## Rotatopes section of wiki

Discussion of shapes with curves and holes in various dimensions.

### Rotatopes section of wiki

I find many formulae to be unknown in the wiki pages of a few rotatera. I hope I can clear those up.

Penteract (I would prefer to consider the irregular case penteractoid or penteroid or whatever else it may be called as the regular one is a special case of this and can always be inferred from this)
Surteron bulk: 2(lbht + lbhw + lbtw + lhtw + bhtw)
Pentavolume: lbhtw

Tesserinder
Surteron bulk: 2πr(rab + rac + rbc + abc)
Pentavolume: πr2abc

Duocyldyinder
Surteron bulk: 2π2ab{ab + (a + b)h}
Pentavolume: π2a2b2h (where a and b are the two radii)

Cubspherinder
Surteron bulk: 4πr2{ab + (2/3)r(a + b)}
Pentavolume:(4/3)πr3ab

Cylspherinder
Surteron bulk: 4π2ab2{a + (2/3)b} (I'm not sure whether I computed this right, though most probably I did)
Pentavolume: (4/3)π2a2b3 (where a is the radius of the circular portion and b is the radius of the spherical portion)

Glominder
Surteron bulk: π2r3(2h + r)
Pentavolume: (1/2)π2r4h

Pentasphere
Surteron bulk: (8/3)π2r4
Pentavolume: (8/15)π2r5

As for the pentasphere, this is what is given in the wiki:
The hypervolumes of a pentasphere are given by:

total edge length = 0
total surface area = 0
total surcell volume = 0
surteron bulk = π2∕2 · r4
pentavolume = π2∕8 · r5

How could this be? This surteron bulk is not even the derivative of the pentavolume w.r.t. the radius. Unless I am very much mistaken, this is true of any number of dimensions and, in addition, we have these methods (quoted from viewtopic.php?f=25&t=1891):
Klitzing wrote:You either can calculate the volume of the unit hyperball recursively:
or explicitely, but separate for even and odd numbers, by means of
Code: Select all
V_(2k)   = pi^k / k!
V_(2k+1) = 2 . k! . (4 pi)^k / (2k+1)!

--- rk

Whew! That finishes all the rotatera!
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### Re: Rotatopes section of wiki

I am using the generally used terminology in the wiki here (Otherwise I would not know whether I should say "surteron bulk" or "surface bulk", as for 4D, "surcell" and "surface" have both been used)
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### Re: Rotatopes section of wiki

I computed for toratopes here: viewtopic.php?f=24&t=1991&start=30

and got the same results for pentasphere as you. I suspect the wiki is just wrong... I didn't work it out for open rotatopes and toratopes, but surfaces and volumes should be easier for them.
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### Re: Rotatopes section of wiki

Cool, this might be a good time to test my surtope algorithm for rotatopes. It's hidden somewhere in the 'Tiger Explained' megathread. The cylspherinder (III)(II) will have a torisphere bound orthogonally to a spheritorus, notated as ((III)I)+((II)II), on the surface. The surchoron bulk of a cylspherinder should then be the sum of the 4-volumes of a spheritorus and a torisphere. Thanks to Marek, we now have a huge database of hypervolumes for surface and bulk. This will make it easier to do the addition. When you get to it, Prashkant, the triocylinder (II)(II)(II) will have three duocylindrical toruses (cyltorinders) bound orthogonally: ((II)I)(II)+((II)I)(II)+((II)I)(II) .

In regards to the names...

An n-dimensional shape has n-1 dimensional shapes as the surface ( as I'm sure you already know). You can use the general suffix -tope in place of any specific name.

A polyhedron is 3D, has 2D faces, named surface
A polychoron is 4D, has 3D cells for the surface, named surcells
A polyteron is 5D, has 4D chora on surface, named surchora
A polypeton is 6D, has 5D tera on surface, named surtera
A polytope is nD, has (n-1)D surtopes

Getting even more specific, we can append: perimeter, area, volume, or bulk, for 1, 2, 3, 4D surfaces, respectively. Anything beyond 4D is just plain 'bulk', as far as I know.
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### Re: Rotatopes section of wiki

ICN5D wrote:An n-dimensional shape has n-1 dimensional shapes as the surface ( as I'm sure you already know). You can use the general suffix -tope in place of any specific name.

A polyhedron is 3D, has 2D faces, named surface
A polychoron is 4D, has 3D cells for the surface, named surcells
A polyteron is 5D, has 4D chora on surface, named surchora
A polypeton is 6D, has 5D tera on surface, named surtera
A polytope is nD, has (n-1)D surtopes

Actually, isn't it like this?
A polyhedron has 2D faces or hedra called surfaces
A polychoron has 3D cells or chora called surcells
A polyteron has 4D tera called surtera
A polypeton has 5D peta called surpeta

What I normally do with the rotatopes that can be expressed as Cartesian products is that I just multiply the formulae for the n-volumes of the Cartesian "factors" to get the n-volume of the required rotatope. As for the surtope n-volume, I take certain methods as follows:

For prisms, I add 2*volume of base to h*surtope volume of base. This always completes the formulae for prisms, which is what most of the rotatopes are.

For rotatopes like cylspherinder (32) I could easily find the pentavolume. The surteron bulk, on the other hand, was a bit troublesome. I was not getting anywhere until I just added as follows:

Volume of sphere*Circumference of circle + Area of circle*Surface area of sphere

Actually, I think this should work for any rotatope. Since a cylinder is the Cartesian product of a circle and a line segment, we can actually get 2πr(r + h) as follows:

Area of circle*Surnullon number of line segment (Always 2) + Circumference of circle*Length of line segment.

I am not good at the toratopes other than rotatopes because of their deformed surfaces.

As for the toruses in triocylinder, I can imagine them in the form of spherated circles. In 4D, we can have two circles with the same centre such that all points in the circumference of one are equidistant from the points in the circumference of the other. In 6D, we can add one more circle to this and spherate the whole thing (as in 3D) to get the triocylinder.

Its surpeton pentavolume is 2π3abc(ab + bc + ca)
Its hexavolume is π3a2b2c2
Where a, b and c are the radii.

Visualising this shape requires something entirely different. I can compute the formulae upto any number of dimensions, but I can't visualise a duocylinder knowing that its surcells are topologically equivalent to tori, but it has no holes.

In regard to the names...
My name is Prashant
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### Re: Rotatopes section of wiki

I know a method for mensuration of rotatopes which is what I have been using to compute the formulae in this thread.
Now I have the following algorithm for the mensuration of toratopes:
1. Find the surtope of the expanded rotatope of the required toratope which is structurally similar to the toratope.
2. Find the n-volume of the surtope.
3. The same formula can be used for finding the n-volume of the toratope.
4. For the surtope volume of the toratope, do not consider the interior region.
I think this has some limitations, which could be overcome if somebody could properly explain the relation between the tiger, the ditorus, the cyltorinder and the triocylinder. I would want to know the nature of the surpeta of the triocylinder.
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Prashantkrishnan
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### Re: Rotatopes section of wiki

Actually, isn't it like this?
A polyhedron has 2D faces or hedra called surfaces
A polychoron has 3D cells or chora called surcells
A polyteron has 4D tera called surtera
A polypeton has 5D peta called surpeta

You could be right, I haven't used that vocabulary in a while.....

In 6D, we can add one more circle to this and spherate the whole thing (as in 3D) to get the triocylinder.

The triocylinder (II)(II)(II) is a cylindrical shape with 3 round rolling sides only. It has no spheration yet, just a product of those three solid disks. Spherating that will end you up with a ((II)(II)(II)) , a triger, which is a toroidal ring in 6D. The spheration process emptied out the rolling sides, leaving behind only the 3-surface edge, which is a single structure as a closed, self-repeating surface. It's the discontinuity of the three separate rolling sides. This 3-surface became embedded with a 3D sphere in every point, making the 6D triger.

I can compute the formulae upto any number of dimensions, but I can't visualise a duocylinder knowing that its surcells are topologically equivalent to tori, but it has no holes.

Duocylinder is a strange entity, indeed. Best way to feel the rolling sides is to observe the 3D cylinder's rolling side. One way to describe it, is as a 'line-torus'. In 3D, this line-donut is a flat rolling side, a tire tread that rolls. A higher dimensional equivalent of that rolling surface can take the form of a 3D circle-torus, which is flat in 4D. A duocylinder has only two of these on the surface. Even though in 3D a torus is an inflated donut with a hole, in 4D, it's a hollow tube-like surface, which can fit another hollow tube around it at 90 degrees, perfectly. Again, it's a very strange thing, that violates 3D logic and thinking. This dual tire tread property is only possible in 4D and above, as far as I know.

I think this has some limitations, which could be overcome if somebody could properly explain the relation between the tiger, the ditorus, the cyltorinder and the triocylinder. I would want to know the nature of the surpeta of the triocylinder.

Both tiger and ditorus are made by a sweeping of the torus around a circle, into 4d, held in different orientations.

Slicing a ditorus into 3D will make two tori laid flat, side by side as one of the possible cuts. Ditorus has two more types of 3D intercepts.

(((II)I)I) Ditorus Cuts in 3D

A tiger, on the other hand, will cut into a vertical column of 2 tori. This toroidal ring is made by embedding a circle into every point of a duocylinder margin ( 2-surface edge )

((II)(II)) Tiger Cuts in 3D

Cyltorinder is also the same as a duocylinder torus. In other words, it's a donut of a duocylinder, made by sweeping a duocylinder around in a circle, into 5D. Or, one could bisecting rotate a torinder ( torus prism) into 5D. Haven't animated that one, yet, but I do have the spherated form of it, the tiger torus (((II)I)(II)) .

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### Re: Rotatopes section of wiki

ICN5D wrote:
Actually, isn't it like this?
A polyhedron has 2D faces or hedra called surfaces
A polychoron has 3D cells or chora called surcells
A polyteron has 4D tera called surtera
A polypeton has 5D peta called surpeta

You could be right, I haven't used that vocabulary in a while.....

That's just one of Wendy's polygloss issues:
She then was going into the naming origins.

"Polyhedron" just means (greek) something, which has "poly" (many) "hedrons" (seats).
Accordingly a "polychoron" then ought to be something, which has "poly" "chora" (by G. Olshevsky new derived grecianisation for 'cells'). Etc.

On the other hand surely any such (mono- = individual) choron (= cell of a polychoron), when considered on its own, happens to be a polyhedron in turn.
And likewise any such (mono)hedron (= seat of a polyhedron), when considered on its own, happens to be a polygon.

--- rk
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### Re: Rotatopes section of wiki

ICN5D wrote: the triocylinder (II)(II)(II) will have three duocylindrical toruses (cyltorinders) bound orthogonally: ((II)I)(II)+((II)I)(II)+((II)I)(II) .

From the animations you gave, I infer that the ditorus and the tiger are topologically equivalent to the surtera of the cyltorinder while the cyltorinder is equivalent to the surpeta of the triocylinder, as you have described here. My algorithm would work for this type of case if we first found the surpeton pentavolume of the triocylinder, then using it the surteron bulk of the cyltorinder and then finally the bulks and the surcell volumes of the tiger and the ditorus.
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### Re: Rotatopes section of wiki

Yes, actually, the cyltorinder has two ortho bound ditoruses on the surface. As for the 5-volume of a cyltorinder, it's the cartesian product of a torus and circle.
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### Re: Rotatopes section of wiki

Does it have a tiger?
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### Re: Rotatopes section of wiki

Nope, just two ditoruses. Similar to the two toruses on the surface of duocylinder, a cyltorinder, which remember is the same as duocylindric torus, has ditoruses. No rotatope has a plain tiger on their surface, other than tiger prism ((II)(II))I. But, they can have a tiger torus, which is equal to cyltorintigroid. Just realized this: a tiger*circle prism, ((II)(II))(II) has a tiger torus ortho bound to a toratiger, notated as (((II)I)(II))+(((II)(II))I).
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### Re: Rotatopes section of wiki

I don't understand then how the triocylinder becomes the extended expanded rotatope of the tiger. As usual, the tiger becomes the most difficult toratope to consider even for my mensuration algorithm At least the algorithm is easy enough for all rotatopes.
Last edited by Prashantkrishnan on Fri Jan 23, 2015 4:33 am, edited 1 time in total.
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### Re: Rotatopes section of wiki

Im not sure what you mean by extension of tiger. The tiger is named for two reasons, one of them being the difficulty level in understanding. Id focus on that one, then move on to the 5D toratopes made by it, then anything 6D. The triocylinder is analogous to a duocylinder in a more complicated way. I have a habit of diving straight into the more wild stuff, before the basics are covered
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### Re: Rotatopes section of wiki

ICN5D wrote:Im not sure what you mean by extension of tiger.

I meant the extended expanded rotatope as given in the list of toratopes in the wiki
Last edited by Prashantkrishnan on Fri Jan 23, 2015 4:32 am, edited 1 time in total.
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### Re: Rotatopes section of wiki

Ah, that makes sense. The generalization of tigroids is as follows:

• A circle (II) is one solid disk. Inflating the 1-surface edge with circle makes torus ((II)I)
• A duocylinder (II)(II) is product of two solid disks. Inflating the 2-surface edge with a circle makes the tiger ((II)(II))
• A triocylinder (II)(II)(II) is product of three solid disks. Inflating the 3-surface edge with sphere makes the triger ((II)(II)(II))
• A tetracylinder (quattrocylinder?) (II)(II)(II)(II) is product of four solid disks. Inflating the 4-surface edge with glome makes tetriger ((II)(II)(II)(II))
• An n-cylinder (II)n is product of n solid disks. Inflating the n-surface edge with n-sphere Sn makes n-tiger ((II)n)

Then, of course, one can have higher than circular-shaped diameters, as higher n-spheres, like the spheritiger ((II)(II)I) , sphere inflated duocylinder edge. There are also sphere*circle prisms, the cylspherinder (III)(II) , where inflating the 3-surface edge with circle makes cylspherintigroid ((III)(II)). Inflating that 3-surface edge with a sphere makes ((III)(II)I). Then, getting into denser nesting, we get shapes like the toritiger: circle inflating 3-surface of tiger (((II)(II))I) , tiger torus as tiger inflating the edge of disk (((II)I)(II)), etc.

I made a list of the toratopes as defined by fiber bundles, which is a sequence of inflating surfaces with n-spheres and n-cylinder edges. It starts with smallest diameter, reads left to right of increasing size. The term C2 means the 2-surface edge of duocylinder, as the Clifford torus, also denoted as [S1*S1] , as product of two ortho disk edges. Fiber bundle sequences may help with visualizing how the toratopes are built. From 2D to 5D:

1-manifold that curves into 2D:
(II) - S1 , the circle

2-Manifolds that curve into 3D:
(III) - S2 , sphere
((II)I) -T2 = S1xS1 , torus

3-Manifolds that curve into 4D:
(IIII) - S3 , glome
((III)I) - S1xS2 , torisphere
((II)II) - S2xS1 , spheritorus
(((II)I)I) - T3 = S1xS1xS1 , ditorus , 3-torus
((II)(II)) - S1xC2 = S1x[S1*S1] , tiger

4-Manifolds that curve into 5D:
(IIIII) - S4 , pentasphere
((II)III) - S3xS1 , glomitorus
((II)(II)I) - S2xC2 , spheritiger
((III)II) - S2xS2 , spherisphere
(((II)I)II) - S2xT2 = S2xS1xS1 , spheriditorus
((III)(II)) - S1x[S2*S1] , cylspherintigroid
(((II)I)(II)) - S1xC2xS1 , tiger torus
((IIII)I) - S1xS3 , toriglome
(((II)II)I) - S1xS2xS1 , torispheritorus
(((II)(II))I) - T2xC2 = S1xS1x[S1*S1] , toritiger
(((III)I)I) - T2xS2 = S1xS1xS2 , ditorisphere
((((II)I)I)I) - T4 = S1xS1xS1xS1 , tritorus, 4-torus
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### Re: Rotatopes section of wiki

Prashantkrishnan wrote:
ICN5D wrote:Im not sure what you mean by extension of tiger.

I meant the extended rotatope as given in the list of toratopes in the wiki
Prashantkrishnan wrote:I don't understand then how the triocylinder becomes the extended rotatope of the tiger. As usual, the tiger becomes the most difficult toratope to consider even for my mensuration algorithm At least the algorithm is easy enough for all rotatopes.

I'm sorry ... What I meant was expanded rotatopes. Anyway, good to see the information about tigroids.

ICN5D wrote:Ah, that makes sense. The generalization of tigroids is as follows:

• A circle (II) is one solid disk. Inflating the 1-surface edge with circle makes torus ((II)I)
• A duocylinder (II)(II) is product of two solid disks. Inflating the 2-surface edge with a circle makes the tiger ((II)(II))
• A triocylinder (II)(II)(II) is product of three solid disks. Inflating the 3-surface edge with sphere makes the triger ((II)(II)(II))
• A tetracylinder (quattrocylinder?) (II)(II)(II)(II) is product of four solid disks. Inflating the 4-surface edge with glome makes tetriger ((II)(II)(II)(II))
• An n-cylinder (II)n is product of n solid disks. Inflating the n-surface edge with n-sphere Sn makes n-tiger ((II)n)

Now I am able to make some sense out of this toratopic notation. Though isn't it more appropriate to consider ((II)) as a tigroid rather than ((II)I) if such a thing does indeed exist? I have read about figures like (I) in another thread. Also, normally we inflate the n-surface edge with (n-1)-sphere while in the first case we are inflating 1-surface edge with circle. (Considering circle as 1-sphere, sphere as 2-sphere, glome as 3-sphere etc.) The pattern is violated.

Then, of course, one can have higher than circular-shaped diameters, as higher n-spheres, like the spheritiger ((II)(II)I) , sphere inflated duocylinder edge. There are also sphere*circle prisms, the cylspherinder (III)(II) , where inflating the 3-surface edge with circle makes cylspherintigroid ((III)(II)). Inflating that 3-surface edge with a sphere makes ((III)(II)I). Then, getting into denser nesting, we get shapes like the toritiger: circle inflating 3-surface of tiger (((II)(II))I) , tiger torus as tiger inflating the edge of disk (((II)I)(II)), etc.

I infer here that when we remove the outermost parantheses of a closed toratope, add I, and put back the parantheses, we give the prefix spheri-(toratope). Also, adding I and closing gives the tori-(toratope). In other words, the sphere can be considered as 'sphericircle' and the torus can be considered as 'toricircle'. And here, we think of inflating the 3-surface edge with a circle. So we can always inflate the m-surface edge with the n-sphere in m + n + 1 dimensions.

I made a list of the toratopes as defined by fiber bundles, which is a sequence of inflating surfaces with n-spheres and n-cylinder edges. It starts with smallest diameter, reads left to right of increasing size. The term C2 means the 2-surface edge of duocylinder, as the Clifford torus, also denoted as [S1*S1] , as product of two ortho disk edges. Fiber bundle sequences may help with visualizing how the toratopes are built. From 2D to 5D:

1-manifold that curves into 2D:
(II) - S1 , the circle

2-Manifolds that curve into 3D:
(III) - S2 , sphere
((II)I) -T2 = S1xS1 , torus

3-Manifolds that curve into 4D:
(IIII) - S3 , glome
((III)I) - S1xS2 , torisphere
((II)II) - S2xS1 , spheritorus
(((II)I)I) - T3 = S1xS1xS1 , ditorus , 3-torus
((II)(II)) - S1xC2 = S1x[S1*S1] , tiger

4-Manifolds that curve into 5D:
(IIIII) - S4 , pentasphere
((II)III) - S3xS1 , glomitorus
((II)(II)I) - S2xC2 , spheritiger
((III)II) - S2xS2 , spherisphere
(((II)I)II) - S2xT2 = S2xS1xS1 , spheriditorus
((III)(II)) - S1x[S2*S1] , cylspherintigroid
(((II)I)(II)) - S1xC2xS1 , tiger torus
((IIII)I) - S1xS3 , toriglome
(((II)II)I) - S1xS2xS1 , torispheritorus
(((II)(II))I) - T2xC2 = S1xS1x[S1*S1] , toritiger
(((III)I)I) - T2xS2 = S1xS1xS2 , ditorisphere
((((II)I)I)I) - T4 = S1xS1xS1xS1 , tritorus, 4-torus

Until 4D, it all seems fine. As for 5D, I'll have to go through it once again. I sometimes wonder whether the existence of more and more notations for the shapes we consider would clarify more or confuse us more. I understand properly only the rotatope notation, from which we can automatically express rotatopes as Cartesian products. Thus the expanded rotatope is what makes toratopes understandable to me. For simpler figures, it always seemed that a toratope is geometrically equivalent to at least one of the surtopes of its expanded rotatopes. This gets confusing when 4D toratopes have 6D expanded rotatopes. I honestly don't see why the triocylinder is the expanded rotatope of the tiger.
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### Re: Rotatopes section of wiki

Though isn't it more appropriate to consider ((II)) as a tigroid rather than ((II)I) if such a thing does indeed exist? I have read about figures like (I) in another thread.

Things like ((II)) and (I) are actually cross sections of a toratope, and don't count as a single, full shape. Making a cut is done by setting certain dimensions to zero, and thus removing them. We're left over with extra diameter terms to interpret as the midsection. Using the sequence for torus ((II)I) (which is not a tigroid), we get ((II)) and ((I)I) for the 2D cuts, making two concentric circles ((II)) and two side by side circles ((I)I). The symbol (I) means two points in a row, as a 1D cut of a full circle (II). You may have also seen these cross section tables, listing all cuts of a toratope:

((II)I) - torus
-----------------
((II)) - 2 circles (II) as concentric pair
((I)I) - 2 circles (II) in 2x1 row
-----------------
((I)) - 4 points in a row

It's a way of using the notation to abstractly derive every intercept of a shape, no matter how complex. It also lets you see how the large intercept arrays are connected together in higher dimensions, once you've gotten the hang of it. The toughest part is connecting a sequence like (((I)I)(I)) to a visual in your mind. An introductory to the concept is elaborated a bit more, here.

Also, normally we inflate the n-surface edge with (n-1)-sphere while in the first case we are inflating 1-surface edge with circle. (Considering circle as 1-sphere, sphere as 2-sphere, glome as 3-sphere etc.)

You know, I actually corrected that, then changed it back . So, to recorrect ...

• An n-cylinder (II)n is product of n solid disks. Inflating the n-surface edge with (n-1)-sphere Sn-1 makes n-tiger ((II)n)

I sometimes wonder whether the existence of more and more notations for the shapes we consider would clarify more or confuse us more.

That's why I'm using no more than two notations. The fiber bundle is another well-known way to describe such things, and may make the toratope notation a bit more clear. Some people may find breaking the shapes down by a linear sequence is easier to grasp than the notation, at first.

I honestly don't see why the triocylinder is the expanded rotatope of the tiger.

Well, it's more like the triocylinder (II)(II)(II) is the expanded rotatope of a duocylinder (II)(II) , as simply a product with one more circle. A triger ((II)(II)(II)) is the expanded toratope of a tiger ((II)(II)). Triocylinder is to triger as duocylinder is to tiger,

(II)(II)(II) -> ((II)(II)(II)) :: (II)(II) -> ((II)(II))

The method that turns duocylinder into tiger is equal to the method of triocylinder to triger. It's the added parentheses around the prism, (II)(II) --> ((II)(II)) that turns it into a toroidal ring.
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ICN5D
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### Re: Rotatopes section of wiki

ICN5D wrote:
I honestly don't see why the triocylinder is the expanded rotatope of the tiger.

Well, it's more like the triocylinder (II)(II)(II) is the expanded rotatope of a duocylinder (II)(II) , as simply a product with one more circle. A triger ((II)(II)(II)) is the expanded toratope of a tiger ((II)(II)). Triocylinder is to triger as duocylinder is to tiger,

(II)(II)(II) -> ((II)(II)(II)) :: (II)(II) -> ((II)(II))

The method that turns duocylinder into tiger is equal to the method of triocylinder to triger. It's the added parentheses around the prism, (II)(II) --> ((II)(II)) that turns it into a toroidal ring.

That's true. What I was talking about was http://hddb.teamikaria.com/wiki/List_of_toratopes. The last column, "Expanded rotatope", is 222 for ((II)(II)).
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Prashantkrishnan
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### Re: Rotatopes section of wiki

Ah, that's an older notation, that I don't think we've used for quite a while. That page, among others, hasn't been updated in a while, either. It's misleading, I know. You'd be best to learn the recent stuff, until one of us goes on there and updates as needed. Personally, I prefer the toratopic notation along with fiber bundles, as they are uniquely different ways of defining the same things. But, that's just me.
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ICN5D
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### Re: Rotatopes section of wiki

ICN5D wrote:Ah, that's an older notation, that I don't think we've used for quite a while. That page, among others, hasn't been updated in a while, either. It's misleading, I know. You'd be best to learn the recent stuff, until one of us goes on there and updates as needed. Personally, I prefer the toratopic notation along with fiber bundles, as they are uniquely different ways of defining the same things. But, that's just me.

Do you mean to say that nowadays expanded rotatopes are not talked of much? I have a use for them:

We know that the expanded rotatope of ((II)I) (torus) is 22 (duocylinder). And also, the surcell volume of the duocylinder is 2π2ab(a+b) where a and b are the two radii. We know that there are two surcells, and surcellwise, we can rewrite the expression as 2π2a2b + 2π2ab2. Both surcells are torus shaped. So we can take one, say 2π2a2b, and say that this is the volume of a torus. We can replace a with r and b with R to get 2π2Rr2. To find the surface area, we can further rewrite the volume formula as πr2 x 2πR. Replacing πr2 with 2πr, we get the surface area as 4π2Rr.

Of course, the formulae for the torus must already be well known, but the formulae for higher dimensional toratopes may not be so well known. So what I am asking is whether we can derive the formulae for the tiger the same way.

I know that it has already been derived in another thread But I am curious to know whether it can be done this way too.
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### Re: Rotatopes section of wiki

Do you mean to say that nowadays expanded rotatopes are not talked of much?

No, they are talked about. It's the specific notation that I don't think is talked about too much anymore. But, the philosphies are still the same, by labeling the triger as an expanded toratope to the tiger.

Of course, the formulae for the torus must already be well known, but the formulae for higher dimensional toratopes may not be so well known. So what I am asking is whether we can derive the formulae for the tiger the same way.

I know that it has already been derived in another thread But I am curious to know whether it can be done this way too.

Well, if so, then you could find the area of a duocylinder margin, then multiply that with the surface of a circle, for the tiger surface volume. And, multiply the circle area with a duocylinder area to get 4D volume. I wonder if that would work?
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ICN5D
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### Re: Rotatopes section of wiki

Well, the area of duocylinder margin is 4 * pi^2 * R1 * R2, which leads to the correct surface and volume of tiger when multiplied by circumference/area of a circle, according to my calculations at viewtopic.php?f=24&t=1991&start=30#p22601.

However, this has to do with special properties of a circle, specifically the fact that a circle doesn't have any intrinsic curvature. I derive 330-tiger later in the thread, and it's more complicated than normal tiger because spheres are intrinsically curved and therefore some corrections to the formula are needed.

The reason is basically that is you take a tube (any shape, in arbitrary dimension, along a line), and bend it, some parts will compress and some will stretch, but the loss and gain will be exactly equal so the total surface and volume will stay the same as for a straight tube, regardless of dimension.
But if you take a sheet (a shape added to each point of a plane, like, say, a cubinder), or something based on a higher-D shape and bend it into something that no longer has Euclidean planar geometry, then the losses and gains will no longer cancel and you must compute what exactly will happen.
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### Re: Rotatopes section of wiki

Ah yes, the spheres. The spheres changed everything. I think we found that all toratopes per dimension, with circle-shaped diameters, have the same volume and surface. And, even the spheres will group into their own equals, as well. You can use that property as a guide, when calculating the volumes and surfaces.
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