## Picture of Toratopes

Discussion of shapes with curves and holes in various dimensions.

### Picture of Toratopes

I was trying to figure out what exactly a Toratope is and I still don't know exactly what a Toratope is. I read the Wiki and saw that there are open and closed Toratopes with each open Toratope correlating with an open Toratope and vice versa. I read about the characteristics of open and closed Toratopes but I'm still confused about open what an open and closed Toratopes. I saw the number of Toratopes for different dimensions but I don't why there are that many Toratopes for each dimension as I am confused as to what a Toratope is.

Could I see some pictures of some pictures of 2d and 3d Toratopes both open and closed? If I saw some pictures of Toratopes I think that might help me understand them better.
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### Re: Picture of Toratopes

An open toratope is a shape with round and flat sides, like a coffee can cylinder shape. The round sides are the rolling sides like a wheel. A cone can also be considered an open toratope, but is more of a rotatope, which contain all but the closed toratopes. Rotatopes are shapes with any combination of round, flat, and pointy parts. Tapertopes are shapes with only flat and pointy parts, like triangles and pyramids.

A closed toratope is a donut or sphere. It is a shape with only one continuous surface like a ball or a ring-like structure with a hole(s). In 3D, there is only one kind of donut, the torus. It's basically an innertube, it has one circular hole, and a circular ring part going around. It is the closed version of a cylinder, which is the only true 3D open toratope. However, when going into 4D, there are more directions to build things with, so you get 4 unique donuts. Each one has a hole(s) and ring(s) part going around it. All four have different combinations of dimensions for their hole(s) or ring(s). And, each one came from a 4D cylinder or other donut type shape. Now, going even higher into 5D gives us another new direction to build with, plus the original 4. So, this allows 11 unique donut-type shapes to exist, with one continuous surface and one or more holes. And even higher into 6D, there are 33 unique donut shapes.

When you went to the wiki, you probably saw something like this:

(II)I - ((II)I)
(III)I - ((III)I)
(II)II - ((II)II)
((II)I)I - (((II)I)I)
(II)(II) - ((II)(II))

This is a notation system that stands for different shapes. The number of ' I ' you see is the number of dimensions it has.

3D
----------------
(II)I - cylinder
((II)I) - torus, has one circular hole, one circular ring

4D
-----------------------

(III)I - spherinder, sphere-prism
((III)I) - torisphere
Torisphere has a spherical hole with a circle wrapped around it. Like a hollow sphere with a thick skin. But, traveling across this skin makes you travel half way around the circle part, into 4D.

Cutting in half:

(II)II - cubinder, square-like cylinder
((II)II) - spheritorus
Spheritorus has a circular hole like 3D donut and a single ring. But the ring was made from a sphere-prism bending around into a circle.

Cutting in half:

((II)I)I - torinder, 3D donut prism, hollow tube-like cylinder
(((II)I)I) - ditorus
Ditorus is the innertube of an innertube. It has two holes, one like 3D donut, and one inside it's main innertube ring.

Cutting in half:

(II)(II) - duocylinder, double cylinder-type shape, has only two rolling sides combined into one from a coffee can cylinder, no flat sides at all like a sphere
((II)(II)) - tiger
Tiger is a very strange and amazing shape. It is like a 3D donut, but two superimposed into one in a very weird way. It has two individual pathways going through the same middle. If it were a wedding ring, it can have two fingers slid through, and they won't touch.

Cutting in half:

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### Re: Picture of Toratopes

I keep saying we should also have rotation pictures

By taking a mid-cut and then rotating the cut plane, one kind of mid-cut can transform into another.
For torisphere, the torus would grow upwards and merge in two concentric spheres, all the time keeping two circles in main plane of torus undisturbed.

For spheritorus, torus would break in two blobs and shrink in two spheres, all the time keeping two circles in a vertical plane undisturbed.

For tiger, the two toruses would somehow cotinuously turn in two toruses in perpendicular direction, keeping four circles in a vertical plane undisturbed.

For ditorus, there's a continuous path between any two of its three cuts.

EDIT: Also, INC5D, the number 33 in 6D includes the sphere, while numbers you give for other dimensions don't.
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### Re: Picture of Toratopes

I never thought about that, but you're right, it would be cool. I guess it would be a matter of switching the cut marker around, and figuring the transition out with the notation. That's a pretty neat way to trace out the shape, when you put it together in your head.

For tiger, the two toruses would somehow continuously turn in two toruses in perpendicular direction, keeping four circles in a vertical plane undisturbed.

Somehow is right! It could be based off the mid-cut of the oblique tiger cross sections. That 4-holed frame would be the mid-shape between transitions. The n-2 cuts seem to be some kind of anchor point for the morphing, in that they can't be moved, interesting.....

Ah, correction: 5 closed toratopes for 4D, 12 closed for 5D, 33 closed for 6D. Have you and wendy figured out the 7D total yet? Is it 90 or 93? There's only one way to truly prove it, and that's a complete enumeration. But, that condensed notation seems to do the trick. I need to learn it to see all of the shapes in them.
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### Re: Picture of Toratopes

ICN5D wrote:I never thought about that, but you're right, it would be cool. I guess it would be a matter of switching the cut marker around, and figuring the transition out with the notation. That's a pretty neat way to trace out the shape, when you put it together in your head.

For tiger, the two toruses would somehow continuously turn in two toruses in perpendicular direction, keeping four circles in a vertical plane undisturbed.

Somehow is right! It could be based off the mid-cut of the oblique tiger cross sections. That 4-holed frame would be the mid-shape between transitions. The n-2 cuts seem to be some kind of anchor point for the morphing, in that they can't be moved, interesting.....

Ah, correction: 5 closed toratopes for 4D, 12 closed for 5D, 33 closed for 6D. Have you and wendy figured out the 7D total yet? Is it 90 or 93? There's only one way to truly prove it, and that's a complete enumeration. But, that condensed notation seems to do the trick. I need to learn it to see all of the shapes in them.

Well, I believe I enumerated it -- also, as Keiji pointed out, it WAS done before and there's a sequence that describes their numbers.

The "anchor point" thing has a very simple explanation: Two hyperplane mid-cuts intersect in a plane, and that's why morphing from one hyperplane cut to another will leave one plane invariant.
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### Re: Picture of Toratopes

ICN5D wrote:Ah, correction: 5 closed toratopes for 4D, 12 closed for 5D, 33 closed for 6D. Have you and wendy figured out the 7D total yet? Is it 90 or 93? There's only one way to truly prove it, and that's a complete enumeration.

Actually, that's not true. You can now easily find the number of toratopes in n dimensions without enumeration. Quoting myself from the Tiger Explained thread:

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

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### Re: Picture of Toratopes

Wow, that sounds incredibly cool . I like the idea of using it with 5 and 6 D cuts, even if they're in higher than 3D. That could illuminate the shapes even better by providing another way to conceptually trace out its structure. I'm imagining a rendering program that does cuts and rotations using whatever familiar notation system. Maybe even render the full arrays of the tigroid cuts. I remember the 330-tiger cut array extremely well, it's the sharpest 6D tigroid in my vision. It would be awesome to see something like the duotorus tiger, or the double tiger array, complete with rotation renders of all axes from all cuts.

The "anchor point" thing has a very simple explanation: Two hyperplane mid-cuts intersect in a plane, and that's why morphing from one hyperplane cut to another will leave one plane invariant.

yep, that makes perfect sense. It will always be an N-1 plane to the cut, when rotating to another. I think I've pretty much nailed down the cut algorithm, safe to say those cut formulas are some trace of evidence So, relocating the removed marker will rotate one cut into another, where both cuts share a common anchor cut, in which they rotate around. Sounds fairly simple. It should also follow a logical path with the anchor, and how it relates in the notation. Now this is something I've been waiting for, something new, unexplored patterns to find with known techniques.
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### Re: Picture of Toratopes

Well, in 5D, you could already see it with the 2D array: horizontal line on the array is one 4D cut, vertical is another 4D cut, and the oblique lines would correspond to cuts gained through rotation. The central 3D cut would be invariant.

It's similar for 6D.
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### Re: Picture of Toratopes

Keiji wrote:
ICN5D wrote:Ah, correction: 5 closed toratopes for 4D, 12 closed for 5D, 33 closed for 6D. Have you and wendy figured out the 7D total yet? Is it 90 or 93? There's only one way to truly prove it, and that's a complete enumeration.

Actually, that's not true. You can now easily find the number of toratopes in n dimensions without enumeration. Quoting myself from the Tiger Explained thread:

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

I checked out that site, it was neat. So many number sequences! Interesting to see how the ' planted tree ' has a numerical pattern, it's very abstract. Also equally mind blowing are the plus one trillion hyperdonuts in 27 dimensions. Good god! But, that doesn't give me the satisfaction of seeing a huge list and picking out the bizarre ones, not the trillion, but a smaller list ( of course ). The freaks that stood out in the zoo. It's just visually appeasing in a very strange way, don't ask. I can't explain it.
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### Re: Picture of Toratopes

Wendy's calculations on the number of toratopes has errors in it. Last night, she dremt the right solution, and it was the same as Marek14's.
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### Re: Picture of Toratopes

Marek14 wrote:Well, in 5D, you could already see it with the 2D array: horizontal line on the array is one 4D cut, vertical is another 4D cut, and the oblique lines would correspond to cuts gained through rotation. The central 3D cut would be invariant.

It's similar for 6D.

Okay, I see what's going on here. The dimensions of the cut array corresponds to the number of reduced dimensions. More reduced dimensions makes the array more complex, one at a time. So, 5D reduced to 3D requires a 2D array to fully represent? I think this is the first time I understood that part about the arrays. I couldn't figure out how you were getting two and three D arrays with 5 and 6D tigroid cuts. Now I understand! That makes it even more attractive to explore 7D tigroids, it would be a 4D array! Represented in a 3D array, of course, it would have to be understood that even this array had an underlying pattern itself. Wow, there would be four mutual ways to move along an axis to morph the cut and do the tiger dance.
Last edited by ICN5D on Wed Mar 05, 2014 7:36 am, edited 1 time in total.
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### Re: Picture of Toratopes

wendy wrote:Wendy's calculations on the number of toratopes has errors in it. Last night, she dremt the right solution, and it was the same as Marek14's.

Wendy, you crack me up. Do you always talk about yourself in the third person? And use the thorn and the strange "twellfty" base number system? Seems to be your thing, along with strange pet names like lecky and magnety. I took me a few tries to decipher those two
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### Re: Picture of Toratopes

ICN5D wrote:
Marek14 wrote:Well, in 5D, you could already see it with the 2D array: horizontal line on the array is one 4D cut, vertical is another 4D cut, and the oblique lines would correspond to cuts gained through rotation. The central 3D cut would be invariant.

It's similar for 6D.

Okay, I see what's going on here. The dimensions of the cut array corresponds to the number of reduced dimensions. More reduced dimensions makes the array more complex, one at a time. So, 5D reduced to 3D requires a 2D array to fully represent? I think this is the first time I understood that part about the arrays. I couldn't figure out how you were getting two and three D arrays with 5 and 6D tigroid cuts. Now I understand! That makes it even more attractive to explore 7D tigroids, it would be a 4D array! Represented in a 3D array, of course, it would have to be understood that even this array had an underlying pattern itself. Wow, there would be four mutual ways to move along an axis to morph the cut and do the tiger dance.

Well, I guess you could represent 7D toratopes as a 3D array of animations
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### Re: Picture of Toratopes

Twelfty isn't that strange. It was used in england until the digital revolution. Thorn was used even more recently, even to this day.

Third person is a way of avoiding first person.

For Lecky and magnety, you should look up your English-American American-English dictionary.
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### Re: Picture of Toratopes

Thinking about the toratope cuts and arrays, it occurs to me that the most important toratopes to visualize are the "base members" of their species. Once you understand those, it's not hard to understand their higher-dimensional analogues.

How to recognize base members? Easily. Base members are exactly those toratopes whose mid-cuts are ALWAYS separated into multiple pieces. These mid-cuts are also usually made of other base toratopes, but not always (triger is a counterexample).

So in 2D, you should understand circle (II), whose mid-cut is two points.
In 3D, torus ((II)I), whose mid-cuts are two separate circles or two concentric circles.

In 4D:
Ditorus (((II)I)I), whose mid-cuts are two toruses lying next to each other or concentric and differing in one of their diameters.
Tiger ((II)(II)), whose mid-cuts are two vertically stacked toruses.

In 5D:
Tritorus ((((II)I)I)I), whose mid-cuts are two ditoruses lying next to each other or concentric and differing in one of their three diameters.
Tiger torus (((II)(II))I), whose mid-cuts are two ditoruses stacked in their medium dimenison or two concentric tigers differing in their minor diameters.
Torus tiger (((II)I)(II)), whose mid-cuts are two displaced tigers, two ditoruses stacked in their minor dimension or two concentric tigers differing in one of their major diameters.

In 6D:
Triger ((II)(II)(II)), whose mid-cuts are two 221-tigers stacked in their minor dimension.
Tetratorus (((((II)I)I)I)I), whose mid-cuts are two tritoruses lying next to each other or concentric and differing in one of their four diameters.
Tiger ditorus ((((II)(II))I)I), whose mid-cuts are two tritoruses stacked in their second-diameter dimension or two concentric tiger toruses differing in their medium or minor diameter.
Torus tiger torus ((((II)I)(II))I), whose mid-cuts are two tiger toruses lying next to each other, two tritoruses stacked in their third-diameter dimension, two concentric tiger toruses differing in one of their major diameters, or two concentric torus tigers differing in their minor diameter.
Ditorus tiger ((((II)I)I)(II)), whose mid-cuts are two torus tigers lying next to each other, two tritoruses stacked in their fourth-diameter dimension or two concentric torus tigers differing in torus major diameter or torus minor diameter.
Tiger tiger (Double tiger) (((II)(II))(II)), whose mid-cuts are two torus tigers stacked in their minor torus dimension or two tiger toruses stacked in their minor dimension.
Duotorus tiger (((II)I)((II)I)), whose mid-cuts are two torus tigers stacked in their circular major dimension or two concentric torus tigers differing in their circular diameter.

In 7D (just the list)
Triger torus (((II)(II)(II))I)
Torus triger (((II)I)(II)(II))
Pentatorus ((((((II)I)I)I)I)I)
Tiger tritorus (((((II)(II))I)I)I)
Torus tiger ditorus (((((II)I)(II))I)I)
Ditorus tiger torus (((((II)I)I)(II))I)
Tritorus tiger (((((II)I)I)I)(II))
Double tiger torus ((((II)(II))(II))I)
Tiger torus tiger ((((II)(II))I)(II))
Duotorus tiger torus ((((II)I)((II)I))I)
Torus double tiger ((((II)I)(II))(II))
Ditorus/torus tiger ((((II)I)I)((II)I))
Tiger/torus tiger (((II)(II))((II)I))
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### Re: Picture of Toratopes

General arrays

Now, let's consider general shape and properties of toratope arrays.

Spheres
Spheres are generally sets of points with given distance from a central point. There is 1 in each dimension from 2 onwards.

A 3D cut of any sphere is always a 3D sphere. And it has only one possible "evolution" when we go from the middle: it will shrink into a point and disappear, tracing a glome in 4D space.

So an array of pentasphere would have just these cuts in both cardinal directions. Array of hexasphere would have it in three. Array of heptasphere would have it in four.

Toruses
Toruses are sets of points with given distance from a sphere. Any dimension n >= 3 has (n-2) toruses.
Toruses have two diameters, major and minor. They can be described by two numbers: (a,b)-torus has a-dimensional basic sphere and additional b dimensions added. Torus is contiguous if a >= 2 and b >= 1.

In 3D, there are 4 possible cuts of toruses:

1: The (3,0) cut. This looks like two concentric spheres. It has two evolutions: (4,0) evolution treats each sphere as a glome, so they both shrink into point. (3,1) evolution treats the whole thing as a torisphere ((3,1)-torus), so they come closer and merge.
In 5D you can have 1 glome evolution and 1 torisphere evolution to create a (4,1)-torus, or two torisphere evolution to create a (3,2)-torus.
In 6D, you can have 2/1 evolutions for (5,1)-torus, 1/2 for (4,2)-torus and 0/3 for (3,3)-torus.
In 7D, you can have 3/1 for (6,1)-torus, 2/2 for (5,2)-torus, 1/3 for (4,3)-torus and 0/4 for (3,4)-torus.

2: The (2,1) cut. This looks like a regular torus. It has a (3,1) evolution that treats it as a torisphere (the central hole becomes filled and then the whole thing shrinks into a point) and a (2,2) evolution that treats it as a spheritorus (the torus becomes thinner until it shrinks into a circle and disappears).
In 5D, you can have 2/0 for (4,1)-torus, 1/1 for (3,2)-torus and 0/2 for (2,3)-torus.
In 6D, you can have 3/0 for (5,1)-torus, 2/1 for (4,2)-torus, 1/2 for (3,3)-torus and 0/3 for (2,4)-torus.
In 7D, you can have 4/0 for (6,1)-torus, 3/1 for (5,2)-torus, 2/2 for (4,3)-torus, 1/3 for (3,4)-torus and 0/4 for (2,5)-torus.

3: The (1,2) cut. This looks like two spheres. It has a (2,2) evolution where the spheres are treated as a spheritorus (they merge and eventually shrink into point) and a (1,3) evolution that treats each of them like a glome (so they shrink into points).
In 5D, you can have 2/0 for (3,2)-torus and 1/1 for (2,3)-torus.
In 6D, you can have 3/0 for (4,2)-torus, 2/1 for (3,3)-torus and 1/2 for (2,4)-torus.
In 7D, you can have 4/0 for (5,2)-torus, 3/1 for (4,3)-torus, 2/2 for (3,4)-torus and 1/3 for (2,5)-torus.

4: The (0,3) cut. This is empty. It has a (1,3) evolution, pair of glomes, where a point appears, grow into sphere and shrinks back, and a (0,4) evolution that is empty.
In 5D, you can have 2/0 for (2,3)-torus.
In 6D, you can have 3/0 for (3,3)-torus and 2/1 for (2,4)-torus.
In 7D, you can have 4/0 for (4,3)-torus, 3/1 for (3,4)-torus and 2/2 for (2,5)-torus.

Ditoruses
Ditoruses are sets of point with given distance from a torus. Any dimension n >= 4 has (n-3)(n-2)/2 ditoruses.
Ditoruses have three diameters, major, middle and minor. They can be described with three numbers: (a,b,c)-ditorus has (a,b)-torus as a base and c additional dimensions. Ditorus is contiguous if a >= 2, b >= 1 and c >= 1.

In 3D, there are 10 possible cuts of ditoruses:
(3,0,0): Four concentric spheres. Evolutions: (4,0,0) - each sphere is treated as a glome, (3,1,0) - the outermost and the innermost sphere is treated as one torisphere and the other two as another, (3,0,1) - two innermost and two outermost spheres are treated as torispheres.
(2,1,0): Two concentric toruses differing in their minor diameter. Evolutions: (3,1,0) - each torus is treated as a torisphere, (2,2,0) - each torus is treated as a spheritorus, (2,1,1) - both toruses are treated as a part of ditorus, they come closer and merge.
(2,0,1): Two concentric toruses differing in their major diameter. Evolutions: (3,0,1) - each torus is treated as a torisphere, (2,1,1) - both toruses are treated as a part of ditorus, they come closer, merge and eventually shrink into a circle, (2,0,2) - each torus is treated as a spheritorus.
(1,2,0): Two pairs of concentric spheres. Evolutions: (2,2,0) - the outer and inner pairs of spheres are treated as spheritoruses, (1,3,0) - each sphere is treated as a glome, (1,2,1) - each pair of concentric spheres is treated as torisphere.
(1,1,1): Two toruses lying in the same plane. Evolutions: (2,1,1) - both toruses are treated as a part of ditorus, they merge together, (1,2,1) - each torus is treated as a torisphere, (1,1,2) - each torus is treated as a spheritorus.
(1,0,2): Four spheres in a line. Evolutions: (2,0,2) - the outer and inner pairs of spheres are treated as spheritoruses, (1,1,2) - pairs of spheres on left and right side are treated as spheritoruses, (1,0,3) - each sphere is treated as a glome.
(0,3,0): Empty. Evolutions: (1,3,0) - a pair of concentric glomes on each side, (0,4,0) - empty, (0,3,1) - empty.
(0,2,1): Empty. Evolutions: (1,2,1) - a torisphere on each side, (0,3,1) - empty, (0,2,2) - empty.
(0,1,2): Empty. Evolutions: (1,1,2) - a spheritorus on each side, (0,2,2) - empty, (0,1,3) - empty.
(0,0,3): Empty. Evolutions: (1,0,3) - a pair of glomes on each side, (0,1,3) - empty, (0,0,4) - empty.

Tigers
Tigers are sets of point with given distance from a cartesian product of two spheres. There's 1 tiger in 4D, each dimension adds (n-2)/2 more, rounded down.
Tigers have three diameters, two major and one minor. They can be described with three numbers: (a,b,c)-tiger has a-dimensional and b-dimensional spheres as a base and c additional dimensions. Tiger is contiguous if a >= 2 and b >= 2. (a,b,c)-tiger and (b,a,c)-tiger are the same figure.

In 3D, there are 6 possible cuts of tigers:

(3,0,0): Empty. Evolutions: (4,0,0) - empty, (3,1,0) - a torisphere on each side, sliced minor dimension first, (3,0,1) - empty.
(2,1,0): Two toruses, verticaly stacked. Evolutions: (3,1,0) - each torus is treated as a torisphere, (2,2,0) - both toruses are treated as a part of a tiger, (2,1,1): each torus is treated as a spheritorus.
(2,0,1): Empty. Evolutions: (3,0,1) - empty, (2,1,1): a spheritorus on each side, sliced minor dimension first, (2,0,2) - empty.
(1,1,1): Four spheres in the vertices of rectangle. Evolutions: (2,1,1) - the top and bottom pairs of spheres are treated as spheritoruses, (1,2,1) - the left and right pairs of spheres are treated as spheritoruses, (1,1,2) - each sphere is treated as a glome.
(1,0,2): Empty. Evolutions: (2,0,2) - empty, (1,1,2): a transversal pair of glomes on each side, (1,0,3) - empty.
(0,0,3): Empty. Evolutions: (1,0,3) - empty, (0,1,3) - empty, (0,0,4) - empty.

This should allow for greater understanding of these categories in any dimension
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### Re: Picture of Toratopes

Tiger/torus tiger (((II)(II))((II)I))

Oh my gosh! That was an unexpected beauty! That's what I mean about seeing the list, the ones that stand out. I guess those types are my favorite, a pure tigroid having two complex tiger symmetry frames. I guess it can be called the 220210-tiger. Don't worry, I won't ask you to spend 4 hours to cut this one down I'm more interested in discovering some sort of immediately identifiable feature about these types, a quick go to for visualizing. Like the case with the 330-tiger: it has two hollow sphere frames, in which the cuts show how both can be interchanged. One hollow sphere becomes the toratope frame, which follows along the other hollow sphere as the tigroid frame. The toratope frame is inflated with a circle, the tigroid frame is inflated with the full toratope.

So, by using this little short cut generalization, a quick map can be made of the toratope along tigoird structures, while maintaining that both can be interchanged with any one of their reduced cuts. The tigroid frame cuts follows the same principle as a circle-inflated toratope cut, but has a whole inflated toratope along its manifold. Cutting far enough will completely separate both hollow frames, in addition to cross-breed hybrid cuts of both superimposed frames. I'm going to meditate on that during my bike ride today. Yep, that's right, I dodge rollerbladers and joggers while doing 6D geometry in my head. Life is great
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### Re: Picture of Toratopes

ICN5D wrote:
Tiger/torus tiger (((II)(II))((II)I))

Oh my gosh! That was an unexpected beauty! That's what I mean about seeing the list, the ones that stand out. I guess those types are my favorite, a pure tigroid having two complex tiger symmetry frames. I guess it can be called the 220210-tiger. Don't worry, I won't ask you to spend 4 hours to cut this one down I'm more interested in discovering some sort of immediately identifiable feature about these types, a quick go to for visualizing. Like the case with the 330-tiger: it has two hollow sphere frames, in which the cuts show how both can be interchanged. One hollow sphere becomes the toratope frame, which follows along the other hollow sphere as the tigroid frame. The toratope frame is inflated with a circle, the tigroid frame is inflated with the full toratope.

So, by using this little short cut generalization, a quick map can be made of the toratope along tigoird structures, while maintaining that both can be interchanged with any one of their reduced cuts. The tigroid frame cuts follows the same principle as a circle-inflated toratope cut, but has a whole inflated toratope along its manifold. Cutting far enough will completely separate both hollow frames, in addition to cross-breed hybrid cuts of both superimposed frames. I'm going to meditate on that during my bike ride today. Yep, that's right, I dodge rollerbladers and joggers while doing 6D geometry in my head. Life is great

Sounds like fun I myself have a "life interest"TV crew following me tomorrow, but sadly, my toratopic achievements would probably be over the head of the viewers
Marek14
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### Re: Picture of Toratopes

Really? That's pretty cool! When can I watch the show? Sadly, the high D stuff is beyond most. But, I believe those cut arrays are something special. They bring those shapes back down to comprehensible terms. These things should be taught in elementary school, when young kids are super absorbent to new concepts. It would raise a whole generation capable of complex reasoning and logical deduction. The new neuron pathways that are made during comprehension have many more uses than high D geometry. I've noticed this lately, when I learn new things.
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ICN5D
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### Re: Picture of Toratopes

Not sure if you'd be able to watch it -- you'd definitely didn't understand the language, though I live in Central Europe.
Marek14
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### Re: Picture of Toratopes

So in 2D, you should understand circle (II), whose mid-cut is two points.
In 3D, torus ((II)I), whose mid-cuts are two separate circles or two concentric circles.

I think I understand what a Toratope is now. So basically a Toratope is a circular shape cut in half so that if you cut a donut, circle, or sphere in half you get a Toratope? Is that the correct definition of a Toratope Marek14?
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anderscolingustafson
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### Re: Picture of Toratopes

anderscolingustafson wrote:
So in 2D, you should understand circle (II), whose mid-cut is two points.
In 3D, torus ((II)I), whose mid-cuts are two separate circles or two concentric circles.

I think I understand what a Toratope is now. So basically a Toratope is a circular shape cut in half so that if you cut a donut, circle, or sphere in half you get a Toratope? Is that the correct definition of a Toratope Marek14?

No, a toratope is just a torus-like shape (torus = doughnut shape). The cutting in half part is only for the purposes of visualization, it has nothing to do with the shape itself. In 3D, the main torus-like shape is the torus itself (i.e., the doughnut), but 4D and above, it is possible to have many different kinds of torus-like shapes (hence the name toratope), including shapes with different kinds of "doughnut holes".
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### Re: Picture of Toratopes

quickfur wrote:
anderscolingustafson wrote:
So in 2D, you should understand circle (II), whose mid-cut is two points.<br abp="656">In 3D, torus ((II)I), whose mid-cuts are two separate circles or two concentric circles.
<br abp="657"><br abp="658">I think I understand what a Toratope is now. So basically a Toratope is a circular shape cut in half so that if you cut a donut, circle, or sphere in half you get a Toratope? Is that the correct definition of a Toratope Marek14?
<br abp="659">No, a toratope is just a torus-like shape (torus = doughnut shape). The cutting in half part is only for the purposes of visualization, it has nothing to do with the shape itself. In 3D, the main torus-like shape is the torus itself (i.e., the doughnut), but 4D and above, it is possible to have many different kinds of torus-like shapes (hence the name toratope), including shapes with different kinds of "doughnut holes".

So do spheres count as Toratopes or does a shape have to have holes in the middle to be a Toratope? Are there Toratopes in 2d and if so what would be some 2d Toratopes?
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anderscolingustafson
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### Re: Picture of Toratopes

I'm sorry, I can't help it. The (((II)(II))((II)I)) is just too darn interesting. I played with some 3D cuts of the 220210:

(((I)(I))((I))) - 110100 - tigric-torus tiger

Now, this one is interesting. I've seen this kind of arrangement before, with () over every ' I " in a torus layout. It's related to a (((I)(I))(I)) 11010-double tiger, 8 torii at vert of cube. By removing the pairs of brackets that make it an octet, a little method I just found, we can derive the base cut as being ((II)(I)) 210-tiger, 2 vert stacked torii. So, these modifying brackets that make it an octet act on this initial vertical column arrangement.

Making (((I)(I))((I))) an octet of 2 vertical stacked torii in vertices of a cube.

(((II)())((I))) - 200100

Okay, this one was tough. There's a tricky sequence of () that can be removed to reveal the initial state, and I eventually found ((II)(I)) to be it, again.

The reduction series went like this:

(((II)())((I))) - 200100
((II)()((I))) - 20100 : removed brackets that make a concentric arrangement of 2 torii
((II)()(I)) - 2010 : removed brackets that make vertical column of 4 torii
((II)(I)) - 210 : vertical column of 2 torii, since the () is empty

That makes the 200100 a vertical column of four groups of 2 concentric torii each.

((()(I))((I)I)) - 010110

And once again, we end up with a ((II)(I)) . But, this one is modified differently:

Reduction tree:
(((I))((I)I)) - 10110 : removed empty ()
((I)((I)I)) - 1110 : removed () that make 4 vert stack torii
((I)(II)) - 120 : removed () that make 2 displaced torii

Which makes 010110 two displaced columns of four torii each.

Now, I'm not sure how to apply the arrays yet. There's more to learn about it, but I'll get it soon. I know it's related to 'what part of what was cut', and the array traces out the missing structure. I'll be back soon with the arrays of these cuts, it's becoming very clear to me how it's made.
Last edited by ICN5D on Thu Mar 06, 2014 6:45 am, edited 1 time in total.
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ICN5D
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### Re: Picture of Toratopes

anderscolingustafson wrote:So do spheres count as Toratopes or does a shape have to have holes in the middle to be a Toratope? Are there Toratopes in 2d and if so what would be some 2d Toratopes?

Yes, spheres are closed toratopes. They have a single smooth surface. Shapes with holes in the middle also are closed toratopes, and also have single smooth surface. A 2D closed toratope is the 2-sphere, a circle. I guess a square is kind of like a 2D open toratope.
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ICN5D
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### Re: Picture of Toratopes

anderscolingustafson wrote:
quickfur wrote:
anderscolingustafson wrote:
So in 2D, you should understand circle (II), whose mid-cut is two points.<br abp="656">In 3D, torus ((II)I), whose mid-cuts are two separate circles or two concentric circles.
<br abp="657"><br abp="658">I think I understand what a Toratope is now. So basically a Toratope is a circular shape cut in half so that if you cut a donut, circle, or sphere in half you get a Toratope? Is that the correct definition of a Toratope Marek14?
<br abp="659">No, a toratope is just a torus-like shape (torus = doughnut shape). The cutting in half part is only for the purposes of visualization, it has nothing to do with the shape itself. In 3D, the main torus-like shape is the torus itself (i.e., the doughnut), but 4D and above, it is possible to have many different kinds of torus-like shapes (hence the name toratope), including shapes with different kinds of "doughnut holes".

So do spheres count as Toratopes or does a shape have to have holes in the middle to be a Toratope? Are there Toratopes in 2d and if so what would be some 2d Toratopes?

I suppose spheres would be trivial toratopes, so in 2D you at least have the circle as a toratope. Most of the interesting things happen in higher dimensions, though, especially 4D and beyond.
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### Re: Picture of Toratopes

ICN5D wrote:I'm sorry, I can't help it. The (((II)(II))((II)I)) is just too darn interesting. I played with some 3D cuts of the 220210:

(((I)(I))((I))) - 110100 - tigric-torus tiger

Now, this one is interesting. I've seen this kind of arrangement before, with () over every ' I " in a torus layout. It's related to a (((I)(I))(I)) 11010-double tiger, 8 torii at vert of cube. By removing the pairs of brackets that make it an octet, a little method I just found, we can derive the base cut as being ((II)(I)) 210-tiger, 2 vert stacked torii. So, these modifying brackets that make it an octet act on this initial vertical column arrangement.

Making (((I)(I))((I))) an octet of 2 vertical stacked torii in vertices of a cube.

Yes, or, in other words, 16 toruses in a 2x2x4 tower.

The possible evolutions in four remaining directions are:

(((Ii)(I))((I))) - This is eight ditoruses arranged in stack 2 in medium dimension x 4 in minor dimension. Evolution looks like the two pairs of toruses in every horizontal layer of the tower go through ditorus cuts.
(((I)(Ii))((I)) - This is eight ditoruses once again, except that it's different pairs of toruses in horizontal layers that merge.
(((I)(I)))((Ii)) - This is eight tigers. One of their circles is one of four circles in xy plane that lie in the vertices of a rectangle, while the second is one of two concentric circles in the zw plane. Evolution is tiger dance for two inner and two outer toruses in each vertical stack.
(((I)(I)))((I)i) - This is also eight tigers, except that the second circle is one of two displaced circles in zw plane, not concentric. Evolution is tiger dance for two top and two bottom toruses in each vertical stack.

(((II)())((I))) - 200100

Okay, this one was tough. There's a tricky sequence of () that can be removed to reveal the initial state, and I eventually found ((II)(I)) to be it, again.

The reduction series went like this:

(((II)())((I))) - 200100
((II)()((I))) - 20100 : removed brackets that make a concentric arrangement of 2 torii
((II)()(I)) - 2010 : removed brackets that make vertical column of 4 torii
((II)(I)) - 210 : vertical column of 2 torii, since the () is empty

That makes the 200100 a vertical column of four groups of 2 concentric torii each.

I'm afraid this is not correct. ANY empty pair of parentheses means an empty cut.

What are this cut's evolutions?

(((II)())((Ii))) and (((II)())((I)i)) are still empty.

(((II)(i))((I)) (2 dimensions make this cut) is different: it's eight ditoruses arranged in a stack 2 in medium dimension x 4 in minor dimension, like one of the evolutions of previous 3D cut. The difference is that this time we're cutting them in the medium direction. This means that we'll encounter one stack of 4 ditoruses from the direction of medium dimension. THIS is the stack of 4 pairs of concentric toruses. Your mistake is that this is not the cut in the middle, but something that gets revelaled a bit further.

((()(I))((I)I)) - 010110

And once again, we end up with a ((II)(I)) . But, this one is modified differently:

Reduction tree:
(((I))((I)I)) - 10110 : removed empty ()
((I)((I)I)) - 1110 : removed () that make 4 vert stack torii
((I)(II)) - 120 : removed () that make 2 displaced torii

Which makes 010110 two displaced columns of four torii each.

Now, I'm not sure how to apply the arrays yet. There's more to learn about it, but I'll get it soon. I know it's related to 'what part of what was cut', and the array traces out the missing structure. I'll be back soon with the arrays of these cuts, it's becoming very clear to me how it's made.

Well, this is also an empty cut. Two of its evolutions, ((()(Ii))((I)I)) and ((()(I))((Ii)I)) are empty once again. The third one, in two dimension, is (((i)(I))((I)I)). From the first cut we know that this is eight tigers with four rectangularly-spaced circles in xy plane and two displaced circles in zw plane. We're cutting through them through a dimension of one of the xy circles. On each side, we'll encounter four tigers and mid-cut of each of them will look as you say - two displaced columns of four torii each. But once again, this will be encountered on each side of the mid-cut, not exactly there.
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### Re: Picture of Toratopes

quickfur wrote:
anderscolingustafson wrote:So do spheres count as Toratopes or does a shape have to have holes in the middle to be a Toratope? Are there Toratopes in 2d and if so what would be some 2d Toratopes?

I suppose spheres would be trivial toratopes, so in 2D you at least have the circle as a toratope. Most of the interesting things happen in higher dimensions, though, especially 4D and beyond.

There's a list of toratopes on the wiki.

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### Re: Picture of Toratopes

Marek14 wrote:This means that we'll encounter one stack of 4 ditoruses from the direction of medium dimension. THIS is the stack of 4 pairs of concentric toruses. Your mistake is that this is not the cut in the middle, but something that gets revelaled a bit further.

On each side, we'll encounter four tigers and mid-cut of each of them will look as you say - two displaced columns of four torii each. But once again, this will be encountered on each side of the mid-cut, not exactly there.

Oh, that's right, they're empty. I haven't done the cuts for some time, warming up to it now. But, there is an improvement in my determining of the emergent cuts outside the empty zone. The method of reducing brackets and recording their effect is a good one. Cool to see I got those right to some degree . I need to practice deriving the arrays from the multiple versions of the same cut. You use the " i " in place of the last removed marker to find the previous shape, then compile them together to form the array. That's a good method. It's a privilege to learn from you, Marek. You certainly are the toratope master.
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ICN5D
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### Re: Picture of Toratopes

Thank you
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