonbisecting rotation of ditorus.

And we're still not done -- its nonempty 3D cuts are:

(((II))(I)) - vertical stack of two major pairs of toruses
(((I)I)(I)) - two vertical stacks of two toruses
(((I))(II)) - vertical stack of four toruses

Second and third of these show torus tiger as "torus along something". In second case, it's "torus along duocylinder margin" and in third case it's "torus along torus".
Why? Well, replace the shape with a point. What do you get? In the third case you'll get four points, which is a trace of torus. In second case, you get 2x2 array of points. There's no toratope with this trace, of course, but it IS a cartesian product of two traces of circles. This also holds for "tiger along circle" and "ditorus along circle" definitions -- if you reduce the shape to points, the cut will be 2 point, a trace of circle.

The trace of torus tiger is ((((I))(I)), a 4x2 array of circles. And 4x2 array of points is cartesian product of traces of circle and torus, thus torus tiger is also "circle along [torus x circle]).

So, for tiger torus (((II)(II))I), we get "ditorus along circle" based on medium stack of two ditoruses (((II)(I))I) and "torus along duocylinder margin" based on 2x2 array of toruses (((I)(I))I). But, of course, we instinctively feel that there should also be a "circle along tiger", don't we?

(As a small aside, note that both tiger torus and torus tigers can be expressed as "torus along duocylinder margin". However, the orientation of the toruses is different.)

The solution is simple. Since torus is "circle along circle" then we can change "torus along duocylinder margin" into "(circle along circle) along duocylinder margin", and by an unexpected application of associative law into "circle along (circle along duocylinder margin)" and thus into "circle along tiger"!

This, however, cannot be done with "torus along duocylinder margin" for torus tiger. The four toruses of its cut can be reduced to circles, but those circles do NOT form the trace of tiger -- instead they form two vertical stacks of two circles. This seems to be a 3D cut of "torus x circle". And we see that "torus x circle" is ALSO "circle along duocylinder margin", if we keep the circle only around one base circle and don't combine them.

So, in final (for now) conclusion, it seems that the "along" suffers from having to take into account various different orientations. Even "circle around circle" has an alternate outcome -- apart from torus, we can also get the duocylinder margin. Torus is if we have circle in xy plane and replace each point by circle in (x~y,z) plane (where x~y refers to some combination of x and y). But if we replace each point by circle in zw plane, we get the duocylinder margin.

[Marek14]

OK, I realized that the approach from previous post is in fact the same thing as previously offered "replacing I's" approach.

For example, tiger as "circle along duocylinder margin" is derived like this:

(II) - circle
Both I's are replaced by circles: ((II)(II)) - tiger.

But tiger can be also described as "torus along circle": ((II)!) -> ((II)(II)) where ! shows the replaced I.
And of course, since torus has two different I's, there's a second "torus along circle": ((!I)I) -> (((II)I)I) - ditorus, which is also "circle along torus", and, in fact, associative "circle along circle along circle" (and the reason for associativity is clear now)
The previous post shows that this definition works pretty well.

So, what would be "tiger along tiger"? It would be the tiger torus tiger ((((II)(II))I)(II)), a 7D toratope.

Torus along sphere x circle margin? There are three different possibilities based on different orientation of torus:
320-tiger 1-torus (((III)(II))I)
31-torus 2-tiger (((III)I)(II))
21-torus 30-tiger (((II)I)(III))

And torus along circle x circle x circle? Why, it's our old friend (((II)(II))(II)), the double tiger!

This allows us to define "A along B" for any two toratopes and "A along B1 x B2 x B3 x ..." in all cases where number of B's doesn't exceed the dimension of A; however, the result is generally non-unique unless A is a sphere or all B are identical and their number equals to the dimension of A.

[ICN5D]

Minor stack of two ditoruses (((II)I)(I))
Major pair of tigers (((II))(II))
Two tigers (((I)I)(II))

The "major pair of tigers" is the slice we need. Here, the circle in "tiger along circle" exists as "middle" circle between differing diameters of the pair.

However, from this we can see that torus tiger is also "ditorus along circle" as it's a nonbisecting rotation of ditorus.

The more I thought about it, and the points you make are turning on the lights. Every way we cut a torus tiger infers that it is tiger along circle. I guess I didn't realize that there could be two orthogonal circles to extrude along. Tiger along circle in the case of (((II)I)(II)) makes a 2x2 array of 4 circles turn into a 4x2 array of 8. This suggests an analogue to the two circles side by side cut of a torus. Except, this time we took a whole tiger, and ran it around the edge of a circle. Also taking a look at the vertical column of 2 ditoruses. This is a minor pair of toruses having a non-intersecting rotation into a minor pair of ditoruses, inferring a tiger along circle, as we see a tiger cut along circle happen. Not to mention the two tigers side by side. That can only happen with a tiger along circle!

And we're still not done -- its nonempty 3D cuts are:

(((II))(I)) - vertical stack of two major pairs of toruses
(((I)I)(I)) - two vertical stacks of two toruses
(((I))(II)) - vertical stack of four toruses

Second and third of these show torus tiger as "torus along something". In second case, it's "torus along duocylinder margin" and in third case it's "torus along torus".

Hmm. I've thought about it some, and I think it's:

(((II))(I)) - cut of major pair of tigers, analogous to pair of circles cut ((II)), shows tiger along circle in the same way
(((I)I)(I)) - side by side of 2 tigers' cut of minor pairs, tiger along circle, analogous to ((I)I) cut
(((I))(II)) - side by side of 2 tigers' other cut, but still in same position as (((I)I)(I)), tiger along circle

The rotation from (((I)I)(I)) to (((I))(II)) shows a tiger rotation x2, side by side, as the major stack of minor pairs morphs into a minor quartet. Along the minor quartet as 1,2,3,4: number 2 and 3 don't interact at all, only 1 with 2, and 3 with 4. In fact, all three 3D cuts show a tiger along circle in three different ways, which is really cool!

Here's that reference animation again, start at 1:40


So, that makes duotorus tiger (((II)I)((II)I) a tiger along duoring ! The tiger has gone along both orthogonal circles, combined into one shape. The 4x4 trace of 16 circles also makes sense, as it is a quartet of quartets of circles. It's a tiger trace in the positions of a larger tiger trace! Oh my, this little conceptual breakthrough is very rewarding! I had a good feel for these shapes, at least I thought. Then came the "A along B" method of description, which breaks major ground in seeing how these things are built. It also shows how 4D works even better, with the rotation stages of a tiger into a duotorus tiger.

So, for tiger torus (((II)(II))I), we get "ditorus along circle" based on medium stack of two ditoruses (((II)(I))I) and "torus along duocylinder margin" based on 2x2 array of toruses (((I)(I))I). But, of course, we instinctively feel that there should also be a "circle along tiger", don't we?

The solution is simple. Since torus is "circle along circle" then we can change "torus along duocylinder margin" into "(circle along circle) along duocylinder margin", and by an unexpected application of associative law into "circle along (circle along duocylinder margin)" and thus into "circle along tiger"!

Yes I do! Which is definitely what it is! I had to visualize the 2x2 circle trace undergoing an inflation of an even more minor diameter, R4 in (((II)(II))I). That makes the tiger's initial R3 minor diameter into the new medium diameter, like a ditorus. And, interestingly enough, we see two ditoruses stacked in their medium dimension .... how about that? God, I love this stuff, seriously. Grokking Galore.

So, in final (for now) conclusion, it seems that the "along" suffers from having to take into account various different orientations. Even "circle around circle" has an alternate outcome -- apart from torus, we can also get the duocylinder margin. Torus is if we have circle in xy plane and replace each point by circle in (x~y,z) plane (where x~y refers to some combination of x and y). But if we replace each point by circle in zw plane, we get the duocylinder margin.

This is true, but we may be able to get around that. With the introduction of " along ortho circle", we can distinguish it between " along circle". But, I guess with a ditorus, we would need three ways to describe this circle. So, perhaps we should identify it by the diameter's hyperplane :

Circle along [...]
• [...] - circle : Torus
• [...] - ortho circle : Duoring / Duocylinder Margin
• [...] - duoring : Tiger
• [...] - torus : Ditorus (((II)I)I)
• [...] - sphere : Torisphere ((III)I)
• [...] - tiger : Toritiger (((II)(II))I)
• [...] - torus-tiger : ditorus-tiger ((((II)I)(II))I)
• [...] - toritiger : ditoritiger ((((II)(II))I)I)

Torus along [...]
• [...] - minor circle : Tiger
• [...] - major circle : Ditorus

Ditorus along [...]
• [...] - minor circle : torus-tiger (((II)I)(II))
• [...] - medium circle : toritiger (((II)(II))I)
• [...] - major circle : tritorus ((((II)I)I)I)

Tiger along [...]
• [...] - major1 circle : torus-tiger (((II)I)(II))
• [...] - major2 circle : torus-tiger (((II)I)(II))
• [...] - minor circle : toritiger (((II)(II))I)


... something to that effect.

So, what would be "tiger along tiger"? It would be the tiger torus tiger ((((II)(II))I)(II)), a 7D toratope.

Really? Well, I guess you're right, given the way a ((((II)(II))I)I) works: it's a torus along tiger. So, by rotating in such a way to make a torus into a tiger ((II)I) ---> ((II)(II)) , then it most likely is tiger along tiger, when considering ((((II)(II))I)(II)) . Wow, amazing! Freakin' crazy, man. I think a list should be made that places the " A along B " definition with all of the others. And of course, how to derive it from the notation.


So, it looks like this system can potentially describe some very high-D toratopes verbally and precisely. How about a real challenge of dissecting something like that 12D (((((II)I)(II))((II)(II)))((II)I)) , or the 17D ((((((((II)I)(II))I)I)(((II)I)I))(((((II)(II))I)I)) ? If not anything cool, how about a torus tiger along torus tiger? I feel it could be the 9D (((((II)I)(II))I)((II)I)) , given the new understanding. We would get (((((I))(I)))((I))) , a 4x2x4 array of major pairs of 64 toruses! Whoa!

[Marek14]

Well, since torus tiger has three different types of dimensions, there are three possibilities for torus tiger along torus tiger, all 9D:

Torus tiger ditorus tiger ((((((II)I)(II))I)I)(II))
Torus triple tiger (((((II)I)(II))(II))(II))
(torus tiger torus)/torus tiger (((((II)I)(II))I)((II)I)) -- this is the one you mentioned

[ICN5D]

So, in light of recent conceptual breakthroughs, I see that your name torus tiger for (((II)I)(II)) can also be, according to the A along B definition, a tigritorus. This name suggests that the shape is a tiger along the edge of a circle, a true torus of a tiger. We can also call the duotorus tiger (((II)I)((II)I)) a tigriduotorus, inferring a tiger along duoring. Does that work?


I also have been thinking about ((((II)I)(II))((II)I)) , and what it would be. I now understand (((((II)I)(II))I)((II)I)) to be a tigritorus along tigritorus. So, what is a ((((II)I)(II))((II)I)) ? It seems like it would be a tigritorus (((II)I)(II)) along cyltorinder margin, since a whole (((II)I)(II)) is in place of a ((!!)((II)I)).

[ICN5D]

Well, since torus tiger has three different types of dimensions, there are three possibilities for torus tiger along torus tiger, all 9D:

Torus tiger ditorus tiger ((((((II)I)(II))I)I)(II))
Torus triple tiger (((((II)I)(II))(II))(II))
(torus tiger torus)/torus tiger (((((II)I)(II))I)((II)I)) -- this is the one you mentioned

I don't understand. How does Torus triple tiger (((((II)I)(II))(II))(II)) work? I see how ((((((II)I)(II))I)I)(II)) and (((((II)I)(II))I)((II)I)) work, they are an entire (((II)I)(II)) in place of one dimension in (((!I)I)(II)) and (((II)I)(!I)). I see it's supposed to go in (((II)!)(II)) , which would make (((II)(((II)I)(II)))(II)). Which could be rearranged to (((((II)I)(II))(II))(II)) ..... oh my god. Marek, what did you do to my brain? I, uh, see how it works, now.

There are three dimensions to embed a torus tiger into another torus tiger, based on which marker was replaced with an entire torus tiger : (((!I)!)(!I)) .These " ! " are the three distinct hyperplane diameters to run a torus tiger along another. Well, I still have yet to see what you say about ((((II)I)(II))((II)I)) , as I feel it's a torus tiger along cyltorinder margin. In this instance of A along B, we have the marker (((II)I)(II)!) as the major shape, inflating (((II)I)(!!)) to make ((((II)I)(II))((II)I)).

[Marek14]

So, in light of recent conceptual breakthroughs, I see that your name torus tiger for (((II)I)(II)) can also be, according to the A along B definition, a tigritorus. This name suggests that the shape is a tiger along the edge of a circle, a true torus of a tiger. We can also call the duotorus tiger (((II)I)((II)I)) a tigriduotorus, inferring a tiger along duoring. Does that work?


I also have been thinking about ((((II)I)(II))((II)I)) , and what it would be. I now understand (((((II)I)(II))I)((II)I)) to be a tigritorus along tigritorus. So, what is a ((((II)I)(II))((II)I)) ? It seems like it would be a tigritorus (((II)I)(II)) along cyltorinder margin, since a whole (((II)I)(II)) is in place of a ((!!)((II)I)).

Not sure it works like that. The (tiger torus)/torus tiger ((((II)I)(II))((II)I)) can be dissected as such (please have a brown paper bag ready in case of blown mind):

(((I!)(II))((II)I)) - tiger/torus tiger along circle
((((II)I)!)((II)I)) - ditorus/torus tiger along circle
((((II)I)(II))(I!)) - torus double tiger along circle

Next level:

(((II)!)((II)I)) - duotorus tiger along torus
(((I!)!)((II)I)) - duotorus tiger along duocylinder margin
(((I!)(II))(I!)) - double tiger along duocylinder margin
((((II)I)!)(I!)) - ditorus tiger along duocylinder margin
((((II)I)(II))!) - torus tiger torus along torus

Third level:

(((II)I)(!!)) - torus tiger along torus x circle
(((II)!)(I!)) - torus tiger along torus x circle
(((I!)!)(I!)) - torus tiger along circle x circle x circle
(((I!)(II))!) - tiger torus along torus x circle
((((II)I)!)!) - tritorus along torus x circle

Fourth level:

((I!)(!!)) - tiger along torus x circle x circle
(((II)I)!) - ditorus along torus tiger
(((II)!)!) - ditorus along torus x torus
(((I!)!)!) - ditorus along circle x circle x circle

Fifth level:
((!!)!) - torus along torus x torus x circle
((I!)!) - torus along torus tiger x circle

Sixth level:
(!!) - circle along torus tiger x torus

[ICN5D]

(((II)I)(!!)) - torus tiger along torus x circle

That's the one, right there! By torus x circle, you also mean ((II)I)(II) (torus*circle)-prism, and when we insert it into the circle parameter of torus tiger (((II)I)(!!)), we inflate only the margin of a cyltorinder.

I see it this way: if double tiger (((II)(II))(II)) is tiger along duoring, where a whole tiger ((II)(II)) is embedded into ((!!)(II)), then the (!!) circle parameter is inflating lower than the N-1 surface, but the N-2 margin. We could say that we put a duocylinder (II)(II) in place of ((!!)(II)) , where we have inflated only the N-2 margin with a tiger. (((II)(II))(II)) will give us four tigers in a 2x2 square array, which when cut further into 3D, we get the 8 toruses in the 2x2x2 cube-array. The four tigers are still in the square, but we see four tiger cuts of minor stacks. The 2x2 array of the tigers is the duoring structure that the whole tiger was run along, from tiger along duocylinder margin.

So, in the case of ((((II)I)(II))((II)I)), we have placed a whole cyltorinder ((II)I)(II) in place of (((II)I)(!!)). This means we inflate the N-2 margin of ((II)I)(II) with a (((II)I)(!!)), making a torus tiger along cyltorinder margin.


And, that's way more ways to break down a ((((II)I)(II))((II)I)) than I thought!!! Holy cow! I see how you do that, now. This has been a highly productive night in comprehension. That system can also tell us other [toratope]-along-[margin] reductions, which is extremely fascinating.

((I!)(!!)) - tiger along torus x circle x circle
(((I!)!)!) - ditorus along circle x circle x circle

Those are amazing. Torus*circle*circle I guess can be the 4D margin of a (torus*duocylinder)-prism? Some kind of duocyltorindyinder ((II)I)(II)(II) .

Ditorus along triocylinder margin, whoa. The " trigering" of a ditorus. But, nope, I think my cranium has been able to contain the expansion force. This will be something I constantly think about all the time, for the next few months or so. At least, until I master the concept, and go into 18D again. You know how I do that


This also means torus squared ((((II)I)((II)I))((II)I)) is a torus tiger along margin of (torus*torus)-prism.

[Marek14]

Basically, in (((II)I)(!!)), each of the ! dimensions is inflating separately, one along the torus part of torus x circle and one along the circle part. It can be taken as replacing (!!) with (((II)I)(II)), but systematically it's probably easier to think about it as replacing one ! with ((II)I) and the other with (II).

This was actually first time I tried to break a toratope down like that, but the algorithm is relatively straightforward and it's another example of usefulness of the toratopic notation

Also notice that we now have an easy way to understand things like torus squared ((((II)I)((II)I))((II)I)) - it's torus along torus^3 prism!

[ICN5D]

Ha! I just made an edit to my post about torus squared! :

"This also means torus squared ((((II)I)((II)I))((II)I)) is a torus tiger along tiger along duoring. Where tiger along duoring is the inflated margin of (torus*torus)-prism "

Or, double tiger along double tiger...

Yeah, this definition is great at breaking them down, and building up the proper notation equation, as well. Now we have a very good understanding of how these wild things are built. How to interpret the wireframe large shapes and the inflating smaller shapes is key, and amazing in the relation to the notation. Totally awesome stuff, right there.

[ICN5D]

Wait a minute, I just caught some things I mentioned. I said both (((II)I)((II)I)) and (((II)(II))(II)) is a tiger along duoring. I guess that's an accurate assumption for both, but the duoring is in a different orientation.

There seem to be four orientations of a duoring path for a tiger, where two are identical to the other two. This gives us two distinct shapes that are equally tiger along duoring. In fact, both 4D cuts are four tigers in a 2x2 square, but the orientation of that square is different.

- The 4x4 circle trace (((I))((I))) is the "vertical square" version, where the duoring is perpendicular to the tigers' duoring frame (((I)I)((I)I))
- The 2x2x2 torus trace (((I)(I))(I)) is the "flat square", the duoring is parallel to one major diameter the tiger's duoring frame (((I)(I))(II))


Also, isn't (((II)(II))((II)I)) a torus tiger along duoring?

[Marek14]

Wait a minute, I just caught some things I mentioned. I said both (((II)I)((II)I)) and (((II)(II))(II)) is a tiger along duoring. I guess that's an accurate assumption for both, but the duoring is in a different orientation.

There seem to be four orientations of a duoring path for a tiger, where two are identical to the other two. This gives us two distinct shapes that are equally tiger along duoring. In fact, both 4D cuts are four tigers in a 2x2 square, but the orientation of that square is different.

- The 4x4 circle trace (((I))((I))) is the "vertical square" version, where the duoring is perpendicular to the tigers' duoring frame (((I)I)((I)I))
- The 2x2x2 torus trace (((I)(I))(I)) is the "flat square", the duoring is parallel to one major diameter the tiger's duoring frame (((I)(I))(II))


Also, isn't (((II)(II))((II)I)) a torus tiger along duoring?

Yes, I've been saying from start that the "along" is generally not unique

So tiger along duoring can be, indeed, both double tiger and duotorus tiger.

And (((II)(II))((II)I)) is indeed torus tiger (((II)I)(II)) along duoring, but there are FOUR different toratopes with this property (on the other hand, this can be also described as tiger along circle^3, and that IS unique ):

Tiger torus tiger ((((II)(II))I)(II)) where you expand both major[torus] dimensions.
Torus double tiger ((((II)I)(II))(II)) where you expand one major[torus] dimension and the medium dimension.
Ditorus/torus tiger ((((II)I)I)((II)I)) where you expand one major[torus] dimension and one major[circle] dimension
Tiger/torus tiger (((II)(II))((II)I)), the one you mentioned, and what's interesting is that this one is torus tiger along duoring in two different ways: either you expand the medium dimension and one major[circle] dimension, OR you can expand both major[circle] dimensions.

[Marek14]

Now, we should shorten "along" to some notation, I'd suggest to write "A along B" as A->B.

So, in 3D:
Torus ((II)I) = circle->circle

In 4D:
Torisphere ((III)I) = circle->sphere
Ditorus (((II)I)I) = circle->circle->circle = circle->torus = torus[major]->circle
Tiger ((II)(II)) = circle->duoring = torus[minor]->circle
Spheritorus ((II)II) = sphere->circle

In 5D:
41-torus ((IIII)I) = circle->glome
311-ditorus (((III)I)I) = circle->circle->sphere = circle->torisphere = torus[major]->sphere
Tritorus ((((II)I)I)I) = circle->circle->circle->circle = circle->ditorus = torus[major]->torus = ditorus[major]->circle
Tiger torus (((II)(II))I) = circle->circle->duoring = circle->tiger = torus[major]->duoring = ditorus[medium]->circle
221-ditorus (((II)II)I) = circle->sphere->circle = circle->spheritorus = torisphere[major]->circle
320-tiger ((III)(II)) = circle->(sphere x circle) = torus[minor]->sphere = torisphere[minor]->circle
Torus tiger (((II)I)(II)) = circle->(torus x circle) = torus[minor]->torus = torus[major/minor]->duoring = ditorus[minor]->circle = tiger->circle
32-torus ((III)II) = sphere->sphere
212-ditorus (((II)I)II) = sphere->circle->circle = sphere->torus = spheritorus[major]->circle
221-tiger ((II)(II)I) = sphere->duoring = spheritorus[minor]->circle
23-torus ((II)III) = glome->circle

[Marek14]

In 6D:
Hexasphere (IIIIII)
51-torus ((IIIII)I) = circle->pentasphere
411-ditorus (((IIII)I)I) = circle->41-torus = torus[major]->glome
3111-tritorus ((((III)I)I)I) = circle->311-ditorus = torus[major]->torisphere = ditorus[major]->sphere
Tetratorus (((((II)I)I)I)I) = circle->tritorus = torus[major]->ditorus = ditorus[major]->torus = tritorus[major]->circle
Tiger ditorus ((((II)(II))I)I) = circle->tiger torus = torus[major]->tiger = ditorus[major]->(circle x circle) = tritorus[secondary]->circle
2211-tritorus ((((II)II)I)I) = circle->221-ditorus = torus[major]->spheritorus = 311-ditorus[major]->circle
320-tiger 1-torus (((III)(II))I) = circle->320-tiger = torus[major]->(sphere x circle) = ditorus[medium]->sphere = 311-ditorus[medium]->circle
Torus tiger torus ((((II)I)(II))I) = circle->torus tiger = torus[major]->(torus x circle) = ditorus[medium]->torus = ditorus[major/medium]->(circle x circle) = tritorus[tertiary]->circle = tiger torus[major]->circle
321-ditorus (((III)II)I) = circle->32-torus = torisphere[major]->sphere
2121-tritorus ((((II)I)II)I) = circle->212-ditorus = torisphere[major]->torus = 221-ditorus[major]->circle
221-tiger 1-torus (((II)(II)I)I) = circle->221-tiger = torisphere[major]->(circle x circle) = 221-ditorus[medium]->circle
231-ditorus (((II)III)I) = circle->23-torus = 41-torus[major]->circle
420-tiger ((IIII)(II)) = circle->(glome x circle) = torus[minor]->glome = 41-torus[minor]->circle
31-torus 20-tiger (((III)I)(II)) = circle->(torisphere x circle) = torus[minor]->torisphere = torus[major/minor]->(sphere x circle) = tiger->sphere = 311-ditorus[minor]->circle
Ditorus tiger ((((II)I)I)(II)) = circle->(ditorus x circle) = torus[minor]->ditorus = torus[major/minor]->(torus x circle) = ditorus[major/minor]->(circle x circle) = tiger->torus = tritorus[minor]->circle = torus tiger[major-torus]->circle
Double tiger (((II)(II))(II)) = circle->(tiger x circle) = torus[minor]->tiger = torus->(circle x circle x circle) = ditorus[medium/minor]->(circle x circle) = tiger[major1/major1]->(circle x circle) = tiger torus[minor]->circle = torus tiger[medium]->circle
22-torus 20-tiger (((II)II)(II)) = circle->(spheritorus x circle) = torus[minor]->spheritorus = torisphere[major/minor]->(circle x circle) = 320-tiger[major-sphere]->circle
42-torus ((IIII)II) = sphere->glome
312-ditorus (((III)I)II) = sphere->torisphere = spheritorus[major]->sphere
2112-tritorus ((((II)I)I)II) = sphere->ditorus = spheritorus[major]->torus = 212-tritorus[major]->circle
220-tiger 2-torus (((II)(II))II) = sphere->tiger = spheritorus[major]->(circle x circle) = 212-ditorus[medium]->circle
222-ditorus (((II)II)II) = sphere->spheritorus = 32-torus[major]->circle
330-tiger ((III)(III)) = circle->(sphere x sphere) = torisphere[minor]->sphere
21-torus 30-tiger (((II)I)(III)) = circle->(torus x sphere) = torus[major/minor]->(circle x sphere) = torisphere[minor]->torus = ditorus[minor]->sphere = 320-tiger[major-circle]->circle
Duotorus tiger (((II)I)((II)I)) = circle->(torus x torus) = torus[major/minor]->(circle x torus) = ditorus[minor]->torus = tiger[major1/major2]->(circle x circle) = torus tiger[major-circle]->circle
321-tiger ((III)(II)I) = sphere->(sphere x circle) = spheritorus[minor]->sphere = 32-torus[minor]->circle
21-torus 21-tiger (((II)I)(II)I) = sphere->(torus x circle) = spheritorus[minor]->torus = spheritorus[major/minor]->(circle x circle) = 212-ditorus[minor]->circle = 221-tiger[major]->circle
33-torus ((III)III) = glome->sphere
213-ditorus (((II)I)III) = glome->torus = 23-torus[major]->circle
Triger ((II)(II)(II)) = sphere->(circle x circle x circle) = spheritorus[minor]->(circle x circle) = 221-tiger[minor]->circle
222-tiger ((II)(II)II) = glome->(circle x circle) = 23-torus[minor]->circle
24-torus ((II)IIII) = pentasphere->circle

[ICN5D]

This new understanding got me thinking about the open toratopes as well. I just figured out that a cyltorinder ((II)I)(II) is ALSO a duocylinder torus!!! Wow! So, we extrude a torus into a prism making torinder ((II)I)I , then lathe around a bisecting hyperplane, parallel to the torus endcaps. This makes cyltorinder ((II)I)(II) , we get a duocylinder-type shape in 5D, composed of 2 orthogonally bound ditoruses. Inflating the margin of a duocylinder torus ((II)I)(II) will inevitably lead to a tiger torus (((II)I)(II))!

The same shape can be made by a bisecting lathe of a cylinder (II)I to make duocylinder (II)(II) , then a non-intersecting rotation around a major diameter hyperplane making duocylinder torus ((II)I)(II) or (II)((II)I) . Absolutely amazing. Then, inflate the ridge mating the two ortho bound ditoruses, and we have a tiger torus (((II)I)(II)), which agrees perfectly with what we have found. Absolutely amazing.

This means (torus*sphere) prism is also cylspherinder torus, ((II)I)(III) and (III)((II)I) from (III)(II). Cylspherinder has a torisphere ortho bound to a spheritorus, and inflating this margin makes ((III)(II)). But, we can have another kind of cylspherinder torus, ((II)II)(II) , the bisecting lathe of a spheritorus prism, making (spheritorus*circle)-prism , cylspheritorinder. A ((II)II)(II) can also be made by a bisecting rotation of a duocylinder torus. If I recall correctly, a bisecting rotation of duocylinder makes cylspherinder. So, this initial duocylinder torus had only the minor shape ( duocylinder ) lathed, while the main circle diameter was held in place. Similar to turning ((II)I) --> ((II)II) , we have turned ((II)I)(II) ---> ((II)II)(II) OR ((II)I)(III) . Absolutely amazing. Love it. Can't stop thinking about it.


And, to go further, I also realized that a triocylinder (II)(II)(II) has two ortho bound duocylinder toruses, made by a bisecting rotation of a duocylinder prism (II)(II)I ---> (II)(II)(II). Inflating the ridge between two ortho bound duocylinder toruses makes the trioring inflate to triger ((II)(II)(II)).

I also had a look at ((((II)(II))(II))(II)), triple tiger. It can be tiger->duoring->duoring , and ditorus[major/medium/minor]->quattroring , the margin of quattrocylinder (II)(II)(II)(II). I'm sure there's an incredible amount of A->B decompositions of triple tiger, these are a few immediately noticeable ones.

[ICN5D]

How about torus squared ((((II)I)((II)I))((II)I)) as circle --> circle --> trioring --> trioring?

Tiger[maj1,2] --> duoring --> trioring ?

[Marek14]

Yes, I think these work

[ICN5D]

I think I've discovered a new algorithm with open toratopes. It allows you to derive the curved surtopes on the surface of the opens. It follows the same principle of "replace X in (xI)" . Check it out:



Consider the duocylinder : (II)(II) , we have (circle1 * circle2)-prism

1) rewrite to (xI)(II) , the first dimension marker, in either factor, will always become " x "

2) take circle2, and move it in place of X , and we get ((II)I) , the first surtope of a torus

3) now, take (II)(xI)

4) take circle1 and move in place of X , and we get ((II)I), the second surtope of a torus

5) steps 1-4 give us ((II)I)+((II)I) , the two orthogonal bound toruses



Now, let's see what (III)(II) cylspherinder has, as (sphere * circle)-prism

1) rewrite into the form (xII)(II)

2) take circle, and move into place of X, giving us ((II)II) , the spheritorus surtope cell

3) now, rewrite to form (III)(xI)

4) take the sphere factor, and move in place of X, making ((III)I), the torisphere surtope cell

5) steps 1-4 give us ((II)II)+((III)I) , the ortho bound spheritorus and torisphere



How about another one, the cyltorinder ((II)I)(II) , as (torus * circle)-prism, aka duocylinder torus

1) rewrite into ((xI)I)(II)

2) take the circle factor, and move into place of X, making (((II)I)I) , the ditorus surtope cell

3) now take form ((II)I)(xI)

4) move the torus factor in place of X, we get (((II)I)I) , the second ditorus surtope cell

5) steps 1-4 give us (((II)I)I)+(((II)I)I) , the two ortho bound ditoruses on the surface of cyltorinder / duocylinder torus
--- Which makes sense, seeing how (II)(II) has ((II)I)+((II)I) on the surface. A torus of such a shape would logically produce (((II)I)I)+(((II)I)I) , which made a torus out of each surtope cell while bound together



Now, how about the cylspherinder torus type-1 ((II)I)(III) , (torus * sphere)-prism

1) rewite to ((xI)I)(III)

2) take the sphere factor, and move in place of X, making (((III)I)I), a ditorisphere surtope cell

3) take form ((II)I)(xII)

4) move the torus factor in place of X, making (((II)I)II) , a spheriditorus surtope cell

5) steps 1-4 make (((III)I)I)+(((II)I)II) , a ditorisphere ortho bound to a spheriditorus



For cylspherinder torus type-2 ((II)II)(II) , (spheritorus * circle)-prism

1) rewrite to ((xI)II)(II)

2) take the circle factor and move into place of X, making (((II)I)II) , a spheriditorus surtope cell

3) rewrite to form ((II)II)(xI)

4) take the spheritorus factor, move in place of X, making (((II)II)I) , a torispheritorus surtope cell

5) Steps 1-4 make (((II)I)II)+(((II)II)I) , a spheriditorus ortho bound to a torispheritorus



What do you think about that?

Also, how about calling the cyltorinder margin a "duoring torus" ? An inflation of a Duoring Torus makes the tiger torus (((II)I)(II)).

[Marek14]

I didn't have time to look at it properly yet, but first question that comes in mind is, why do you always replace the first marker?

In ((II)I)(III), why using it as ((xI)I)(III) and not as ((II)I)(xII)?

[ICN5D]

I didn't have time to look at it properly yet, but first question that comes in mind is, why do you always replace the first marker?

In ((II)I)(III), why using it as ((xI)I)(III) and not as ((II)I)(xII)?

Well, you do end up using both formats, to derive both surtopes. It doesn't matter which you choose first, as long as the first marker in either factor gets replaced. I'm not sure exactly why this is, but following the system gets you all of the correct surtopes I used my own notation system to get the same results a few years ago, so I followed the rules that worked out in this notation. It's like you have to cross-breed the factors together, and step down 1 dimension for each surtope.

[Marek14]

I didn't have time to look at it properly yet, but first question that comes in mind is, why do you always replace the first marker?

In ((II)I)(III), why using it as ((xI)I)(III) and not as ((II)I)(xII)?

Well, you do end up using both formats, to derive both surtopes. It doesn't matter which you choose first, as long as the first marker in either factor gets replaced. I'm not sure exactly why this is, but following the system gets you all of the correct surtopes I used my own notation system to get the same results a few years ago, so I followed the rules that worked out in this notation. It's like you have to cross-breed the factors together, and step down 1 dimension for each surtope.

What if you use something like ((III)(II))(II), though -- there's no unique "first marker" to choose in the first part.