ICN5D wrote: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 wrote: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
ICN5D wrote: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 ?
ICN5D wrote: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?
ICN5D wrote: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 wrote: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.
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 wrote:Marek14 wrote: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 wrote:Duocylinder: max(x^2+y^2,z^2+t^2) == 1. Cylinder crosscut, circle footprint. Free rolling in 1 dimension.
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.
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.
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.
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?).
ICN5D wrote:Marek14 wrote: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
ICN5D wrote: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?
((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 wrote: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.
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
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
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ICN5D wrote: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
ICN5D wrote: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
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