The Tiger Explained

[ICN5D]

I just figured out the surface elements of the cylspherinder. Crazy thing is, I didn't even need to consult the algorithms to do it. I can visualize it completely, now.

The cylspherinder can be made by the cartesian product of a sphere and a circle, or by rotating a sphere-prism into 5-D. Starting with a sphere, we extrude it into a prism, the spherinder. We now have two flat sphere ends, connected by a glomohedrix prism. This glomohedrix prism is the linear connection between both of the surfaces of the sphere-ends. Similar to the glomolatrix prism ( line-torus), that connects the circle ends of a cylinder together. If we then rotate this sphere prism around, in the traditional prism way into N+1, we will trace out the sphere ends into a new sphere torus, the spheritorus. The connecting glomohedrix prism will become rotated in a way where only the subshape ( minor radius ) is modified. My name for the glomohedrix prism is a line-globus, meaning "line that has been extruded along the 2-plane of a sphere". This name helps me identify the connecting torus as being linear ( from the initial extrusion ), between the sphere ends. When spun, linear turns into a circular, and by spinning only the minor line part of the spherical major radius, we end up with a circle, embedded into the 2-plane of a sphere, the Torisphere.

So, I found that interesting, that the cylspherinder's surface is a torisphere orthogonally bounded to a spheritorus, its toratopic dual. Is there a name for such a property, to have toratopic duals as surface elements?

[Marek14]

Well, since cylspherinder is a cartesian product, it has a "sharp" element somewhere, which is the product of surface of sphere and a circular line. Two 4D elements grow out of that, a ball x circle with a topology of toraspherinder and a sphere x disk.

Cuts of cylspherinder (III)(II) are as follows:

4D cuts:
Three cuts of form (II)(II). These are duocylinders.
Two cuts of form (III)(I). These are spherinders.

3D cuts:
Three cuts of form (I)(II). These are cylinders. In plane, shape has circular symmetry and as you go from the center, the height of cylinder shrinks until it becomes a circle and disappears.
Six cuts of form (II)(I). These are also cylinders. However, in one direction, the diameter shrinks until the cylinder becomes a line and disappears, and in the other direction, the height shrinks.
One cut of form (III)(). This is a sphere. Graph has circular symmetry and sphere stays the same until you cross a circular boundary where it disappears abruptly.

[ICN5D]

Hmm, I don't really sense any sharp elements on it, only two curved surfaces. Now, I do sense sharp edges, where the torisphere meets the spheritorus, at a 90 degree angle. A spherated margin of this would result in the cylspherintigroid, but you know that. Is that what you mean? By sharp element? Where the "sphere innertube" meets up with the " spherical torus " ?

[Marek14]

Hmm, I don't really sense any sharp elements on it, only two curved surfaces. Now, I do sense sharp edges, where the torisphere meets the spheritorus, at a 90 degree angle. A spherated margin of this would result in the cylspherintigroid, but you know that. Is that what you mean? By sharp element? Where the "sphere innertube" meets up with the " spherical torus " ?

Yes, that's it. In my third 3D cut, you can notice a discontinuity - that's it.

[ICN5D]

Cool, nice to see I'm on the right track. The cut algorithm really is something else! It illuminates some very tough shapes, in better light. Like the cylspherintigroid, you know the cuts crazy well. That challenges my mind. This year, it is my goal to become intimately familiar with all of the 5D and 6D toratopes. Are the cuts of 5-D in 4-D, or have you reduced them into 3-D?

-Philip

[Marek14]

Well, first you make up the 4D cuts, then reduce them into 3D. 6D shapes could be also reduced to a 3D array of 3D cuts.

Let's try an example: a ((II)(II)(II)), triple tiger (a triger?).

5D cuts: all look the same (but they are really three separate kinds, 2+2+2), they have form ((II)(II)(I)): they look like two toraduocyldyinders or 221-tigers in my notation, which are displaced in 5th dimension (dimension perpendicular to both of their base planes).

4D cuts:
3 cuts (actually 1+1+1) of form ((II)(II)()): they are empty.
12 cuts (4+4+4) of form ((II)(I)(I)): They are quadruplets of toracubinders (or 22-toruses) in a rectangle configuration whose main planes are perpendicular to that rectangle.

3D cuts:
12 cuts (2+2+2+2+2+2) of form ((II)(I)()).
The cut itself is empty, but every of the 3 coordinates will add one more dimension, I'll mark that new dimension with lowercase i:
((II)(Ii)()): This, as we've seen, is also empty.
((II)(I)(i)): This is the quadruplet of toracubinders. Since two dimensions result in the same form, we conclude that the array will have cylindrical symmetry.
The axis of the cylinder is the first cut, and so it's empty. Moving from the axis, we'll encounter a pair of toracubinders in whatever direction (the axis passes between the two pairs). Because of the position of the lowercase "i", we can see that we'll encounter them along one of their "minor" dimensions. They will look like pair of circles, one above the other, blow into toruses, then shrink into circles and disappear.

Not entirely sure what happens when you go in axial direction from these pictures, but pairs of toruses would most likely merge.

Then there are 8 cuts of form ((I)(I)(I)).

This cut looks like 8 spheres in vertices of a cube. (This is analogical to a 2D cut of tiger which looks like four circles.).

The three axes expand this to ((Ii)(I)(I)), ((I)(Ii)(I)) and ((I)(I)(Ii)). You see that the form is always the same, but its orientation is different. All of these are toracubinder quartets.

A toracubinder has a cut that looks like two displaced spheres. This shows us what happens on the axes here: four pairs of spheres will behave as Cassini oval rotations, merge, and eventually disappear -- but WHICH pairs that will be will be different for every of the three cardinal directions.

So, here's some homework for you: Try to make 2D arrays of 2D cuts for the 4D toratopes, both open and closed. These arrays can be drawn on paper, so they can be understood more easily.

[ICN5D]

Homework! All right! Haven't had that in forever. I'll work on that, definitely.

[ICN5D]

One way to generate tiger naturally, together with other toratopes in 4D, is simply to consider rotations.

If we start with a circle in 2D, there are two different ways to rotate it: either around a bisecting line, or around a non-intersecting line. First way results in a sphere, second in torus.

Now, there are, similarly, two ways to rotate a sphere. If we rotate it around a bisecting plane, we get a glome. But if we rotate it around a non-intersecting plane, we get (211).

There are, then, FOUR ways to rotate a torus, since it has two distinct kinds of coordinate planes.

If we put torus's main circle in xy plane, then rotation around xy plane (and parallel) is different than rotation around xz or yz. Thus, we can produce four different shapes by rotating a torus:

xy, bisecting: (211)
xy, non-intersecting: (22) - TIGER
xz, bisecting: (31)
xz, non-intersecting: ((21)1) - tetratorus

The general rule for rotating a toratope is like this:

1. Reduce it to unary notation (all ones). Tiger would be ((11)(11)).
2. You rotate around a coordinate hyperplane (or a parallel plane). It has all the dimensions (ones) of the figure - except for one. Replace that one with X. In our case, we can replace any of the four, and get an equivalent representation:

((X1)(11))

3. If you rotated around a BISECTING plane, replace X with 11. If you rotated around a NON-INTERSECTING PLANE, replace X with (11)

((111)(11)) - bisecting
(((11)1)(11)) - non-intersecting

4. Collapse it back
(32) - bisecting
((21)2) - non-intersecting

That's very interesting, right there. I saw this a while back, but it was impossible to see. Now, tonight, I finally saw all of the rotations of a torus. That's pure knowledge going into my brain. And it's incredible. Can every toratope be made by the rotation of another?

[Marek14]

Every closed toratope, yes, since every one must contain either (II) or II.

Pentasphere - bisecting glome: (IIII) -> (XIII) -> (IIIII)
Toratesserinder - nonbisecting glome (IIII) -> (XIII) -> ((II)III) or bisecting toracubinder ((II)II) -> ((II)XI) -> ((II)III)
Toraduocyldyinder - nonbisecting toracubinder ((II)II) -> ((II)XI) -> ((II)(II)I)
Toracubspherinder - bisecting toracubinder ((II)II) -> ((XI)II) -> ((III)II) or bisecting toraspherinder ((III)I) -> ((III)X) -> ((III)II)
Toracubtorinder - nonbisecting toracubinder ((II)II) -> ((XI)II) -> (((II)I)II) or bisecting ditorus (((II)I)I) -> (((II)I)X) -> (((II)I)II)
Cylspherintigroid - bisecting tiger ((II)(II)) -> ((XI)(II)) -> ((III)(II)) or nonbisecting toraspherinder ((III)I) -> ((III)X) -> ((III)(II))
Cyltorintigroid - nonbisecting tiger ((II)(II)) -> ((XI)(II)) -> (((II)I)(II)) or nonbisecting ditorus (((II)I)I) -> (((II)I)X) -> (((II)I)(II))
Toraglominder - bisecting toraspherinder ((III)I) -> ((XII)I) -> ((IIII)I)
Cylindrical ditorus - nonbisecting toraspheringer ((III)I) -> ((XII)I) -> (((II)II)I) or bisecting ditorus (((II)I)I) -> (((II)X)I) -> (((II)II)I)
Tigric torus - nonbisecting ditorus (((II)I)I) -> (((II)X)I) -> (((II)(II))I)
Spheric ditorus - bisecting ditorus (((II)I)I) -> (((XI)I)I) -> (((III)I)I)
Tritorus - nonbisecting ditorus (((II)I)I) -> (((XI)I)I) -> ((((II)I)I)I)

[ICN5D]

Marek, what is the craziest, most complicated toratope that you ever tried to cut/understand/visualize? You go all the way to 6D easily, so I'm interested.

And, what do you mean by "closed toratope". Are there any other kinds of toratopes ?

[Marek14]

Marek, what is the craziest, most complicated toratope that you ever tried to cut/understand/visualize? You go all the way to 6D easily, so I'm interested.

And, what do you mean by "closed toratope". Are there any other kinds of toratopes ?

Well, I think I didn't go higher than 6D. As for closed and opne, I follow the notation of http://teamikaria.com/hddb/wiki/List_of_toratopes

[ICN5D]

That's cool, 6-D has quite the zoo of shapes anyway! I'm glad you detailed them, they'll hurt my brain at first, but I'll see them at some point. Did you create the notation, or just the cut and rotate algorithms? Do you have any other cool tricks that can be done with the notation?

[Marek14]

That's cool, 6-D has quite the zoo of shapes anyway! I'm glad you detailed them, they'll hurt my brain at first, but I'll see them at some point. Did you create the notation, or just the cut and rotate algorithms? Do you have any other cool tricks that can be done with the notation?

I think that I made the first version of the notation, but it was a long time ago.

[ICN5D]

So, how about for the other coordinate planes, with the other rotations? I suppose the same rotation that turns a torus into a tiger could be applied to a toratesserinder. And, of course, the more complex, the more distinct coordinate planes it would have. This would make for MANY rotations of just one of them. Have you looked into those ways, too?

[Marek14]

So, how about for the other coordinate planes, with the other rotations? I suppose the same rotation that turns a torus into a tiger could be applied to a toratesserinder. And, of course, the more complex, the more distinct coordinate planes it would have. This would make for MANY rotations of just one of them. Have you looked into those ways, too?

Unfortunately, if you do it with open toratopes, you get outside of system. As an easy example, imagine a square rotated through a vertical nonbisecting line. You'd get a torus with square cross-section which is not included in open or closed toratope list.

Reminds me of my old graphotope system: you can start from arbitrary graph on n nodes, assign a dimension to each one, and construct a shape whose cross-cut is a square if the two dimensions of the plane are not joined by an edge, and a circle if they are. Those have some nice properties: for example you can "roll" them between two nodes joined with an edge, a cross-cut is constructed by simple removing of a node and by removing a node and all connected nodes, you can figure how a footprint of the shape would look.

For example, in 3D, you have four possible graphotopes: a cube, a cylinder, a sphere, and a "dome" or "crind" with equation max(x^2,y^2) + z^2 = 1.

All cube projections are square crosscut and square footprint - since the footprint has the same dimension as crosscut, it follows that a cube can't roll.
All sphere projections are circular crosscut and point footprint - footprint is 2 dimensions lower, so sphere can roll in 2 directions.
Cylinder projections are either circular crosscut and circular footprint - when the cylinder stands - or a square crosscut and a line footprint when it lies down. Since the difference is 1 dimension, you see that it can only roll in 1 direction.

Now, for all these rolls, rolling in a coordinate direction traces a line in graph, but it was between two identical nodes. This is not the case with dome.

The two projections of dome are square crosscut with point footprint and a circular crosscut with line footprint. If you roll it from first orientation in any of the 2 direction, you'll arrive in the second orientation which has only 1 direction to roll, and continuing there will lead back to first orientation.

[ICN5D]

Unfortunately, if you do it with open toratopes, you get outside of system. As an easy example, imagine a square rotated through a vertical nonbisecting line. You'd get a torus with square cross-section which is not included in open or closed toratope list.

I noticed that, there is no square torus or cube torus. They are all round, with round things impregnated into other round things. I like to describe the rolling side of a cylinder as a "line-torus", basically a hollow tube. Then, for a spherinder, the lacing is a "line-toraspherinder", which helps me distinguish it as being a linear attachment between the surface of two spheres. Then, if I were to rotate this spherinder into 5D, I simply add a spin operator to the line part, and voila, I get the circle-toraspherinder, where the word "circle" can be dropped (given that this surcell does a bisecting rotation). The flat sphere endcaps will have a non-bisecting rotation, becoming a toracubinder. Which will nicely add up to the cylspherinder! I came across this method on my own, before I learned your notation well.Yours is still a more powerful system, being closely related to the actual equations. This makes it easier to cut and rotate toratopes, since the mathematical way is more clearly reflected in the notation.

But, I'm sure the tesseract-toraglominder can be expressed in your toratope notation. It will probably show the same properties with rotations and cuts, when applied. I wonder what it would look like, a little expanded to make room for other cross-cut toratopes?


Did you mess around with the closed toratopes, using the other coordinate plane rotations? I'm curious about other tiger-like ( polytigroid? ) shapes with parallel non-bisecting. It would be interesting to see how many ways a single shape can be made, through a combination of them. But, it's possible that you already did, and I haven't noticed yet

The two projections of dome are square crosscut with point footprint and a circular crosscut with line footprint. If you roll it from first orientation in any of the 2 direction, you'll arrive in the second orientation which has only 1 direction to roll, and continuing there will lead back to first orientation.

That's strange, I never looked into that one. It has a weird rolling ability, especially when comparing it to the cylinder. Are there any 4 or 5D versions of this? I suppose using two flat spheres could be analogous, or perhaps some crazy combination I can't even think of yet.



By the way, I'm still working on the homework. The 2D cuts of 4D toratopes are a challenge. A good challenge! I'm trying to adapt some new ways to define cuts and rotations in the notation I've been using. But, for now, I'm still converting into the other system to understand how it works. I didn't forget about it!

--Philip


ps: the "triger" is a great name! It must be so, it's too good to pass up.

[Marek14]

Unfortunately, if you do it with open toratopes, you get outside of system. As an easy example, imagine a square rotated through a vertical nonbisecting line. You'd get a torus with square cross-section which is not included in open or closed toratope list.

I noticed that, there is no square torus or cube torus. They are all round, with round things impregnated into other round things. I like to describe the rolling side of a cylinder as a "line-torus", basically a hollow tube. Then, for a spherinder, the lacing is a "line-toraspherinder", which helps me distinguish it as being a linear attachment between the surface of two spheres. Then, if I were to rotate this spherinder into 5D, I simply add a spin operator to the line part, and voila, I get the circle-toraspherinder, where the word "circle" can be dropped (given that this surcell does a bisecting rotation). The flat sphere endcaps will have a non-bisecting rotation, becoming a toracubinder. Which will nicely add up to the cylspherinder! I came across this method on my own, before I learned your notation well.Yours is still a more powerful system, being closely related to the actual equations. This makes it easier to cut and rotate toratopes, since the mathematical way is more clearly reflected in the notation.

But, I'm sure the tesseract-toraglominder can be expressed in your toratope notation. It will probably show the same properties with rotations and cuts, when applied. I wonder what it would look like, a little expanded to make room for other cross-cut toratopes?

Well, maybe if you added square brackets [] to the notation to show cartesian product. Even then, it's problematic, since the rotations are not the basic operations of the notation. They are just something that flows out of it.

Did you mess around with the closed toratopes, using the other coordinate plane rotations? I'm curious about other tiger-like ( polytigroid? ) shapes with parallel non-bisecting. It would be interesting to see how many ways a single shape can be made, through a combination of them. But, it's possible that you already did, and I haven't noticed yet

Basically, every unique point where there are two I's next to each other is one rotation. For example, the torus sequence (torus, ditorus, tritorus, ...) all have only 1 possible way to be made using rotation.

The two projections of dome are square crosscut with point footprint and a circular crosscut with line footprint. If you roll it from first orientation in any of the 2 direction, you'll arrive in the second orientation which has only 1 direction to roll, and continuing there will lead back to first orientation.

That's strange, I never looked into that one. It has a weird rolling ability, especially when comparing it to the cylinder. Are there any 4 or 5D versions of this? I suppose using two flat spheres could be analogous, or perhaps some crazy combination I can't even think of yet.

Yes, as I said, you can start with ANY graph. For 4D, for example, you have the following:

Tesseract: max(x^2,y^2,z^2,t^2) == 1. Projection is cube crosscut, cube footprint. No rolling.

Cubinder: max(x^2+y^2,z^2,t^2) == 1. Projection 1 is cylinder crosscut, cylinder footprint. No rolling. Projection 2 is cube crosscut, square footprint. Free rolling in 1 dimension.

Dominder: max(max(x^2,y^2)+z^2,t^2) == 1. A dome prism. Projection 1 is dome crosscut, dome footprint. No rolling. Projection 2 is cylinder crosscut, square footprint. Roll in 1 dimension to Projection 3. Projection 3 is cube crosscut, line footprint. Roll in 2 dimensions to Projection 2.
Duocylinder: max(x^2+y^2,z^2+t^2) == 1. Cylinder crosscut, circle footprint. Free rolling in 1 dimension.

Tridome: max(x^2,y^2,z^2)+t^2 == 1. Projection 1 is dome crosscut, square footprint. Roll in 1 dimension to Projection 2. Projection 2 is cube crosscut, point footprint. Roll in 3 dimensions to Projection 1.
Spherinder: max(x^2+y^2+z^2,t^2) == 1. Projection 1 is sphere crosscut, sphere footprint. No rolling. Projection 2 is cylinder crosscut, vertical line footprint. Free rolling in 2 dimensions.
Longdome: not easily expressible as equation. Projection 1 is dome crosscut, circle footprint. Roll in 1 Dimension to Projection 2. Projection 2 is cylinder crosscut, horizontal line footprint. Free roll in 1 dimension. Roll in 2nd dimension to Projection 1.

Spheridome: max(x^2+y^2,z^2)+t^2 == 1. Projection 1 is sphere crosscut, circle footprint. Roll in 1 dimension to Projection 3. Projection 2 is dome crosscut, horizontal line footprint. Free roll in 1 dimension. Roll in 2nd dimension to Projection 3. Projection 3 is cylinder crosscut, point footprint. Roll in 1 dimension to Projection 1. Free roll in 2nd/3rd dimension to Projection 2.
Cyclodome: max(x^2,y^2)+max(z^2,y^2) == 1. Projection is dome crosscut, vertical line footprint. Roll in 2 dimensions to differently oriented instance of same projection.

Semiglome: max(x^2,y^2)+z^2+t^2 == 1. Projection 1 is sphere crosscut, line footprint. Free roll in 2 dimensions to Projection 2. Projection 2 is dome crosscut, point projection. Free roll in 1 dimension. Roll in 2nd or 3rd dimension to Projection 1.

Glome: x^2+y^2+z^2+t^2 == 1. Projection is sphere crosscut, point footprint. Free roll in 3 dimensions.

From this description, you can see a few things.

1: You can build these from lower shapes. If you have equation a == 1 and you attach another node (t dimension, say) to graph, unconnected to anything, you'll get max(a,t^2) == 1. If you attach another node, connected to everything, you'll get a + t^2 == 1. Similarly for attaching multiple nodes at once, as long as they are all unconnected or all connected to all previous nodes.
2. By switching addition and maximum, you'll get a "dual" shape based on a complementary graph. For example, cyclodome, whose graph is 4-cycle, is complementary to duocylinder whose graph is two K2's.
3. However, not all shapes can be made with this simple construction. Longdome is based on line of 4 nodes -- this is a self-complementary graph, so any representation must lead to the same shape when addition and maximum is switched. I think it has no "simplest" equation but several equations that use coordinates multiple times and reduce, in the end, to the same shape.

By the way, I'm still working on the homework. The 2D cuts of 4D toratopes are a challenge. A good challenge! I'm trying to adapt some new ways to define cuts and rotations in the notation I've been using. But, for now, I'm still converting into the other system to understand how it works. I didn't forget about it!

--Philip


ps: the "triger" is a great name! It must be so, it's too good to pass up.

And yet you misspelt it

[ICN5D]

Duocylinder: max(x^2+y^2,z^2+t^2) == 1. Cylinder crosscut, circle footprint. Free rolling in 1 dimension.

I remember reading somewhere that a duocylinder can roll along 2 simultaneous linear directions. How does that work with this circle footprint? It seems very similar to a sphere's rolling ability, but as a combination of two linears. I see the duocylinder has having only two rolling sides, oriented at right angles of each other. There is no flat sides to place it on, no matter what, it will be on a rolling side. Which would make parallel parking way easier.

Longdome: not easily expressible as equation. Projection 1 is dome crosscut, circle footprint. Roll in 1 Dimension to Projection 2. Projection 2 is cylinder crosscut, horizontal line footprint. Free roll in 1 dimension. Roll in 2nd dimension to Projection 1.

Longdome is based on line of 4 nodes -- this is a self-complementary graph, so any representation must lead to the same shape when addition and maximum is switched. I think it has no "simplest" equation but several equations that use coordinates multiple times and reduce, in the end, to the same shape.

That one sounds very strange! I can't visualize it yet.

Cyclodome: max(x^2,y^2)+max(z^2,y^2) == 1. Projection is dome crosscut, vertical line footprint. Roll in 2 dimensions to differently oriented instance of same projection.

Is this what you get by a bisecting rotation of a dome into 4D?

Spheridome: max(x^2+y^2,z^2)+t^2 == 1. Projection 1 is sphere crosscut, circle footprint. Roll in 1 dimension to Projection 3. Projection 2 is dome crosscut, horizontal line footprint. Free roll in 1 dimension. Roll in 2nd dimension to Projection 3. Projection 3 is cylinder crosscut, point footprint. Roll in 1 dimension to Projection 1. Free roll in 2nd/3rd dimension to Projection 2.

This one sounds like the 2 bisecting spheres with a skin wrapped around them. The spherical equivalent to the crind.

These domes are more abstract than the other shapes I've been able to see. Though, they do follow predictable patterns. Thank you for detailing them, it sheds light on how they would feel in the hands.

I've been thinking lately on the duocylinder margin. It feels like it's one continuous 90 degree edge, that bounds the two rolling sides. Almost like it's made from 4 circular edges, all linked together, attached by a point. There would be no separation between them, just one single, crazy-looking sharp edge.

Also, I'm starting to see the cylspherintigroid. Once the rotation method that turns a torus into a tiger was made clear, it's easily repeatable with a toraspherinder. Simply hold the toraspherinder flat, and do the same non-bisecting rotation into 5D. The margin of a cylspherinder has the same feeling as a dome-like rolling ability.

Well, first you make up the 4D cuts, then reduce them into 3D. 6D shapes could be also reduced to a 3D array of 3D cuts.

Let's try an example: a ((II)(II)(II)), triple tiger (a triger?).

Did I misspell it? My bad

[Marek14]

Duocylinder: max(x^2+y^2,z^2+t^2) == 1. Cylinder crosscut, circle footprint. Free rolling in 1 dimension.

I remember reading somewhere that a duocylinder can roll along 2 simultaneous linear directions. How does that work with this circle footprint? It seems very similar to a sphere's rolling ability, but as a combination of two linears. I see the duocylinder has having only two rolling sides, oriented at right angles of each other. There is no flat sides to place it on, no matter what, it will be on a rolling side. Which would make parallel parking way easier.

Well, the circle footprint can be rolled in direction perpendicular to the circle, but of course you can also roll the circle along a line in its own plane

Longdome: not easily expressible as equation. Projection 1 is dome crosscut, circle footprint. Roll in 1 Dimension to Projection 2. Projection 2 is cylinder crosscut, horizontal line footprint. Free roll in 1 dimension. Roll in 2nd dimension to Projection 1.

Longdome is based on line of 4 nodes -- this is a self-complementary graph, so any representation must lead to the same shape when addition and maximum is switched. I think it has no "simplest" equation but several equations that use coordinates multiple times and reduce, in the end, to the same shape.

That one sounds very strange! I can't visualize it yet.

Cyclodome: max(x^2,y^2)+max(z^2,y^2) == 1. Projection is dome crosscut, vertical line footprint. Roll in 2 dimensions to differently oriented instance of same projection.

Is this what you get by a bisecting rotation of a dome into 4D?

Well, when trying to visualize bisecting rotations of these shapes, the algorithm seems to be as follows:

0: Pick a node (x)
1: Replicate the node x (add second node y connected to the same nodes the first one is)
2: Join x and y.

When using this algorithm on the dome, it seems that the result is a spheridome or a semiglome (depending on direction of rotation). Cyclodome doesn't seem to be a rotation.

Spheridome: max(x^2+y^2,z^2)+t^2 == 1. Projection 1 is sphere crosscut, circle footprint. Roll in 1 dimension to Projection 3. Projection 2 is dome crosscut, horizontal line footprint. Free roll in 1 dimension. Roll in 2nd dimension to Projection 3. Projection 3 is cylinder crosscut, point footprint. Roll in 1 dimension to Projection 1. Free roll in 2nd/3rd dimension to Projection 2.

This one sounds like the 2 bisecting spheres with a skin wrapped around them. The spherical equivalent to the crind.

These domes are more abstract than the other shapes I've been able to see. Though, they do follow predictable patterns. Thank you for detailing them, it sheds light on how they would feel in the hands.

I've been thinking lately on the duocylinder margin. It feels like it's one continuous 90 degree edge, that bounds the two rolling sides. Almost like it's made from 4 circular edges, all linked together, attached by a point. There would be no separation between them, just one single, crazy-looking sharp edge.

Also, I'm starting to see the cylspherintigroid. Once the rotation method that turns a torus into a tiger was made clear, it's easily repeatable with a toraspherinder. Simply hold the toraspherinder flat, and do the same non-bisecting rotation into 5D. The margin of a cylspherinder has the same feeling as a dome-like rolling ability.

Well, first you make up the 4D cuts, then reduce them into 3D. 6D shapes could be also reduced to a 3D array of 3D cuts.

Let's try an example: a ((II)(II)(II)), triple tiger (a triger?).

Did I misspell it? My bad

Oh, sorry about that, forgot I made that name.

[ICN5D]

Okay, so, been cutting some toratopes lately. I was thinking about the triger, and its 3-D cuts. I can see how one of the 2-D cuts of a tiger is four circles in vertices of a square. Now, I can see how the triger would have eight sphere intercepts, in 3-D. So, according to this sequence, a tetratiger ((II)(II)(II)(II)) would make sixteen glome intercepts, which would still end up slicing into spheres, in vertices of a tesseract. Then, the pentatiger ((II)(II)(II)(II)(II)) would make 32 intercepts in 3-D, which would be pentaspheres in nature, but slice into spheres, in vertices of a geoteron. Do these intercepts cluster around and make a hollow sphere-like thing, or do they fill up the whole volume within the n-cube vertices, identical to a 3-D projection?


Can you walk me through the cuts of the 330-tiger?

[Marek14]

Okay, so, been cutting some toratopes lately. I was thinking about the triger, and its 3-D cuts. I can see how one of the 2-D cuts of a tiger is four circles in vertices of a square. Now, I can see how the triger would have eight sphere intercepts, in 3-D. So, according to this sequence, a tetratiger ((II)(II)(II)(II)) would make sixteen glome intercepts, which would still end up slicing into spheres, in vertices of a tesseract. Then, the pentatiger ((II)(II)(II)(II)(II)) would make 32 intercepts in 3-D, which would be pentaspheres in nature, but slice into spheres, in vertices of a geoteron. Do these intercepts cluster around and make a hollow sphere-like thing, or do they fill up the whole volume within the n-cube vertices, identical to a 3-D projection?

Well, the coordinate axes will exhibit similar behaviour, where only the edges will be filled. I think that the axes within the projection will always stay empty, so it will be a hollow structure no matter where you go.

Can you walk me through the cuts of the 330-tiger?

((III)(III))
5D cuts: 3 cuts of form ((II)(III)) (cylspherintigroid or 320-tiger) and 3 cuts of form ((III)(II)), same form, but differently oriented.
4D cuts: 3 cuts of form ((I)(III)) (two toraspherinders displaced in 4th dimension), 3 cuts of same form ((III)(I)), 9 cuts of form ((II)(II)) (tiger).
3D cuts: 1 cut of form (()(III)), 1 cut of same form ((III)()), 9 cuts of form ((II)(I)), 9 cuts of same form ((I)(II)).

((III)()) cut: This cut is empty. Chart has spherical symmetry. When going outwards from center, you'll see a sphere splitting in two concentric spheres which then remerge and disappear.
((II)(I)) cut: This cut is two parallel toruses. Chart has cylindrical symmetry (of course orientation of this symmetry has nothing to do with cylindrical symmetry of the individual graphs -- it matches because of the high symmetry of the shape). When going outward from the axis of the chart, you get the cuts of tiger, so the toruses will merge. When you go along the axis of the chart, each torus behaves as cut of toraspherinder, filling the hole and then disappearing.

[ICN5D]

Cool, thanks for that! I feel that if I start with something with a lot of symmetry, it will be easier to visualize. This one is basically the double-spherical version of the tiger, and all of the same symmetries will be reflected. It's neat to see the cylspherintigroid come out out of it, as an analog to a sphere cross section. I see how the cut algorithm works better with each walk-through.

((III)()) cut: This cut is empty. Chart has spherical symmetry. When going outwards from center, you'll see a sphere splitting in two concentric spheres which then remerge and disappear.

Does this spherical symmetry mean that you can travel any direction outwards and hit the radius when the concentric spheres split?

((II)(I)) cut: This cut is two parallel toruses. Chart has cylindrical symmetry (of course orientation of this symmetry has nothing to do with cylindrical symmetry of the individual graphs -- it matches because of the high symmetry of the shape). When going outward from the axis of the chart, you get the cuts of tiger, so the toruses will merge. When you go along the axis of the chart, each torus behaves as cut of toraspherinder, filling the hole and then disappearing.

So, the same two toruses share two unique symmetries, depending on the axis? That's mind boggling. How are you defining outward vs along the axis of the chart? How are you establishing the direction of the tiger-like cuts vs toraspherinder cuts?

I think I understand it, now. The two toruses are an analog to the cross sections of a sphere, so moving up or down will change the major radius of both. But moving side-ways will be the tiger-symmetry, and the two will merge like a tiger does. That's amazing! I'd love to see it rendered, all of the different axes and symmetries. I'm going to take a stab at it tomorrow, another cut breakdown.

[Marek14]

Cool, thanks for that! I feel that if I start with something with a lot of symmetry, it will be easier to visualize. This one is basically the double-spherical version of the tiger, and all of the same symmetries will be reflected. It's neat to see the cylspherintigroid come out out of it, as an analog to a sphere cross section. I see how the cut algorithm works better with each walk-through.

((III)()) cut: This cut is empty. Chart has spherical symmetry. When going outwards from center, you'll see a sphere splitting in two concentric spheres which then remerge and disappear.

Does this spherical symmetry mean that you can travel any direction outwards and hit the radius when the concentric spheres split?

Yes.

((II)(I)) cut: This cut is two parallel toruses. Chart has cylindrical symmetry (of course orientation of this symmetry has nothing to do with cylindrical symmetry of the individual graphs -- it matches because of the high symmetry of the shape). When going outward from the axis of the chart, you get the cuts of tiger, so the toruses will merge. When you go along the axis of the chart, each torus behaves as cut of toraspherinder, filling the hole and then disappearing.

So, the same two toruses share two unique symmetries, depending on the axis? That's mind boggling. How are you defining outward vs along the axis of the chart? How are you establishing the direction of the tiger-like cuts vs toraspherinder cuts?

I think I understand it, now. The two toruses are an analog to the cross sections of a sphere, so moving up or down will change the major radius of both. But moving side-ways will be the tiger-symmetry, and the two will merge like a tiger does. That's amazing! I'd love to see it rendered, all of the different axes and symmetries. I'm going to take a stab at it tomorrow, another cut breakdown.

Outward vs. along the axis: This can be explained as following:

((II)(I)), when moving along an axis, will result in either ((IIi)(I)) -- one axis, or ((II)(Ii)) -- two axes. But since two different axes lead to exactly same thing (including orientations), it follows that ANY direction in the plane defined by these two axes will look the same. So the chart has cylindrical symmetry. This follows from the fact that 330-tiger is a rotation of cylspherintigroid: a cylspherintigroid has a 2D chart with tiger cuts on one axes and two parallel spherinders cut on the other, and this one is this chart rotated along the "parallel spherinders" axis into third dimension.

[Marek14]

Had a bit of a walk down memory lane through old posts here. Found that PWrong was actually the first one to encounter the tiger, though it was just an oddity in parametric equations for him and it took some time before we made sense of it. I also found my very first post here where I introduced graphotopes (shapes like longdome I've mentioned here recently).

So, anyway, this time I tried to envision the block toruses.

In 3D, you can build a simple torus from blocks -- just take a 1x3x3 cuboid and take the middle block out. You get something which is topologically equivalent to torus.

If you take the same shape and lift it in 4D (so it will be a 1x1x3x3 hypercuboid with middle block missing), you'll get an analogue of toracubinder. Toraspherinder can be built like a 1x3x3x3 cuboid with middle block missing.

How would a block rendition of tiger look like?

It could look like this:

Code: Select all

OOO|OOO|OOO
O O|O O|O O
OOO|OOO|OOO
---+---+---
OOO|   |OOO
O O|   |O O
OOO|   |OOO
---+---+---
OOO|OOO|OOO
O O|O O|O O
OOO|OOO|OOO


This is a 2D array of 2D layers.

And ditorus might look like this:

Code: Select all

OOOOOOOOO
OOOOOOOOO
OOOOOOOOO
---------
OOOOOOOOO
O       O
OOOOOOOOO
---------
OOOOOOOOO
O OOOOO O
OOOOOOOOO
---------
OOO   OOO
O O   O O
OOO   OOO
---------
OOOOOOOOO
O OOOOO O
OOOOOOOOO
---------
OOOOOOOOO
O       O
OOOOOOOOO
---------
OOOOOOOOO
OOOOOOOOO
OOOOOOOOO
---------


In 3D, this would be a hollow torus, but in 4D it will represent a ditorus.

[ICN5D]

That's neat, it looks very experimental and early on, trying to make sense of things. I am learning that understanding cuts and their symmetry is the best way to visualize the toratopes. They are much more abstract with the holes, and trying to understand it in its entirety first is quite a learning curve. The three basic 4D toratopes should come first, then the tiger. All that's left is to apply these cut patterns and cross-breed them together. One can almost see the holes when watching things split and applying the symmetry.

On that topic, I spent the day visualizing the 330-tiger. Not ALL of my time, but little moments that were free! It definitely is a perfect cross between a sphere and a tiger. I see how the plane that slides between the two stacked toruses has the tiger-like symmetry, in a sphere-like way. The hole between the two toruses is shaped like a sphere. It makes sense when moving up or down, watching the two toruses collapse their main circle, like the cross sections of a sphere. But, I'm curious as to what happens after the two toruses collapse into spheres. It seems like the spheres should merge into one, then collapse to a point. This would reflect the symmetry of when the two concentric spheres split and merge in the other cut.

EDIT: Actually, I think I see what's happening to the two spheres. Let's say at origin, the two toruses are laid flat on XY, separated along Z. Moving along the XY plane will obey tiger-symmetry, and merge the toruses. Moving along Z will obey toraspherinder symmetry, and the two toruses will collapse their main circles into spheres. But, moving further along Z will never merge them, only shrink to points and disappear. One would have to move along Z, collapse to spheres, THEN move along XY to merge them into one sphere.

I also imagined an oblique slicing: a sphere splits into two, while pinching in the middle, and separating while expanding their main circle, into two vertically stacked toruses.

This makes me wonder what else has yet to be found. Perhaps some bizarre equation for a yet to be explored toratope? I guess only +7D is the answer to that one!

[ICN5D]

Okay, Marek, bringing out the big guns, now

I don't even know where to begin with these. I see how the sphere symmetry is reflected in the 330-tiger. But, as for how to begin with a torus-shape tiger symmetry, or even a tiger shaped tiger symmetry, I don't know.

(((II)I)((II)I)) - 21210-duotorus tiger : maybe has a crazy jumble of ditoruses?

(((II)(II))(II)) - 22020-double tiger : maybe has four tigers ( vertical stack toruses ) in vertices of a square, for 3D cut?

If you can cut these crazy things, then I seriously genuflect to thee

[Marek14]

Okay, Marek, bringing out the big guns, now

I don't even know where to begin with these. I see how the sphere symmetry is reflected in the 330-tiger. But, as for how to begin with a torus-shape tiger symmetry, or even a tiger shaped tiger symmetry, I don't know.

(((II)I)((II)I)) - 21210-duotorus tiger : maybe has a crazy jumble of ditoruses?

(((II)(II))(II)) - 22020-double tiger : maybe has four tigers ( vertical stack toruses ) in vertices of a square, for 3D cut?

If you can cut these crazy things, then I seriously genuflect to thee

Let's see, then:

Duotorus tiger (((II)I)((II)I))
5D cuts:
2 (((I)I)((II)I)) and 2 similar (((II)I)((I)I)): two cyltorintigroids displaced in dimension of their circle.
1 (((II))((II)I)) and 1 similar (((II)I)((II))): two cyltorintigroids differing in their circle radius (cyltorintigroid has 4 radii: major and minor radius of its constituent torus, radius of its constituent circle, which define the 3D (torus x circle) surface in 5D and its own minor radius)

4D cuts:
A: 1 ((()I)((II)I)) and 1 similar (((II)I)(()I)): empty. This will figure in 3 different 3D cuts.
B: 2 (((I))((II)I)) and 2 similar (((II)I)((I))): four ditoruses displaced in a line along their minor dimension. This will figure in 3 different 3D cuts.
C: 4 (((I)I)((I)I)): four tigers displaced in vertices of rectangle whose each dimension lies in one of the tigers' circles. This will figure in 2 different 3D cuts.
D: 2 (((I)I)((II))) and 2 similar (((II))((I)I)): four tigers forming two pairs displaced in dimension of one circle. Each pair differs in the diameter connected to the OTHER circle. This will figure in 3 different 3D cuts.
E: 1 (((II))((II))): four tigers. All are in same place, but they have all 4 possible combination of 2 values for each circular radius. This will figure in just 1 3D cut.

3D cuts:
1 ((())((II)I)) and 1 similar (((II)I)(())):
The center of ((())((II)I)) is empty. Its three axes are (((i))((II)I)), (((i))((II)I)), and ((()i)((II)I)). This means that the chart has cylindrical symmetry; axial direction on chart will be always empty because it's cut A. Equatorial directions are cut B: four ditoruses. Ditorus sliced in the minor direction is two cocircular toruses, so equatorial slice of this type will be a torus separating in two cocirculars, merging and disappearing... and then the same thing once again.
What happens if you go to the point midway between these two outer toruses and THEN go upwards in axial direction?
Well, then you're tracking the cyltorintigroid (((II)I)(II)) that was rotated (around nonbisecting line in circle dimension) to create the duotorus tiger.
Cyltorintigroid has a 4D cut corresponding to two ditoruses displaced in the minor dimension: ((II)I)(I)). Since the 3D cut between the two ditoruses is empty, it must be the 3D cut ((II)I)()).
This means that if we go "upwards" from here, we'll get the SAME cut -- once again, a torus will appear in each direction, separate in two cocirculars and merges.
This gives us a powerful image of how this chart looks:
It's a HOLLOW TORUS!
Points on inner and outer border of the torus correspond to 3D cuts with just one torus. The closer to the centre of the wall between the two toruses, the more the two cocircular toruses will be separated. Points outside of these two toruses have empty cuts.

2 ((()I)((I)I)) and 2 similar (((I)I)(()I)):
The center of ((()I)((I)I)) is, once again, empty. Its three axes are (((i)I)((I)I)), (((i)I)((I)I)), and ((()I)((Ii)I)). Once again, then, we have a cylindrical symmetry for the chart. And axial direction is once again cut A, and therefore completely empty. But this time, the equatorial direction isn't cut B but rather cut C: four displaced tigers.
Now, how will a cut of these 4 tigers appear? Well, since the lowercase i in this case is in a dimension they are displaced in, it means we'll encounter just two tigers when going in any direction (good thing too since they are hungry!).
So we have to examine the ((I)((I)I)) cut of ((II)((I)I): two displaced tigers. And we find that this cut is just four toruses in rectangle, two stacks of two toruses next to each other.
In equatorial direction of our cut, we'll therefore see a pair of coplanar circles transforming into -- by this point hopefully well-known -- pairs of parallel toruses typical for tiger sections.
Now, the cylindrical symmetry of the chart tells us that this is another rotation of cyltorintigroid. This time, we have to find a 4D cut corresponding to two displaced tigers, and this is (((I)I)(II)). The middle section we got in our cut is (((I)I)(I)), which means that once again, going "upwards" from that gives us the same sequence. So the overall shape of this 3D chart is...
A TORUS!
This time, the middle part of the torus is comprised of the maximally separated quartets of toruses and the wall is border case of two coplanar circles.

1 ((()I)((II))) and 1 similar (((II))(()I)):
And once again, the center of ((()I)((II))) is empty. This time, the three axes are (((i)I)((II))), (((i)I)((II))), and ((()I)((II)i)), so we once again have cylindrical symmetry of the chart and the axial direction is once again cut A and thus empty. The equatorial direction, though, is cut D, the weird assembly of two pairs of twin tigers.
Since the middle cut is in the middle between the pairs, we will once again encounter just two tigers when going in any direction (although even then, surrounded by tigers is not a nice way to go). This time we're going for ((I)((II))) cut of two tigers differing in one of their major diameters, ((II)((II))).
What is this cut? Why, it's four toruses once again -- but this time in different configurations: There are two concentric toruses differing in their major diameters, and then they are repeated in a parallel plane.
So when going from the center of the chart in equatorial direction, we'll encounter two circles, but this time concentric, and each of them will once again go through all the tiger sections.
Does cyltorintigroid have a 3D cut of this type? Why, it does -- (((II)I)(II)) -> (((II))(I)). This time, though, we'll encounter something different when we'll go "upwards" from its center: our equatorial direction is (((II))(Ii)), two concentric tigers, so the axial direction must be (((II)i)(I)), or two ditoruses displaced in their minor dimension. So going "upwards" will have each pair of concentric toruses go through ditorus sections.
The chart as a whole is still toroidal, though -- once again, it's 2D chart of cyltorintigroid rotated around a nonbisecting axis.

4 (((I))((I)I)) and 4 similar (((I)I)((I))):
Here, in (((I))((I)I)) we have -- finally! -- center that is NOT empty. The center is eight -- yes, eight -- toruses assembled in two vertical stacks of four. The axes are (((Ii))((I)I)), (((I)i)((I)I)), and (((I))((Ii)I)), this is cuts D, C and B.
The D-cut direction are four tigers from previous cut. This time, though, we see mid-cuts of all four at once, and so, when moving in this direction, they four pairs of toruses will perform the "tiger dance".
But which four pairs? Well, since in this direction the tigers form two twin pairs differing in a major diameter, we see that the pairs are two inner toruses and two outer toruses of each stack. So the inner toruses will merge and disappear before their outer brothers, but both stacks will behave in an identical way.
What about the C-cut direction? This cut, too, is four tigers, but displaced in a rectangle. So in this direction, it will be top two and bottom two toruses of each stack that perform the tiger dance.
And the B-cut direction? This one is four ditoruses. This time, each of the four HORIZONTAL pairs of toruses will go through a series of ditorus cuts.
Just as illustration, D and C cuts together form a 2D chart of (((Ii)i)((I)I)) two cyltorintigroids displaced in circle dimension (so anywhere in this plane the two vertical stacks will stay separated since each corresponds to one cyltorintigroid).
D and B cuts together form a 2D chart of (((Ii))((Ii)I), two concentric cyltorintigroids differing in circle radius, and C and B cuts together form a 2D chart of (((I)i)((Ii)I)) -- once again two cyltorintigroids separated in circle dimension, though this time it's the top and bottom quartet of toruses that will stay separate throughout the plane.

2 (((I))((II))) and 2 similar (((II))((I))):
Like in the previous case, the center of (((I))((II))) is eight toruses, though in a different configuration: it's a vertical stack of four pairs of concentric toruses differing in their major diameter. The three axes are (((Ii))((II))), (((I)i)((II))), and (((I))((II)i)), cuts E, D and B.
The E-cut are four tigers differing in their two major diameters. If we move in it, the outer and inner stack of toruses will stay separate, but the inner and outer pairs of each will merge in tiger dance.
The D-cut are two pairs of tigers differing in their major diameter. If we move in it, the outer and inner stack of toruses will once again stay separate, but this time the tiger dance will be performed by the top and bottom pairs of each.
The B-cut are four ditoruses, and in this direction, each of the four pairs of concentric toruses will merge according to ditorus cuts.
E and D cuts together form a 2D chart of (((Ii)i)((II)), two cylspherintigroids differing in their circle diameter. Outer and inner stack of toruses stays separate throughout that plane.
E and B cuts together form a 2D chart of (((Ii))((II)i)), once again two cylspherintigroids differing in their circle diameter.
D and B cuts together form a 2D chart of (((I)i)((II)i)), two cylspherintigroids displaced in their circle dimension. Top and bottom group of toruses stays separate throughout that plane.

So, that's duotorus tiger and now for the double tiger (((II)(II))(II)):

5D cuts:
2 (((I)(II))(II)) and 2 similar (((II)(I))(II)): two cylspherintigroids displaced in their minor torus dimension.
2 (((II)(II))(I)): two toraduocyldyinders displaced in their minor dimension.

4D cuts:
A: 1 ((()(II))(II)) and 1 similar (((II)())(II)): empty. Will figure in 2 3D cuts.
B: 4 (((I)(I))(II)): four tigers in vertices of square lying in plane of one of their circles. Will figure in 2 3D cuts.
C: 4 (((I)(II))(I)) and 4 similar (((II)(I))(I)): four ditoruses in vertices of rectangle lying in their medium and minor dimensions. Will figure in 3 3D cuts.
D: 1 (((II)(II))()): empty. Will figure in 1 3D cut.

3D cuts:

2 ((()(I))(II)) and 2 similar (((I)())(II)):
The center of ((()(I))(II)) is empty. The axes are (((i)(I))(II)), (((i)(I))(II)), and ((()(Ii))(II)). Chart has cylindrical symmetry with axial cut A and equatorial cuts B.
With cylindrical symmetry and empty axial cut, our first question will be: what's rotated here? Double tiger can be formed as a rotation in 2 different ways, both nonbisecting: either rotate a cyltorintigroid ((II)I)(II)) in minor torus dimension, or rotate toraduocyldyinder (((II)(II))I) in its minor dimension.
Well, the equatorial cuts are B, four tigers in vertices of square (if we have sufficiently symmetrical double tiger). Since the mid-cut is empty, we'll find two displaced tigers on each side (or maybe they'll find us!). Cyltorintigroid has a 4D cut of 2 displaced tigers (((I)I)(II)) and the equatorial direction corresponds to (((I)i)(II)), meaning that when we get to the center, it will look like (((I))(II)): a vertical stack of four toruses. When approaching from the center of chart, this will be a tiger dance starting with two parallel circles, each becoming its own pair of parallel toruses.
2D chart of cyltorintigroid with this mid-cut is (((Ii)i)(II)), so one direction (our equatorial) is two displaced tigers and the other direction ("upwards") is two tigers differing in one major diameter. So when going upwards from the cyltorintigroid mid-cut, we'll see tiger dance performed by two inner toruses, followed by the two outer toruses.
The chart as a whole is toroidal -- it's torus formed by a 2D chart of cyltorintigroid.

2 ((()(II))(I)) and 2 similar (((II)())(I)):
The center of ((()(II))(I)) is once again empty. The axes are (((i)(II))(I)), (((i)(II))(I)), and ((()(II))(Ii)), cylindrical symmetry with axial cut A and equatorial cuts C.
Once again, this looks like some sort of toroid. Equatorial cuts are C, four ditoruses, but we'll only encounter two in every direction. Lucky us, they are much less dangerous than tigers!
By eliminating the whole group (i), we can see how the middle of our encounter with two ditoruses will look: (((II))(I)), which is a vertical stack of two pairs of concentric toruses differing in their major diameter. This is a 3D cut of cyltorintigroid (((II)i)(Ii)). If we go from equatorial direction, we'll go through its (((II)i)(I)) cut, seeing two parallel circles, each splitting in two concentric toruses. If we go from the four toruses of this mid-cut "upwards", we'll mimic the (((II))(Ii)) cut, seeing tiger dance performed by outer and inner pair of toruses at the same time.
Toroidal chart once again.

8 (((I)(I))(I)):
The center of (((I)(I))(I)) is fortunately not empty -- but what is it? It turns out it's eight toruses in the vertices of a cube. The axes are (((Ii)(I))(I)), (((I)(Ii))(I)) and (((I)(I))(Ii)). First two are C-cuts (but in different orientation, so the chart doesn't have cylindrical symmetry) and the third one is a B-cut.
The C-cuts are four ditoruses and we see all of them as four pairs of coplanar toruses. But four toruses in vertices of a square can be paired in two different ways -- and that is why the two C-cuts are different, each of them pairs them differently. So in these directions, four pairs of coplanar toruses merge.
In the B-cut direction, it's different. B-cut is four tigers, and in this direction it will be the 4 vertical stacks that merge in tiger dance.
The plane formed by both C-cuts is a 2D chart of (((Ii)(Ii))(I)): two toraduocyldyinders displaced in their minor dimension. The top and bottom quartet will never merge here.
The plane formed by one C-cut and the B-cut is a 2D chart of (((Ii)(I))(Ii)): two cylspherintigroids displaced in their minor torus dimension. Here, toruses once again form two quartets that stay separate throughout the plane.

2 (((I)(II))()) and 2 similar (((II)(I))()):
The center of (((I)(II))()) is empty and its axes are (((Ii)(II))()), (((I)(II))(i)), and (((I)(II))(i)): we have a cylindrical symmetry with axial cut D and equatorial cuts C.
So the axial cut is once again empty, but the rotation is completely different: this time we'll be rotating a toraduocyldyinder, not a cylspherintigroid.
The equatorial cut is four ditoruses (we'll find two in every direction) and the middle of these two (obtained by eliminating (i)) is (((I)(II))). This is a vertical stack of two pairs of cocircular toruses (differing in their minor diameter).
Toraduocyldyinder (((II)(II))I) has a 3D cut of this type: (((II)(Ii))i). In equatorial direction, we have (((II)(I))i), so there we'll see (going from the center of the chart) vertical stack of two toruses which then both split into an outer and inner torus, then remerge. If we go from the central cut of this (stack of two pairs of cocircular toruses) upwards, we'll trace the other toraduocyldyinder cut (((II)(Ii))) which are two tigers differing in ther minor diameter. So there will be tiger dance performed by both the outer and the inner pair of toruses (outer merge, then inner merge, then inner disappear, then outer disappear). Chart as a whole is toroidal rotation of toraduocyldyinder 2D chart.

And that's all! Too me about 2 hours to write...

[Marek14]

Seeing all this, I believe I finally found an answer to an old poem:

Tyger, tyger, burning bright
In the forests of the night,
What immortal hand or eye
Could frame thy fearful symmetry?

After wrangling with the symmetry of tigers, I believe that the answer is "mine"

[ICN5D]

Oh my gosh. That's amazing. I believe I owe you something..... <bows on one knee>. No doubt you have tamed the tiger.

But, seriously, I picked those two because they are probably the hardest ones possible I really appreciate the fact that you spent 2 hours slicing and dicing them for me. I think you can tell I have a desire to learn these shapes. And, you have a desire to teach them to anyone who wants to learn. You're a fun person to interact with, Marek. That post is going to be a permanent reference for those shapes, too. It's not just to display the algorithm at work, but all of the information that can be derived out of it.

I think I see how these cartesian products work. If we have a basic form of ((N)(M)), the surface of shape (M) is the tiger symmetry and (N) is the (N)-torus that embeds in the tiger frame. So, ((II)(II)) has two circles (II), where the first one becomes the regular torus ((II)I), and the second one becomes the circular shaped tiger symmetry. This gives us two vertically stacked toruses ( tiger-symmetry) like the cut of a hollow circle. This is also reflected in ((III)(III)), where we have a torispherinder ((III)I) that obeys tiger symmetry around a hollow sphere. This makes for the cuts of a torispherinder along with cuts of a spherical tiger. There is a strange double manifold thing going on here, but not in a linear construction. It's more in a combined way, that truly reflects the cartesian product between the two manifolds. For the double tiger, I predicted the eight toruses in vertices of a cube! How awesome is that? I guess that's what I meant by tiger-cuts in vertices of a square.

1) So, this means that (((II)I)((II)I)) has the cuts of a ditorus along with the cuts of a torus-shaped tiger?

2) And, for the double tiger, we have the cuts of a tiger torus along with the cuts of a hollow circle?

3) Does ((III)((II)I)) make something different than (((II)I)(III)) ? A 3210-tiger vs a 2130-tiger? It seems like it would be possible to interchange the shape of the tiger-symmetry with that of the crosscut. Rather than a torispherinder + torus-shaped tiger, we could have a ditorus + spherical-tiger. Is that possible?

--Philip

[Marek14]

Oh my gosh. That's amazing. I believe I owe you something..... <bows on one knee>. No doubt you have tamed the tiger.

But, seriously, I picked those two because they are probably the hardest ones possible I really appreciate the fact that you spent 2 hours slicing and dicing them for me. I think you can tell I have a desire to learn these shapes. And, you have a desire to teach them to anyone who wants to learn. You're a fun person to interact with, Marek. That post is going to be a permanent reference for those shapes, too. It's not just to display the algorithm at work, but all of the information that can be derived out of it.

Yes, I am.

I think I see how these cartesian products work. If we have a basic form of ((N)(M)), the surface of shape (M) is the tiger symmetry and (N) is the (N)-torus that embeds in the tiger frame. So, ((II)(II)) has two circles (II), where the first one becomes the regular torus ((II)I), and the second one becomes the circular shaped tiger symmetry. This gives us two vertically stacked toruses ( tiger-symmetry) like the cut of a hollow circle. This is also reflected in ((III)(III)), where we have a torispherinder ((III)I) that obeys tiger symmetry around a hollow sphere. This makes for the cuts of a torispherinder along with cuts of a spherical tiger. There is a strange double manifold thing going on here, but not in a linear construction. It's more in a combined way, that truly reflects the cartesian product between the two manifolds. For the double tiger, I predicted the eight toruses in vertices of a cube! How awesome is that? I guess that's what I meant by tiger-cuts in vertices of a square.

Well, in ((N)(M)), the two are interchangeable. Better way to think of it is to imagine a cartesian product of surfaces of (N) and (M) and then taking a set of points that have set distance from this shape.

1) So, this means that (((II)I)((II)I)) has the cuts of a ditorus along with the cuts of a torus-shaped tiger?

Well, its 5D cuts are only torus tigers, but ditorus appears in 4D cuts.

2) And, for the double tiger, we have the cuts of a tiger torus along with the cuts of a hollow circle?

The base 5D cuts are tiger torus and toraduocyldyinder.

3) Does ((III)((II)I)) make something different than (((II)I)(III)) ? A 3210-tiger vs a 2130-tiger? It seems like it would be possible to interchange the shape of the tiger-symmetry with that of the crosscut. Rather than a torispherinder + torus-shaped tiger, we could have a ditorus + spherical-tiger. Is that possible?

--Philip

Yes, these are interchangeable. Basically, this shape is, in my notation, 2130-torus tiger. You start by taking a torus in hyperplane xyz and sphere in hyperplane wvu, make their cartesian product, then search for points on certain distance from this object. The object exists in 6D, but it only has 4 inner dimensions since it's just a product of surfaces.

5D cuts are:
3 ((II)((II)I)): a cyltorintigroid
2 ((III)((I)I)): two cylspherintigroids displaced in circular dimension
1 ((III)((II))): two cylspherintigroids differing in their circular diameter

4D cuts are:
A: 3 ((I)((II)I)): two ditoruses displaced in minor dimension, featuring in 3 3D cuts.
B: 6 ((II)((I)I)): two displaced tigers, featuring in 3 3D cuts.
C: 3 ((II)((II))): two tigers differing in a major diameter, featuring in 2 3D cuts.
D: 1 ((III)(()I)): empty cut, featuring in 2 3D cuts.
E: 2 ((III)((I))): four toraspherinders displaced in minor dimension, featuring in 2 3D cuts.

Now, I don't have time to parse 3D cuts today morning, so what I'm going to do is to just list them here and see if you can figure out the rest
1 (()((II)I)) (empty) with axes A, A, A
6 ((I)((I)I)) (four toruses in two vertical stacks of two) with axes B, B, A
3 ((I)((II))) (vertical stack of two concentric toruses) with axes C, C, A
3 ((II)(()I)) (empty) with axes D, B, B
6 ((II)((I))) (vertical stack of four toruses) with axes E, C, B
1 ((III)(())) (empty) with axes E, E, D

The first and the last are of particular importance as they completely separate the spherical and torus part.