Another Kind of Toratope.

Discussion of shapes with curves and holes in various dimensions.

Another Kind of Toratope.

Many of the toratopes can be thought of as a hollow N-sphere (or a product of them) thickened by a solid n-sphere (or another toratope derived by this procedure) - we can write this as A@B (A thickened by B), where the torus is circle@circle, torisphere = sphere@circle, spheritorus = circle@sphere, tiger = duoring@circle, and ditorus = circle@(circle@circle). Now lets open the floodgates! Lets let A be any uniform continuum, which is a manifold where all of its points are congruent. This would include n-spheres, their products, coiling spirals that curve back on themselves, their products - get the picture . We can now have coilitoruses, trifoil knot toruses and so forth. Imagine taking a square and instead of curving both the x and y axis back on itself to get the duoring (circle x circle), we coil it five times and then curve it back on itself not once, but twice - to get a 5-2 coil on the x axis and then coil it 3 times on the y axis before curving it back on itself - call this A. 'A' would be a two dimensional manifold coiling around in 8 dimensions, so we need to thicken it with at least a 6-D shape - lets use the triger and call this B - but why stop there - lets take the triger out for a spin. As we traverse the locations of the square which got turned into the twisted continuum A - the triger (B) could undergo a triple swirl rotation. As we go across the x axis of the square, the triger ((II)(II)(II)) could rotate like this ((ab)(ac)(bc)) and the y axis could cause it to spin like ((cb)(ba)(ca)) where the a's, b's and c's shows how the dimensions pair up in the triple rotation, we can call the spin of the triger 'C'- imagine what A@B(with spin C) looks like. These could be quite scary.
Whale Kumtu Dedge Ungol.
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Re: Another Kind of Toratope.

You might notice that the tiger is the swirled torus. You can take a swirl dodecahedron, and run a torus-like hole up the middle of it. Whereas the tiger has two holes, the hollowed out swirl dodecahedron has twelve, and the swirl-icosahedron-torus has twenty. You can have hundreds of toric holes.

It gets worse. What do you do when you take something like the octagonny, and replace all the faces and the octahedrons, with empty space. Kind of like a spherated edge frame with the triangles filled in?

The 'pancake toruses' are frightening enough, and all of the torotopes are pancake toruses. These work like this. In 2d, there are no connected figures with holes connected to the outside.

If you now make a pancake, which is in effect, a thin prism of 2d, you can only make it circular, but you can poke circular holes in it. The number of circular holes is in 3D, the 'genus' or kind of thing. All solids with three holes are topologically equal to a pancake with three holes in it. You can make the pancake into a hollow sphere by shrinking the outline of the pancake into a circle. So a hollow sphere with N holes let into it, gives a pancake with N-1 holes.

In four dimensions, both the pancake shape and the holes let into it, can themselves have holes. So you are not restricted to a sphere-shaped pancake: it can be a torus-shape, or something with two holes let in. Inside the pancake, you can drop not just spherical holes, but even torus-shaped holes, and so forth. And these can also be knotted (like a trifoil knot), or even linked (like a chain of links). The tiger gives a torus-shaped pancake, with a single tube cut around inside the tyre. Philip would appreciate it if you said that the tiger-shape is what you get between the tire and the tube of a bicycle wheel.

You can suppose *any number* of tubes live in the torus. This is what i was talking about with the swirl-icosahedron-torus (19 tubes), or swirl-dodecahedron-torus (11 tubes). And it's topologically different if the 19 tubes of the torus are given a 360-degree twist as you go around the torus.

The edge-frame of eight cubes, in three dimensions (ie just the 27 vertices and the 54 edges), is not a pancake hole in three dimensions, but you can use it as a stamp to make holes in the 4d pancake. You can have two of these, forming a body-centred cubic.

And four dimensions, like every dimension that supports holes, have non-pancake holed things too.
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wendy
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Re: Another Kind of Toratope.

Hm, being a Polyhedron dude, perhaps you could tell me what are the simplest analogues of Stewart toroids for the 5 4D toratopes?

The thickening is a more general operation, yes. The reason basic toratopes use spheres is that this eliminates necessity to orient the "thickening" shape.

Here's another fun thing to do:

A ditorus and a tiger can be both constructed by taking a torus and rotating it around a non-intersecting plane (vertical for ditorus, horizontal for tiger). This means that a ditorus and tiger could be transformed into each other by taking a torus and continuously moving that rotation plane...
Marek14
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Re: Another Kind of Toratope.

It really does not follow that because two comb products have the same surface, they have the same interior.

The sphere-circle torus ((ii)ii) and the circle-sphere torus ((iii)i), both have the same surface (ie the round side of a spherinder), but they are hooked up in 'hose' and 'sock' forms respectively. The sphere-circle torus can hold inside a sphere-surface, which can not shrink to zero, but not a circle, while the circle-sphere comb can hold a circle, but not a sphere.

The tri-circular combs, such as the duotorus and the tiger, are different. One could render these in a covering of cubes, eight to a corner. For example, the tiger might yield 5*12*12 = 720 cubes.
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wendy
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Re: Another Kind of Toratope.

Marek14 wrote:Hm, being a Polyhedron dude, perhaps you could tell me what are the simplest analogues of Stewart toroids for the 5 4D toratopes?

I haven't explored 4-D Stewart toroids before, this could make another interesting group of polychora to investigate like the CRFs.

wendy wrote:You might notice that the tiger is the swirled torus. You can take a swirl dodecahedron, and run a torus-like hole up the middle of it. Whereas the tiger has two holes, the hollowed out swirl dodecahedron has twelve, and the swirl-icosahedron-torus has twenty. You can have hundreds of toric holes.

It gets worse. What do you do when you take something like the octagonny, and replace all the faces and the octahedrons, with empty space. Kind of like a spherated edge frame with the triangles filled in?

The 'pancake toruses' are frightening enough, and all of the torotopes are pancake toruses. These work like this. In 2d, there are no connected figures with holes connected to the outside.

If you now make a pancake, which is in effect, a thin prism of 2d, you can only make it circular, but you can poke circular holes in it. The number of circular holes is in 3D, the 'genus' or kind of thing. All solids with three holes are topologically equal to a pancake with three holes in it. You can make the pancake into a hollow sphere by shrinking the outline of the pancake into a circle. So a hollow sphere with N holes let into it, gives a pancake with N-1 holes.

In four dimensions, both the pancake shape and the holes let into it, can themselves have holes. So you are not restricted to a sphere-shaped pancake: it can be a torus-shape, or something with two holes let in. Inside the pancake, you can drop not just spherical holes, but even torus-shaped holes, and so forth. And these can also be knotted (like a trifoil knot), or even linked (like a chain of links). The tiger gives a torus-shaped pancake, with a single tube cut around inside the tyre. Philip would appreciate it if you said that the tiger-shape is what you get between the tire and the tube of a bicycle wheel.

I would love to explore a 4 dimensional cave system with all of these topologies hidden inside . Compared to 3-D, 4-D is like Pandora's box of bizarre shapes, symmetries, and topologies.
Whale Kumtu Dedge Ungol.
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Re: Another Kind of Toratope.

You really should try out John McIntosh's 4D Blocks game. The first part of it is a 4D maze where you'll get to experience firsthand, just how hard it is to navigate in 4D. Even when I maintain a fixed vertical orientation, I still find myself completely lost 85% of the time. I guess my brain just doesn't grok a floor that is 3D in extent! You can tweak the maze generation parameters to make loops and stuff, kinda like toratopes. Lemme just warn you that even a non-looping maze is hard enough to navigate; add loops, and you're in for a real mindbender.

And once you're ready to move on to the real stuff, you can look at some of the 4D sceneries that comes with the game, some of which include the cutest little 4D train that can move around in tracks. I tried following the train around once -- it's far, far harder than it looks!!! One time I followed the train into a tunnel and got lost inside (it's not even a branching tunnel!) for a good long while, 'cos I just couldn't wrap my brain around the 24 orientations I could be in. (That's while maintaining a fixed vertical orientation... if you don't do that, it becomes 192 orientations, and you'll be completely, totally lost like 120% of the time. )
quickfur
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Re: Another Kind of Toratope.

Polyhedron Dude wrote:
Marek14 wrote:Hm, being a Polyhedron dude, perhaps you could tell me what are the simplest analogues of Stewart toroids for the 5 4D toratopes?

I haven't explored 4-D Stewart toroids before, this could make another interesting group of polychora to investigate like the CRFs.

Just review one general family of the easier ones of the Steward toroids, as being described here. In fact they are all based on a single degenerate segmentochoron: xx3ox4xx&#x (= sirco || girco). That one in fact acts as a double cover of girco: one base is girco itself, and the "other side" is its euclidean decomposition into the "opposite" base (sirco) and all the lacing bits (tricues, squacues, and cubes). You obviously then can drill holes into girco by rejecting sirco and either all cubes or all tricues or all squacues.

The same techniques clearly apply in any other dimension as well. You just would have to start with any degenerate segmentoteron (as a 4D euclidean decomposition). This in fact is just what my recently added new page was intended for! Just look down for the flat segmentotera.

Sure, Steward was considering non-intersecting stuff only (kind of looking for "convex" degenerate segmentotopes, ). So obviously the there provided cases
would serve. Likewise all the ones mentioned within post p21653. But there are lots of other such degenerate segmentotera (as I'm just digging out currently).

--- rk
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