The Tiger Explained

[ICN5D]

Checking my understanding...

For the 4D torii
((II)I)I)
1. (II)=Start with circle
2. ((II)I)=Extend each point on the circle with a given fixed radius and add 1 dimension=spherate circle in 3D (as if you spherate it in 2D you only end up with a hollow annulus or concentric circles=((II)) )=torus
3.((II)I)I)=Spherate the torus in 3+1=4D=Ditorus

((III)I)
1.(III)=Start with 2-sphere (not 3-ball)
2. ((III)I)=Spherate 2-sphere in 3+1=4D=Torisphere

When you see a (II) to ((II)I) transition, think of only the SURFACE of the original circle getting inflated by another circle, represented by ((...)I), where the (...) term is taking the place of an entire " I " inside (II). So, we're literally embedding a whole circle into the surface of another. Same with ((III)I), we have a sphere initially (III) that turns into ((III)I), where we have inflated only the surface of the sphere, with a circle to make a torisphere. Same way works with ((II)II), we have an original circle (II), and we use a whole sphere ((...)II) to inflate the disk's edge, into a spheritorus. And for the ditorus, we start with a torus ((II)I), and inflate the edge with a circle to get (((II)I)I), where an entire torus is within ((...)I). Now, for the duocylinder (II)(II), spherating it will hollow out the shape down to its edge, and inflate this ridge with a circle, where two entire circles are in the place of a " I " term in ((...)(...)) = (II) . For the (III)(III), we have the cartesian product of two solid spheres. For ((III)(III)), it now becomes the cartesian product of two hollow spheres, that got inflated with a circle.


For the position of the radii in the toratopes, they are always major radius at innermost pair of (), as in for the ditorus (((II)I)I) : (((major)middle)minor)

(I) - 2 points at vertices of line
(I)(I) - 4 points at vert of square
(I)(I)(I) - 8 pts at vert of cube
(I)(I)(I)(I) - 16 pts at vert of tesseract
(I)(I)(I)(I)(I) - 32 pts at vert of penteract

(I) - 2 pts at vert of line
((I)) - 4 pts along line, restricted to one dim
(((I))) - 8 pts along line
((((I)))) - 16 pts along line

(II) - circle
((II)) - 2 concentric circles
(((II))) - 4 concentric circles
((((II)))) - 8 concentric circles

(II) - circle
((I)I) - 2 circles along line
(((I)I)) - 4 circles along line, restricted to one dimension
((((I)I))) - 8 circles along line
(((((I)I)))) - 16 circles along line

((I)(I)) - 4 circles at vert of square
(((I)(I))) - 4 pairs of 2 circles at vert of square
((((I)(I)))) - 4 pairs of 4 circles at vert of square
(((((I)(I))))) - 4 pairs of 8 circles at vert of square


(III) - sphere
((III)) - 2 concentric spheres
(((III))) - 4 concentric spheres
((((III)))) - 8 concentric spheres

(III)
((I)II) - 2 spheres along line
(((I)II)) - 4 spheres along line
((((I)II))) - 8 spheres along line

((I)(I)I) - four spheres at vert of square
(((I)(I)I)) - four pairs of 2 spheres at vert of square
((((I)(I)I))) - four pairs of 4 spheres at vert of square

((I)(I)(I)) - eight spheres at vert of cube
(((I)(I)(I)))) - eight pairs of 2 spheres at vert of cube
((((I)(I)(I)))) - eight pairs of 4 spheres at vert of cube

(((II)I)I) - ditorus
(((I)I)I) - 2 torii along line
(((II))I) - 2 concentric torii
(((II)I)) - 2 cocircular torii

((II)I) - torus
(II)(I) - cartesian product with circle and ortho hollow line, 2 vertical stacked circles
((II)(I)) - spherate, inflate edges of circles with circles, makes 2 vertical stacked torii
(((II)(I))) - spherate, 2 pairs of 2 vertical stacked torii, along a line
((((II)(I))))) - 4 pairs of 2 vertical stacked torii along a line

((II)I) - torus
(((II)I)) - 2 cocircular torii
((((II)I))) - 4 cocircular torii
(((((II)I)))) - 8 cocircular torii

(((I)I)I) - 2 torii along line
((((I)I)I)) - 4 torii at vert of square
(((((I)I)I))) - 8 torii at vert of cube

(((II))I) - 2 concentric torii
((((II)))I) - 4 concentric torii
(((((II))))I) - 8 concentric torii

(((II))I) - 2 concentric torii
((((II))I)) - 2 pairs of 2 concentric torii along line
(((((II))I))) - 4 pairs of 2 concentric torii along a line
((((((II))I)))) - 8 pairs of 2 concentric torii along a line

(((((II)))I)) - 2 pairings of 4 concentric torii along line
((((((II)))I))) - 4 pairings of 4 concentric torii along line

(((I)I)I) - 2 displaced torii along a line
((((I))I)I) - 4 torii along a line
(((((I)))I)I) - 8 torii along a line

((((I)I))I) - not quite sure, I may have some wrong. It's 3:27 am, and time to go to bed.

-- Philip

[Marek14]

(II) - circle
((I)I) - 2 circles along line
(((I)I)) - 4 circles along line, restricted to one dimension
((((I)I))) - 8 circles along line
(((((I)I)))) - 16 circles along line

Actually, no. It should be:
(((I)I)) - 2 pairs of concentric cirles
((((I)I))) - 2 groups of 4 concentric cirles
(((((I)I)))) - 2 groups of 8 concentric cirles

So, what's 4, 8 and 16 circles along line? It's (((I))I), ((((I)))I) and (((((I))))I)

(III)
((I)II) - 2 spheres along line
(((I)II)) - 4 spheres along line
((((I)II))) - 8 spheres along line

Same problem, these would be two groups of concentric spheres. Spheres along line would have forms like (((I))II), ((((I)))II) and (((((I))))II)

((II)I) - torus
(II)(I) - cartesian product with circle and ortho hollow line, 2 vertical stacked circles
((II)(I)) - spherate, inflate edges of circles with circles, makes 2 vertical stacked torii
(((II)(I))) - spherate, 2 pairs of 2 vertical stacked torii, along a line
((((II)(I))))) - 4 pairs of 2 vertical stacked torii along a line

Actually, (((II)(I))) would be two pairs of cocircular toruses, vertically stacked. ((((II)(I))))) would then be two groups of 4 cocircular toruses, vertically stacked.

2 pairs of 2 vertical stacked toruses would be (((I)I)(I)). Four pairs would be ((((I))I)(I))

(((I)I)I) - 2 torii along line
((((I)I)I)) - 4 torii at vert of square
(((((I)I)I))) - 8 torii at vert of cube

((((I)I)I)) would be two cocircular pairs along line. (((((I)I)I))) would be two cocircular quartets along line.

4 toruses at vert of square is (((I)(I))I). 8 at vert of cube would be (((I)(I))(I))

(((II))I) - 2 concentric torii
((((II))I)) - 2 pairs of 2 concentric torii along line
(((((II))I))) - 4 pairs of 2 concentric torii along a line
((((((II))I)))) - 8 pairs of 2 concentric torii along a line

((((II))I)) is four toruses which are concentric and/or cocircular (basically, if the major radii are a1,a2 and the minor are b1,b2, these four toruses have all four combinations, a1b1, a1b2, a2b1, a2b2)
(((((II))I))) uses four minor radiuses and has 8 toruses, ((((((II))I)))) uses 8 minor radiuses and has 16 toruses in total.

2 pairs of 2 concentric toruses along line would be ((((I)I))I). 4 pairs would be (((((I)I)))I), 8 pairs would be ((((((I)I))))I)

(((((II)))I)) - 2 pairings of 4 concentric torii along line
((((((II)))I))) - 4 pairings of 4 concentric torii along line

(((((II)))I)) is 8 toruses in combination with 4 major and 2 minor radii. ((((((II)))I))) is 16 toruses in combination of 4 major and 4 minor.

2 groups of 4 concentric toruses along line would be (((((I)I)))I). 4 groups would be ((((((I)I))))I)

((((I)I))I) - not quite sure, I may have some wrong. It's 3:27 am, and time to go to bed.
-- Philip

Let's analyze ((((I)I))I):

(I): two points.
((I)I): two displaced circles
(((I)I)): two pairs of concentric circles
((((I)I))I): two pairs of concentric toruses.

[ICN5D]

Thanks Marek. Those parentheses are tricky. I'm still trying to nail down what happens when the same radius keeps getting spherated over and over. I'm trying to see the pattern of the repeated attribute. You may be familiar with it , but all of the parentheses blend together, and I lose track of them in the complex ones. Have you ever thought of a way to slim down the number of brackets, like in the concentric arrangements? Perhaps a subscript for the number of overlapping () ? Nothing too complicated?

((((II))I)) is four toruses which are concentric and/or cocircular (basically, if the major radii are a1,a2 and the minor are b1,b2, these four toruses have all four combinations, a1b1, a1b2, a2b1, a2b2)

Not sure what that would look like. Sounds interesting, though. It's a new type of arrangement that I'm not familiar with. So, are these 2 concentric groups of 2 cocircular torii? I don't see any difference if they would be 2 cocircular groups of 2 concentric, as in they would be the same.

[Marek14]

Thanks Marek. Those parentheses are tricky. I'm still trying to nail down what happens when the same radius keeps getting spherated over and over. I'm trying to see the pattern of the repeated attribute. You may be familiar with it , but all of the parentheses blend together, and I lose track of them in the complex ones. Have you ever thought of a way to slim down the number of brackets, like in the concentric arrangements? Perhaps a subscript for the number of overlapping () ? Nothing too complicated?

You mean like ((I)) could be written as (I)₂?

[Keiji]

Sounds like a good idea - and things like ((II)(II)) could be written as ((II)²) or even ((I²)²).

[ICN5D]

Yes, it could be a way to simplify them, especially when getting into arrangements of 8 or more cut shapes. ((((II)(I)))) can be ((II)(I))₃

[ICN5D]

Sounds like a good idea - and things like ((II)(II)) could be written as ((II)²) or even ((I²)²).

Hmm, this one ((I²)²) reflects the 220-tiger numerical notation. Interesting. It would be easy to convert, and then apply cuts.


Then, how could you represent ((I)(I)(I))? It seems like it would be irreducible, or maybe ((I)³)? Superscript for repeated terms, subscript for repeated brackets () ?

[Marek14]

Could work, though for work with cuts and such you'd still probably have to expand it to avoid mistakes...

[Keiji]

Then, how could you represent ((I)(I)(I))? It seems like it would be irreducible, or maybe ((I)³)? Superscript for repeated terms, subscript for repeated brackets () ?

Yes, exactly

[ICN5D]

Cool! Then, we can make neat generalizations like ((I)ⁿ)_m = 2ⁿ groups of 2^(m-1) concentric n-spheres at vertices of an n-cube.

And, such and so forth. A directly followable math for toratopes and cuts. I'm sure these patterns can be applied elsewhere, too.

[ICN5D]

This got me thinking more about those simple formulas. God bless, this always happens at 3:00 am for me. How about some more? In analogy, " 2⁰ concentric ... " means just one shape by itself.

((I)_aI^b)_c == 2^(a-1) groups of 2^(c-1) concentric (b+1)-spheres along a line

(I^a)_b == 2^(b-1) concentric a-spheres

(((I)_aI^b)_cI^d)_e == to be continued....

[ICN5D]

I came up with a few more of those generic toratope cut formulas. What do you all think about them? I think it's an awesome idea, and worth exploring. Based on what Marek and I have been discussing, a lot of them can be made with what's been already detailed. In the " 2⁰ groups of ... " , this means only one group, not one shape by itself.

(I)^a == 2^a points at vert of an a-cube

(((I)I)I)_a == 2 groups of 2^(a-1) cocircular torii along a line

((I²)(I))_a == 2 groups of 2^(a-1) cocircular torii, vertical stacked

(((I)_aI)(I))_b == 2^(a-1) groups of 2^(b-1) cocircular torii, vertical stacked

(((I)²)I) == 4 torii at vert of square, not sure how to generalize these other than packing on more () around the preexisting

(((I)²)(I)) == 8 torii at vert of cube, same as above

(((I)_aI)I) == 2^(a-1) torii along a line

(((I²)_aI) == 2^(a-1) concentric torii

(((I²)I)_a == 2^(a-1) cocircular torii

(((I)_aI)_bI)_c == 2^(a-1) groups of 2^(b-1) concentric groupings of 2^(c-1) cocircular torii along a line

(((I)_aI)_b(I)_c)_d == 2^(a-1) groups of 2^(b-1) concentric groupings of 2^(c-1) vertical groupings of 2^(d-1) cocircular torii along a line

((I^a)I^b) == a,b-torus

((I^a)^b) == a*b-tiger



After a while of getting familiar with the system, the cuts and descriptions can be cross-combined, or even have a math applied to them from the numerical toratope notation.


So, what is ((II)((I)))? I see it as four vertically stacked torii. Maybe this could be ((II)(I)_a) = 2^(a-1) torii in a vertical column

I now see the relationship with ((II)I) and (((I)(I))(I)), where every " I " term in the torus was spherated, making eight points, then the cartesian products were mathematically embedded in the torus. So, this would mean that for a spheritorus ((II)II), doing the same would be (((I)(I))(I)(I)) which means 16 spheritoruses in vert of a tesseract. And for the 32-torus ((III)II) , (((I)(I)(I))(I)(I)) means 32 spheritorispheres at the vertices of a geoteron. This would also make the ((((I)(I))(I))(((I)(I))(I)))) mean 64 duotorus tigers at the vertices of a hexeract, which is the cut of an incredibly complicated 12D tigroid.

-- Philip

[Marek14]

Yup, seems that you got all correct

[wendy]

I've been trying to work out the torotope notation, and i think i understand it as a series of three processes, two of which are connected directly to the comb product. If this is right, it will no doubt explain the assorted confusion caused by slightly different use of words like 'spheration'.

Simple Toruses

For this, a 'simple torus' is one where all of the open brackets occur before the first close bracket.

Something like the brick products eg [xyz], gives a product where 'x', 'y' and 'z' carry the size, and [] is how they are combined. Since in the bracketotope notation, one might represent these as eg [III]. The torotope notation looks similar, but is read entirely differently.

In the torutope notation, it is the brackets that carry the size, and the lines give the dimensions where the thing might be let expand. A single axis X might carry several different parameters. We can use different kinds of brackets eg [], (), {}, <>, to make it easier which set of brackets is meant, and use coordinate letters for the 'I' values.

Code: Select all

                              8
        +---------------------]
                              |
                    4         |      12
                    (---------+-------)
                    |                 |
                2   |   6        10   |  14
                {---+---}         {---+---}
                                                      8 4 2
   [#]  [#]     [#######]         [#######]     {([ x ] ) } line bi-shell
   --------------------------------------------------------
   (#(  (#(     )#######)         )#######)     {([xy ] ) }  circle bi-shell
   (#(  )#)     (#######(         }#######)     {([x  ]y) }  two circle-shells
   (#)  (#)     (#######)         (#######)     {([ x ] )y}  four circles



The diagram above shows the +x side for a three-parameter torotope (((x))). The inner brackets [] are applied first, creates a pair of points at +8, -8. This makes a line segment or 1d sphere. The second brackets (), creates a shell, or thickened surface. This replaces the surface by a shell, from +4 to +12, and -4 to -12, which is a thickened dyad-surface. When applied to circles, this is an analus, but it is better to stick to the compound (sphere)-shell. The third set of brackets {} makes a shell of the shell, so we get a 'bi-shell', or a pair of concentric shells. The diagram shows, for example, a dyadic bi-shell.

When we add 'Y' to the mix, we can do it inside any bracket. This means the first appearance of Y would be on a circle of 8, 4 or 2. An appearance at eg, 4, would make the 8-circle happen in 1D, giving +8, -8, but it is at these points that we draw circles of radius 4. But the 2 is applied after the circles in the XY planes are created, giving rise to two circle-shells.

Comb Products

The comb is a pondering (or dimension-reducing) product. When it is applied, one constructs a 'normal' or orthogonal to the surface, of length as the diameter, and turns this into a sphere.

For example, [xy] makes a circle of radius 8. If we want to turn this into a torus, we add a second circle (nz} of diameter 4. This creates a second circle, where the axis n is orthogonal to the surface of the circle in xy, and z is perpendicular to z. This is ([xy]z) = [xy] ## (nz)

Were a fourth element {n} applied, this would replace the surface of the torus of length 2, but there is only one dimension, so we are left with a dyad seperating inside from out. This figure would be {([xy]z)} = [xy] ## (nz) ## {n}. This makes a torus shell.

There is a direct conversion between simple-torus in torotope notation and as a product. In essence, one adds an additional 'n' to each set of I's, (including empty brackets), to get eg

((12)3) -> (12) ## (n3) bi-cicle comb (3-torus)
((12)) -> (12) ## (n) circle-diad comb = circle shell.
((123)45)) -> (123) ## (n45) = bi-spheric comb
(((12)3)4) -> (12) ## (n3) ## (n4) = tri-circular comb.

One will note that that a dyad-product applied before a solid figure, eg [x]##(yz) = ([x]yz) produces multiple copies in the direction of the axis in the initial product, so this produces two spheres, ie dyad ## sphere = two spheres.

Applied after the solid, it hollows out the solid and creates a new surface inside, so a sphere ## dyad makes a sphere-shell.

Where the innermost figure is empty, eg ([]x), the figure is empty. Literally, it has no presence in the space x. This is important when we slice these things up.

Of Hoses and Socks

The hose-operator is an 'outer' process. In the code-box above, we would apply a hose, by shifting everything right (by adding 16), and then adding a new "outer" operator <> to replicate everything at +16, -16.

The sock operator is an 'inner' process, "covering" the old surface. In the diagram above, we would replace the line-segment 2-6 with a dyad-shell, with bits at 1-3 (covering 2), and 5-7 (covering 6)

By the hose operator <> of size 16, {([x])} becomes {([<x>])}; ie [x]##(n)##{n} gives <x>##[n]##(n)##{n}

By the sock operator <> of size 1, {([x])} becomes <{([x])}>; ie [x]##(n)##{n} gives [x]##(n)##{n}##<n>

When we apply this to a circle-circle operation. Suppose we have a circle of rad 4 (x,y), ie (xy).

The hose operator applied with [nz], would prefix (xy) as [nz] ## (xy). But since n must be orthogonal to a pre-existing surface, we swap n with x, to get [xz] ## (ny). This would create a circle in the xz plane of radius 8, crossing at x=+8, -8. Around these in the xy plane, we draw circles of radius 4. In fact, at every point on the circle [xy]. we draw a perpendicular plane and draw a circle radius 4, whereever the circle crosses the plane. It keeps the same section, but is now a hose-shape thing.

The sock operator is inside, so for our radius 4 circle (x,y), take the comb with a radius 2 {n,z} to give:

(xy) ## {nz} = {(xy)z}.

At each point on the surface of x,y, we draw a perpendicular n, of radius 2 (ie length 4), and then use this as a diameter of a circle in the plane set forth by the lines n and z. This makes a torus.

Compound Toruses

For this discussion, the 'compound torus' contains at least one section where a close bracket occurs before an open bracket. The simplest real figure with this is the tiger ((wx)(yz)). Lower dimensional figures exist, but are degenerate.

The 'abuttal operator' here is what might be construed as a 'right-comb' product. This produces an undistorted cartesian product of the surfaces of the two elements in product. One is not really going to enumerate the figure in question, because ideally, one would set this in a set of brackets. We shall suppose that the product is &&.

For example, the surface of the right-comb of two dyads [x] && (y) would give the four vertices of a generalised rectangle (the edges are the diameters given by [] and (). ). One would then suppose that it is a '1-dimensional solid' or 'latrid' that connects and is bounded by these four points. The next step is to enclose these with brackets, eg [x]&&(y) ## {n}. This would create four circles, of radius {}, whose centres are located at x,y all-change-sign.

In practice, it is best to assume that the right-comb product happens, and not to worry too much about what sort of neetan might have that as a surface, since the next step is to enclose the surface.

However, one must be mindful, that in something like <[wx](yz)>, that [] and () represent orthogonal sets of the parameters as in the diagram 1, (eg, r, s) and that the points produced in the product, are surrounded by layers t.

Something like <[[ab]c](de)> would represent the cartesian product of a torus-surface [[ab]c] and a circle-surface (de), which is then spherated in five dimensions.

Of Slices

The sections taken by placing one or more of the coordinates to zero, amounts to simply removing those letters. If all of the letters inside an inner set of brackets are removed, the slice is shown to be empty. This means that it has no presence in that space.

For example, the torus ([xy]z), sliced to x=0, y=0, gives ([]z). Because the innermost element has no surface (being a point), the cartesian product of surfaces is empty, and the torus has no presence on the line x=0, y=0.

Setting y,z to 0 gives ([x]), which is a dyad-shell. Literally, it gives [x]##(n). This gives, two line segments place symmetrically around the origin.

For something like the tiger <[wx](yz)>, setting w=0, gives <[x](yz)>. The inner bits represent a dyad [x] and a circle (yz). The abbutal makes a cartesian product of the surfaces, ie the two circles on a cylinder. The final act is to replace these circles (yz) with ((yz)r): toruses.

Setting w, z to 0, gives <[x](y)> This gives by [x](y), the product of the surfaces of the two dyads, and by <>, four circles in square formation. This is different to four circles in a line <([xy])>. The three parameters are preserved, by the radius of the circles, and the dimensions of the x-y coordinate where the figure falls.

Rotating the space first in the wx plane, would cause [x] to rotate into [wx], the second axis yz makes (y) into (yz). The <> size is preserved throughout.

[Marek14]

You use some different words, wendy, but yes, it seems all correct.

[wendy]

Thanks. It took me a while to figure out the deep mistery. Now we can see if it can be built on. The 'abuttal product' is the secret here.

On the sholders of giants, one can see further.

[wendy]

Names for the torotopes

The "simple" torotopes, being those whose opening brackets all preceed any closing brackets, are simple combs, and we can read off the torotope-notation directly into a comb. On the other hand, combs are more general, since they are not restricted to round things: a hexagon is a kind of circle, and an tetrahedron is a kind of sphere.

The way combs are read, is that one reads the sections from the largest to the smallest, and then attaches 'comb' to the end of this. For a simple torotope, one has to read an extra 'I' for each but the innermost brackets. This gives then the correct sections at each point.

For example, the torus ((ii)i) gives ii, ji comb, that is a circle-circle comb, or bi-circular comb.

A di-torus (((ii)i)i) gives ii, i=ji, ji that is, a circle-circle-circle comb, or tri-circle comb.

One is not restricted to circles and spheres. Lining up the torus with a 8*3 array of squares gives an octagon-triangle comb. This is roughly an octagon-shaped wheel made out of triangular-shaped tubing,

The tiger is not a simple comb, although it is a tri-circle comb in form, the nature of the product is that one can not stop at the bi-circular comb on the way. This is because it is a right-comb, and we ought need something fancier to approach this class of name.

The idea of calling the tiger a 'bi-circle tiger', as much as much as one might regard the inner bits (ii)(ii) as the parameters of the product, causes some problems, because it is actually tri-parametric. The circle that we use to expand the prism-product of the circle-surfaces out is a third parameter. That's why it has three sets of brackets. Since ultimately, the outer brackets enclose two objects () (), it would be a circle, and thus any polygon. A name on the line of bi-circle-tiger does not address the need for this third polygon.

But we shall work on it.

[ICN5D]

I have no idea how you figured that out, wendy, but it's pretty cool. So, there's 95 toratopes in 7D, huh? Do you have a quick systematic way to enumerate them all? As in, not through the painstaking task of deriving by hand? I'd like to see them, I'll bet there's some strange ones hiding out in 7D. How many are in 8D?

[Marek14]

Well, it can be computed by taking all partitions of the number of dimensions except for partition into 1's, and then multiplying each with a number greater than 2.

For example, for closed toratopes (if you include open, the number is doubled):

1D - 1 - 1
2D - 2 - 1
3D - 3, 21 - 2
4D - 4, 31(x2), 22, 211 - 5
5D - 5, 41(x5), 32(x2), 311(x2), 221, 2111 - 12
6D - 6, 51(x12), 42(x5), 411(x5), 33(x3), 321(x2), 3111(x2), 222, 2211, 21111 - 33
7D - 7, 61(x33), 52(x12), 511(x12), 43(x10), 421(x5), 4111(x5), 331(x3), 322(x2), 3211(x2), 31111(x2), 2221, 22111, 211111 - 90
8D - 8, 71(x90), 62(x33), 611(x33), 53(x24), 521(x12), 5111(x12), 44(x15), 431(x10), 422(x5), 4211(x5), 41111(x5), 332(x3), 3311(x3), 3221(x2), 32111(x2), 311111(x2), 2222, 22211, 221111, 2111111 - 261

[wendy]

It's actually 93, because i counted two of them twice.

The calculations for it are here.

In essence, one breaks the notation to a series of nested 'simple rototopes', which count as an 'I' in a larger one. It's then a matter of plotting out the trees as in the post, giving them the weights, and then filling in powers of 2 for each weight.

I think Marek14 missed three at the final tree. He counts one there, where i get four.

[ICN5D]

Wow, I see some cool number patterns in there. So, the number series is the numerical toratope notation, like 211-ditorus? I see many familiar ones in there, but I don't understand how there can be so many of the same number sequence.


4D - 4, 31(x2), 22, 211 - 5

4- (III)
31 - ((III)I) ----> I see the normal notation for 31-torus
31 - ((II)II) ?? ----> I'm not sure how you get that from 31, not even sure what 31(x2) means, though it
22 - ((II)(II))
211 - (((II)I)I)

[Marek14]

Wow, I see some cool number patterns in there. So, the number series is the numerical toratope notation, like 211-ditorus? I see many familiar ones in there, but I don't understand how there can be so many of the same number sequence.


4D - 4, 31(x2), 22, 211 - 5

4- (III)
31 - ((III)I) ----> I see the normal notation for 31-torus
31 - ((II)II) ?? ----> I'm not sure how you get that from 31, not even sure what 31(x2) means, though it
22 - ((II)(II))
211 - (((II)I)I)

Not really. It's:

4 - (IIII)
31 - ((III)I) and (((II)I)I) since both of these have a parenthesis with three I's in it and one free I
22 - ((II)(II))
211 - ((II)II)

And final partition, 1111 is omitted since it would be (IIII) again.

[Keiji]

The number of closed (equivalently, open) toratopes in n dimensions is http://oeis.org/A000669. See Toratope#Counting_toratopes.

[Marek14]

It seems strange, Wendy, since our calculations differ even for lower dimension - we both count 33 in 6D, but you have 11 in 5D, 4 in 4D and 1 in 3D which suggests that you don't count spheres as toratopes as I do. I think that's a mistake since spheres often appear as elements of toratopes and this way you have to use them when they are, but omit them if there's nothing else. It's a bit inconsistent.

So, let's see. In 3D we have torus ((II)I).

In 4D we have ((III)I), ((II)II) and (((II)I)I). These three belong to your weight 1. The tiger ((II)(II)) is weight 2.

In 5D you have 7 weight 1. Strange, I would think 8:
((II)III) - my 2111 group
((III)II) and (((II)I)II) - my 311 group (2 members)
((IIII)I), (((III)I)I), (((II)II)I), ((((II)I)I)I) and (((II)(II))I) - my 41 group (5 members)

And the tigerlikes with two subordinate sets of parentheses are
((III)(II)) and (((II)I)(II)) - my 32 group (2 members)
((II)(II)I) - my 221 group

So there is a small discrepancy, I guess you put (((II)(II))I) into second column.

In 6D, the weight 1 would be:
((II)IIII) - my 21111 group
((III)III) and (((II)I)III) - my 3111 group (2 members)
((IIII)II), (((III)I)II), (((II)II)II), ((((II)I)I)II) and (((II)(II))II) - my 411 group (5 members)
((IIIII)I), (((IIII)I)I), ((((III)I)I)I), (((((II)I)I)I)I), ((((II)(II))I)I), ((((II)II)I)I), (((III)(II))I), ((((II)I)(II))I), (((III)II)I), ((((II)I)II)I), (((II)(II)I)I), (((II)III)I) - my 51 group (12 members)

That's 20 instead of your 15.

The tigers would be:
((IIII)(II)), (((III)I)(II)), (((II)II)(II)), ((((II)I)I)(II)) and (((II)(II))(II)) - my 42 group (5 members)
((III)(III)), ((III)((II)I)) and (((II)I)((II)I)) - my 33 group (3 members)
((III)(II)I) and (((II)I)(II)I) - my 321 group (2 members)
((II)(II)II) - my 2211 group

So I count 11.

The trigers would be
((II)(II)(II)) - my 222 group

In total 20+11+1 = 32 +1 for the sphere = 33

And for 7D... I'd better compress the notation for that. In compressed notation, (III...) is marked by n and I by 1, so tiger is (22)

No parenthesis on second level (1 toratope):
7 group - (7) - 1 member

One set of parentheses on second level (53 toratopes):
61 group: (61), ((51)1), (((41)1)1), ((((31)1)1)1), (((((21)1)1)1)1), ((((22)1)1)1), ((((211)1)1)1), (((32)1)1), ((((21)2)1)1), (((311)1)1), ((((21)11)1)1), (((221)1)1), (((2111)1)1), ((42)1), (((31)2)1), ((((21)1)2)1), (((22)2)1), (((211)2)1), ((411)1), (((31)11)1), ((((21)1)11)1), (((22)11)1), (((211)11)1), ((33)1), ((3(21))1), (((21)(21))1), ((321)1), (((21)21)1), ((3111)1), (((21)111)1), ((222)1), ((2211)1), ((21111)1) - 33 members
511 group: (511), ((41)11), (((31)1)11), ((((21)1)1)11), (((22)1)11), (((211)1)11), ((32)11), (((21)2)11), ((311)11), (((21)11)11), ((221)11), ((2111)11) - 12 members
4111 group: (4111), ((31)111), (((21)1)111), ((22)111), ((211)111) - 5 members
31111 group: (31111), ((21)1111) - 2 members
211111 group: (211111) - 1 member

Two sets of parentheses on second level (33 toratopes):
52 group: (52), ((41)2), (((31)1)2), ((((21)1)1)2), (((22)1)2), (((211)1)2), ((32)2), (((21)2)2), ((311)2), (((21)11)2), ((221)2), ((2111)2) - 12 members
43 group: (43), (4(21)), ((31)3), ((31)(21)), (((21)1)3), (((21)1)(21)), ((22)3), ((22)(21)), ((211)3), ((211)(21)) - 10 members
421 group: (421), ((31)21), (((21)1)21), ((22)21), ((211)21) - 5 members
331 group: (331), (3(21)1), ((21)(21)1) - 3 members
3211 group: (3211), ((21)211) - 2 members
22111 group: (22111) - 1 member

Three sets of parentheses on second level (3 toratopes):
322 group: (322), ((21)22) - 2 members
2221 group: (2221) - 1 member

Total 3+33+53+1 = 90

The 7D toratopes can be divided into species as follows (each species correspond to a certain rooted tree):

Sphere species: (7) - 1 member
Torus species: (61), (511), (4111), (31111), (211111) - 5 members
Ditorus species: ((51)1), ((411)1), ((3111)1), ((21111)1), ((41)11), ((311)11), ((2111)11), ((31)111), ((211)111), ((21)1111) - 10 members
Tiger species: (52), (43), (421), (331), (3211), (22111) - 6 members
Tritorus species: (((41)1)1), (((311)1)1), (((2111)1)1), (((31)11)1), (((211)11)1), (((21)111)1), (((31)1)11), (((211)1)11), (((21)11)11), (((21)1)111) - 10 members
Tiger torus species: ((42)1), ((33)1), ((321)1), ((2211)1), ((32)11), ((221)11), ((22)111) - 7 members
Torus tiger species: ((41)2), ((311)2), ((2111)2), ((31)3), ((211)3), ((31)21), ((211)21), (4(21)), (3(21)1), ((21)211) - 10 members
Triger species: (322), (2221) - 2 members
Tetratorus species: ((((31)1)1)1), ((((211)1)1)1), ((((21)11)1)1), ((((21)1)11)1), ((((21)1)1)11) - 5 members
Tiger ditorus species: (((32)1)1), (((221)1)1), (((22)11)1), (((22)1)11) - 4 members
Torus tiger torus species: (((31)2)1), (((211)2)1), ((3(21))1), (((21)21)1), (((21)2)11) - 5 members
Ditorus tiger species: (((31)1)2), (((211)1)2), (((21)11)2), (((21)1)3), (((21)1)21) - 5 members
Triger torus species: ((222)1) - 1 member
Tiger tiger species: ((32)2), ((221)2), ((22)3), ((22)21) - 4 members
Duotorus tiger species: ((31)(21)), ((211)(21)), ((21)(21)1) - 3 members
Torus triger species: ((21)22) - 1 member
Tetratorus species: (((((21)1)1)1)1) - 1 member
Tiger tritorus species: ((((22)1)1)1) - 1 member
Torus tiger ditorus species: ((((21)2)1)1) - 1 member
Ditorus tiger torus species: ((((21)1)2)1) - 1 member
Tritorus tiger species: ((((21)1)1)2) - 1 member
Tiger tiger torus species: (((22)2)1) - 1 member
Tiger torus tiger species: (((22)1)2) - 1 member
Duotorus tiger torus species: (((21)(21))1) - 1 member
Torus tiger tiger species: (((21)2)2) - 1 member
Ditorus/torus tiger species: (((21)1)(21)) - 1 member
Tiger/torus tiger species: ((22)(21)) - 1 member

[wendy]

In 5D, i make 11, without the sphere, as

group of 7: iiii.i iii.ii iii.i.i ii.iii iii.ii.i iii.i.ii ii.i.i.i that is 41, 32, 311, 23, 221, 212, 2111, being simple.
group of 4; (2,11), and (1,21), ie (i(ii)(ii)), ((iii)(ii)) and (((ii)i)(ii)), and (((ii)(ii))i))

In 5D you have 7 weight 1. Strange, I would think 8:
((II)III) - my 2111 group
((III)II) and (((II)I)II) - my 311 group (2 members)
((IIII)I), (((III)I)I), (((II)II)I), ((((II)I)I)I) and (((II)(II))I) - my 41 group (5 members)

And the tigerlikes with two subordinate sets of parentheses are
((III)(II)) and (((II)I)(II)) - my 32 group (2 members)
((II)(II)I) - my 221 group

So there is a small discrepancy, I guess you put (((II)(II))I) into second column.

Here a = (ii), b0 = (iii), b1 = ((ii)i)

W: 2,11 [(III)(II)I] and ([(II)(II)]I) gives (aai) and ((aa)i) as the outer group
W: 1,21 [(III)(II)] and [((II)I)(II)] gives (b1 a) as the outer group

6D

I'm getting 33 without the sphere, not with it. So i'm picking up one that you're not.

In 6D, the weight 1 would be:
((II)IIII) - my 21111 group
((III)III) and (((II)I)III) - my 3111 group (2 members)
((IIII)II), (((III)I)II), (((II)II)II), ((((II)I)I)II) and (((II)(II))II) - my 411 group (5 members)
((IIIII)I), (((IIII)I)I), ((((III)I)I)I), (((((II)I)I)I)I), ((((II)(II))I)I), ((((II)II)I)I), (((III)(II))I), ((((II)I)(II))I), (((III)II)I), ((((II)I)II)I), (((II)(II)I)I), (((II)III)I) - my 51 group (12 members)

That's 20 instead of your 15.

411 (((II)(II))II) , 51 ((((II)(II))I)I), (((III)(II))I), ((((II)I)(II))I), (((II)(II)I)I), are actually counted elsewhere. They all feature the string ")(" , and thus are not simple torotopes. Resolving the brackets we get ((AA)ii), (((AA)i)i), ((B0,A)i), and ((AA1)1), which is 3,11, 3,11, 2,21 and 3,11 respectively. The missing 3,11 is ((AA11) which is in your 2211 group.

C/- ((AA)ii), (((AA)i)i), ((B0,A)1), ((B1, A)1), (AA1)1), being 3,11, 3,11, 2,21, 2,21, and 3,11 respectively
42: (C0,A), (C1, A), (C2, A), (C3, A) are my 2,21 group, are 1,31 and ((AA)A) is a 11111 group.
33: (B0,B0), (B0,B1) and (B1,B1) correspond to 1,22
321: (B0,A,1) and (B1,A,1) are part of 2,21 group
2211: (AAii) is 3,11
222 (AAA)

3,11 (4), 1,31 (4), 1,22 (3), 2,111 (2)

The actual catalog of 2,21 is (AB1), ((AB)1), ((A1),B) and ((B1)A), which is more than we have counted

AB1 ((ii)(iii)i), ((ii)((ii)i))
AB/1 (((ii)(iii))i), and ((ii)((ii)i))i)
A1/B ((ii)i))(iii)), (((ii)i)((ii)i))
B1/A ((iii)i))(ii)) (((ii)i)i)(ii)))

So the missing animal here is a 1,1,1,1,1 gives (((11)(11))(ii))

Looking at the fact that there are eight 2,21's rather than four as counted, it might be supposed the torotope count for 6D is actually 36.

So it looks like a refinement of the counting rules are in order. I suspect something is array, but i can't put my finger on it at this stage. It may have something to do with the loose '1', because some of the (AB/1) class reslove to (BB) etc. May need to reweight the graphs so this does not happen.

[Marek14]

Maybe you're counting some shapes twice? It certainly looks like my list is complete...

[wendy]

It looks like it is. Specifically, when there are loose nodes in the tower, they can be found in two different ways. But last night's dream and the blurb under the OEIS seems to indicate that Marek14's method is indeed correct.

[ICN5D]

I thought about some toratopes today. They are the:

((((II)(II))(II))(II)) - 2202020 - double-tiger tiger

(((II)(II))((II)(II))) - 2202200 - duotiger

I'm definitely not ready for these two yet, but I know of their existence now. It's another frontier I haven't tried out, in 8D. But, they are certainly some fascinating ones, aren't they? The complex tigroid symmetries are stacked up in ways I haven't explored yet. Though, I'm not as frightened of double-tiger tiger as I am of duotiger. 2202200 is amazing from it's cartesian product of two tigers, then spherated again. It's the inflation of a 6D margin of two orthogonal bound tigers


Then, of course, there's these two, which play on the duotorus tiger property:


((((II)I)((II)I))(II)) - 2121020 - duotorus-tiger tiger

((((II)I)(II))((II)I)) - 2120210 - cyltorintigroid-torus tiger


Not even going to try. Not yet. Gotta master 21210 duotorus tiger first, THEN position the telescope. Though, once again, I'm not as afraid of 2121020. For the other one, a cyltorintigroid is complex enough, let alone multiplied by a torus, then spherated. Not to mention all of these 8D tigroids will require a 5D cut array if reduced to 3D cuts. It would probably be better to stop at 4 or 5D cuts, since at that point they are relatively simple. One would be a vertical stack of 4 cyltorintigroids displaced along the 6 or 7D axis. At that point, they're not so complex, but it does require more time to warm up to 5D toratopes. But, I suppose that once you go that far, might as well cut down the cyltorintigroids. At least knowing of the initial vertical column of 4 would be a good reference. Hmm, doesn't seem so hard after all......


Ohhh, man. So, I wrote quite a bit of good stuff. And, I forgot to ctrl+copy all that I wrote. So, when I get auto logged out, the text field empties and doesn't save what I wrote. And, I forgot to save it. At least SOME of it survived death. I'll try to xenoplast the lost vitals in, the best that I can.


The last tigroid I wanted to explore was the (((III)I)((II)II)), the 31220 torisphere-spheritorus tiger. I'm going to make an attempt at a new deductive reasoning logic with the notation.

• One of it's 6D cuts is a single duotorus tiger (((II)I)((II)I)), with a 2D cut array from (((IIi)I)((II)Ii)).

(((IIi)I)((II)I))
Moving along the first cut axis will collapse a major radius to zero, into a ((III)((II)I)) sphere-torus tiger, then shrink to a torus and disappear.

(((II)I)((II)Ii))
The second cut axis will collapse a minor radius to zero, making a cyltorintigroid (((II)I)(II)), then disappear suddenly




• One of its 5D cuts are 2 concentric cylspherintigroids in a vert column of 2 (((III))((II))), with a 3D cut array from (((III)i)((II)ii)

(((III)i)((II)))
Moving along the first cut axis will merge the concentric groupings into a vertical column of 2 cylspherintigroids ((III)((II)))

(((III))((II)ii))
Moving along this 2D cut-plane will merge the vertical column grouping into 2 concentric cylspherintigroids (((III))(II))




• One of its 4D cuts are a quartet of tigers in vertices of a square (((I)I)((I)I)), with a 4D cut array from (((Iii)I)((Ii)Ii))

(((Iii)I)((I)I))
This is a 2D cut-plane that will merge the quartet into 2 tigers displaced along a line ((II)((I)I)), then shrink and disappear.

(((I)I)((Ii)I))
Moving along the third cut axis will merge the quartet into another line of 2 (((I)I)(II)), orthogonal to the first cut-plane's line

(((I)I)((I)Ii))
Moving along the final fourth axis will collapse the quartet of tigers into a quartet of toruses (((I)I)(I)), which corresponds to a cut evolution of the sphere minor radius of a spheritorus: moving out will collapse the torus into a thin ring. In this case, one of the major radii of the tigers collapsed to zero, leaving behind toruses. It deflated the displaced circles into two points along a line. How curious


I feel like I'm getting the hang of it.

[Marek14]

I thought about some toratopes today. They are the:

((((II)(II))(II))(II)) - 2202020 - double-tiger tiger

(((II)(II))((II)(II))) - 2202200 - duotiger

I'm definitely not ready for these two yet, but I know of their existence now. It's another frontier I haven't tried out, in 8D. But, they are certainly some fascinating ones, aren't they? The complex tigroid symmetries are stacked up in ways I haven't explored yet. Though, I'm not as frightened of double-tiger tiger as I am of duotiger. 2202200 is amazing from it's cartesian product of two tigers, then spherated again. It's the inflation of a 6D margin of two orthogonal bound tigers

The best way to consider these might be through 4D cuts.
((((Ii)(Ii))(Ii))(Ii)) is sixteen ditoruses, stacked in 2x2x2x2 tesseract.
Each of the four evolutions will merge eight pairs of ditoruses arranged in particular direction.
In ((((Ii)(I))(I))(I)) and ((((I)(Ii))(I))(I)), the merged shapes will be tritoruses.
In ((((I)(I))(Ii))(I)), the merged shapes will be tiger toruses.
In ((((I)(I))(I))(Ii)), the merged shapes will be torus tigers.

(((Ii)(Ii))((Ii)(Ii)))) is sixteen tigers based on duocylinders formed by 4 rectangularly-arranged circles in xy plane and 4 rectangularly-arranged circles in zw plane.
All 4 evolutions will play out the same: eight pairs of tigers arranged in particular direction will merge. The merged shapes will be torus tigers.

The last tigroid I wanted to explore was the (((III)I)((II)II)), the 31220 torisphere-spheritorus tiger. I'm going to make an attempt at a new deductive reasoning logic with the notation.

• One of it's 6D cuts is a single duotorus tiger (((II)I)((II)I)), with a 2D cut array from (((IIi)I)((II)Ii)).

(((IIi)I)((II)I))
Moving along the first cut axis will collapse a major radius to zero, into a ((III)((II)I)) sphere-torus tiger, then shrink to a torus and disappear.

Not entirely.
If you collapse a major radius of a toratope, you won't get a neat picture. There is no intermediate stage where the shape would equal ((III)((II)I)), you will just get something with the same topology.

(((II)I)((II)Ii))
The second cut axis will collapse a minor radius to zero, making a cyltorintigroid (((II)I)(II)), then disappear suddenly

Corrent.

• One of its 5D cuts are 2 concentric cylspherintigroids in a vert column of 2 (((III))((II))), with a 3D cut array from (((III)i)((II)ii)

Not sure if "vert column" is the correct description here. It's basically a group of 4 concentric cylspherintigroids with 4 combinations of two possible values for each major diameter.

(((III)i)((II)))
Moving along the first cut axis will merge the concentric groupings into a vertical column of 2 cylspherintigroids ((III)((II)))

(((III))((II)ii))
Moving along this 2D cut-plane will merge the vertical column grouping into 2 concentric cylspherintigroids (((III))(II))

• One of its 4D cuts are a quartet of tigers in vertices of a square (((I)I)((I)I)), with a 4D cut array from (((Iii)I)((Ii)Ii))

(((Iii)I)((I)I))
This is a 2D cut-plane that will merge the quartet into 2 tigers displaced along a line ((II)((I)I)), then shrink and disappear.

(((I)I)((Ii)I))
Moving along the third cut axis will merge the quartet into another line of 2 (((I)I)(II)), orthogonal to the first cut-plane's line

(((I)I)((I)Ii))
Moving along the final fourth axis will collapse the quartet of tigers into a quartet of toruses (((I)I)(I)), which corresponds to a cut evolution of the sphere minor radius of a spheritorus: moving out will collapse the torus into a thin ring. In this case, one of the major radii of the tigers collapsed to zero, leaving behind toruses. It deflated the displaced circles into two points along a line. How curious

I feel like I'm getting the hang of it.

[ICN5D]

The best way to consider these might be through 4D cuts.
((((Ii)(Ii))(Ii))(Ii)) is sixteen ditoruses, stacked in 2x2x2x2 tesseract.
Each of the four evolutions will merge eight pairs of ditoruses arranged in particular direction.
In ((((Ii)(I))(I))(I)) and ((((I)(Ii))(I))(I)), the merged shapes will be tritoruses.
In ((((I)(I))(Ii))(I)), the merged shapes will be tiger toruses.
In ((((I)(I))(I))(Ii)), the merged shapes will be torus tigers.

(((Ii)(Ii))((Ii)(Ii)))) is sixteen tigers based on duocylinders formed by 4 rectangularly-arranged circles in xy plane and 4 rectangularly-arranged circles in zw plane.
All 4 evolutions will play out the same: eight pairs of tigers arranged in particular direction will merge. The merged shapes will be torus tigers.

Yeah, I started thinking about them later after the post. I was able to pick out the middle-most cut of (((II)(II))((II)(II))), as the 16 tigers in vert of tesseract. Cutting any further will make an empty set, but moving out will make a sudden appearance of 8 torii in vertices of cube that divide and do the tiger dance, then disappear. Then, of course, the (((II)(II))(II))(II)) is simply the double tiger along rim of tiger, so one of its 6D cuts is four double tigers at verts of square. But then, this could be cut down into a quartet of quartets of tigers in a 2x2x4 flat array.

Not entirely.
If you collapse a major radius of a toratope, you won't get a neat picture. There is no intermediate stage where the shape would equal ((III)((II)I)), you will just get something with the same topology.

I wasn't sure about that one. I was drawing inspiration from one of the momentary forms of a torisphere cut collapsing into a sphere, then to a point. I was thinking of that sphere that existed for a short while, though remaining mindful that it only appears this way as the hole fills in.

Not sure if "vert column" is the correct description here. It's basically a group of 4 concentric cylspherintigroids with 4 combinations of two possible values for each major diameter.

You're right It's not a hollow circle (I) to make two points. It is 4 concentric pairs, that have two separate groups in their two main diameters. Moving out will merge them into one along a 2-plane and solo axis.

That was fun, I'll try some more later.