## Publishing an article on rotatopes and toratopes

Discussion of shapes with curves and holes in various dimensions.

### Publishing an article on rotatopes and toratopes

So as I said in my comeback thread, I'd love it if we could all work on publishing some of the material here. Publishing in a journal is obviously a very different prospect than a forum post or a wiki article. The biggest difference is the size constraints and the audience. The shorter the article is, the more likely we are to get it published, and we can publish in a higher quality journal. The articles I've published or submitted have all been about 20 pages or less. So we wouldn't include huge tables up to 10 dimensions, although we will include a link to the forum or the wiki. On the other hand we want it to be extremely clear, and written mostly in conventional notation, except where we invent our own notation. We also need to check that nothing we've invented hasn't already been published with different names.

So what do we want to publish first? Even if we ignore toratopes and just look at rotatopes, there might actually be enough material there for one article. Then toratopes could be in a later article. Alternatively we could do rotatopes and toratopes together and have a bit less information about each.

Here's a list of topics we could write about. Probably not all of these will be included. For some there might be enough material for a full article.
Definition of rotatopes
Naming system and notation, including the implicit equations
Parametric equations
Counting rotatopes (it's just the partition function)
Frames (e.g. the 1-frame, 2-frame and 3-frame of a cylinder are "two circles", "cylinder surface" and "solid cylinder" respectively)
Cuts
Visualisations

Definition of toratopes, including the operation that I think was called "spheration" but maybe there's another name for it now.
Naming system and notation, including the implicit equations
Parametric equations
Counting toratopes (this has a more complicated formula that I came up with and Keiji rediscovered more recently).
Homology groups (the algebraic topology way of dealing with "holes". I did a lot of this back in the day)
Cuts
Visualisations PWrong
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### Re: Publishing an article on rotatopes and toratopes

On the other hand we want it to be extremely clear, and written mostly in conventional notation, except where we invent our own notation. We also need to check that nothing we've invented hasn't already been published with different names.

I've done a decent amount of searching around for similar stuff, but I couldn't find any. A good sign. Plus, the toratopic notation is still unknown, another good sign. But, I've been spreading the word on Reddit this week all about it. There's a ton of non-believers, while some are truly impressed with it.

So what do we want to publish first? Even if we ignore toratopes and just look at rotatopes, there might actually be enough material there for one article. Then toratopes could be in a later article. Alternatively we could do rotatopes and toratopes together and have a bit less information about each.

I'm inclined to go with both rotatopes and toratopes, since they're so closely related. Ultimately, toratopes are constructed from the edges of rotatopes. What do you think of the bundling terminology thread? I came across similar definitions on Wikipedia, and I took to the translation of toratopes. It'll work for rotatopes, too. I also developed a surtope algorithm to derive the surface elements of them, as well. My knowledge is highly focused and narrow to specific things, where you've been through a LOT more schooling.

I think referencing A000669 for the combinatorics of both rotatopes and toratopes is good. There's a decent formula for it up on the OEIS. Maybe list them up to 6D would be a good stopping point in the article, since in 7D it gets to 90 of them.

All of these seem great to have in the article, I'm afraid! I'm not sure how to pick out the most general introduction for both.

Definition of both
Naming system and notation, including the implicit equations
Parametric equations
Counting rotatopes (it's just the partition function)
Frames (e.g. the 1-frame, 2-frame and 3-frame of a cylinder are "two circles", "cylinder surface" and "solid cylinder" respectively)
Cuts
Visualisations

Though, maybe a first article should go through the :

Definition of rotatopes
Naming system and notation, including the implicit equations
Counting rotatopes (it's just the partition function)

at first. This would detail the theory and number sequence behind construction. Then a later one to elaborate on what was listed in the first, would go into more visual detail with cross sections and visualizing into 3D, along with the equations:

Frames (e.g. the 1-frame, 2-frame and 3-frame of a cylinder are "two circles", "cylinder surface" and "solid cylinder" respectively)
Cuts
Visualisations

Or.... maybe elaborate on just one of them, rotatope then toratope, with more detail. I'm undecided. A first one on open toratopes ( rotatopes) , then a follow-up on toratopes, which elaborates on the inflated margins of previoulsy mentioned opens. A000669 will equally detail both, so it seems like one then the other would work. I have much more experience with illustrating toratopes, not so much rotatopes.

I think "spheration" has turned into "inflation", defined by a circle bundle over some Clifford manifold or other n-sphere. Some high-D toratopes are made by inflating with more than just a sphere, like a whole 3-torus or tiger, we've discovered. So, I feel the 'bundle over' term is good, if not 'inflation'.

So, there's some ideas. It's 2am, I need to go to sleep now!
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### Re: Publishing an article on rotatopes and toratopes

Well, I would include implicit equations as well as parametric -- they are probably easier for making cuts.

And we could start with the ancient attempt to find 4D toratopes -- we show how the 3D cuts evolve into torisphere, spheritorus and ditorus, show that one simple 3D cut is missing from these, and introduce the tiger. Really, the true exploration of toratopes started with the discovery of tiger (if I remember it correctly, PWrong was the first one to stumble upon it, but I was the one who gave it the name and realized that it's actually crucial element of the whole system).

I think we'll also need a link to simple arrays of the 12 5D toratopes.

Advanced topics would then be the focus on "skinny" toratopes (those with no "extra" dimensions so any coordinate cut splits them in two), and their classification via trace into species, genera and orders.

BTW, ICN5D, I'm looking at reddit now and you say you're in contact with the 3DCalc's author? How does it go?

EDIT: Took more gander at the comments. Seems that some users are not taking this forum seriously because of some doubtful science/philosophy in other parts of the forum/site. That's why I try to keep myself in the Geometry subforum... Marek14
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### Re: Publishing an article on rotatopes and toratopes

Marek wrote:And we could start with the ancient attempt to find 4D toratopes -- we show how the 3D cuts evolve into torisphere, spheritorus and ditorus, show that one simple 3D cut is missing from these, and introduce the tiger. Really, the true exploration of toratopes started with the discovery of tiger (if I remember it correctly, PWrong was the first one to stumble upon it, but I was the one who gave it the name and realized that it's actually crucial element of the whole system).

That's a cool history lesson! I like it. It will detail what little was known, then as you said, introduce the missing ingredient : the tiger. This is a great way to start it.

I think we'll also need a link to simple arrays of the 12 5D toratopes.

I can do the visuals if needed/wanted. Along with representing the notation and cut algorithm. Which is something I've been meaning to get around to, anyways....

BTW, ICN5D, I'm looking at reddit now and you say you're in contact with the 3DCalc's author? How does it go?

It goes well! He ( Paul Seeburger ) is just as blown away with my application of his program. I posted many of the exploration functions, upon request. He's about to add more adjustable parameters within the year. That'll make +6D rotate+translate functions wayyyy easier to handle, and require fewer unique functions for each toratope. I can't wait for that! I'm asking him for 14 parameters, for use with 10D functions. His program is crucial for me, since it requires no script commands. I just simply derive the equation, make it 3D, input explore functions, then graph. It's that simple.

EDIT: Took more gander at the comments. Seems that some users are not taking this forum seriously because of some doubtful science/philosophy in other parts of the forum/site. That's why I try to keep myself in the Geometry subforum... Ridiculous, isn't it? I completely understand if you don't want to get involved. That's okay. It's just like what Gallileo experienced, when he said the earth went around the sun. Anytime a fundamentally new concept is introduced into the academic mainstream, there will always be skeptics. Those who will not believe. But, the vast majority consensus really likes it, and wants more. The math-inclined on /r/mathpics are amazed at the simplicity and power of the notation. It's elegant, to be truthful. But, still soooo new. No matter, though. I've spent the last 14 years overcoming objection, by working in a retail store and selling bicycles! When I'm done posting here, I'm going on there to defend this awesome forum. You all are my friends, and I know many of you have extremely high intellect. It's a bit of a melting pot of great minds, this place. I don't care what they think, because I know better.

Can't win them all, though. Not at first. When this paper hits the mainstream, they will go from " Why are you doing this? " to " How did you do that? " . Just watch. I'm excited for the future, and hopefully have provided a worthy contribution.
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### Re: Publishing an article on rotatopes and toratopes

ICN5D wrote:
EDIT: Took more gander at the comments. Seems that some users are not taking this forum seriously because of some doubtful science/philosophy in other parts of the forum/site. That's why I try to keep myself in the Geometry subforum... Ridiculous, isn't it? I completely understand if you don't want to get involved. That's okay. It's just like what Gallileo experienced, when he said the earth went around the sun. Anytime a fundamentally new concept is introduced into the academic mainstream, there will always be skeptics. Those who will not believe. But, the vast majority consensus really likes it, and wants more. The math-inclined on /r/mathpics are amazed at the simplicity and power of the notation. It's elegant, to be truthful. But, still soooo new. No matter, though. I've spent the last 14 years overcoming objection, by working in a retail store and selling bicycles! When I'm done posting here, I'm going on there to defend this awesome forum. You all are my friends, and I know many of you have extremely high intellect. It's a bit of a melting pot of great minds, this place. I don't care what they think, because I know better.

Can't win them all, though. Not at first. When this paper hits the mainstream, they will go from " Why are you doing this? " to " How did you do that? " . Just watch. I'm excited for the future, and hopefully have provided a worthy contribution.

Well, what I meant was that the posters in question might be correct that other parts of the site might not really be "up to scrutiny". The introduction and such are all from 2003 -- looks like the forum is the only thing that advanced since then. Basically, as for legitimate math advances, we might only have toratopes and CRF project, but on the other hands, at least these are no quack math: we have models and equations and they work; and we're not really trying to give them some extra-mathematical meaning. It's certainly not fair to condemn us as "massive cranks" based on mistakes or misunderstandings in a 10-year old document that predecesses all of us While there might be legitimate doubts about meaning or importance of things we discovered here, there can be no such doubt about their existence. For example, the CRF project discovered a hitherto unknown class of shapes using J91's and J92's and that is a fact.

I'd say that what we're doing is that we're doing discrete mathematics in higher-D spaces, i.e. counting distinct varieties as opposed to studying continuous variables. I would guess that we tend to be people who find discrete thinking easier.
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### Re: Publishing an article on rotatopes and toratopes

Now, for the "sanitized" history of toratopes (it should be documented in the ancient threads of the toratope forum). PWrong, feel free to suggest additions/alterations, I probably don't remember everything.

1. Originally, we knew of three 4D toratopes: Torisphere, spheritorus and ditorus. We started from parametric equations, but for theoretical purposes, we can define these three as sets of points:

Torisphere is a set of points in 4D with specific distance from the surface of a sphere.
Spheritorus is a set of points in 4D with specific distance from a circle.
Ditorus is a set of points in 4D with specific distance from the surface of a torus.

Now, if we add the glome (4-sphere) to the mix and define it as a set of points in 4D with specific distance from a given point, we have basis for the first iteration of the theory.

In this, each toratope is defined as set of points in particular dimension with specific distance from the surface of a lower-dimensional toratope.

Each toratope will now lead to exactly two toratopes in next higher dimension: one where you re-use its generating toratope and just increase the dimension, and one where you use the toratope for generating the new one. So a sphere leads into glome and torisphere and torus leads into spheritorus and ditorus.

The resulting theory is simple, the number of toratopes simply doubles every dimension. Note that even in 6D, the 16 simple toratopes form a slight majority of all toratopes, only in 7D the others start to prevail.

2. Tiger and extension of theory

Now we start cutting the 4D toratopes in half by coordinate hyperplanes. We focus on the cuts where the toratope is split in two, and then we reverse-engineer those.

When the cut results in two toratopes, they are "almost identical", they differ either in a single coordinate of their center (so we have two translated copies) or in a value of single diameter.
So two spheres can appear displaced (a cut of spheritorus) or as a concentric pair (a cut of torisphere), and two toruses can appear displaced horizontally, displaced vertically, as a major pair and as a minor pair.
Here, it would be noted that all of these, EXCEPT the vertical displacement, appear as cuts of ditorus. The three rotations of ditorus morphing between these can be shown.

Now a tiger is introduced as a "missing" parametric configuration and it's shown that its cuts are all of this missing type. The result is that our definition of toratopes has to be amended to also admit sets of points with a given distance from a CARTESIAN PRODUCT of surfaces of two or more lower-dimensional toratopes.

This is the major result which is the basis for all the rest of toratopic theory.

Then we can show 4D toratopes with various combinations of sizes (in particular, that two major diameters of tiger are completely independent and can be equal or different).

Only after this, we can start talking about 5D toratopes which are still sparse enough to allow for individual discussion and which can be shown through the use of arrays. And here is a good place to bring in the toratopic notation and the relation between toratopes and partitions including the A000669 sequence, which will serve as a proof that the list of 12 5D toratopes is exhaustive. Then the 7 "basic" 6D cases can be mentioned in passing (tetratorus, tiger ditorus, torus tiger torus, ditorus tiger, double tiger, duotorus tiger and triger). For higher dimensions, we'll just mention the total numbers and give some links.
(Note that they might exist additional shapes we don't count among the basic group, like the "mantis" and "spider" which are obtained by different generalization of tiger, but whose exact equations I've been unable to determine so far.)

This all concerns just the closed toratopes as the open ones are simply various types of prisms. The problem here is that introducing both types at once leads to confusion (as you can see in "The tiger doesn't exist" thread): since there is 1-1 correspondence between both types in the notation, it naturally invites ideas of some operation that transforms an open toratope into corresponding closed one, but that is a red herring which can lead the reader down the wrong path.

Closed toratopes are also "simpler" in the way that it usually doesn't really matter if you're talking about their surface or bulk, as the rendered pictures will be the same in both cases; this, however, is not true for open toratopes. I'd mention those only as an aside.
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### Re: Publishing an article on rotatopes and toratopes

I'm still worried that torotopes are the thin edge of the wedge, and that ye can have tigers with six or twelve holes in them. Bower's 'swirl-prisms' produce ample exposure that a swirl-prism can be made to have a very large number of toric holes in them.
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### Re: Publishing an article on rotatopes and toratopes

wendy wrote:I'm still worried that torotopes are the thin edge of the wedge, and that ye can have tigers with six or twelve holes in them. Bower's 'swirl-prisms' produce ample exposure that a swirl-prism can be made to have a very large number of toric holes in them.

I encountered this problem when thinking about the mantis and spider, but I don't think it's that troublesome. Toratopes are recursively defined. The simple toratopes (toruses, ditoruses etc.) in the original system might be called "Rank 1 toratopes", the current system might be called "Rank 2 toratopes". If it can be extended into "Rank 3 toratopes" (and I think the possibility exists), it looks likely that those will have much more complicated equations and behaviour, so Rank 2 will still be useful as a step to understand them.
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### Re: Publishing an article on rotatopes and toratopes

Very concise, I like the history lesson. Ultimately, it's the evolution of the theories behind it, leading to the rest of them. Another idea I had, was maybe going into detail regarding the Villarceau sections of the toratopes, and how they relate to the V-section of a torus. Deriving the plane equations for them is above my academic ability, but I have and can discover them in the plotter. Those also seem to be worth mentioning, if not at first, then maybe later.

I really like the idea of relating how those tangent locations can inscribe a polytope, in place of their vertices. That is fundamentally cool, whatever it may mean someday. Some toratopes have similar tangent locations, like the 12-tangent slice of the double tiger and triger, both of which inscribe an icosahedron. If the V-section of a torus corresponds to Hopf circles, maybe V-sections for higher toratopes are related to higher Hopf fibrations? Something to think about.
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### Re: Publishing an article on rotatopes and toratopes

Here's a thought: could we use toratopic notation to derive surface/volume formulas for toratopes?

My guess would be that the surface could be derived from rotations (as each toratope in 3D or higher is easily expressible as rotation). If we do enough, we might find some relationship.

Volume should be easy -- just integration of surface formula over the minor diameter of the figure (since the volume is basically infinite stacking of surfaces of all minor diameters up until certain value). Surface, I think, is the more important value here.

If we can achieve this, the article should get much bigger potential.

First question: Does this:
http://en.wikipedia.org/wiki/Pappus%27s ... id_theorem
generalize easily into higher dimensions?
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### Re: Publishing an article on rotatopes and toratopes

That's a great idea. I've been wanting to explore that theorem with the toratopes. It seems that to generalize the surface that gets expanded by a circle, the "set of points" , it would be the edge of the open toratope that the closed one came from came from. This edge would be the complex way the circle took, during its revolution sequence. Or, more specifically, the unique clifford torus shape. I like it!
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### Re: Publishing an article on rotatopes and toratopes

Let's have a look at tiger, considered as revolution of torus into vertical dimension.

The torus (sqrt(x^2 + y^2) - R)^2 + z^2 = r^2 will be cut into z-level stripes which look like concentric circles. For given z, it will be circles

x^2 + y^2 = (R + sqrt(r^2 - z^2))^2
and
x^2 + y^2 = (R - sqrt(r^2 - z^2))^2

The circumferences (if we ignore the singular case z^2 = r^2) are
2 pi (R + sqrt(r^2 - z^2))
and
2 pi (R - sqrt(r^2 - z^2)),
which can be summed to:
4 pi R

But if we sum this for a normal torus, and we just have multiply this by the z-range (which is 2r), we get 8 pi R r, which doesn't correspond to torus surface formula 4 pi^2 R r, so I probably made a mistake somewhere. Where does the second pi come from in this construction?
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### Re: Publishing an article on rotatopes and toratopes

I wouldn't spend too much time on the history. The best way to learn about toratopes won't be the same as the way we discovered them. The important things to note is who discovered what concepts, and that they were all discovered here. With the toratope notation defined the way it is now, the tiger isn't really that special. It's probably an accident of history that we discovered it last and had to change our notation to include it. At least that's how I remember it. The historical details can be found here in the forum, or someone could write a page on the website or the wiki about it.

One thing we need is a lot of citations of other journal articles. Some of our shapes are well known in mathematics (for example the duocylinder is well known as the Clifford torus, and our "ditorus" is actually called a 3-torus), and others may have been discovered but not widely used. I suspect that someone somewhere has at least come up with the concept of a spherinder before, probably calling it I x S^2. I'm not so sure about the cubinder.

This all concerns just the closed toratopes as the open ones are simply various types of prisms. The problem here is that introducing both types at once leads to confusion (as you can see in "The tiger doesn't exist" thread): since there is 1-1 correspondence between both types in the notation, it naturally invites ideas of some operation that transforms an open toratope into corresponding closed one, but that is a red herring which can lead the reader down the wrong path.

This is definitely worth mentioning. In fact we should probably check carefully that such an operation does not exist. Or if it does exist we should investigate it. Do we have a name for these pairs?
As well as this (I'm not sure if this has been mentioned already), we should mention how some closed toratopes are homotopic (the can be deformed into) to some open toratopes, for example the torus (21) is homotopic to the duocylinder 22, and the three shapes 222, (21)2 and ((21)1) are all homotopic. We should figure out a scheme for which toratopes are homotopic and prove that it works. This will tie in to the homology groups thing.

I'll explain the basic idea to my supervisor in the next few days and see what he thinks. Then I'll let you all know when I can get started.
Last edited by PWrong on Tue Jul 29, 2014 8:24 am, edited 1 time in total. PWrong
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### Re: Publishing an article on rotatopes and toratopes

For volumes and surface areas, the new Mathematica 10 could be extremely useful. It treats regions as first class citizens, so we might be able to define toratopes recursively and calculate their volumes.
http://www.wolfram.com/mathematica/new-in-10/geometric-computation/ PWrong
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### Re: Publishing an article on rotatopes and toratopes

Here's another important question:
For which values of the radii do these shapes intersect themselves? This is probably fairly easy to answer but it's worth knowing. PWrong
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### Re: Publishing an article on rotatopes and toratopes

You can regard these figures as a kind of prism product of simple torii.

For example, take a circle, and 'spherate' its surface. You get a torus of great radius R_0 and small radius R_1. The torus ((ii)i)i) us a spheration of a torus, so you have R_0, R_1, R_2. Evidently there is going to be kissing if R_0 = R_1+R_2+... or R_1 = R_2+...

In the case of a tiger ((ii)(ii)), there are two inner radii (ii) and (II), the overlap by the outer (....) will occur if that radius is as large as the smaller of (ii) and (II). That is, one treats a tiger as being the smallest of the several radii, and apply the formula above.
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### Re: Publishing an article on rotatopes and toratopes

PWrong wrote:Here's another important question:
For which values of the radii do these shapes intersect themselves? This is probably fairly easy to answer but it's worth knowing.

Well, this can be answered through rotations. Every toratope is a rotation of lower-dimensional toratope, bisecting or non-bisecting.

With bisecting rotations, the new shape is always the same "species" as the old, so the same rules for radii are retained. For nonbisecting rotations, the plane/hyperplane to rotate around doesn't intersect the toratope and the new radius is distance between the hyperplane and the center.

Now, we can define a "maximum outreach" for each dimension (most distant point on the axis that belongs to the toratope) as a sum of radius belonging to that dimension and all radii belonging to superior parenthesis pairs.

For a torus ((II)I), with dimensions ((xy)z) and radii ((R)r), maximum outreach for x and y is R+r and maximum outreach for z is just r. If we rotate the torus through major dimension into ditorus (((II)I)I), the new radius, corresponding to (II), must be greater than the major-dimension outreach of the original torus, otherwise the plane you rotate around would intersect the torus. This means that if we mark the new radii as (((S)R)r), then S > R+r.

If we rotate torus through minor dimension into a tiger, the maximum outreach of the minor dimension is only r, and so the condition for new diameter is only that it has to be greater than r.
((R)(S)r) -> S > r

This leads to a general rule for toratopes: a radius associated with any pair of parenthesis must be greater than sum of all radiuses in higher-level parentheses. Let's illustrate it on torus tiger (((II)I)(II)):

r4 is minor diameter, and so it's not constrained.
r3 is on lower level than r4, and so it must hold r3 > r4.
r2 is also on lower level than r4, so it must hold r2 > r4.
r1 is on lower level than r2 and r4, so it must hold r1 > r2 + r4.
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### Re: Publishing an article on rotatopes and toratopes

How much detail should be gone into with toratopic notation? Should there be cross section tables with each shape? Or like a term glossary with a specific shape and all of its different arrangements? How about the theory?

There's the various clifford tori as the common surface between open and closed. The edge of one is the wireframe of the other, before spheration. It would seem that the sequence of Cliffords and n-spheres is the actual toratope combinatorics. And, what differentiates open from closed is the set of points from the edge. With using and describing the Cliffords, we can easily describe very high-D toratopes with a simple sequence of them. As in, the generic structure of toratopes comes from the generic structure of their cliffords.

Take the tiger as the most basic example: we have the first carteisan product of surfaces, the Clifford 2-torus, related to (II)(II) and ((II)(II)). We can call the 2-frame of the duocylinder [S^1]^2, if no such notation exists (or, does it?). The implicit equation for the cliffords are easy, as a toratope with its minor diameter removed. By using these basic Cliffords as building blocks, the arrays and structure of base-species toratopes can be easily defined.

So, as a general rule,

(II)(II) - duocylinder , 2-frame = [S^1]^2
(II)(II)(II) - (circle^3) prism , 3-frame = [S^1]^3
(II)(II)(II)(II) - (circle^4) prism , 4-frame = [S^1]^4
(II)(II)(II)(II)(II) - (circle^5) prism , 5-frame = [S^1]^5
(II)(II)(II)(II)(II)(II) - (circle^6) prism , 6-frame = [S^1]^6

(circle^n) has n-frame of [S^1]^n

((II)(II)) - tiger , 1-sphere over the Clifford 2-torus = S^1 x [S^1]^2
((II)(II)(II)) - triger , 2-sphere over the Clifford 3-torus = S^2 x [S^1]^3
((II)(II)(II)(II)) - tetriger , 3-sphere over the Clifford 4-torus = S^3 x [S^1]^4
((II)(II)(II)(II)(II)) - pentatiger , 4-sphere over the Clifford 5-torus = S^4 x [S^1]^5
((II)(II)(II)(II)(II)(II)) - hexatiger , 5-sphere over the Clifford 6-torus = S^5 x [S^1]^6

n-tiger is (n-1)-sphere over clifford n-torus = S^(n-1) x [S^1]^n

All Clifford tori have the lowest cut of 0D points in an n-cube array. This makes them all equal sized diameters:

[S^1]^2 = intercepts 2-plane as 2x2 square array of 4 points
[S^1]^3 = intercepts 3-plane as 2x2x2 cube array of 8 points
[S^1]^4 = intercepts 4-plane as 2x2x2x2 tesseract array of 16 points
[S^1]^5 = intercepts 5-plane as 2x2x2x2x2 5-cube array of 32 points
[S^1]^6 = intercepts 6-plane as 2x2x2x2x2x2 6-cube array of 64 points
[S^1]^7 = intercepts 7-plane as 2x2x2x2x2x2x2 7-cube array of 128 points
[S^1]^8 = intercepts 8-plane as 2x2x2x2x2x2x2x2 8-cube array of 256 points
[S^1]^9 = intercepts 9-plane as 2x2x2x2x2x2x2x2x2 9-cube array of 512 points

[S^1]^n = cuts to n-plane as n-cube array of 2^n points

When you get to something like a (((II)I)(II)), it ends up being a torus with a tiger cross-cut, which I think is called tiger bundle over a circle. This makes the cyltorinder ((II)I)(II) also a torus with a duocylinder cross-cut. The 3-frame is a clifford torus bundle over a circle. And for a (((II)I)I)(II) , we get a duocylindric ditorus , or duocyliner over a torus. The way this nested circle works in higher-D toratopes, are that you end up with nested clifford tori combined with T^n 's.

So, take a duotorus tiger, for instance: (((II)I)((II)I)) , what we get is a circle bundle over medium clifford torus over major clifford torus. There are two differently sized clifford tori, playing the role of equal sized medium and major diameters. If we use the inner-most circle parameters, we can establish the major diameters. I belive Marek helped with that one, by establishing the rule about freely adjustable diameters. These would be the major most, and could be any size without self-intersecting. I feel that the number of these inner-most major diameters would be the major clifford n-torus. Then, the next level down would be the medium/secondary clifford n-torus.

(((II)I)(II)) - tiger over circle = S^1 x [S^1]^2 x S^1
(((II)I)((II)I)) - tiger over clifford torus = S^1 x [S^1]^2 x [S^1]^2
((((II)I)I)((II)I)) - tiger over clifford torus over circle = S^1 x [S^1]^2 x [S^1]^2 x S^1
((((II)I)I)(((II)I)I)) - tiger over clifford torus over clifford torus = S^1 x [S^1]^2 x [S^1]^2 x [S^1]^2

(((II)I)(II)(II)) - 2-sphere over clifford 3-torus over circle = S^1 x [S^1]^3 x S^1
(((II)I)((II)I)(II)) - 2-sphere over clifford 3-torus over clifford torus = S^1 x [S^1]^3 x [S^1]^2
(((II)I)((II)I)((II)I)) - 2-sphere over clifford 3-torus over clifford 3-torus = S^1 x [S^1]^3 x [S^1]^3

((((II)I)I)((II)I)((II)I)) - 2-sphere over clifford 3-torus over clifford 3-torus over circle = S^1 x [S^1]^3 x [S^1]^3 x S^1
((((II)I)I)(((II)I)I)((II)I)) - 2-sphere over clifford 3-torus over clifford 3-torus over clifford torus = S^1 x [S^1]^3 x [S^1]^3 x [S^1]^2
((((II)I)I)(((II)I)I)(((II)I)I)) - 2-sphere over clifford 3-torus over clifford 3-torus over clifford 3-torus = S^1 x [S^1]^3 x [S^1]^3 x [S^1]^3

((II)I) - 2-torus
(((II)(II))I) - 2-torus over clifford 2-torus = T^2 x [S^1]^2
(((II)(II))(II)) - 2-torus over clifford 3-torus = T^2 x [S^1]^3

(((II)I)I) - 3-torus
((((II)(II))I)I) - 3-torus over clifford 2-torus = T^3 x [S^1]^2
((((II)(II))(II))I) - 3-torus over clifford 3-torus = T^3 x [S^1]^3
((((II)(II))(II))(II)) - 3-torus over clifford 4-torus = T^3 x [S^1]^4

((((II)I)I)I) - 4-torus
(((((II)(II))I)I)I) - 4-torus over clifford 2-torus = T^3 x [S^1]^2
(((((II)(II))(II))I)I) - 4-torus over clifford 3-torus = T^3 x [S^1]^3
(((((II)(II))(II))(II))I) - 4-torus over clifford 4-torus = T^3 x [S^1]^4
(((((II)(II))(II))(II))(II)) - 4-torus over clifford 5-torus = T^3 x [S^1]^5

All the large complex arrays come from the clifford combinations, before the small-shape toratope inflates it:

[S^1]^2 x S^1 = intercepts 2-plane as 4x2 square array of 8 points
[S^1]^2 x [S^1]^2 = intercepts 2-plane as 4x4 square array of 16 points
[S^1]^2 x [S^1]^2 x S^1 = intercepts 2-plane as 8x4 square array of 32 points
[S^1]^2 x [S^1]^2 x [S^1]^2 = intercepts 2-plane as 8x8 square array of 64 points

[S^1]^3 x S^1 = intercepts 3-plane as 4x2x2 square array of 16 points
[S^1]^3 x [S^1]^2 = intercepts 3-plane as 4x4x2 square array of 32 points
[S^1]^3 x [S^1]^3 = intercepts 3-plane as 4x4x4 square array of 64 points

[S^1]^3 x [S^1]^3 x S^1 = intercepts 3-plane as 8x4x4 square array of 64 points
[S^1]^3 x [S^1]^3 x [S^1]^2 = intercepts 3-plane as 8x8x4 square array of 128 points
[S^1]^3 x [S^1]^3 x [S^1]^3 = intercepts 3-plane as 8x8x8 square array of 256 points

I also came up with a surtope algorithm here and here for the opens, a little while ago. It might be useful in some way for double-checking or deriving surface hypervolumes.

The list so far, using A || B for prisms , and [A + B] for orthogonally bound curved cells

Code: Select all
`Surtopes of the Open Toratopes--------------------------------2D:II - [line || line]+[line || line]3D:III - [sqr || sqr] + [sqr || sqr] + [sqr || sqr] = [sqr || sqr]^3(II)I - [circle || circle] + [ line-->circle ]4D:IIII - [cube || cube]^4(II)II - [(II)I || (II)I]^2 + [square-->circle](II)(II) - ((II)I)+((II)I)(III)I - [(III) || (III)] + [line-->sphere]((II)I)I - cylinder-->circle , [((II)I) || ((II)I)] + [line-->torus] 5D:IIIII - [tess || tess]^5(II)III - [[(II)II || (II)II]^3 + [cube-->circle](II)(II)I - [(II)(II) || (II)(II)] + [((II)I)I+((II)I)I]  (III)II - [(III)I || (III)I]^2 + [square-->sphere]((II)I)II - cubinder-->circle , [((II)I)I || ((II)I)I]^2 + [square-->torus] (III)(II) - ((III)I)+((II)II)((II)I)(II) - duocylinder-->circle , (((II)I)I)+(((II)I)I) (IIII)I - [(IIII) || (IIII)] + [line-->glome]((II)II)I - spherinder-->circle , [((II)II) || ((II)II)] + [line-->spheritorus] ((II)(II))I - cylinder-->duoring , [((II)(II)) || ((II)(II))] + [line-->tiger]((III)I)I - cylinder-->sphere , [((III)I) || ((III)I)] + [line-->torisphere](((II)I)I)I - cylinder-->torus , [(((II)I)I) || (((II)I)I)] + [line-->ditorus]6D:IIIIII - [penteract || penteract]^6(II)IIII - [(II)III || (II)III]^4 + [tesseract-->circle] (II)(II)II - [(II)(II)I || (II)(II)I]^2 + [((II)I)II+((II)I)II] (II)(II)(II) - ((II)I)(II)+((II)I)(II)+((II)I)(II)(III)III - [[(III)II || (III)II]^3 + [cube-->sphere]((II)I)III - tesserinder-->circle , [((II)I)II || ((II)I)II]^3 + [cube-->torus] (III)(II)I - [(III)(II) || (III)(II)] + [((III)I)I+((II)II)I]  ((II)I)(II)I - [((II)I)(II) || ((II)I)(II)] + [(((II)I)I)I+(((II)I)I)I]  (III)(III) - ((III)II)+((III)II)((II)I)(III) - (((III)I)I)+(((II)I)II) Type-1 , (((II)I)II)+(((II)II)I) Type-2((II)I)((II)I) - ((((II)I)I)I)+((((II)I)I)I)(IIII)II - [(IIII)I || (IIII)I]^2 + [square-->glome]((II)II)II - cubspherinder-->circle , [((II)II)I || ((II)II)I]^2 + [square-->sphere-->circle]((II)(II))II - cubinder-->duoring , [((II)(II))I || ((II)(II))I]^2 + [square-->tiger]((III)I)II - cubinder-->sphere , [((III)I)I || ((III)I)I]^2 + [square-->circle-->sphere](((II)I)I)II - cubinder-->torus , [(((II)I)I)I || (((II)I)I)I]^2 + [square-->torus-->circle](IIII)(II) - ((II)III)+((IIII)I)((II)II)(II) - (((II)I)II)+(((II)II)I)((II)(II))(II) - (((II)I)(II))+(((II)(II))I) ((III)I)(II) - (((II)II)I)+(((III)I)I)(((II)I)I)(II) - ((((II)I)I)I)+((((II)I)I)I)(IIIII)I - [(IIIII) || (IIIII)] + [line-->pentasphere]((II)III)I - glominder-->circle , [((II)III) || ((II)III)] + [line-->glome-->circle] ((II)(II)I)I - spherinder-->duoring , [((II)(II)I) || ((II)(II)I)] + [line-->sphere-->duoring]((III)II)I - spherinder-->sphere , [((III)II) || ((III)II)] + [line-->sphere-->sphere] (((II)I)II)I - spherinder-->torus , [(((II)I)II) || (((II)I)II)] + [line-->sphere-->torus] ((III)(II))I - cylinder-->(sphere x circle) , [((III)(II)) || ((III)(II))] + [line-->cylspherintigroid](((II)I)(II))I - cylinder-->duoring-->circle , [(((II)I)(II)) || (((II)I)(II))] + [line-->tiger torus]((IIII)I)I - cylinder-->glome , [((IIII)I) || ((IIII)I)] + [line-->circle-->glome] (((II)II)I)I - cylinder-->spheritorus , [(((II)II)I) || (((II)II)I)] + [line-->torus-->sphere-->circle] (((II)(II))I)I - cylinder-->tiger , [(((II)(II))I) || (((II)(II))I)] + [line-->torus-->duoring](((III)I)I)I - cylinder-->torisphere , [(((III)I)I) || (((III)I)I)] + [line-->ditorus-->sphere]((((II)I)I)I)I - cylinder-->ditorus , [((((II)I)I)I) || ((((II)I)I)I)] + [line-->torus-->torus]`
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ICN5D
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### Re: Publishing an article on rotatopes and toratopes

Of the notation, because ye are doing a lot of reduction into sections, it would probably be useful to explain it in stages. In the early part, it would serve to use letters here, and different curlies, so that the reader can follow what is going on.

1. Pre-tigers. These are where all the open brackets are before any close brackets. What you might do here is show how eg

(ii) is a circle, (iii) is a sphere, etc. Then the idea that ( (..)i) turns the surface of () into a circle (eg ((ii)i) turns the circle into a torus)
likewise ((iii)i) does the same thing for a sphere.

2. Tigers. The kernel here is "sibling brackets", or two sets of () in the same set, eg [(ii)(ii)].

3. Sections. One drops letters to view the section in a particular subspace, so eg [(wx){yz}] gives [(x){y}]. This gives four circles in a rectangle of edges() and {}, the diameter is []. The sides of the rectangles are cross-sections of a circle of diameter () and a circle of diameter {}.
The dream you dream alone is only a dream
the dream we dream together is reality.

\(\LaTeX\ \) at https://greasyfork.org/en/users/188714-wendy-krieger wendy
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### Re: Publishing an article on rotatopes and toratopes

I suspect some people might be annoyed or disheartened by some of the things I'm about to say, but I should say them anyway.

How much detail should be gone into with toratopic notation? Should there be cross section tables with each shape? Or like a term glossary with a specific shape and all of its different arrangements? How about the theory?

Personally I think it should be almost all theory. Details should be fleshed out for all the 4D shapes, and we could perhaps have a table for the 5D toratopes and maybe a few higher than that, without too many details. Then we should have general formulas with proofs for facts about general toratopes in any dimension.

As for the names, we should think carefully about them, because I'm not sure how mathematicians will react to them. Personally I think we should definitely keep the name "tiger" even though it's a bit out there, because it's special to us as a community. But we have to be aware that if other mathematicians start using these shapes they will likely call them something else. The most common way for a new idea to get named is this: one mathematician comes up with an idea in a paper, then another mathematician cites the paper and gives the idea a name (usually naming it after the author of the first paper). Some of the names we might want to give up entirely before we publish.

For example, "ditorus", to be perfectly honest, is a terrible name for several reasons:
1. There's not much about it that's "2-like", except that you're applying the "spherate with a circle" operation twice to a base circle. But then there's three circles.
2. Many other toratopes could equally be called a ditorus for the same reason.
3. The standard name for it in mathematics is the "3-torus", except that is also used for a (which is also a bad name given that any toratope has an equal claim to that name).
4. It could be confused with the double torus.

Of the notation, because ye are doing a lot of reduction into sections, it would probably be useful to explain it in stages.

On the subject of how best to explain it, we have to keep in mind that we're explaining this to experts. When you explain this stuff on the forum or a website, the risk is that people will think "I don't understand this , it's too hard". With mathematicians, the risk is that they'll think one or more of the following:

1. "This notation is unfamiliar where it could have been familiar"
2. "This is just showing off"
3. "These people don't know what they're doing so I can't trust their proofs"
4. "This information is unnecessary or uninteresting"
5. "This fact or shape is already well-known but they weren't aware of that"
6. "This is just a trivial calculation"
7. "This name is too fanciful or doesn't match the description well enough"

I had a LOT of stuff in my thesis that I cut out because my supervisor warned me the examiners wouldn't appreciate it. I did include an appendix of Mathematica notebooks with some examples that I thought were really cool, and I suspect most of you would think so too. But my supervisor didn't read it, saying "some people might find this interesting. I don't, and the examiners probably won't read it either". So we have to be aware of that attitude, and make sure our paper is as interesting and concise as possible. Most mathematicians will be interested in new ideas, theorems and proofs. Names and tables of numbers might sadly be totally ignored. But we can and will link to the forum and the wiki in the bibliography for readers who are interested in that side of things. PWrong
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### Re: Publishing an article on rotatopes and toratopes

The conversation ISN5D had on Reddit recently supports your claims -- if we're going to present this to professionals, we have to understand that they will hold us to much higher standard than people on the forum. As for names, as long as we keep tiger, I'm cool with it The lists of names I've been posting were really more for my benefit, as they allowed me to look at the structure from another vantage point.
Marek14
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### Re: Publishing an article on rotatopes and toratopes

Yeah, that's what I've been wondering, is exactly how trivial and overlapping the notation is. Posting this on reddit was probably the largest exposure it had, and its application. One would have to do a lot of searching through an obscure forum to find toratopic notation. Try google searching some notation sequence, and you'll see what I mean.

But, I don't know, it still seems that the notation has value. After all, it does represent real mathematics, in a highly condensed form, as implicit surface equations. I think the novel approach comes from representing the implicit surface in as few symbols as possible. And, it computes! There's a lot of complex analysis that can be condensed, using computation algorithms with the notation. Deriving the lower-D intercepts can be a grueling process with a polynomial. The cut algorithm can tell you the cross section (and thus set of points) of every rank1,2 toratope, in any dimension. That's pretty awesome. The surtope algorithm can compute the n-frame surface of all open toratopes, and allow you to add up their hypervolumes. As a generalization, it seems that any n-dimensional open/closed toratope has no lower than a 2-frame. Where, n-1 frames are the surface, and n-2 down to the 2-frame are the combo of clifford tori.

We can go into detail with the equation deriving method, and from there, elaborate on rotate and translate control of the 3D hyperplane ( maybe just a little ). The visuals can serve as an additional proof of concept, when we graph the functions based on the notation. That is, showing how a 90 deg turns go from axial midcut to another, which effectively swaps the position(s) of dimension markers. Even the empty cuts show up as predicted. If the 4D cut of (((II)I)((II)I)) can be a square of four tigers (((I)I)((I)I)) , one can move the 3D cut across and through those four many different ways to show how it's really four tigers. And, one can even make the gradual transition to the cut of four concentric tigers (((II))((II))), while cutting through.

I've done a lot of google scholar searching for "multidimensional toroid" , but all I find is CPU related stuff, and algorithms. Nothing to do with discrete surfaces from hyperdonuts in euclidean space. Just about all maths describing them are using a polynomial, not an implicit equation. Apparently, it requires some rather complex math to derive a 5D polynomial from an implicit. So, maybe toratopic notation and its algorithms have an application in solving hyperradicals? Since high-D toratopes can be built from multiple inflation sequences, the polynomial has several solutions. All the different cuts are the lower-D plane intercepts, and also have many solutions. Maybe providing that as well would help? But, that may be getting off topic.

It could be just a neat application for recreational math. Or, it could be something new, as a workaround to explore some really complex math in easier terms. It's still extremely cool, though!

EDIT: Another noteworthy thing : There are many number sequences that apply to high-D geometry. What's been discovered here, is that toratopic notation is the application to A000669. Has there been any others for this one?

EDIT2 : Correction: Well, I guess a circle has a 1-frame, forgot about that one! So " any n-dimensional open/closed toratope has no lower than a 1-frame "
Last edited by ICN5D on Thu Jul 31, 2014 1:22 am, edited 1 time in total.
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ICN5D
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### Re: Publishing an article on rotatopes and toratopes

I agree we should keep the notation. Although I thought we used to say things like ((21)(22)) rather than (((II)I)((II)(II)))?

As well as the spheration operation, we can describe toratopes as different ways of folding down higher dimensional rotatopes to lower dimensions. For example the torus (21) is a Clifford torus 22 folded into 3D. (22), ((21)1), (21)2 and maybe others are all folded down versions of 222. The folding down actions are homeomorphisms and we could probably write them explicitly. PWrong
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### Re: Publishing an article on rotatopes and toratopes

How does the folding down work? Is it like a physical transformation? Or describing a rotation that was undone?
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### Re: Publishing an article on rotatopes and toratopes

PWrong wrote:I agree we should keep the notation. Although I thought we used to say things like ((21)(22)) rather than (((II)I)((II)(II)))?

We did, yes. But the question is, why should 1 mean "I" and not "(I)" when all other numbers include their own parenteses? The current notation is basically helpful for cuts and we don't explore many toratopes so large that it would be completely unreadable.
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### Re: Publishing an article on rotatopes and toratopes

I talked to my supervisor today. He seemed mildly interested, and asked me to write up the basic idea and send it to him. I have another thing I need to publish before I start on that though.

How does the folding down work? Is it like a physical transformation? Or describing a rotation that was undone?

I'm sure there'd be a homeomorphism (continuous map with a continuous inverse) between them, and I suspect there'd be a simple homotopy (a deformation) between them.

If you take the Clifford torus:
x^2 + y^2 = R^2
z^2 + w^2 = r^2,
to fold down we need to move the w axis somewhere else, namely the normal vector of the circle in the xy plane. So w is replaced with (sqrt(x^2 + y^2) - R)^2 to make
z^2 + (sqrt(x^2 + y^2) - R)^2 = r^2.
That's the basic idea. What we need to do to find a homotopy is find the parametric equations of each shape, and then bring in a new time parameter t, such that when t = 0 we have the first shape, and when t = 1 we have the second shape. PWrong
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### Re: Publishing an article on rotatopes and toratopes

PWrong wrote:I talked to my supervisor today. He seemed mildly interested, and asked me to write up the basic idea and send it to him. I have another thing I need to publish before I start on that though.

How does the folding down work? Is it like a physical transformation? Or describing a rotation that was undone?

I'm sure there'd be a homeomorphism (continuous map with a continuous inverse) between them, and I suspect there'd be a simple homotopy (a deformation) between them.

If you take the Clifford torus:
x^2 + y^2 = R^2
z^2 + w^2 = r^2,
to fold down we need to move the w axis somewhere else, namely the normal vector of the circle in the xy plane. So w is replaced with (sqrt(x^2 + y^2) - R)^2 to make
z^2 + (sqrt(x^2 + y^2) - R)^2 = r^2.
That's the basic idea. What we need to do to find a homotopy is find the parametric equations of each shape, and then bring in a new time parameter t, such that when t = 0 we have the first shape, and when t = 1 we have the second shape.

That seems related to a question I was pondering:

Ditorus and tiger are both nonbisecting rotations of torus. That means there should be a way to start with ditorus and move the rotation plane around the shape until it transforms into tiger.
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### Re: Publishing an article on rotatopes and toratopes

That's a good point. Maybe we should try to find explicit homotopies between all of the 4D toratopes whenever they exist.

EDIT: Actually the obvious way to do it is this.
Let (x,y,z,w) = f(θ_1, θ_2, θ_3) be the parametric equations for the tiger. Let (x,y,z,w) g(θ_1, θ_2, θ_3) be the parametric equations for the tiger. Then the homotopy is
h(θ_1, θ_2, θ_3, t) = (1 -t) f(θ_1, θ_2, θ_3) + t g(θ_1, θ_2, θ_3).

This simply follows a straight line from each point in the ditorus to each point in the tiger. Then we just have to prove that this map is continuous. I'm not sure this is the same as moving the plane of rotation though. That might depend on the precise parametrisation we use.

To show that two shapes are NOT homotopic, we need the homotopy groups. PWrong
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### Re: Publishing an article on rotatopes and toratopes

PWrong wrote:That's a good point. Maybe we should try to find explicit homotopies between all of the 4D toratopes whenever they exist.

EDIT: Actually the obvious way to do it is this.
Let (x,y,z,w) = f(θ_1, θ_2, θ_3) be the parametric equations for the tiger. Let (x,y,z,w) g(θ_1, θ_2, θ_3) be the parametric equations for the tiger. Then the homotopy is
h(θ_1, θ_2, θ_3, t) = (1 -t) f(θ_1, θ_2, θ_3) + t g(θ_1, θ_2, θ_3).

This simply follows a straight line from each point in the ditorus to each point in the tiger. Then we just have to prove that this map is continuous. I'm not sure this is the same as moving the plane of rotation though. That might depend on the precise parametrisation we use.

To show that two shapes are NOT homotopic, we need the homotopy groups.

Not much time at the moment, but I guess that if you have a 3D shape

x = f(t,u,v)
y = g(t,u,v)
z = h(t,u,v)

then I think you can manipulate the equation to create a rotation. And if you know parametric equations of torus, then linear combinations of them can make arbitrary rotation of torus. So from these two, it should be possible.
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### Re: Publishing an article on rotatopes and toratopes

OK, specifically...

Let's say that we rotate around plane xy, so z will get mixed with w.

A point [x,y,z] will become a circle [x,y,z*cos(v),z*sin(v)].

x = f(t,u)
y = g(t,u)
z = h(t,u)

will then become

x = f(t,u)
y = g(t,u)
z = h(t,u)*cos(v)
w = h(t,u)*sin(v)

Normal equations for torus are

x = r*cos(t)*cos(u) + R*cos(t)
y = r*cos(t)*sin(u) + R*cos(t)
z = r*sin(t)

Now, if we shift the torus in z direction, we get:

x = r*cos(t)*cos(u) + R*cos(t)
y = r*cos(t)*sin(u) + R*cos(t)
z = r*sin(t) + S

The rotation will then result in a tiger

x = r*cos(t)*cos(u) + R*cos(t)
y = r*cos(t)*sin(u) + R*cos(t)
z = r*sin(t)*cos(v) + S*cos(v)
w = r*sin(t)*sin(v) + S*sin(v)

Now, we can rotate the basic torus around y axis by angle fi:

x = r*cos(t)*cos(u)*cos(fi) + R*cos(t)*cos(fi) + r*sin(t)*sin(fi)
y = r*cos(t)*sin(u) + R*cos(t)
z = r*cos(t)*cos(u)*sin(fi) + R*cos(t)*sin(fi) - r*sin(t)*cos(fi)

This will be once again offset through z:

x = r*cos(t)*cos(u)*cos(fi) + R*cos(t)*cos(fi) + r*sin(t)*sin(fi)
y = r*cos(t)*sin(u) + R*cos(t)
z = r*cos(t)*cos(u)*sin(fi) + R*cos(t)*sin(fi) - r*sin(t)*cos(fi) + S

and rotated:

x = r*cos(t)*cos(u)*cos(fi) + R*cos(t)*cos(fi) + r*sin(t)*sin(fi)
y = r*cos(t)*sin(u) + R*cos(t)
z = r*cos(t)*cos(u)*cos(v)*sin(fi) + R*cos(t)*cos(v)*sin(fi) - r*sin(t)*cos(v)*cos(fi) + S*cos(v)
w = r*cos(t)*cos(u)*sin(v)*sin(fi) + R*cos(t)*sin(v)*sin(fi) - r*sin(t)*sin(v)*cos(fi) + S*sin(v)

with fi = pi/2, this will reduce to

x = r*sin(t)
y = r*cos(t)*sin(u) + R*cos(t)
z = r*cos(t)*cos(u)*cos(v) + R*cos(t)*cos(v) - S*cos(v)
w = r*cos(t)*cos(u)*sin(v) + R*cos(t)*sin(v) - S*sin(v)

which are parametric equations of ditorus.
Marek14
Pentonian

Posts: 1148
Joined: Sat Jul 16, 2005 6:40 pm

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