Publishing an article on rotatopes and toratopes

Discussion of shapes with curves and holes in various dimensions.

Re: Publishing an article on rotatopes and toratopes

Postby ICN5D » Fri Aug 01, 2014 6:11 am

Look what I found:

http://www.mdpi.com/1099-4300/15/10/4285

It's all about the notation combinatorics, and a detailed analysis on it. They probably didn't know about the geometry application.
in search of combinatorial objects of finite extent
ICN5D
Pentonian
 
Posts: 1047
Joined: Mon Jul 28, 2008 4:25 am
Location: Orlando, FL

Re: Publishing an article on rotatopes and toratopes

Postby ICN5D » Sat Aug 02, 2014 7:09 pm

Been thinking of polynomials lately. Actually quite a bit. I think I have a neat way to go about writing explicts. But, later, after work.
in search of combinatorial objects of finite extent
ICN5D
Pentonian
 
Posts: 1047
Joined: Mon Jul 28, 2008 4:25 am
Location: Orlando, FL

Re: Publishing an article on rotatopes and toratopes

Postby Marek14 » Sun Aug 03, 2014 2:24 pm

Surfaces and volumes of toratopes:

First of all, it's easier to compute the volumes -- the surface is then just volume derived by minor diameter (because a volume of toratope can be understood as infinite sum of thin surfaces).

Starting from circle (area pi*r^2, circumference 2*pi*r).

Sphere, according to wikipedia (http://en.wikipedia.org/wiki/Sphere) can be computed through integration as sum of pi*y^2 where y^2 = r^2 - x^2, leading to classic formula 4/3*pi*r^3 with derivation 4*pi*r^2.

Now, we can consider torus as sliced vertically into rings. Each ring has outer diameter (R + y) and inner diameter (R - y) where x^2 + y^2 = r^2.

The area of ring is area of large circle minus area of small circle:
pi*(R + y)^2 - pi*(R - y)^2
pi*((R + y)^2 - (R - y)^2)
pi*(R^2 + 2Ry + y^2 - R^2 +2Ry - y^2)
pi*(4Ry)
pi*(4R*sqrt(r^2-x^2))

And the integral for x from -r to r is 2*pi^2*R*r^2

Surface derivation is 4*pi^2*R*r -- both correspond to wikipedia formulas for torus, so it seems we're on the right track.

Now onward to 4D toratopes:

Glome - we can work similarly to sphere:
Each slice is 4/3*pi*y^3 = 4/3*pi*(r^2 - x^2)^(3/2)
Volume = 1/2*pi^2*r^4
Surface = 2*pi^2*r^3

Torisphere - we use slices in the shape of hollow balls:
Each slice is 4/3*pi*(R + y)^3 - 4/3*pi*(R - y)^3
Which is:
4/3*pi*(R^3 + 3R^2y + 3Ry^2 + y^3 - R^3 + 3R^2y - 3Ry^2 +y^3)
4/3*pi*(6R^2y + 2y^3)
8/3*pi*(3R^2 + y^2)*y
8/3*pi*(3R^2 + r^2 - x^2)*sqrt(r^2-x^2)
This gives volume of torisphere as pi^2*r^2*(r^2 + 4R^2).
This is substantially different from formula shown on our own wiki page (8/3*pi^2*R*r^3), but I have a reason to think this formula is wrong: laically speaking, bulk of torisphere is small circle (r^2) smeared along the surface of big sphere (R^2), so why would the formula have R in first power and r in third?
Surface of torisphere would then be r-derivation, i.e. 4*pi^2*r*(r^2 + 2R^2)

Spheritorus - here we use torus slices:
Each slice is 2*pi^2*R*y^2 -> 2*pi^2*R*(r^2 - x^2)
Volume is 8/3*pi^2*R*r^3 -- the wiki formula for torisphere.
Surface, then, is 8*pi^2*R*r^2, again corresponding to what wiki uses for torisphere.

Ditorus - in this case we'll use slices shaped like hollow torus. We'll use radii R, r and s:
Each slice is 2*pi^2*R*(r + y)^2 - 2*pi^2*R*(r - y)^2
2*pi^2*R*(r^2 + 2ry + y^2 - r^2 + 2ry - y^2)
2*pi^2*R*(4ry)
8*pi^2*R*r*sqrt(s^2 - x^2)

Result is 4*pi^3*R*r*s^2 -- same as on our wiki.
Surface will be 8*pi^3*R*r*s

Now, let's look if spheritorus and ditorus formulas can be decomposed according to Pappus's centroid theorem.
8/3*pi^2*R*r^3 = 4/3*pi*r^3 * 2*pi*R
4*pi^3*R*r*s^2 = 2*pi^2*r*s^2 * 2*pi*R

So it looks like this theorem still works in 4D. Torisphere gets a different, ugly formula because it cannot be expressed as nonbisecting rotation.

OK, and what about tiger?

Tiger, unlike the others, has no "nice" slices whose volume could be always easily computed. But, if Pappus' centroid theorem holds, then any nonbisecting rotation of torus would have the same surface and volume, regardless of orientation, and so a tiger with major diameters R1 and R2 and minor diameter r would use the same formulas as ditorus:
Volume: 4*pi^3*R1*R2*r^2
Surface: 8*pi^3*R1*R2*r

Now let's consider a tiger as a collection of various duocylinder margins.
A duocylinder margin is cartesian product of two circles (with radii R1 and R2), so naturally its surface will be 4*pi^2*R1*R2.
Now look:
4*pi^3*R1*R2*r^2 = 4*pi^2*R1*R2 * pi*r^2
In every point of the margin there is a circle, which is perpendicular to two dimensions of margin at that point (although in different points of the margin, the orientation is different).

So, main points of this post:
1. Torisphere formulas on our wiki actually belong to spheritorus.
2. Pappus' centroid theorem seems to hold in higher dimensions. Is there a proof of this? If yes, we need to find and cite it. If not, we might try to supply it on our own, and that would be a major result.
3. In 3D, the theorem considers constant area moving along the curve with the important thing being that it's always perpendicular to the curve at every point. It then says that the surface/volume of figure such created is equivalent to simple cartesian product of the area and curve. This could be extended to considering both the area and the curve to have more dimensions. But apparently it only works if the inner curvature of both is zero. Duocylinder margin has Euclidean geometry and works, but sphere has not, and so torisphere's bulk isn't simply 4*pi*R^2 * pi*r^2 = 4*pi^2*R^2*r^2 and its surface isn't simply 4*pi*R^2 * 2*pi*r = 8*pi^2*R^2*r. But these expressions DO appear in the true formulas!
pi^2*r^2*(r^2 + 4R^2) = (4*pi^2*R^2*r^2) + (pi^2*r^4)
4*pi^2*r*(r^2 + 2R^2) = (8*pi^2*R^2*r) + (4*pi^2*r^3)
I suspect there's a correction term which compensates the geometry. The centroid theorem no longer holds exactly if we're moving on a curved surface.
4. If the centroid theorem works as I suggest, we have an easy (or at least doable) way to compute surfaces and volumes of arbitrary toratopes.

As I say, the most important thing now is the Pappus' centroid theorem. If we can figure out or obtain a proof that it works in higher dimensions and/or that the curve can be replaced by flat surface, we'll obtain a really significant result.
Marek14
Pentonian
 
Posts: 1099
Joined: Sat Jul 16, 2005 6:40 pm

Re: Publishing an article on rotatopes and toratopes

Postby Marek14 » Sun Aug 03, 2014 5:16 pm

Thinking about it, it's possible that the centroid theorem can be extended in a trivial way -- you can cut the 3D figure you want to rotate into strips like you want to integrate, then exchange every strip with a 2D strip of the same numerical area as is the 3D strip's volume. Then you use the centroid theorem on the resulting 2D figure...
Marek14
Pentonian
 
Posts: 1099
Joined: Sat Jul 16, 2005 6:40 pm

Re: Publishing an article on rotatopes and toratopes

Postby ICN5D » Mon Aug 04, 2014 5:38 am

Marek wrote: Pappus' centroid theorem seems to hold in higher dimensions. Is there a proof of this? If yes, we need to find and cite it.



Does this help?

http://mathoverflow.net/questions/60416 ... eneralized


I'm thinking the bundling sequence is the surface of revolution, which there are several different ways in +5D. A good thing might be a general way to inflate a small shape toratope with a clifford torus. Toratopes seem to follow a concise sequence of smallest shape inflating larger and larger surface, of either n-spheres, n-tori, and/or clifford tori .
in search of combinatorial objects of finite extent
ICN5D
Pentonian
 
Posts: 1047
Joined: Mon Jul 28, 2008 4:25 am
Location: Orlando, FL

Re: Publishing an article on rotatopes and toratopes

Postby Marek14 » Mon Aug 04, 2014 6:14 am

Trying for 5D:

Pentasphere (R)
Volume of glome = 1/2*pi^2*R^4
Using glome slices 1/2*pi^2*(R^2-x^2)^2
Volume: 8/15*pi^2*R^5
Surface: 8/3*pi^2*R^4

41-torus ((R)r)
Volume of hollow glome = 1/2*pi^2*(R + r)^4 - 1/2*pi^2*(R - r)^4 = 4*pi^2*R*r*(R^2 + r^2)
Using hollow glome slices 4*pi^2*R*sqrt(r^2 - x^2)*(R^2 + r^2 - x^2)
Volume: 1/2*pi^3*R*r^2*(4R^2 + 3r^2)
Surface: 2*pi^3*R*r*(2R^2 + 3r^2)

311-ditorus (((R)r)s)
Volume of hollow torisphere = pi^2*(r + s)*(4R^2 + (r + s)^2) - pi^2*(r - s)*(4R^2 + (r - s)^2) = 8*pi^2*r*s*(2R^2 + r^2 + s^2)
Using hollow torisphere slices 8*pi^2*r*sqrt(s^2 - x^2)*(2R^2 + r^2 + s^2 - x^2)
Volume: pi^3*r*s^2*(8R^2 + 4r^2 + 3s^2)
Surface: 4*pi^3*r*s*(4R^2 + 2r^2 + 3s^2)

Tritorus ((((R)r)s)r)
Volume of hollow ditorus = 4*pi^3*R*r*(s + t)^2 - 4*pi^3*R*r*(s - t)^2 = 16*pi^3*R*r*s*t
Using hollow ditorus slices 16*pi^3*R*r*s*sqrt(t^2 - x^2)
Volume: 8*pi^4*R*r*s*t^2
Surface: 16*pi^4*R*r*s*t
Tritorus as rotation of ditorus:
Volume = 4*pi^3*r*s*t^2 * 2*pi*R = 8*pi^4*R*r*s*t^2
Surface = 8*pi^3*r*s*t * 2*pi*R = 16*pi^4*R*r*s*t

Tiger torus (((R1)(R2)r)s)
Volume of hollow tiger = same as hollow ditorus = 16*pi^3*R1*R2*r*s
Using hollow tiger slices 16*pi^3*R1*R2*r*sqrt(s^2 - x^2)
Volume: 8*pi^4*R1*R2*r*s^2
Surface: 16*pi^4*R1*R2*r*s
Tiger torus as rotation of ditorus:
Volume = 4*pi^3*R1*r*s^2 * 2*pi*R2 = 8*pi^4*R1*R2*r*s^2
Surface = 8*pi^3*R1*r*s * 2*pi*R2 = 16*pi^4*R1*R2*r*s

221-ditorus (((R)r)s)
Volume of hollow spheritorus = 8/3*pi^2*R*(r + s)^3 - 8/3*pi^2*R*(r - s)^3 = 16/3*pi^2*R*s*(3r^2 + s^2)
Using hollow spheritorus slices 16/3*pi^2*R*sqrt(s^2 - x^2)*(3r^2 + s^2 - x^2)
Volume: 2*pi^3*R*s^2*(4r^2 + s^2)
Surface: 8*pi^3*R*s*(2r^2 + s^2)
221-ditorus as rotation of torisphere:
Volume = pi^2*s^2*(4r^2 + s^2) * 2*pi*R = 2*pi^3*R*s^2*(4r^2 + s^2)
Surface = 4*pi^2*s*(2r^2 + s^2) * 2*pi*R = 8*pi^3*R*s*(2r^2 + s^2)

320-tiger ((R1)(R2)r)
We'll compute it as rotation of torisphere, giving it same formulas as 221-ditorus.
Volume: 2*pi^3*R2*r^2*(4R1^2 + r^2)
Surface: 8*pi^3*R2*r*(2R1^2 + r^2)
If we try to look at circle expansion of sphere*circle cartesian product, we'll get:
4*pi*R1^2 * 2*pi*R2 * pi*r^2 = 8*pi^3*R1^2*R2*r^2, which is part of volume. A curvature correction of 2*pi^3*R2*r^4 has to be added.

Torus tiger (((R1)r)(R2)s)
Torus tiger can be considered either a rotation of tiger or a rotation of ditorus
Rotation of tiger:
Volume: 4*pi^3*r*R2*s^2 * 2*pi*R1 = 8*pi^4*R1*R2*r*s^2
Surface: 8*pi^3*r*R2*s * 2*pi*R1 = 16*pi^4*R1*R2*r*s
Rotation of ditorus:
Volume: 4*pi^3*R1*r*s^2 * 2*pi*R2 = 8*pi^4*R1*R2*r*s^2
Surface: 8*pi^3*R1*r*s * 2*pi*R2 = 16*pi^4*R1*R2*r*s

32-torus ((R)r)
Volume of torisphere = pi^2*r^2*(4*R^2 + r^2)
Using torisphere slices pi^2*(r^2 - x^2)*(4*R^2 + (r^2 - x^2))
Volume: 16/15*pi^2*r^3*(5*R^2 + r^2)
Surface: 16/3*pi^2*r^2*(3*R^2 + r^2)

212-ditorus (((R)r)s)
Volume of ditorus = 4*pi^3*R*r*s^2
Using ditorus slices 4*pi^3*R*r*(s^2 - x^2)
Volume: 16/3*pi^3*R*r*s^3
Surface: 16*pi^3*R*r*s^2
212-ditorus as rotation of spheritorus:
Volume: 8/3*pi^2*r*s^3 * 2*pi*R = 16/3*pi^3*R*r*s^3
Surface: 8*pi^2*r*s^2 * 2*pi*R = 16*pi^3*R*r*s^2

221-tiger ((R1)(R2)r)
Volume of tiger = 4*pi^3*R1*R2*r^2
Using tiger slices 4*pi^3*R1*R2*(r^2 - x^2)
Volume: 16/3*pi^3*R1*R2*r^3
Surface: 16*pi^3*R1*R2*r^2
221-tiger as rotation of spheritorus:
Volume: 8/3*pi^2*R1*r^3 * 2*pi*R2 = 16/3*pi^3*R1*R2*r^3
Surface: 8*pi^2*R1*r^2 * 2*pi*R2 = 16*pi^3*R1*R2*r^2

23-torus ((R)r)
Volume of spheritorus = 8/3*pi^2*R*r^3
Using spheritorus slices 8/3*pi^2*R*(r^2 - x^2)^(3/2)
Volume: pi^3*R*r^4
Surface: 4*pi^3*R*r^3
23-torus as rotation of glome:
Volume: 1/2*pi^2*r^4 * 2*pi*R = pi^3*R*r^4
Surface: 2*pi^2*r^3 * 2*pi*R = 4*pi^3*R*r^3

So, what have we learned:
1. Skinny toratopes have extraordinarily simple formulas since they can be just composed as a series of rotation.
2. Presence of higher-dimensional spheres complicates the formulas.
3. The formula might tell us something about whether the geometry of surface is curved.
4. We can create equivalency class on toratopes based on formulas, where two different rotations of same toratope are put into the same class.

Now, I've done this until 5D. So far each shape could be composed in one or more of three ways:

Slices integration, which works for toratopes which stay whole when you cut them in a minor dimension.
Hollow slices integration, which works for toratopes with separate into pairs when you cut them in a minor dimension.
Rotation + Pappus' centroid theorem, which "works" (still unproven) for toratopes containing (II) string -- and so far gives same results as previous two when applicable on the same toratope.

But in 6D, we get into trouble because of the existence of 330-tiger ((III)(III)). The 330-tiger has no minor dimension so it can't be cut through it and it's not a non-bisecting rotation of any toratope.

Can anyone suggest a technique that would work for volume/surface of this figure?
Last edited by Marek14 on Sat Aug 09, 2014 12:01 pm, edited 3 times in total.
Marek14
Pentonian
 
Posts: 1099
Joined: Sat Jul 16, 2005 6:40 pm

Re: Publishing an article on rotatopes and toratopes

Postby ICN5D » Mon Aug 04, 2014 6:46 am

Well, yes, I think. If you take the cartesian product of the surface of two spheres, (4pi*r1^2)(4pi*r2^2) = 8pi^4*r1^2*r2^2

Then, multiply with the area of a circle, (8pi^4*r1^2*r2^2)*pi*r3^2 = 8*pi^5*r1^2*r2^2*r3^2

So, volume of ((III)(III)) should be 8*pi^5*r1^2*r2^2*r3^2

Actually, I rather like the fact that all you have to do is multiply the volumes and surfaces together!
in search of combinatorial objects of finite extent
ICN5D
Pentonian
 
Posts: 1047
Joined: Mon Jul 28, 2008 4:25 am
Location: Orlando, FL

Re: Publishing an article on rotatopes and toratopes

Postby Marek14 » Mon Aug 04, 2014 6:52 am

ICN5D wrote:Well, yes, I think. If you take the cartesian product of the surface of two spheres, (4pi*r1^2)(4pi*r2^2) = 8pi^4*r1^2*r2^2

Then, multiply with the area of a circle, (8pi^4*r1^2*r2^2)*pi*r3^2 = 8*pi^5*r1^2*r2^2*r3^2

So, volume of ((III)(III)) should be 8*pi^5*r1^2*r2^2*r3^2

Actually, I rather like the fact that all you have to do is multiply the volumes and surfaces together!


Nope, that wouldn't work. See my derivation of 320-tiger where I tried just that and found that it only gives part of the full volume.
Marek14
Pentonian
 
Posts: 1099
Joined: Sat Jul 16, 2005 6:40 pm

Re: Publishing an article on rotatopes and toratopes

Postby ICN5D » Mon Aug 04, 2014 7:02 am

Well, dang. You're right. That's weird, it seems like a great way to do it, though. It worked so well with the tiger. What's that curvature correction you spoke of?
in search of combinatorial objects of finite extent
ICN5D
Pentonian
 
Posts: 1047
Joined: Mon Jul 28, 2008 4:25 am
Location: Orlando, FL

Re: Publishing an article on rotatopes and toratopes

Postby Marek14 » Mon Aug 04, 2014 8:35 am

ICN5D wrote:Well, dang. You're right. That's weird, it seems like a great way to do it, though. It worked so well with the tiger. What's that curvature correction you spoke of?


It's like this: A curve like circle can be curved however you want, but if you were a 1D being living on the curve, you'd have no way to find out the curvature -- a line simply has only one possible geometry. But there are multiple geometries in 2D. Tiger is based on product of two circles, i.e. two curves -- this means the product has normal Euclidean geometry. A sphere, however, has spherical geometry which is different. A triangle drawn on duocylinder margin will have sum of angles 180 degrees; a triangle drawn on sphere will have a larger sum.

Alternately, consider a ring with outer diameter R + r and inner diameter R - r. The outer circumference is 2*pi*R + 2*pi*r, the inner circumference is 2*pi*R - 2*pi*r. If you add them together, you get 4*pi*R, meaning that the total circumference of ring depends ONLY on its average diameter and not on its thickness.

However, this doesn't work for a hollow sphere: the outer surface would be 4*pi*(R^2 + 2Rr + r^2) and inner would be 4*pi*(R^2 - 2Rr + r^2); summing of these will give us 4*pi*R^2 + 4*pi*r^2 so total surface of hollow sphere DOES depend on the thickness. The "curvature correction" is the extra 4*pi*r^2 term in addition to surface of medium sphere 4*pi*R^2.

In other words: if you think of surface (or even volume) of torus as a set of parallel circles, then these circles will always occur in pairs whose circumference will "cancel" each other, simplifying the result -- but torisphere which is similarly made up of parallel spherical shells cannot have them paired in this way, and that's why the result is more complicated.
Marek14
Pentonian
 
Posts: 1099
Joined: Sat Jul 16, 2005 6:40 pm

Re: Publishing an article on rotatopes and toratopes

Postby ICN5D » Tue Aug 05, 2014 3:23 am

Hmmm. How about multiplying the surface of two spheres together making 4D margin, then establishing an outer, middle, and inner sphere*sphere margin?


On another note, check this out:

http://pages.uoregon.edu/njp/su.pdf

Very interesting! So, it's true that there is a combinatoric pattern in polynomials, from 'hypertoric varieties'. Toratopic notation and its algorithms are probably going to revolutionize this field. Just sayin' :)
in search of combinatorial objects of finite extent
ICN5D
Pentonian
 
Posts: 1047
Joined: Mon Jul 28, 2008 4:25 am
Location: Orlando, FL

Re: Publishing an article on rotatopes and toratopes

Postby Marek14 » Tue Aug 05, 2014 7:13 am

Actually, maybe we could use the ((III)()) cut of 330-tiger. This looks like 3D array of hollow spheres of mid-radius R1 whose half-separation is r at distance R2 from origin in the other 3 coordinates and shrinks to zero at distances R2 + r and R2 - r. At distance R2 + x and R2 - x, the spheres have radii R1 + y and R1 + y where x^2 + y^2 = r^2

Hollow sphere's volume is 4/3*pi*(R1 + y)^3 - 4/3*pi*(R1 - y)^3 = 8/3*pi*y*(3R1^2 + y^2).
This can be multiplied by surface of second sphere at the distance. This is 4*pi*(R2 + x)^2 where x goes from -r to r.

So we integrate the product of both, 8/3*pi*y*(3R1^2 + y^2) * 4*pi*(R2 + x)^2, replacing y with sqrt(r^2 - x^2):

8/3*pi*sqrt(r^2 - x^2)*(3R1^2 + r^2 - x^2) * 4*pi*(R2 + x)^2 = 32/3*pi^2*(R2 + x)^2*sqrt(r^2 - x^2)*(3R1^2 + r^2 - x^2)

We get a pretty wild volume formula: 2/3*pi^3*r^2*(24*R1^2*R2^2 + 6*r^2*(R1^2 + R2^2) + r^4) -- but notice that it DOES contain the naive 4*pi*R1^2 * 4*pi*R2^2 * pi*r^2 = 16*pi^3*R1^2*R2^2*r^2 = 2/3*pi^3*r^2 * 24*R1^2*R2^2, so it looks like we are on the right track. Also note that the formula is symmetrical in R1 and R2 -- necessary feature, considering that both spheres in 330-tiger are interchangeable.

The surface comes out as 4*pi^3*r*(8*R1^2*R2^2 + 4*r^2*(R1^2 + R2^2) + r^4).

Let's try out this new calculation on lower tigers:

220-tiger
Ring with area 4*pi*R1*y
Circle with circumference 2*pi*(R2 + x)
4*pi*R1*sqrt(r^2 - x^2) * 2*pi*(R2 + x) = 8*pi^2*R1*(R2 + x)*sqrt(r^2 - x^2)
Volume: 4*pi^3*R1*R2*r^2 -- exactly corresponding to previously computed formula.

320-tiger
Ring with area 4*pi*R1*y
Sphere with surface 4*pi*(R2 + x)^2
4*pi*R1*sqrt(r^2 - x^2) * 4*pi*(R2 + x)^2 = 16*pi^2*R1*(R2 + x)^2*sqrt(r^2 - x^2)
Volume: 2*pi^3*R1*r^2*(4*R2^2 + r^2) -- exactly corresponding to previously computed formula

320-tiger, try 2
Spherical shell with volume 8/3*pi*y*(3R1^2 + y^2)
Circle with circumference 2*pi*(R2 + x)
8/3*pi*sqrt(r^2 - x^2)*(3R1^2 + r^2 - x^2) * 2*pi*(R2 + x) = 16/3*pi^2*(R2 + x)*sqrt(r^2 - x^2)*(3*R1^2 + r^2 - x^2)
Volume: 2*pi^3*R2*r^2*(4*R1^2 + r^2) -- also works (only R1 and R2 are switched compared to previous computation).

So, again, this needs a professional attention to rigorously prove it (I'm not very good in calculus), but it SEEMS to be a method how to compute a toratope with arbitrary (II...I) string. Moreover, we can use it for (II) string without relying on (so far unproven) extension of Pappus' centroid theorem.

Let's try this for a triger ((II)(II)(II)).

Here we don't have hollow anything but whole tigers of volume 4*pi^3*R1*R2*y^2 along rings of circumference 2*pi*(R3 + x)
4*pi^3*R1*R2*(r^2 - x^2) * 2*pi*(R3 + x) = 8*pi^4*R1*R2*(R3 + x)*(r^2 - x^2)
Volume: 32/3*pi^4*R1*R2*R3*r^3
Surface: 32*pi^4*R1*R2*R3*r^2

On the other hand, Pappus' calculation as rotation of 221-tiger is:
16/3*pi^3*R1*R2*r^3 * 2*pi*R3 = 32/3*pi^4*R1*R2*R3*r^3

So it seems to work.

If we try for a 3330-triger ((III)(III)(III)), we get:
330-tigers of volume 2/3*pi^3*y^2*(24*R1^2*R2^2) + 6*y^2*(R1^2 + R2^2) + y^4),
smeared along spheres of surface 4*pi*(R3 + x)^2:
2/3*pi^3*(r^2 - x^2)*(24*R1^2*R2^2 + 6*(r^2 - x^2)*(R1^2 + R2^2) + (r^2 - x^2)^2) * 4*pi*(R3 + x)^2 = 8/3*pi^4*(r^2 - x^2)*(24*R1^2*R2^2 + 6*(r^2 - x^2)*(R1^2 + R2^2) + (r^2 - x^2)^2)*(R3 + x)^2
and the volume is
256/945*pi^4*r^3*(315*R1^2*R2^2*R3^2 + 63*r^2*(R1^2*R2^2 + R1^2*R3^2 + R2^2*R3^2) + 9*r^4*(R1^2 + R2^2 + R3^2) + r^6)
with surface
256/105*pi^4*r^2*(105*R1^2*R2^2*R3^2 + 35*r^2*(R1^2*R2^2 + R1^2*R3^2 + R2^2*R3^2) + 7*r^4*(R1^2 + R2^2 + R3^2) + r^6)

Note that the formulas are once again symmetrical with regard to R1, R2 and R3 who are all interchangeable for this toratope.
Fun!
Marek14
Pentonian
 
Posts: 1099
Joined: Sat Jul 16, 2005 6:40 pm

Re: Publishing an article on rotatopes and toratopes

Postby Marek14 » Tue Aug 05, 2014 12:56 pm

Let's try complete 6D analysis.

1. Hexasphere (IIIIII), (r)
Volume of pentasphere = 8/15*pi^2*r^5
Using pentasphere slices 8/15*pi^2*(r^2 - x^2)^(5/2)
Volume = 1/6*pi^3*r^6
Surface = pi^3*r^5

2. 51-torus ((IIIII)I), ((R)r)
Volume of hollow pentasphere = 8/15*pi^2*(R + r)^5 - 8/15*pi^2*(R - r)^5 = 16/15*pi^2*r*(5*R^4 + 10*R^2*r^2 + r^4)
Hollow pentasphere slices 16/15*pi^2*sqrt(r^2 - x^2)*(5*R^4 + 10*R^2*(r^2 - x^2) + (r^2 - x^2)^2)
Volume = 1/3*pi^3*r^2*(8*R^4 + 12*R^2*r^2 + r^4)
Surface = 2/3*pi^3*r*(8*R^4 + 24*R^2*r^2 + 3*r^4)

3. 411-ditorus (((IIII)I)I), (((R)r)s)
Volume of hollow 41-torus = 1/2*pi^3*R*(r + s)^2*(4R^2 + 3(r + s)^2) - 1/2*pi^3*R*(r - s)^2*(4R^2 + 3(r - s)^2) = 4*pi^3*R*r*s*(2*R^2 + 3*r^2 + 3*s^2)
Hollow 41-torus slices 4*pi^3*R*r*sqrt(s^2 - x^2)*(2*R^2 + 3*r^2 + 3*(s^2 - x^2))
Volume = 1/2*pi^4*R*r*s^2*(8*R^2 + 12*r^2 + 9*s^2)
Surface = 2*pi^4*R*r*s*(4*R^2 + 6*r^2 + 9*s^2)

4. 3111-tritorus ((((III)I)I)I), ((((R)r)s)t)
Volume of hollow 311-ditorus = pi^3*r*(s + t)^2*(8R^2 + 4r^2 + 3(s + t)^2) - pi^3*r*(s - t)^2*(8R^2 + 4r^2 + 3(s - t)^2) = 8*pi^3*r*s*t*(4*R^2 + 2*r^2 + 3*s^2 + 3*t^2)
Hollow 311-ditorus slices 8*pi^3*r*s*sqrt(t^2 - x^2)*(4*R^2 + 2*r^2 + 3*s^2 + 3*(t^2 - x^2))
Volume = pi^4*r*s*t^2*(16*R^2 + 8*r^2 + 12*s^2 + 9*t^2)
Surface = 4*pi^4*r*s*t*(8*R^2 + 4*r^2 + 6*s^2 + 9*t^2)

5. Tetratorus (((((II)I)I)I)I), (((((R)r)s)t)u)
Volume of hollow tritorus = 8*pi^4*R*r*s*(t + u)^2 - 8*pi^4*R*r*s*(t - u)^2 = 32*pi^4*R*r*s*t*u
Hollow tritorus slices 32*pi^4*R*r*s*t*sqrt(u^2 - x^2)
Volume = 16*pi^5*R*r*s*t*u^2
Surface = 32*pi^5*R*r*s*t*u
Tetratorus as rotation of tritorus:
Volume = 8*pi^4*r*s*t*u^2 * 2*pi*R = 16*pi^5*R*r*s*t*u^2
Surface = 16*pi^4*r*s*t*u * 2*pi*R = 32*pi^5*R*r*s*t*u

6. Tiger ditorus ((((II)(II))I)I), ((((R1)(R2)r)s)t)
Volume of hollow tiger torus = same as hollow tritorus = 32*pi^4*R1*R2*r*s*t
Hollow tiger torus slices 32*pi^4*R1*R2*r*s*sqrt(t^2 - x^2)
Volume = 16*pi^5*R1*R2*r*s*t^2
Surface = 32*pi^5*R1*R2*r*s*t
Tiger ditorus as rotation of tritorus:
Volume = 8*pi^4*R1*r*s*t^2 * 2*pi*R2 = 16*pi^5*R1*R2*r*s*t^2
Surface = 16*pi^4*R1*r*s*t * 2*pi*R2 = 32*pi^5*R1*R2*r*s*t

7. 2211-tritorus ((((II)II)I)I), ((((R)r)s)t)
Volume of hollow 221-ditorus = 2*pi^3*R*(s + t)^2*(4r^2 + (s + t)^2) - 2*pi^3*R*(s - t)^2*(4r^2 + (s - t)^2) = 16*pi^3*R*s*t*(2*r^2 + s^2 + t^2)
Hollow 221-ditorus slices 16*pi^3*R*s*sqrt(t^2 - x^2)*(2*r^2 + s^2 + (t^2 - x^2))
Volume = 2*pi^4*R*s*t^2*(8*r^2 + 4*s^2 + 3*t^2)
Surface = 8*pi^4*R*s*t*(4*r^2 + 2*s^2 + 3*t^2)
2211-tritorus as rotation of 311-ditorus:
Volume = pi^3*s*t^2*(8r^2 + 4s^2 + 3t^2) * 2*pi*R = 2*pi^4*R*s*t^2*(8*r^2 + 4*s^2 + 3*t^2)
Surface = 4*pi^3*s*t*(4r^2 + 2s^2 + 3t^2) * 2*pi*R = 8*pi^4*R*s*t*(4*r^2 + 2*s^2 + 3*t^2)

8. 320-tiger 1-torus (((III)(II))I), (((R1)(R2)r)s)
Volume of hollow 320-tiger = same as hollow 221-ditorus = 16*pi^3*R2*r*s*(2*R1^2 + r^2 + s^2)
Hollow 320-tiger slices 16*pi^3*R2*r*sqrt(s^2 - x^2)*(2*R1^2 + r^2 + (s^2 - x^2))
Volume = 2*pi^4*R2*r*s^2*(8*R1^2 + 4*r^2 + 3*s^2)
Surface = 8*pi^4*R2*r*s*(4*R1^2 + 2*r^2 + 3*s^2)
320-tiger 1-torus as rotation of 311-ditorus:
Volume = pi^3*r*s^2*(8R1^2 + 4r^2 + 3s^2) * 2*pi*R2 = 2*pi^4*R2*r*s^2*(8*R1^2 + 4*r^2 + 3*s^2)
Surface = 4*pi^3*r*s*(4R1^2 + 2r^2 + 3s^2) * 2*pi*R2 = 8*pi^4*R2*r*s*(4*R1^2 + 2*r^2 + 3*s^2)

9. Torus tiger torus ((((II)I)(II))I), ((((R1)r)(R2)s)t)
Volume of hollow torus tiger = same as hollow tritorus/hollow tiger torus = 32*pi^4*R1*r*R2*s*t
Hollow torus tiger slices 32*pi^4*R1*r*R2*s*sqrt(t^2 - x^2)
Volume = 16*pi^5*R1*r*R2*s*t^2
Surface = 32*pi^5*R1*r*R2*s*t
Torus tiger torus as rotation of tiger torus:
Volume = 8*pi^4*r*R2*s*t^2 * 2*pi*R1 = 16*pi^5*R1*r*R2*s*t^2
Surface = 16*pi^4*r*R2*s*t * 2*pi*R1 = 32*pi^5*R1*r*R2*s*t
Torus tiger torus as rotation of tritorus:
Volume = 8*pi^4*R1*r*s*t^2 * 2*pi*R2 = 16*pi^5*R1*r*R2*s*t^2
Surface = 16*pi^4*R1*r*s*t * 2*pi*R2 = 32*pi^5*R1*r*R2*s*t

10. 321-ditorus (((III)II)I), (((R)r)s)
Volume of hollow 32-torus = 4/15*pi^2*(r + s)^3*(5R^2 + 4(r + s)^2) - 4/15*pi^2*(r - s)^3*(5R^2 + 4(r - s)^2) = 8/15*pi^2*s*(15*R^2*r^2 + 5*R^2*s^2 + 20*r^4 + 40*r^2*s^2 + 4*s^4)
Hollow 32-torus slices 8/15*pi^2*sqrt(s^2 - x^2)*(15*R^2*r^2 + 5*R^2*(s^2 - x^2) + 20*r^4 + 40*r^2*(s^2 - x^2) + 4*(s^2 - x^2)^2)
Volume = 1/3*pi^3*s^2*(12*R^2*r^2 + 3*R^2*s^2 + 16*r^4 + 24*r^2*s^2 + 2*s^4)
Surface = 4/3*pi^3*s*(6*R^2*r^2 + 3*R^2*s^2 + 8*r^4 + 24*r^2*s^2 + 3*s^4)

11. 2121-tritorus ((((II)I)II)I), ((((R)r)s)t)
Volume of hollow 212-ditorus = 16/3*pi^3*R*r*(s + t)^3 - 16/3*pi^3*R*r*(s - t)^3 = 32/3*pi^3*R*r*t*(3*s^2 + t^2)
Hollow 212-ditorus slices 32/3*pi^3*R*r*sqrt(t^2 - x^2)*(3*s^2 + (t^2 - x^2))
Volume = 4*pi^4*R*r*t^2*(4*s^2 + t^2)
Surface = 16*pi^4*R*r*t*(2*s^2 + t^2)
2121-tritorus as rotation of 221-ditorus:
Volume = 2*pi^3*r*t^2*(4s^2 + t^2) * 2*pi*R = 4*pi^4*R*r*t^2*(4*s^2 + t^2)
Surface = 8*pi^3*r*t*(2s^2 + t^2) * 2*pi*R = 16*pi^4*R*r*t*(2*s^2 + t^2)

12. 221-tiger 1-torus (((II)(II)I)I), (((R1)(R2)r)s)
Volume of hollow 221-tiger = same as hollow 212-ditorus = 32/3*pi^3*R1*R2*s*(3*r^2 + s^2)
Hollow 221-tiger slices 32/3*pi^3*R1*R2*sqrt(s^2 - x^2)*(3*r^2 + (s^2 - x^2))
Volume = 4*pi^4*R1*R2*s^2*(4*r^2 + s^2)
Surface = 16*pi^4*R1*R2*s*(2*r^2 + s^2)
221-tiger 1-torus as rotation of 221-ditorus:
Volume = 2*pi^3*R1*s^2*(4r^2 + s^2) * 2*pi*R2 = 4*pi^4*R1*R2*s^2*(4*r^2 + s^2)
Surface = 8*pi^3*R1*s*(2r^2 + s^2) * 2*pi*R2 = 16*pi^4*R1*R2*s*(2*r^2 + s^2)

13. 231-ditorus (((II)III)I), (((R)r)s)
Volume of hollow 23-torus = pi^3*R*(r + s)^4 - pi^3*R*(r - s)^4 = 8*pi^3*R*r*s*(r^2 + s^2)
Hollow 23-torus slices 8*pi^3*R*r*sqrt(s^2 - x^2)*(r^2 + (s^2 - x^2))
Volume = pi^4*R*r*s^2*(4*r^2 + 3*s^2)
Surface = 4*pi^4*R*r*s*(2*r^2 + 3*s^2)
231-ditorus as rotation of 41-torus:
Volume = 1/2*pi^3*r*s^2*(4r^2 + 3s^2) * 2*pi*R = pi^4*R*r*s^2*(4*r^2 + 3*s^2)
Surface = 2*pi^3*r*s*(2r^2 + 3s^2) * 2*pi*R = 4*pi^4*R*r*s*(2*r^2 + 3*s^2)

14. 420-tiger ((IIII)(II)), ((R1)(R2)r)
Smearing hollow glome with volume 4*pi^2*R1*y*(R1^2 + y^2)
over circle with circumference 2*pi*(R2 + x)
4*pi^2*R1*sqrt(r^2 - x^2)*(R1^2 + r^2 - x^2) * 2*pi*(R2 + x) = 8*pi^3*R1*(R2 + x)*sqrt(r^2 - x^2)*(R1^2 + r^2 - x^2)
Volume = pi^4*R1*R2*r*(4*R1^2 + 3*r^2)
Surface = 4*pi^4*R1*R2*(2*R1^2 + 3*r^2)
Or:
Smearing ring with area 4*pi*R2*y
over glome with surface 2*pi^2*(R1 + x)^3
4*pi*R2*sqrt(r^2 - x^2) * 2*pi^2*(R1 + x)^3 = 8*pi^3*R2*(R1 + x)^2*sqrt(r^2 - x^2)
Volume = pi^4*R1*R2*r*(4*R1^2 + 3*r^2)
Surface = 4*pi^4*R1*R2*(2*R1^2 + 3*r^2)
420-tiger as rotation of 41-torus:
Volume = 1/2*pi^3*R1*r^2*(4R1^2 + 3r^2) * 2*pi*R2 = pi^4*R1*R2*r*(4*R1^2 + 3*r^2)
Surface = 2*pi^3*R1*r*(2R1^2 + 3r^2) * 2*pi*R2 = 4*pi^4*R1*R2*(2*R1^2 + 3*r^2)

15. 31-torus 20-tiger (((III)I)(II)), (((R1)r)(R2)s)
Smearing torisphere shell with volume 8*pi^2*r*y*(2*R1^2 + r^2 + y^2)
over circle with circumference 2*pi*(R2 + x)
8*pi^2*r*sqrt(s^2 - x^2)*(2*R1^2 + r^2 + s^2 - x^2) * 2*pi*(R2 + x) = 16*pi^3*r*(R2 + x)*sqrt(s^2 - x^2)*(2*R1^2 + r^2 + s^2 - x^2)
Volume = 2*pi^4*r*R2*s^2*(8*R1^2 + 4*r^2 + 3*s^2)
Surface = 8*pi^4*r*R2*s^2*(4*R1^2 + 2*r^2 + 3*s^2)
Or:
Smearing ring with area 4*pi*R2*y
over torisphere with surface 4*pi^2*(r + x)*(2*R1^2 + (r + x)^2)
4*pi*R2*sqrt(s^2 - x^2) * 4*pi^2*(r + x)*(2*R1^2 + (r + x)^2) = 16*pi^3*R2*(r + x)*sqrt(s^2 - x^2)*(2*R1^2 + (r + x)^2)
Volume = 2*pi^4*r*R2*s^2*(8*R1^2 + 4*r^2 + 3*s^2)
Surface = 8*pi^4*r*R2*s^2*(4*R1^2 + 2*r^2 + 3*s^2)
31-torus 20-tiger as rotation of 311-ditorus:
Volume = pi^3*r*s^2*(8R1^2 + 4r^2 + 3s^2) * 2*pi*R2 = 2*pi^4*r*R2*s^2*(8*R1^2 + 4*r^2 + 3*s^2)
Surface = 4*pi^3*r*s*(4R1^2 + 2r^2 + 3s^2) * 2*pi*R2 = 8*pi^4*r*R2*s*(4*R1^2 + 2*r^2 + 3*s^2)

16. Ditorus tiger ((((II)I)I)(II)), ((((R1)r)s)(R2)t)
Smearing ditorus shell with volume 16*pi^3*R1*r*s*y
over circle with circumference 2*pi*(R2 + x)
16*pi^3*R1*r*s*sqrt(t^2 - x^2) * 2*pi*(R2 + x) = 32*pi^4*R1*r*s*(R2 + x)*sqrt(t^2 - x^2)
Volume = 16*pi^5*R1*r*s*R2*t^2
Surface = 32*pi^5*R1*r*s*R2*t
Or:
Smearing ring with area 4*pi*R2*y
over ditorus with surface 8*pi^3*R1*r*(s + x)
4*pi*R2*sqrt(t^2 - x^2) * 8*pi^3*R1*r*(s + x) = 32*pi^4*R1*r*R2*(s + x)*sqrt(t^2 - x^2)
Volume = 16*pi^5*R1*r*s*R2*t^2
Surface = 32*pi^5*R1*r*s*R2*t
Ditorus tiger as rotation of torus tiger:
Volume = 8*pi^4*r*s*R2*t^2 * 2*pi*R1 = 16*pi^5*R1*r*s*R2*t^2
Surface = 16*pi^4*r*s*R2*t * 2*pi*R1 = 32*pi^5*R1*r*s*R2*t
Ditorus tiger as rotation of tritorus:
Volume = 8*pi^4*R1*r*s*t^2 * 2*pi*R2 = 16*pi^5*R1*r*s*R2*t^2
Surface = 16*pi^4*R1*r*s*t * 2*pi*R2 = 32*pi^5*R1*r*s*R2*t

17. Double tiger (((II)(II))(II)), (((R1)(R2)r)(R3)s)
Smearing tiger shell 16*pi^3*R1*R2*r*y
over circle with circumference 2*pi*(R3 + x)
16*pi^3*R1*R2*r*sqrt(s^2 - x^2) * 2*pi*(R3 + x) = 32*pi^4*R1*R2*r*(R3 + x)*sqrt(s^2 - x^2)
Volume = 16*pi^5*R1*R2*r*R3*s^2
Surface = 32*pi^5*R1*R2*r*R3*s
Or:
Smearing ring with area 4*pi*R3*y
over tiger with surface 8*pi^3*R1*R2*(r + x)
4*pi*R3*sqrt(s^2 - x^2) * 8*pi^3*R1*R2*(r + x) = 32*pi^4*R1*R2*R3*(r + x)*sqrt(s^2 - x^2)
Volume = 16*pi^5*R1*R2*r*R3*s^2
Surface = 32*pi^5*R1*R2*r*R3*s
Double tiger as rotation of torus tiger:
Volume = 8*pi^4*R1*r*R3*s^2 * 2*pi*R2 = 16*pi^5*R1*R2*r*R3*s^2
Surface = 16*pi^4*R1*r*R3*s * 2*pi*R2 = 32*pi^5*R1*R2*r*R3*s
Double tiger as rotation of tiger torus:
Volume = 8*pi^4*R1*R2*r*s^2 * 2*pi*R3 = 16*pi^5*R1*R2*r*R3*s^2
Surface = 16*pi^4*R1*R2*r*s * 2*pi*R3 = 32*pi^5*R1*R2*r*R3*s

18. 22-torus 20-tiger (((II)II)(II)), (((R1)r)(R2)s)
Smearing spheritorus shell with volume 16/3*pi^2*R1*y*(3r^2 + y^2)
over circle with circumference 2*pi*(R2 + x)
16/3*pi^2*R1*sqrt(s^2 - x^2)*(3r^2 + s^2 - x^2) * 2*pi*(R2 + x) = 32/3*pi^3*R1*(R2 + x)*sqrt(s^2 - x^2)*(3*r^2 + s^2 - x^2)
Volume = 4*pi^4*R1*R2*s^2*(4*r^2 + s^2)
Surface = 16*pi^4*R1*R2*s*(2*r^2 + s^2)
Or:
Smearing ring with area 4*pi*R2*y
over spheritorus with surface 8*pi^2*R1*(r + x)^2
4*pi*R2*sqrt(s^2 - x^2) * 8*pi^2*R1*(r + x)^2 = 32*pi^3*R1*R2*(r + x)^2*sqrt(s^2 - x^2)
Volume = 4*pi^4*R1*R2*s^2*(4*r^2 + s^2)
Surface = 16*pi^4*R1*R2*s*(2*r^2 + s^2)
22-torus 20-tiger as rotation of 320-tiger:
Volume = 2*pi^3*R2*s^2*(4r^2 + s^2) * 2*pi*R1 = 4*pi^4*R1*R2*s^2*(4*r^2 + s^2)
Surface = 8*pi^3*R2*s*(2r^2 + s^2) * 2*pi*R1 = 16*pi^4*R1*R2*s*(2*r^2 + s^2)
22-torus 20-tiger as rotation of 221-ditorus:
Volume = 2*pi^3*R1*s^2*(4r^2 + s^2) * 2*pi*R2 = 4*pi^4*R1*R2*s^2*(4*r^2 + s^2)
Surface = 8*pi^3*R1*s*(2r^2 + s^2) * 2*pi*R2 = 16*pi^4*R1*R2*s*(2*r^2 + s^2)

19. 42-torus ((IIII)II), ((R)r)
Volume of 41-torus = 1/2*pi^3*R*r^2*(4R^2 + 3r^2)
Using 41-torus slices 1/2*pi^3*R*(r^2 - x^2)*(4R^2 + 3(r^2 - x^2))
Volume = 8/15*pi^3*R*r^3*(5*R^2 + 3*r^2)
Surface = 8*pi^3*R*r^2*(R^2 + r^2)

20. 312-ditorus (((III)I)II), (((R)r)s)
Volume of 311-ditorus = pi^3*r*s^2*(8R^2 + 4r^2 + 3s^2)
Using 311-ditorus slices pi^3*r*(s^2 - x^2)*(8R^2 + 4r^2 + 3(s^2 - x^2))
Volume = 16/15*pi^3*r*s^3*(10*R^2 + 5*r^2 + 3*s^2)
Surface = 16*pi^3*r*s^2*(2*R^2 + r^2 + s^2)

21. 2112-tritorus ((((II)I)I)II), ((((R)r)s)t)
Volume of tritorus = 8*pi^4*R*r*s*t^2
Using tritorus slices 8*pi^4*R*r*s*(t^2 - x^2)
Volume = 32/3*pi^4*R*r*s*t^3
Surface = 32*pi^4*R*r*s*t^2
2112-tritorus as rotation of 212-ditorus:
Volume = 16/3*pi^3*r*s*t^3 * 2*pi*R = 32/3*pi^4*R*r*s*t^3
Surface = 16*pi^3*r*s*t^2 * 2*pi*R = 32*pi^4*R*r*s*t^2

22. 220-tiger 2-torus (((II)(II))II), (((R1)(R2)r)s)
Volume of tiger torus = 8*pi^4*R1*R2*r*s^2
Using tiger torus slices 8*pi^4*R1*R2*r*(s^2 - x^2)
Volume = 32/3*pi^4*R1*R2*r*s^3
Surface = 32*pi^4*R1*R2*r*s^2
220-tiger 2-torus as rotation of 212-ditorus:
Volume = 16/3*pi^3*R1*r*s^3 * 2*pi*R2 = 32/3*pi^4*R1*R2*r*s^3
Surface = 16*pi^3*R1*r*s^2 * 2*pi*R2 = 32*pi^4*R1*R2*r*s^2

23. 222-ditorus (((II)II)II), (((R)r)s)
Volume of 221-ditorus = 2*pi^3*R*s^2*(4r^2 + s^2)
Using 221-ditorus slices 2*pi^3*R*(s^2 - x^2)*(4r^2 + s^2 - x^2)
Volume = 32/15*pi^3*R*s^3*(5*r^2 + s^2)
Surface = 32/3*pi^3*R*s^2*(3*r^2 + s^2)
222-ditorus as rotation of 32-torus:
Volume = 4/15*pi^2*s^3*(5*r^2 + 4*s^2) * 2*pi*R = 8/15*pi^3*R*s^3*(5*r^2 + 4*s^2)
Surface = 4/3*pi^2*s^2*(3*r^2 + 4*s^2) * 2*pi*R = 8/3*pi^3*R*s^2*(3*r^2 + 4*s^2)
Note the discrepancy. There is probably a mistake somewhere, but I'm having problems finding it.
Can someone try to find the problem? I think I'm done for today :)

Groups:
Toratopes can be separated into groups that share a single set volume/surface formulas per dimension (and in each dimension it's just previous dimension * 2*pi*(new radius)). Each group is formed by a basic toratope (which is not a nonbisecting rotation of anything, therefore doesn't contain the (II) string) and all higher-dimensional toratopes formed by nonbisecting rotations.
Up to 6D, the groups are:
Point/circle group: Circle | torus | ditorus, tiger | tritorus, tiger torus, torus tiger | tetratorus, tiger ditorus, torus tiger torus, ditorus tiger, double tiger, duotorus tiger
Sphere group: Sphere | spheritorus | 212-ditorus, 221-tiger | 2112-tritorus, 220-tiger 2-torus, 21-torus 21-tiger, triger
Glome group: Glome | 23-torus | 213-ditorus, 222-tiger
Torisphere group: Torisphere | 221-ditorus, 320-tiger | 2121-tritorus, 221-tiger 1-torus, 22-torus 20-tiger, 21-torus 30-tiger
Pentasphere group: Pentasphere | 24-torus
41-torus group: 41-torus | 231-ditorus, 420-tiger
311-ditorus group: 311-ditorus | 2211-tritorus, 320-tiger 1-torus, 31-torus 20-tiger
32-torus group: 32-torus | 222-ditorus, 321-tiger
Hexasphere group: Hexasphere
51-torus grou*p: 51-torus
411-ditorus group: 411-ditorus
3111-tritorus group: 3111-tritorus
321-ditorus group: 321-ditorus
42-torus group: 42-torus
312-ditorus group: 312-ditorus
330-tiger group: 330-tiger
33-torus group: 33-torus
Last edited by Marek14 on Mon Aug 11, 2014 5:09 am, edited 2 times in total.
Marek14
Pentonian
 
Posts: 1099
Joined: Sat Jul 16, 2005 6:40 pm

Re: Publishing an article on rotatopes and toratopes

Postby ICN5D » Tue Aug 05, 2014 9:24 pm

Wow, awesome work, Marek. Also noted is the ridiculous huge surface and volume equations for 3330-tiger. Finding stronger generalizations here may be new and worthy stuff.
in search of combinatorial objects of finite extent
ICN5D
Pentonian
 
Posts: 1047
Joined: Mon Jul 28, 2008 4:25 am
Location: Orlando, FL

Re: Publishing an article on rotatopes and toratopes

Postby Marek14 » Sat Aug 09, 2014 12:01 pm

OK, I realized where's the problem with 222-ditorus: my calculations for 32-torus are wrong.

32-torus ((R)r)
Volume of torisphere = pi^2*r^2*(R^2 + r^2)
Using torisphere slices pi^2*(r^2 - x^2)*(R^2 + (r^2 - x^2))
Volume: 4/15*pi^2*r^3*(5R^2 + 4r^2)
Surface: 4/3*pi^2*r^2*(3R^2 + 4r^2)


But the volume of torisphere is actually pi^2*r^2*(4*R^2 + r^2), so...

32-torus ((R)r)
Volume of torisphere = pi^2*r^2*(4*R^2 + r^2)
Using torisphere slices pi^2*(r^2 - x^2)*(4*R^2 + (r^2 - x^2))
Volume: 16/15*pi^2*r^3*(5*R^2 + r^2)
Surface: 16/3*pi^2*r^2*(3*R^2 + r^2)

I edited my post with 5D calculations.

Now...

23. 222-ditorus (((II)II)II), (((R)r)s)
Volume of 221-ditorus = 2*pi^3*R*s^2*(4r^2 + s^2)
Using 221-ditorus slices 2*pi^3*R*(s^2 - x^2)*(4r^2 + s^2 - x^2)
Volume = 32/15*pi^3*R*s^3*(5*r^2 + s^2)
Surface = 32/3*pi^3*R*s^2*(3*r^2 + s^2)
222-ditorus as rotation of 32-torus:
Volume = 16/15*pi^2*s^3*(5*r^2 + s^2) * 2*pi*R = 32/15*pi^3*R*s^3*(5*r^2 + s^2)
Surface = 16/3*pi^2*s^2*(3*r^2 + s^2) * 2*pi*R = 32/3*pi^3*R*s^2*(3*r^2 + s^2)

and the calculations agree.

Let's continue in 6D:

24. 330-tiger ((III)(III)), ((R1)R2)r)
I've already calculated this:
Smearing hollow sphere with volume 8/3*pi*y*(3R1^2 + y^2)
over sphere with surface 4*pi*(R2 + x)^2
8/3*pi*sqrt(r^2 - x^2)*(3R1^2 + r^2 - x^2) * 4*pi*(R2 + x)^2 = 32/3*pi^2*(R2 + x)^2*sqrt(r^2 - x^2)*(3R1^2 + r^2 - x^2)
Volume = 2/3*pi^3*r^2*(24*R1^2*R2^2 + 6*r^2*(R1^2 + R2^2) + r^4)
Surface = 4*pi^3*r*(8*R1^2*R2^2 + 4*r^2*(R1^2 + R2^2) + r^4)

25. 21-torus 30-tiger (((II)I)(III)), (((R1)r)(R2)s)
Smearing torus shell with volume 8*pi^2*R1*r*y
over sphere with surface 4*pi*(R2 + x)^2
8*pi^2*R1*r*sqrt(s^2 - x^2) * 4*pi*(R2 + x)^2 = 32*pi^3*R1*r*sqrt(s^2 - x^2)*(R2 + x)^2
Volume = 4*pi^4*R1*r*s^2*(4*R2^2 + s^2)
Surface = 16*pi^4*R1*r*s*(2*R^2 + s^2)
Or:
Smearing hollow sphere with volume 8/3*pi*y*(3R2^2 + y^2)
over torus with surface 4*pi^2*R1*(r + x)
8/3*pi*sqrt(s^2 - x^2)*(3R2^2 + s^2 - x^2) * 4*pi^2*R1*(r + x) = 32/3*pi^3*R1*sqrt(s^2 - x^2)*(3R2^2 + s^2 - x^2)*(r + x)
Volume = 4*pi^4*R1*r*s^2*(4*R2^2 + s^2)
Surface = 16*pi^4*R1*r*s*(2*R^2 + s^2)
21-torus 30-tiger as rotation of 320-tiger:
Volume = 2*pi^3*r*s^2*(4*R2^2 + s^2) * 2*pi*R1 = 4*pi^4*R1*r*s^2*(4*R2^2 + s^2)
Surface = 8*pi^3*r*s*(2*R2^2 + s^2) * 2*pi*R1 = 16*pi^4*R1*r*s*(2*R^2 + s^2)

26. Duotorus tiger (((II)I)((II)I)), (((R1)r1)((R2)r2)s)
Smearing torus shell with volume 8*pi^2*R1*r1*y
over torus with surface 4*pi^2*R2*(r2 + x)
8*pi^2*R1*r1*sqrt(s^2 - x^2) * 4*pi^2*R2*(r2 + x) = 32*pi^4*R1*R2*r1*sqrt(s^2 - x^2)*(r2 + x)
Volume = 16*pi^5*R1*r1*R2*r2*s^2
Surface = 32*pi^5*R1*r1*R2*r2*s
Duotorus tiger as rotation of torus tiger:
Volume = 8*pi^4*R1*r1*r2*s^2 * 2*pi*R2 = 16*pi^5*R1*r1*R2*r2*s^2
Surface = 16*pi^4*R1*r1*r2*s * 2*pi*R2 = 32*pi^5*R1*r1*R2*r2*s

27. 321-tiger ((III)(II)I), ((R1)(R2)r)
Volume of 320-tiger = 2*pi^3*R2*r^2*(4R1^2 + r^2)
Using 320-tiger slices 2*pi^3*R2*(r^2 - x^2)*(4R1^2 + r^2 - x^2)
Volume = 32/15*pi^3*R2*r^3*(5*R1^2 + r^2)
Surface = 32/3*pi^3*R2*r^2*(3*R1^2 + r^2)
321-ditorus as rotation of 32-torus:
Volume = 16/15*pi^2*r^3*(5*R1^2 + r^2) * 2*pi*R2 = 32/15*pi^3*R2*r^3*(5*R1^2 + r^2)
Surface = 16/3*pi^2*r^2*(3*R1^2 + r^2) * 2*pi*R2 = 32/3*pi^3*R2*r^2*(3*R1^2 + r^2)

28. 21-torus 21-tiger (((II)I)(II)I), (((R1)r)(R2)s)
Volume of torus tiger = 8*pi^4*R1*R2*r*s^2
Using torus tiger slices 8*pi^4*R1*R2*r*(s^2 - x^2)
Volume = 32/3*pi^4*R1*r*R2*s^3
Surface = 32*pi^4*R1*r*R2*s^2
21-torus 21-tiger as rotation of 212-ditorus:
Volume = 16/3*pi^3*R1*r*s^3 * 2*pi*R2 = 32/3*pi^4*R1*r*R2*s^3
Surface = 16*pi^3*R1*r*s^2 * 2*pi*R2 = 32*pi^4*R1*r*R2*s^2
21-torus 21-tiger as rotation of 221-tiger:
Volume = 16/3*pi^3*r*R2*s^3 * 2*pi*R1 = 32/3*pi^4*R1*r*R2*s^3
Surface = 16*pi^3*r*R2*s^2 * 2*pi*R1 = 32*pi^4*R1*r*R2*s^2

29. 33-torus ((III)III), ((R)r)
Volume of 32-torus = 16/15*pi^2*r^3*(5*R^2 + r^2)
Using 32-torus slices 16/15*pi^2*(r^2 - x^2)^(3/2)*(5*R^2 + r^2 - x^2)
Volume = 1/3*pi^3*r^4*(6*R^2 + r^2)
Surface = 2*pi^3*r^3*(4*R^2 + r^2)

30. 213-ditorus (((II)I)III), (((R)r)s)
Volume of 212-ditorus = 16/3*pi^3*R*r*s^3
Using 212-ditorus slices = 16/3*pi^3*R*r*(s^2 - x^2)^(3/2)
Volume = 2*pi^4*R*r*s^4
Surface = 8*pi^4*R*r*s^3
213-ditorus as rotation of 23-torus:
Volume = pi^3*r*s^4 * 2*pi*R = 2*pi^4*R*r*s^4
Surface = 4*pi^3*r*s^3 * 2*pi*R = 8*pi^4*R*r*s^3

31. Triger ((II)(II)(II)), ((R1)(R2)(R3)r)
Smearing tigers of volume 4*pi^3*R1*R2*y^2
over circle of circumference 2*pi*(R3 + x)
4*pi^3*R1*R2*(r^2 - x^2) * 2*pi*(R3 + x) = 8*pi^4*R1*R2*(r^2 - x^2)*(R3 + x)
Volume = 32/3*pi^4*R1*R2*R3*r^3
Surface = 32*pi^4*R1*R2*R3*r^2
Triger as rotation of 221-tiger:
Volume = 16/3*pi^3*R1*R2*r^3 * 2*pi*R3 = 32/3*pi^4*R1*R2*R3*r^3
Surface = 16*pi^3*R1*R2*r^2 * 2*pi*R3 = 32*pi^4*R1*R2*R2*r^2

32. 222-tiger ((II)(II)II), ((R1)(R2)r)
Volume of 221-tiger = 16/3*pi^3*R1*R2*r^3
Using 221-tiger slices 16/3*pi^3*R1*R2*(r^2 - x^2)^(3/2)
Volume = 2*pi^4*R1*R2*r^4
Surface = 8*pi^4*R1*R2*r^3
222-tiger as rotation of 23-torus
Volume = pi^3*R1*r^4 * 2*pi*R2 = 2*pi^4*R1*R2*r^4
Surface = 4*pi^3*R1*r^3 * 2*pi*R2 = 8*pi^4*R1*R2*r^3

33. 24-torus ((II)IIII) ((R)r)
Volume of 23-torus = pi^3*R*r^4
Using 23-torus slices pi^3*R*(r^2 - x^2)^2
Volume = 16/15*pi^3*R*r^5
Surface = 16/3*pi^3*R*r^4
24-torus as rotation of pentasphere:
Volume = 8/15*pi^2*r^5 * 2*pi*R = 16/15*pi^3*R*r^5
Surface = 8/3*pi^2*r^4 * 2*pi*R = 16/3*pi^3*R*r^4

So the 6D cases are complete :)
Marek14
Pentonian
 
Posts: 1099
Joined: Sat Jul 16, 2005 6:40 pm

Re: Publishing an article on rotatopes and toratopes

Postby Marek14 » Mon Aug 11, 2014 5:45 am

As for 7D, it's probably no longer useful to take it case-by-case, instead we can look into groups.

For all these, R, r, s, t, u correspond to the radii of "mother figure" while r1, r2, etc. are additional radii introduced by rotations.

Circle group: Pentatorus, tiger tritorus, torus tiger ditorus, ditorus tiger torus, double tiger torus, duotorus tiger torus, tritorus tiger, tiger torus tiger, torus double tiger, ditorus/torus tiger, tiger/torus tiger
Volume = 32*pi^6*r1*r2*r3*r4*r5*r^2
Surface = 64*pi^6*r1*r2*r3*r4*r5*r

Sphere group: 21112-tetratorus, 220-tiger 12-ditorus, 21-torus 20-tiger 2-torus, 211-ditorus 21-tiger, 220-tiger 21-tiger, 21-torus 21-torus 1-tiger, torus triger
Volume = 64/3*pi^5*r1*r2*r3*r^3
Surface = 64*pi^5*r1*r2*r3*r^2

Glome group: 2113-tritorus, 220-tiger 3-torus, 21-torus 22-tiger, 2221-triger
Volume = 4*pi^5*r1*r2*r3*r^4
Surface = 16*pi^5*r1*r2*r3*r^3

Torisphere group: 21121-tetratorus, 220-tiger 21-ditorus, 21-torus 21-tiger 1-torus, triger torus, 212-ditorus 20-tiger, 221-tiger 20-tiger, 211-ditorus 30-tiger, 220-tiger 30-tiger, 22-torus 21-torus 0-tiger
Volume = 8*pi^5*r1*r2*r^2*(4*R^2 + r^2)
Surface = 32*pi^5*r1*r2*r*(2*R^2 + r^2)

Pentasphere group: 214-ditorus, 223-tiger
Volume = 32/15*pi^4*r1*r2*r^5
Surface = 32/3*pi^4*r1*r2*r^4

41-torus group: 2131-tritorus, 222-tiger 1-torus, 23-torus 20-tiger, 21-torus 40-tiger
Volume = 2*pi^5*r1*r2*r^2*(4*R^2 + 3*r^2)
Surface = 8*pi^5*r1*r2*r*(2*R^2 + 3*r^2)

311-ditorus group: 21211-tetratorus, 221-tiger 11-ditorus, 22-torus 20-tiger 1-torus, 21-torus 30-tiger 1-torus, 221-ditorus 20-tiger, 320-tiger 20-tiger, 31-torus 21-torus 0-tiger
Volume = 4*pi^5*r1*r2*r*s^2*(8R^2 + 4r^2 + 3s^2)
Surface = 16*pi^5*r1*r2*r*s*(4R^2 + 2r^2 + 3s^2)

32-torus group: 2122-tritorus, 221-tiger 2-torus, 22-torus 21-tiger, 21-torus 31-tiger, 3220-triger
Volume = 64/15*pi^4*r1*r2*r^3*(5*R^2 + r^2)
Surface = 64/3*pi^4*r1*r2*r^2*(3*R^2 + r^2)

Hexasphere group: 25-torus
Volume = 1/3*pi^4*r1*r^6
Surface = 2*pi^4*r1*r^5

51-torus group: 241-ditorus, 520-tiger
Volume = 2/3*pi^4*r1*r^2*(8*R^4 + 12*R^2*r^2 + r^4)
Surface = 4/3*pi^4*r1*r*(8*R^4 + 24*R^2*r^2 + 3*r^4)

411-ditorus group: 2311-tritorus, 420-tiger 1-torus, 41-torus 20-tiger
Volume = pi^5*r1*R*r*s^2*(8*R^2 + 12*r^2 + 9*s^2)
Surface = 4*pi^5*r1*R*r*s*(4*R^2 + 6*r^2 + 9*s^2)

3111-tritorus group: 22111-tetratorus, 320-tiger 11-ditorus, 31-torus 20-tiger 1-torus, 311-ditorus 20-tiger
Volume = 2*pi^5*r1*r*s*t^2*(16*R^2 + 8*r^2 + 12*s^2 + 9*t^2)
Surface = 8*pi^5*r1*r*s*t*(8*R^2 + 4*r^2 + 6*s^2 + 9*t^2)

321-ditorus group: 2221-tritorus, 321-tiger 1-torus, 32-torus 20-tiger
Volume = 2/3*pi^4*r1*s^2*(12*R^2*r^2 + 3*R^2*s^2 + 16*r^4 + 24*r^2*s^2 + 2*s^4)
Surface = 8/3*pi^4*r2*s*(6*R^2*r^2 + 3*R^2*s^2 + 8*r^4 + 24*r^2*s^2 + 3*s^4)

42-torus group: 232-ditorus, 421-tiger
Volume = 16/15*pi^4*r1*R*r^3*(5*R^2 + 3*r^2)
Surface = 16*pi^4*r1*R*r^2*(R^2 + r^2)

312-ditorus group: 2212-tritorus, 320-tiger 2-torus, 31-torus 21-tiger
Volume = 32/15*pi^4*r1*r*s^3*(10*R^2 + 5*r^2 + 3*s^2)
Surface = 32*pi^4*r1*r*s^2*(2*R^2 + r^2 + s^2)

330-tiger group: 22-torus 30-tiger
Volume = 4/3*pi^4*r1*r^2*(24*R1^2*R2^2 + 6*r^2*(R1^2 + R2^2) + r^4)
Surface = 8*pi^4*r1*r*(8*R1^2*R2^2 + 4*r^2*(R1^2 + R2^2) + r^4)

33-torus group: 223-ditorus, 322-tiger
Volume = 2/3*pi^4*r1*r^4*(6*R^2 + r^2)
Surface = 4*pi^4*r1*r^3*(4*R^2 + r^2)

New 7D groups:
Heptasphere group: Heptasphere (IIIIIII), (r)
Volume of hexasphere = 1/6*pi^3*r^6
Using hexasphere slices 1/6*pi^3*(r^2 - x^2)^3
Volume = 16/105*pi^3*r^7
Surface = 16/15*pi^3*r^6

61-torus group: 61-torus ((IIIIII)I), ((R)r)
Volume of hollow hexasphere = 1/6*pi^3*(R + r)^6 - 1/6*pi^3*(R - r)^6 = 2/3*pi^3*R*r*(3*R^4 + 10*R^2*r^2 + 3*r^4)
Hollow hexasphere slices 2/3*pi^3*R*sqrt(r^2 - x^2)*(3*R^4 + 10*R^2*(r^2 - x^2)+ 3*(r^2 - x^2)^2)
Volume = 1/8*pi^4*R*r^2*(8*R^4 + 20*R^2*r^2 + 5*r^4)
Surface = 1/4*pi^4*R*r*(8*R^4 + 40*R^2*r^2 + 15*r^4)

511-ditorus group: 511-ditorus (((IIIII)I)I), (((R)r)s)
Volume of hollow 51-torus = 1/3*pi^3*(r + s)^2*(8*R^4 + 12*R^2*(r + s)^2 + (r + s)^4) - 1/3*pi^3*(r - s)^2*(8*R^4 + 12*R^2*(r - s)^2 + (r - s)^4) = 4/3*pi^3*r*s*(8*R^4* + 24*R^2*r^2 + 24*R^2*s^2 + 3*r^4 + 10*r^2*s^2 + 3*s^4)
Hollow 51-torus slices: 4/3*pi^3*r*sqrt(s^2 - x^2)*(8*R^4 + 24*R^2*r^2 + 24*R^2*(s^2 - x^2) + 3*r^4 + 10*r^2*(s^2 - x^2) + 3*(s^2 - x^2)^2)
Volume = 1/12*pi^4*r*s^2*(64*R^4 + 192*R^2*r^2 + 144*R^2*s^2 + 24*r^4 + 60*r^2*s^2 + 15*s^4)
Surface = 1/6*pi^4*r*s*(64*R^4 + 192*R^2*r^2 + 288*R^2*s^2 + 24*r^4 + 120*r^2*s^2 + 45*s^4)

4111-tritorus group: 4111-tritorus ((((IIII)I)I)I), ((((R)r)s)t)
Volume of hollow 411-ditorus = 1/2*pi^4*R*r*(s + t)^2*(8*R^2 + 12*r^2 + 9*(s + t)^2) - 1/2*pi^4*R*r*(s - t)^2*(8*R^2 + 12*r^2 + 9*(s - t)^2) = 4*pi^4*R*r*s*t*(4*R^2 + 6*r^2 + 9*s^2 + 9*t^2)
Hollow 411-ditorus slices: 4*pi^4*R*r*s*sqrt(t^2 - x^2)*(4*R^2 + 6*r^2 + 9*s^2 + 9*(t^2 - x^2))
Volume = 1/2*pi^5*R*r*s*t^2*(16*R^2 + 24*r^2 + 36*s^2 + 27*t^2)
Surface = 2*pi^5*R*r*s*t*(8*R^2 + 12*r^2 + 18*s^2 + 27*t^2)

31111-tetratorus group: 31111-tetratorus (((((III)I)I)I)I), (((((R)r)s)t)u)
Volume of hollow 3111-tritorus = pi^4*r*s*(t + u)^2*(16*R^2 + 8*r^2 + 12*s^2 + 9*(t + u)^2) - pi^4*r*s*(t - u)^2*(16*R^2 + 8*r^2 + 12*s^2 + 9*(t - u)^2) = 8*pi^4*r*s*t*u*(8*R^2 + 4*r^2 + 6*s^2 + 9*t^2 + 9*u^2)
Hollow 3111-tritorus slices: 8*pi^4*r*s*t*sqrt(u^2 - x^2)*(8*R^2 + 4*r^2 + 6*s^2 + 9*t^2 + 9*(u^2 - x^2))
Volume = pi^5*r*s*t*u^2*(32*R^2 + 16*r^2 + 24*s^2 + 36*t^2 + 27*u^2)
Surface = 4*pi^5*r*s*t*u*(16*R^2 + 8*r^2 + 24*s^2 + 18*t^2 + 27*u^2)

3211-tritorus group: 3211-tritorus ((((III)II)I)I), ((((R)r)s)t)
Volume of hollow 321-ditorus = 1/3*pi^3*(s + t)^2*(12*R^2*r^2 + 3*R^2*(s + t)^2 + 16*r^4 + 24*r^2*(s + t)^2 + 2*(s + t)^4) - 1/3*pi^3*(s - t)^2*(12*R^2*r^2 + 3*R^2*(s - t)^2 + 16*r^4 + 24*r^2*(s - t)^2 + 2*(s - t)^4) = 8/3*pi^3*s*t*(6*R^2*r^2 + 3*R^2*s^2 + 3*R^2*t^2 + 8*r^4 + 24*r^2*s^2 + 24*r^2*t^2 + 3*s^4 + 10*s^2*t^2 + 3*t^4)
Hollow 321-ditorus slices: 8/3*pi^3*s*sqrt(t^2 - x^2)*(6*R^2*r^2 + 3*R^2*s^2 + 3*R^2*(t^2 - x^2) + 8*r^4 + 24*r^2*s^2 + 24*r^2*(t^2 - x^2) + 3*s^4 + 10*s^2*(t^2 - x^2) + 3*(t^2 - x^2)^2)
Volume = 1/6*pi^4*s*t^2*(48*R^2*r^2 + 24*R^2*s^2 + 18*R^2*t^2 + 64*r^4 + 192*r^2*s^2 + 144*r^2*t^2 + 24*s^4 + 60*s^2*t^2 + 15*t^4)
Surface = 1/3*pi^4*s*t*(48*R^2*r^2 + 24*R^2*s^2 + 36*R^2*t^2 + 64*r^4 + 192*r^2*s^2 + 288*r^2*t^2 + 24*s^4 + 120*s^2*t^2 + 45*t^4)

421-ditorus group: 421-ditorus (((IIII)II)I), (((R)r)s)
Volume of hollow 42-torus = 8/15*pi^3*R*(r + s)^3*(5*R^2 + 3*(r + s)^2) - 8/15*pi^3*R*(r - s)^3*(5*R^2 + 3*(r - s)^2) = 16/15*pi^3*R*s*(15*R^2*r^2 + 5*R^2*s^2 + 15*r^4 + 30*r^2*s^2 + 3*s^4)
Hollow 42-torus slices: 16/15*pi^3*R*sqrt(s^2 - x^2)*(15*R^2*r^2 + 5*R^2*(s^2 - x^2) + 15*r^4 + 30*r^2*(s^2 - x^2) + 3*(s^2 - x^2)^2)
Volume = pi^4*R*s^2*(8*R^2*r^2 + 2*R^2*s^2 + 8*r^4 + 12*r^2*s^2 + s^4)
Surface = 2*pi^4*R*s*(8*R^2*r^2 + 4*R^2*s^2 + 8*r^4 + 24*r^2*s^2 + 3*s^4)

3121-tritorus group: 3121-tritorus ((((III)I)II)I), ((((R)r)s)t)
Volume of hollow 312-ditorus: 16/15*pi^3*r*(s + t)^3*(10*R^2 + 5*r^2 + 3*(s + t)^2) - 16/15*pi^3*r*(s - t)^3*(10*R^2 + 5*r^2 + 3*(s - t)^2) = 32/15*pi^3*r*t*(30*R^2*s^2 + 10*R^2*t^2 + 15*r^2*s^2 + 5*r^2*t^2 + 15*s^4 + 30*s^2*t^2 + 3*t^4)
Hollow 312-ditorus slices: 32/15*pi^3*r*sqrt(t^2 - x^2)*(30*R^2*s^2 + 10*R^2*(t^2 - x^2) + 15*r^2*s^2 + 5*r^2*(t^2 - x^2) + 15*s^4 + 30*s^2*(t^2 - x^2) + 3*(t^2 - x^2)^2)
Volume = 2*pi^4*r*t^2*(16*R^2*s^2 + 4*R^2*t^2 + 8*r^2*s^2 + 2*r^2*t^2 + 8*s^4 + 12*s^2*t^2 + t^4)
Surface = 4*pi^4*r*t*(16*R^2*s^2 + 8*R^2*t^2 + 8*r^2*s^2 + 4*r^2*t^2 + 8*s^4 + 24*s^2*t^2 + 3*t^4)

330-tiger 1-torus group: 330-tiger 1-torus (((III)(III))I), (((R1)(R2)r)s)
Volume of hollow 330-tiger: 2/3*pi^3*(r + s)^2*(24*R1^2*R2^2 + 6*(r + s)^2*(R1^2 + R2^2) + (r + s)^4) - 2/3*pi^3*(r - s)^2*(24*R1^2*R2^2 + 6*(r - s)^2*(R1^2 + R2^2) + (r - s)^4) = 8/3*pi^3*r*s*(24*R1^2*R2^2 + 12*r^2*(R1^2 + R2^2) + 12*s^2*(R1^2 + R2^2) + 3*r^4 + 10*r^2*s^2 + 3*s^4)
Hollow 330-tiger slices: 8/3*pi^3*r*sqrt(s^2 - x^2)*(24*R1^2*R2^2 + 12*r^2*(R1^2 + R2^2) + 12*(s^2 - x^2)*(R1^2 + R2^2) + 3*r^4 + 10*r^2*(s^2 - x^2) + 3*(s^2 - x^2)^2)
Volume = 1/2*pi^4*r*s^2*(64*R1^2*R2^2 + 32*r^2*(R1^2 + R2^2) + 24*s^2*(R1^2 + R2^2) + 8*r^4 + 20*r^2*s^2 + 5*s^4)
Surface = pi^4*r*s*(64*R1^2*R2^2 + 32*r^2*(R1^2 + R2^2) + 48*s^2*(R1^2 + R2^2) + 8*r^4 + 40*r^2*s^2 + 15*s^4)

331-ditorus group: 331-ditorus (((III)III)I), (((R)r)s)
Volume of hollow 33-torus = 1/3*pi^3*(r + s)^4*(6*R^2 + (r + s)^2) - 1/3*pi^3*(r - s)^4*(6*R^2 + (r - s)^2) = 4/3*pi^3*r*s*(12*R^2*r^2 + 12*R^2*s^2 + 3*r^4 + 10*r^2*s^2 + 3*s^4)
Hollow 33-torus slices: 4/3*pi^3*r*sqrt(s^2 - x^2)*(12*R^2*r^2 + 12*R^2*(s^2 - x^2)+ 3*r^4 + 10*r^2*(s^2 - x^2) + 3*(s^2 - x^2)^2)
Volume = 1/4*pi^4*r*s^2*(32*R^2*r^2 + 24*R^2*s^2 + 8*r^4 + 20*r^2*s^2 + 5*s^4)
Surface = 1/2*pi^4*r*s*(32*R^2*r^2 + 48*R^2*s^2 + 8*r^4 + 40*r^2*s^2 + 15*s^4)

52-torus group: 52-torus ((IIIII)II), ((R)r)
Volume of 51-torus = 1/3*pi^3*r^2*(8*R^4 + 12*R^2*r^2 + r^4)
Using 51-torus slices: 1/3*pi^3*(r^2 - x^2)*(8*R^4 + 12*R^2*(r^2 - x^2) + (r^2 - x^2)^2)
Volume = 32/315*pi^3*r^3*(35*R^4 + 42*R^2*r^2 + 3*r^4)
Surface = 32/15*pi^3*r^2*(5*R^4 + 10*R^2*r^2 + r^4)

412-ditorus group: 412-ditorus (((IIII)I)II), (((R)r)s)
Volume of 411-ditorus = 1/2*pi^4*R*r*s^2*(8*R^2 + 12*r^2 + 9*s^2)
Using 411-ditorus slices: 1/2*pi^4*R*r*(s^2 - x^2)*(8*R^2 + 12*r^2 + 9*(s^2 - x^2))
Volume = 8/15*pi^4*R*r*s^3*(10*R^2 + 15*r^2 + 9*s^2)
Surface = 8*pi^4*R*r*s^2*(2*R^2 + 3*r^2 + 3*s^2)

3112-tritorus group: 3112-tritorus ((((III)I)I)II), ((((R)r)s)t)
Volume of 3111-tritorus = pi^4*r*s*t^2*(16*R^2 + 8*r^2 + 12*s^2 + 9*t^2)
Using 3111-tritorus slices: pi^4*r*s*(t^2 - x^2)*(16*R^2 + 8*r^2 + 12*s^2 + 9*(t^2 - x^2))
Volume = 16/15*pi^4*r*s*t^3*(20*R^2 + 10*r^2 + 15*s^2 + 9*t^2)
Surface = 16*pi^4*r*s*t^2*(4*R^2 + 2*r^2 + 3*s^2 + 3*t^2)

322-ditorus group: 322-ditorus (((III)II)II), (((R)r)s)
Volume of 321-ditorus = 1/3*pi^3*s^2*(12*R^2*r^2 + 3*R^2*s^2 + 16*r^4 + 24*r^2*s^2 + 2*s^4)
Using 321-ditorus slices: 1/3*pi^3*(s^2 - x^2)*(12*R^2*r^2 + 3*R^2*(s^2 - x^2) + 16*r^4 + 24*r^2*(s^2 - x^2) + 2*(s^2 - x^2)^2)
Volume = 16/315*pi^3*s^3*(105*R^2*r^2 + 21*R^2*s^2 + 140*r^4 + 168*r^2*s^2 + 12*s^4)
Surface = 16/15*pi^3*s^2*(15*R^2*r^2 + 5*R^2*s^2 + 20*r^4 + 40*r^2*s^2 + 4*s^4)

430-tiger group: 430-tiger ((IIII)(III)), ((R1)(R2)r)
Smearing hollow sphere with volume 8/3*pi*sqrt(r^2 - x^2)*(3*R2^2 + (r^2 - x^2)) over a glome with surface 2*pi^2*(R1 + x)^3
8/3*pi*sqrt(r^2 - x^2)*(3*R2^2 + (r^2 - x^2)) * 2*pi^2*(R1 + x)^3 = 16/3*pi^3*sqrt(r^2 - x^2)*(3*R2^2 + (r^2 - x^2))*(R1 + x)^3
Volume = pi^4*R1*r^2*(8*R1^2*R2^2 + 2*R1^2*r^2 + 6*R2^2*r^2 + r^4)
Surface = 2*pi^4*R1*r*(8*R1^2*R2^2 + 4*R1^2*r^2 + 12*R2^2*r^2 + 3*r^4)

31-torus 30-tiger group: 31-torus 30-tiger (((III)I)(III)), (((R1)r)(R2)s)
Smearing hollow sphere with volume 8/3*pi*sqrt(s^2 - x^2)*(3*R2^2 + (s^2 - x^2)) over a torisphere with surface 4*pi^2*(r + x)*(2*R1^2 + (r + x)^2)
8/3*pi*sqrt(s^2 - x^2)*(3*R2^2 + (s^2 - x^2)) * 4*pi^2*(r + x)*(2*R1^2 + (r + x)^2) = 32/3*pi^3*sqrt(s^2 - x^2)*(3*R2^2 + (s^2 - x^2))*(r + x)*(2*R1^2 + (r + x)^2)
Volume = 2*pi^4*r*s^2*(16*R1^2*R2^2 + 4*R1^2*s^2 + 8*r^2*R2^2 + 2*r^2*s^2 + 6*R2^2*s^2 + s^4)
Surface = 4*pi^4*r*s*(16*R1^2*R2^2 + 8*R1^2*s^2 + 8*r^2*R2^2 + 4*r^2*s^2 + 12*R2^2*s^2 + 3*s^4)

43-torus group: 43-torus ((IIII)III), ((R)r)
Volume of 42-torus = 8/15*pi^3*R*r^3*(5*R^2 + 3*r^2)
Using 42-torus slices: 8/15*pi^3*R*(r^2 - x^2)^(3/2)*(5*R^2 + 3*(r^2 - x^2))
Volume = 1/2*pi^4*R*r^4*(2*R^2 + r^2)
Surface = pi^4*R*r^3*(4*R^2 + 3*r^2)

313-ditorus group: 313-ditorus (((III)I)III), (((R)r)s)
Volume of 312-ditorus = 16/15*pi^3*r*s^3*(10*R^2 + 5*r^2 + 3*s^2)
Using 312-ditorus slices: 16/15*pi^3*r*(s^2 - x^2)^(3/2)*(10*R^2 + 5*r^2 + 3*(s^2 - x^2))
Volume = pi^4*r*s^4*(4*R^2 + 2*r^2 + s^2)
Surface = 2*pi^4*r*s^3*(8*R^2 + 4*r^2 + 3*s^2)

331-tiger group: 331-tiger ((III)(III)I), ((R1)(R2)r)
Volume of 330-tiger = 2/3*pi^3*r^2*(24*R1^2*R2^2 + 6*r^2*(R1^2 + R2^2) + r^4)
Using 330-tiger slices: 2/3*pi^3*(r^2 - x^2)*(24*R1^2*R2^2 + 6*(r^2 - x^2)*(R1^2 + R2^2) + (r^2 - x^2)^2)
Volume = 64/105*pi^3*r^3*(35*R1^2*R2^2 + 7*r^2*(R1^2 + R2^2) + r^4)
Surface = 64/15*pi^3*r^2*(15*R1^2*R2^2 + 5*r^2*(R1^2 + R2^2) + r^4)

34-torus group: 34-torus ((III)IIII), ((R)r)
Volume of 33-torus = 1/3*pi^3*r^4*(6*R^2 + r^2)
Using 33-torus slices: 1/3*pi^3*(r^2 - x^2)^2*(6*R^2 + (r^2 - x^2))
Volume = 32/105*pi^3*r^5*(7*R^2 + r^2)
Surface = 32/15*pi^3*r^4*(5*R^2 + r^2)
Marek14
Pentonian
 
Posts: 1099
Joined: Sat Jul 16, 2005 6:40 pm

Re: Publishing an article on rotatopes and toratopes

Postby wendy » Mon Aug 11, 2014 7:18 am

If these things are combs, then one might expect a surface that is a product of surfaces, with some sort of content term, and the volume might then be something like r.dS. Have you thought of dividing out the product of surfaces, and see what you get elsewise?
The dream you dream alone is only a dream
the dream we dream together is reality.
User avatar
wendy
Pentonian
 
Posts: 1805
Joined: Tue Jan 18, 2005 12:42 pm
Location: Brisbane, Australia

Re: Publishing an article on rotatopes and toratopes

Postby Marek14 » Mon Aug 11, 2014 7:21 am

wendy wrote:If these things are combs, then one might expect a surface that is a product of surfaces, with some sort of content term, and the volume might then be something like r.dS. Have you thought of dividing out the product of surfaces, and see what you get elsewise?


Generally, when the surfaces have non-Euclidean geometry, various extra terms start to accumulate.
Marek14
Pentonian
 
Posts: 1099
Joined: Sat Jul 16, 2005 6:40 pm

Re: Publishing an article on rotatopes and toratopes

Postby ICN5D » Tue Aug 12, 2014 2:52 am

Wow. All base -species toratopes have the same exact volume and surface. This means, when the minor diameter is all the same, the deflated n-frames are equally sized, too. Hmm, curious. And, furthermore, a direct relation to the presence of +2-spheres. I wonder, is there a bisecting rotation equivalent to Pappus' centroid theorem?
in search of combinatorial objects of finite extent
ICN5D
Pentonian
 
Posts: 1047
Joined: Mon Jul 28, 2008 4:25 am
Location: Orlando, FL

Re: Publishing an article on rotatopes and toratopes

Postby Marek14 » Tue Aug 12, 2014 5:39 am

ICN5D wrote:Wow. All base -species toratopes have the same exact volume and surface. This means, when the minor diameter is all the same, the deflated n-frames are equally sized, too. Hmm, curious. And, furthermore, a direct relation to the presence of +2-spheres. I wonder, is there a bisecting rotation equivalent to Pappus' centroid theorem?


Well, not ALL base-species toratopes are equal -- see triger -- but all the "binary" ones are.

As for bisecting rotation equivalent -- actually, you can use the centroid theorem in reverse. For example:

Area of half-circle = 1/2*pi*r^2
Volume of sphere = 4/3*pi*r^3

From Pappus' centroid theorem:
1/2*pi*r^2 * 2*pi*x = 4/3*pi*r^3
pi^2*r^2*x = 4/3*pi*r^3
pi*x = 4/3*r
x = 4/(3*pi)*r (roughly 0,425 r)

The x you get shows you the position of half-circle's centroid.
Marek14
Pentonian
 
Posts: 1099
Joined: Sat Jul 16, 2005 6:40 pm

Re: Publishing an article on rotatopes and toratopes

Postby ICN5D » Tue Aug 12, 2014 6:02 am

That makes sense, actually. The triger is built from a 2-sphere! So, you're thinking binary, trinary, quaternary, etc for number of major diameters?
in search of combinatorial objects of finite extent
ICN5D
Pentonian
 
Posts: 1047
Joined: Mon Jul 28, 2008 4:25 am
Location: Orlando, FL

Re: Publishing an article on rotatopes and toratopes

Postby Marek14 » Tue Aug 12, 2014 6:43 am

ICN5D wrote:That makes sense, actually. The triger is built from a 2-sphere! So, you're thinking binary, trinary, quaternary, etc for number of major diameters?


Well, "binary" in this sense means "binary tree", i.e. every set of brackets contains exactly two objects, either two more sets of brackets or two I's or set of brackets and an I.
Marek14
Pentonian
 
Posts: 1099
Joined: Sat Jul 16, 2005 6:40 pm

Re: Publishing an article on rotatopes and toratopes

Postby ICN5D » Wed Aug 20, 2014 5:20 am

Implicit surface equations 2D thru 6D

2D:
(II) - x^2 + y^2 - R1^2 = 0

3D:
(III) - x^2 + y^2 + z^2 - R1^2 = 0
((II)I) - (sqrt(x^2 + y^2) - R1)^2 + z^2 - R2^2 = 0

4D:
(IIII) - x^2 + y^2 + z^2 + w^2 - R1^2 = 0
((II)II) - (sqrt(x^2 + y^2) - R1)^2 + z^2 + w^2 - R2^2 = 0
((II)(II)) - (sqrt(x^2 + y^2) - R1a)^2 + (sqrt(z^2 + w^2) - R1b)^2 - R2^2 = 0
((III)I) - (sqrt(x^2 + y^2 + z^2) - R1)^2 + w^2 - R2^2 = 0
(((II)I)I) - (sqrt((sqrt(x^2 + y^2) - R1)^2 + z^2) - R2)^2 + w^2 - R3^2 = 0

5D:
(IIIII) - x^2 + y^2 + z^2 + w^2 + v^2 - R1^2 = 0
((II)III) - (sqrt(x^2 + y^2) - R1)^2 + z^2 + w^2 + v^2 - R2^2 = 0
((II)(II)I) - (sqrt(x^2 + y^2) - R1a)^2 + (sqrt(z^2 + w^2) - R1b)^2 + v^2 - R2^2 = 0
((III)II) - (sqrt(x^2 + y^2 + z^2) - R1)^2 + w^2 + v^2 - R2^2 = 0
(((II)I)II) - (sqrt((sqrt(x^2 + y^2) - R1)^2 + z^2) - R2)^2 + w^2 + v^2 - R3^2 = 0
((III)(II)) - (sqrt(x^2 + y^2 + z^2) - R1a)^2 + (sqrt(w^2 + v^2) - R1b)^2 - R2^2 = 0
(((II)I)(II)) - (sqrt((sqrt(x^2 + y^2) - R1a)^2 + z^2) - R2)^2 + (sqrt(w^2 + v^2) - R1b)^2 - R3^2 = 0
((IIII)I) - (sqrt(x^2 + y^2 + z^2 + w^2) - R1)^2 + v^2 - R2^2 = 0
(((II)II)I) - (sqrt((sqrt(x^2 + y^2) - R1)^2 + z^2 + w^2) - R2)^2 + v^2 - R3^2 = 0
(((II)(II))I) - ((sqrt(x^2 + y^2) - R1a)^2 + (sqrt(z^2 + w^2) - R1b)^2 - R2)^2 + v^2 - R3^2 = 0
(((III)I)I) - (sqrt((sqrt(x^2 + y^2 + z^2) - R1)^2 + w^2) - R2)^2 + v^2 - R3^2 = 0
((((II)I)I)I) - (sqrt((sqrt((sqrt(x^2 + y^2) - R1)^2 + z^2 ) - R2)^2 + w^2) - R3)^2 + v^2 - R4^2= 0

6D:
(IIIIII) - x^2 + y^2 + z^2 + w^2 + v^2 + u^2 - R1^2 = 0
((II)IIII) - (sqrt(x^2 + y^2) - R1)^2 + z^2 + w^2 + v^2 + u^2 - R2^2 = 0
((II)(II)II) - (sqrt(x^2 + y^2) - R1a)^2 + (sqrt(z^2 + w^2) - R1b)^2 + v^2 + u^2 - R2^2 = 0
((II)(II)(II)) - (sqrt(x^2 + y^2) - R1a)^2 + (sqrt(z^2 + w^2) - R1b)^2 + (sqrt(v^2 + u^2) - R1c)^2 - R2^2 = 0
((III)III) - (sqrt(x^2 + y^2 + z^2) - R1)^2 + w^2 + v^2 + u^2 - R2^2 = 0
(((II)I)III) - (sqrt((sqrt(x^2 + y^2) - R1)^2 + z^2) - R2)^2 + w^2 + v^2 + u^2 - R3^2 = 0
((III)(II)I) - (sqrt(x^2 + y^2 + z^2) - R1a)^2 + (sqrt(w^2 + v^2) - R1b)^2 + u^2 - R2^2 = 0
(((II)I)(II)I) - (sqrt((sqrt(x^2 + y^2) - R1a)^2 + z^2) - R2)^2 + (sqrt(w^2 + v^2) - R1b)^2 + u^2 - R3^2 = 0
((III)(III)) - (sqrt(x^2 + y^2 + z^2) - R1a)^2 + (sqrt(w^2 + v^2 + u^2) - R1b)^2 - R2^2 = 0
(((II)I)(III)) - (sqrt((sqrt(x^2 + y^2) - R1a)^2 + z^2) - R2)^2 + (sqrt(w^2 + v^2 + u^2) - R1b)^2 - R3^2 = 0
(((II)I)((II)I)) - (sqrt((sqrt(x^2 + y^2) - R1a)^2 + z^2) - R2a)^2 + (sqrt((sqrt(w^2 + v^2) - R1b)^2 + u^2) - R2b)^2 - R3^2 = 0
((IIII)II) - (sqrt(x^2 + y^2 + z^2 + w^2) - R1)^2 + v^2 + u^2 - R2^2 = 0
(((II)II)II) - (sqrt((sqrt(x^2 + y^2) - R1)^2 + z^2 + w^2) - R2)^2 + v^2 + u^2 - R3^2 = 0
(((II)(II))II) - ((sqrt(x^2 + y^2) - R1a)^2 + (sqrt(z^2 + w^2) - R1b)^2 - R2)^2 + v^2 + u^2 - R3^2 = 0
(((III)I)II) - (sqrt((sqrt(x^2 + y^2 + z^2) - R1)^2 + w^2) - R2)^2 + v^2 + u^2 - R3^2 = 0
((((II)I)I)II) - (sqrt((sqrt((sqrt(x^2 + y^2) - R1)^2 + z^2 ) - R2)^2 + w^2) - R3)^2 + v^2 + u^2 - R4^2= 0
((IIII)(II)) - (sqrt(x^2 + y^2 + z^2 + w^2) - R1a)^2 + (sqrt(v^2 + u^2) - R1b)^2 - R2^2 = 0
(((II)II)(II)) - (sqrt((sqrt(x^2 + y^2) - R1a)^2 + z^2 + w^2) - R2)^2 + (sqrt(v^2 + u^2) - R1b)^2 - R3^2 = 0
(((II)(II))(II)) - (sqrt((sqrt(x^2 + y^2) - R1a)^2 + (sqrt(z^2 + w^2) - R1b)^2) - R2)^2 + (sqrt(v^2 + u^2) - R1c)^2 - R3^2 = 0
(((III)I)(II)) - (sqrt((sqrt(x^2 + y^2 + z^2) - R1a)^2 + w^2) - R2)^2 + (sqrt(v^2 + u^2) - R1b)^2 - R3^2 = 0
((((II)I)I)(II)) - (sqrt((sqrt((sqrt(x^2 + y^2) - R1a)^2 + z^2) - R2)^2 + w^2) - R3)^2 + (sqrt(v^2 + u^2) - R1b)^2 - R4^2 = 0
((IIIII)I) - (sqrt(x^2 + y^2 + z^2 + w^2 + v^2) - R1)^2 + u^2 - R2^2 = 0
(((II)III)I) - (sqrt((sqrt(x^2 + y^2) - R1)^2 + z^2 + w^2 + v^2) - R2)^2 + u^2 - R3^2 = 0
(((II)(II)I)I) - ((sqrt(x^2 + y^2) - R1a)^2 + (sqrt(z^2 + w^2) - R1b)^2 + v^2 - R2)^2 + u^2 - R3^2 = 0
(((III)II)I) - (sqrt((sqrt(x^2 + y^2 + z^2) - R1)^2 + w^2 + v^2) - R2)^2 + u^2 - R3^2 = 0
((((II)I)II)I) - (sqrt((sqrt((sqrt(x^2 + y^2) - R1)^2 + z^2 ) - R2)^2 + w^2 + v^2) - R3)^2 + u^2 - R4^2= 0
(((III)(II))I) - ((sqrt(x^2 + y^2 + z^2) - R1a)^2 + (sqrt(w^2 + v^2) - R1b)^2 - R2)^2 + u^2 - R3^2 = 0
((((II)I)(II))I) - ((sqrt((sqrt(x^2 + y^2) - R1a)^2 + z^2) - R2)^2 + (sqrt(w^2 + v^2) - R1b)^2 - R3)^2 + u^2 - R4^2 = 0
(((IIII)I)I) - (sqrt((sqrt(x^2 + y^2 + z^2 + w^2) - R1)^2 + v^2) - R2)^2 + u^2 - R3^2 = 0
((((II)II)I)I) - (sqrt((sqrt((sqrt(x^2 + y^2) - R1)^2 + z^2 + w^2) - R2)^2 + v^2) - R3)^2 + u^2 - R4^2= 0
((((II)(II))I)I) - (sqrt(((sqrt(x^2 + y^2) - R1a)^2 + (sqrt(z^2 + w^2) - R1b)^2 - R2)^2 + v^2) - R3)^2 + u^2 - R4^2 = 0
((((III)I)I)I) - (sqrt((sqrt((sqrt(x^2 + y^2 + z^2) - R1)^2 + w^2 ) - R2)^2 + v^2) - R3)^2 + u^2 - R4^2= 0
(((((II)I)I)I)I) - (sqrt((sqrt((sqrt((sqrt(x^2 + y^2) - R1)^2 + z^2) - R2)^2 + w^2) - R3)^2 + v^2) - R4)^2 + u^2 - R5^2 = 0

EDIT 8/20 : Re-established the major diameters, according to the free majors
Last edited by ICN5D on Wed Aug 20, 2014 10:29 pm, edited 1 time in total.
in search of combinatorial objects of finite extent
ICN5D
Pentonian
 
Posts: 1047
Joined: Mon Jul 28, 2008 4:25 am
Location: Orlando, FL

Re: Publishing an article on rotatopes and toratopes

Postby Marek14 » Wed Aug 20, 2014 5:31 am

Thanks, ICN5D!
Marek14
Pentonian
 
Posts: 1099
Joined: Sat Jul 16, 2005 6:40 pm

Re: Publishing an article on rotatopes and toratopes

Postby ICN5D » Wed Aug 20, 2014 10:27 pm

Sure. Since you went into 7D, I'll have to reciprocate. I also edited the above list, changing the diameter hierarchy according to the Free Major Diameters rule :

7D Implicit Equations:

1. (IIIIIII) - x^2 + y^2 + z^2 + w^2 + v^2 + u^2 + t^2 - R1^2 = 0
2. ((IIIIII)I) - (sqrt(x^2 + y^2 + z^2 + w^2 + v^2 + u^2) - R1)^2 + t^2 - R2^2 = 0
3. (((IIIII)I)I) - (sqrt((sqrt(x^2 + y^2 + z^2 + w^2 + v^2) - R1)^2 + u^2) - R2)^2 + t^2 - R3^2 = 0
4. ((((IIII)I)I)I) - (sqrt((sqrt((sqrt(x^2 + y^2 + z^2 + w^2) - R1)^2 + v^2 ) - R2)^2 + u^2) - R3)^2 + t^2 - R4^2 = 0
5. (((((III)I)I)I)I) - (sqrt((sqrt((sqrt((sqrt(x^2 + y^2 + z^2) - R1)^2 + w^2) - R2)^2 + v^2) - R3)^2 + u^2) - R4)^2 + t^2 - R5^2 = 0
6. ((((((II)I)I)I)I)I) - (sqrt((sqrt((sqrt((sqrt((sqrt(x^2 + y^2) - R1)^2 + z^2) - R2)^2 + w^2) - R3)^2 + v^2) - R4)^2 + u^2) - R5)^2 + t^2 - R6^2 = 0
7. (((((II)(II))I)I)I) - (sqrt((sqrt(((sqrt(x^2 + y^2) - R1a)^2 + (sqrt(z^2 + w^2) - R1b)^2 - R2)^2 + v^2) - R3)^2 + u^2) - R4)^2 + t^2 - R5^2 = 0
8. (((((II)II)I)I)I) - (sqrt((sqrt((sqrt((sqrt(x^2 + y^2) - R1)^2 + z^2 + w^2) - R2)^2 + v^2) - R3)^2 + u^2) - R4)^2 + t^2 - R5^2 = 0
9. ((((III)(II))I)I) - (sqrt(((sqrt(x^2 + y^2 + z^2) - R1a)^2 + (sqrt(w^2 + v^2) - R1b)^2 - R2)^2 + u^2) - R3)^2 + t^2 - R4^2 = 0
10. (((((II)I)(II))I)I) - (sqrt(((sqrt((sqrt(x^2 + y^2) - R1a)^2 + z^2) - R2)^2 + (sqrt(w^2 + v^2) - R1b)^2 - R3)^2 + u^2) - R4)^2 + t^2 - R5^2 = 0
11. ((((III)II)I)I) - (sqrt((sqrt((sqrt(x^2 + y^2 + z^2) - R1)^2 + w^2 + v^2) - R2)^2 + u^2) - R3)^2 + t^2 - R4^2 = 0
12. (((((II)I)II)I)I) - (sqrt((sqrt((sqrt((sqrt(x^2 + y^2) - R1)^2 + z^2) - R2)^2 + w^2 + v^2) - R3)^2 + u^2) - R4)^2 + t^2 - R5^2 = 0
13. ((((II)(II)I)I)I) - (sqrt(((sqrt(x^2 + y^2) - R1)^2 + (sqrt(z^2 + w^2) - R2)^2 + v^2 - R3)^2 + u^2) - R4)^2 + t^2 - R5^2 = 0
14. ((((II)III)I)I) - (sqrt((sqrt((sqrt(x^2 + y^2) - R1)^2 + z^2 + w^2 + v^2) - R2)^2 + u^2) - R3)^2 + t^2 - R4^2 = 0
15. (((IIII)(II))I) - ((sqrt(x^2 + y^2 + z^2 + w^2) - R1)^2 + (sqrt(v^2 + u^2) - R2)^2 - R3)^2 + t^2 - R4^2 = 0
16. ((((III)I)(II))I) - ((sqrt((sqrt(x^2 + y^2 + z^2) - R1a)^2 + w^2) - R2)^2 + (sqrt(v^2 + u^2) - R1b)^2 - R3)^2 + t^2 - R4^2 = 0
17. (((((II)I)I)(II))I) - ((sqrt((sqrt((sqrt(x^2 + y^2) - R1a)^2 + z^2) - R2)^2 + w^2) - R3)^2 + (sqrt(v^2 + u^2) - R1b)^2 - R4)^2 + t^2 - R5^2 = 0
18. ((((II)(II))(II))I) - ((sqrt((sqrt(x^2 + y^2) - R1a)^2 + (sqrt(z^2 + w^2) - R1b)^2) - R2)^2 + (sqrt(v^2 + u^2) - R1c)^2 - R3)^2 + t^2 - R4^2 = 0
19. ((((II)II)(II))I) - ((sqrt((sqrt(x^2 + y^2) - R1a)^2 + z^2 + w^2) - R2)^2 + (sqrt(v^2 + u^2) - R1b)^2 - R3)^2 + t^2 - R4^2 = 0
20. (((IIII)II)I) - (sqrt((sqrt(x^2 + y^2 + z^2 + w^2) - R1)^2 + v^2 + u^2) - R2)^2 + t^2 - R3^2 = 0
21. ((((III)I)II)I) - (sqrt((sqrt((sqrt(x^2 + y^2 + z^2) - R1)^2 + w^2) - R2)^2 + v^2 + u^2) - R3)^2 + t^2 - R4^2 = 0
22. (((((II)I)I)II)I) - (sqrt((sqrt((sqrt((sqrt(x^2 + y^2) - R1)^2 + z^2) - R2)^2 + w^2) - R3)^2 + v^2 + u^2) - R4)^2 + t^2 - R5^2 = 0
23. ((((II)(II))II)I) - (sqrt(((sqrt(x^2 + y^2) - R1a)^2 + (sqrt(z^2 + w^2) - R1b)^2 - R2)^2 + v^2 + u^2) - R3)^2 + t^2 - R4^2 = 0
24. ((((II)II)II)I) - (sqrt((sqrt((sqrt(x^2 + y^2) - R1)^2 + z^2 + w^2 ) - R2)^2 + v^2 + u^2) - R3)^2 + t^2 - R4^2 = 0
25. (((III)(III))I) - ((sqrt(x^2 + y^2 + z^2) - R1a)^2 + (sqrt(w^2 + v^2 + u^2) - R1b)^2 - R2)^2 + t^2 - R3^2 = 0
26. ((((II)I)(III))I) - ((sqrt((sqrt(x^2 + y^2) - R1)^2 + z^2) - R2a)^2 + (sqrt(w^2 + v^2 + u^2) - R2b)^2 - R3)^2 + t^2 - R4^2 = 0
27. ((((II)I)((II)I))I) - ((sqrt((sqrt(x^2 + y^2) - R1a)^2 + z^2) -R2a)^2 + (sqrt((sqrt(w^2 + v^2) - R1b)^2 + u^2) - R2b)^2 - R3)^2 + t^2 - R4^2 = 0
28. (((III)(II)I)I) - ((sqrt(x^2 + y^2 + z^2) - R1a)^2 + (sqrt(w^2 + v^2) - R1b)^2 + u^2 - R2)^2 + t^2 - R3^2 = 0
29. ((((II)I)(II)I)I) - ((sqrt((sqrt(x^2 + y^2) - R1)^2 + z^2) - R2a)^2 + (sqrt(w^2 + v^2) - R2b)^2 + u^2 - R3)^2 + t^2 - R4^2 = 0
30. (((III)III)I) - (sqrt((sqrt(x^2 + y^2 + z^2) - R1)^2 + w^2 + v^2 + u^2) - R2)^2 + t^2 - R3^2 = 0
31. ((((II)I)III)I) - (sqrt((sqrt((sqrt(x^2 + y^2) - R1)^2 + z^2 ) - R2)^2 + w^2 + v^2 + u^2) - R3)^2 + t^2 - R4^2= 0
32. (((II)(II)(II))I) - ((sqrt(x^2 + y^2) - R1a)^2 + (sqrt(z^2 + w^2) - R1b)^2 + (sqrt(v^2 + u^2) - R1c)^2 - R2)^2 + t^2 - R3^2 = 0
33. (((II)(II)II)I) - ((sqrt(x^2 + y^2) - R1)^2 + (sqrt(z^2 + w^2) - R2)^2 + v^2 + u^2 - R3)^2 + t^2 - R4^2 = 0
34. (((II)IIII)I) - (sqrt((sqrt(x^2 + y^2) - R1)^2 + z^2 + w^2 + v^2 + u^2) - R2)^2 + t^2 - R3^2 = 0
35. ((IIIII)(II)) - (sqrt(x^2 + y^2 + z^2 + w^2 + v^2) - R1a)^2 + (sqrt(u^2 + t^2) - R1b)^2 - R2^2 = 0
36. (((IIII)I)(II)) - (sqrt((sqrt(x^2 + y^2 + z^2 + w^2) - R1a)^2 + v^2) - R2)^2 + (sqrt(u^2 + t^2) - R1b)^2 - R3^2 = 0
37. ((((III)I)I)(II)) - (sqrt((sqrt((sqrt(x^2 + y^2 + z^2) - R1a)^2 + w^2) - R2)^2 + v^2) - R3)^2 + (sqrt(u^2 + t^2) - R1b)^2 - R4^2 = 0
38. (((((II)I)I)I)(II)) - (sqrt((sqrt((sqrt((sqrt(x^2 + y^2) - R1a)^2 + z^2 ) - R2)^2 + w^2) - R3)^2 + v^2) - R4)^2 + (sqrt(u^2 + t^2) - R1b)^2 - R5^2 = 0
39. ((((II)(II))I)(II)) - (sqrt(((sqrt(x^2 + y^2) - R1a)^2 + (sqrt(z^2 + w^2) - R1b)^2 - R2)^2 + v^2) - R3)^2 + (sqrt(u^2 + t^2) - R1c)^2 - R4^2 = 0
40. ((((II)II)I)(II)) - (sqrt((sqrt((sqrt(x^2 + y^2) - R1a)^2 + z^2 + w^2) - R2)^2 + v^2) - R3)^2 + (sqrt(u^2 + t^2) - R1b)^2 - R4^2 = 0
41. (((III)(II))(II)) - (sqrt((sqrt(x^2 + y^2 + z^2) - R1a)^2 + (sqrt(w^2 + v^2) - R1b)^2) - R2)^2 + (sqrt(u^2 + t^2) - R1c)^2 - R3^2 = 0
43. (((III)II)(II)) - (sqrt((sqrt(x^2 + y^2 + z^2) - R1a)^2 + w^2 + v^2) - R2)^2 + (sqrt(u^2 + t^2) - R1b)^2 - R3^2 = 0
44. ((((II)I)II)(II)) - (sqrt((sqrt((sqrt(x^2 + y^2) - R1a)^2 + z^2) - R2)^2 + w^2 + v^2) - R3)^2 + (sqrt(u^2 + t^2) - R1b)^2 - R4^2 = 0
45. (((II)(II)I)(II)) - (sqrt((sqrt(x^2 + y^2) - R1a)^2 + (sqrt(z^2 + w^2) - R1b)^2 + v^2) - R2)^2 + (sqrt(u^2 + t^2) - R1c)^2 - R3^2 = 0
46. (((II)III)(II)) - (sqrt((sqrt(x^2 + y^2) - R1a)^2 + z^2 + w^2 + v^2) - R2)^2 + (sqrt(u^2 + t^2) - R1b)^2 - R3^2 = 0
47. ((IIIII)II) - (sqrt(x^2 + y^2 + z^2 + w^2 + v^2) - R1)^2 + u^2 + t^2 - R2^2 = 0
48. (((IIII)I)II) - (sqrt((sqrt(x^2 + y^2 + z^2 + w^2) - R1)^2 + v^2) - R2)^2 + u^2 + t^2 - R3^2 = 0
49. ((((III)I)I)II) - (sqrt((sqrt((sqrt(x^2 + y^2 + z^2) - R1)^2 + w^2) - R2)^2 + v^2) - R3)^2 + u^2 + t^2 - R4^2 = 0
50. (((((II)I)I)I)II) - (sqrt((sqrt((sqrt((sqrt(x^2 + y^2) - R1)^2 + z^2) - R2)^2 + w^2) - R3)^2 + v^2) - R4)^2 + u^2 + t^2 - R5^2 = 0
51. ((((II)(II))I)II) - (sqrt(((sqrt(x^2 + y^2) - R1a)^2 + (sqrt(z^2 + w^2) - R1b)^2 - R2)^2 + v^2) - R3)^2 + u^2 + t^2 - R4^2 = 0
52. ((((II)II)I)II) - (sqrt((sqrt((sqrt(x^2 + y^2) - R1)^2 + z^2 + w^2) - R2)^2 + v^2) - R3)^2 + u^2 + t^2 - R4^2 = 0
53. (((III)(II))II) - ((sqrt(x^2 + y^2 + z^2) - R1a)^2 + (sqrt(w^2 + v^2) - R1b)^2 - R2)^2 + u^2 + t^2 - R3^2 = 0
54. ((((II)I)(II))II) - ((sqrt((sqrt(x^2 + y^2) - R1a)^2 + z^2) - R2)^2 + (sqrt(w^2 + v^2) - R1b)^2 - R3)^2 + u^2 + t^2 - R4^2 = 0
55. (((III)II)II) - (sqrt((sqrt(x^2 + y^2 + z^2) - R1)^2 + w^2 + v^2) - R2)^2 + u^2 + t^2 - R3^2 = 0
56. ((((II)I)II)II) - (sqrt((sqrt((sqrt(x^2 + y^2) - R1)^2 + z^2) - R2)^2 + w^2 + v^2) - R3)^2 + u^2 + t^2 - R4^2 = 0
57. (((II)(II)I)II) - ((sqrt(x^2 + y^2) - R1a)^2 + (sqrt(z^2 + w^2) - R1b)^2 + v^2 - R2)^2 + u^2 + t^2 - R3^2 = 0
58. (((II)III)II) - (sqrt((sqrt(x^2 + y^2) - R1)^2 + z^2 + w^2 + v^2) - R2)^2 + u^2 + t^2 - R3^2 = 0
59. ((IIII)(III)) - (sqrt(x^2 + y^2 + z^2 + w^2) - R1a)^2 + (sqrt(v^2 + u^2 + t^2) - R1b)^2 - R2^2 = 0
60. (((II)I)(IIII)) - (sqrt((sqrt(x^2 + y^2) - R1a)^2 + z^2) - R2)^2 + (sqrt(w^2 + v^2 + u^2 + t^2) - R1b)^2 - R3^2 = 0
61. (((III)I)(III)) - (sqrt((sqrt(x^2 + y^2 + z^2) - R1a)^2 + w^2) - R2)^2 + (sqrt(v^2 + u^2 + t^2) - R1b)^2 - R3^2 = 0
62. (((III)I)((II)I)) - (sqrt((sqrt(x^2 + y^2 + z^2) - R1a)^2 + w^2) -R2a)^2 + (sqrt((sqrt(v^2 + u^2) - R1b)^2 + t^2) - R2b)^2 - R3^2 = 0
63. ((((II)I)I)(III)) - (sqrt((sqrt((sqrt(x^2 + y^2) - R1a)^2 + z^2) - R2)^2 + w^2) - R3)^2 + (sqrt(v^2 + u^2 + t^2) - R1b)^2 - R4^2 = 0
64. ((((II)I)I)((II)I)) - (sqrt((sqrt((sqrt(x^2 + y^2) - R1a)^2 + z^2) - R2a)^2 + w^2) - R3)^2 + (sqrt((sqrt(v^2 + u^2) - R1b)^2 + t^2) - R2b)^2 - R4^2 = 0
65. (((II)(II))(III)) - (sqrt((sqrt(x^2 + y^2) - R1a)^2 + (sqrt(z^2 + w^2) - R1b)^2) - R2)^2 + (sqrt(v^2 + u^2 + t^2) - R1c)^2 - R3^2 = 0
66. (((II)(II))((II)I)) - (sqrt((sqrt(x^2 + y^2) - R1a)^2 + (sqrt(z^2 + w^2) - R1b)^2) - R3)^2 + (sqrt((sqrt(v^2 + u^2) - R1c)^2 + t^2) - R2)^2 - R4^2 = 0
67. (((II)II)(III)) - (sqrt((sqrt(x^2 + y^2) - R1a)^2 + z^2 + w^2) - R2)^2 + (sqrt(v^2 + u^2 + t^2) - R1b)^2 - R3^2 = 0
68. (((II)II)((II)I)) - (sqrt((sqrt(x^2 + y^2) - R1a)^2 + z^2 + w^2) - R2a)^2 + (sqrt((sqrt(v^2 + u^2) - R1b)^2 + t^2) - R2b)^2 - R3^2 = 0
69. ((IIII)(II)I) - (sqrt(x^2 + y^2 + z^2 + w^2) - R1a)^2 + (sqrt(v^2 + u^2) - R1b)^2 + t^2 - R2^2 = 0
70. (((III)I)(II)I) - (sqrt((sqrt(x^2 + y^2 + z^2) - R1)^2 + w^2) - R2a)^2 + (sqrt(v^2 + u^2) - R2b)^2 + t^2 - R3^2 = 0
71. ((((II)I)I)(II)I) - (sqrt((sqrt((sqrt(x^2 + y^2) - R1a)^2 + z^2) - R2)^2 + w^2) - R3)^2 + (sqrt(v^2 + u^2) - R1b)^2 + t^2 - R4^2 = 0
72. (((II)(II))(II)I) - (sqrt((sqrt(x^2 + y^2) - R1a)^2 + (sqrt(z^2 + w^2) - R1b)^2) - R2)^2 + (sqrt(v^2 + u^2) - R1c)^2 + t^2 - R3^2 = 0
73. (((II)II)(II)I) - (sqrt((sqrt(x^2 + y^2) - R1a)^2 + z^2 + w^2) - R2)^2 + (sqrt(v^2 + u^2) - R1b)^2 + t^2 - R3^2 = 0
74. ((IIII)III) - (sqrt(x^2 + y^2 + z^2 + w^2) - R1)^2 + v^2 + u^2 + t^2 - R2^2 = 0
75. (((III)I)III) - (sqrt((sqrt(x^2 + y^2 + z^2) - R1)^2 + w^2) - R2)^2 + v^2 + u^2 + t^2 - R3^2 = 0
76. ((((II)I)I)III) - (sqrt((sqrt((sqrt(x^2 + y^2) - R1)^2 + z^2) - R2)^2 + w^2) - R3)^2 + v^2 + u^2 + t^2 - R4^2= 0
77. (((II)(II))III) - ((sqrt(x^2 + y^2) - R1a)^2 + (sqrt(z^2 + w^2) - R1b)^2 - R2)^2 + v^2 + u^2 + t^2 - R3^2 = 0
78. (((II)II)III) - (sqrt((sqrt(x^2 + y^2) - R1)^2 + z^2 + w^2) - R2)^2 + v^2 + u^2 + t^2 - R3^2 = 0
79. ((III)(III)I) - (sqrt(x^2 + y^2 + z^2) - R1a)^2 + (sqrt(w^2 + v^2 + u^2) - R1b)^2 + t^2 - R2^2 = 0
80. (((II)I)(III)I) - (sqrt((sqrt(x^2 + y^2) - R1a)^2 + z^2) - R2)^2 + (sqrt(w^2 + v^2 + u^2) - R1b)^2 + t^2 - R3^2 = 0
81. (((II)I)((II)I)I) - (sqrt((sqrt(x^2 + y^2) - R1a)^2 + z^2) -R2a)^2 + (sqrt((sqrt(w^2 + v^2) - R1b)^2 + u^2) - R2b)^2 + t^2 - R3^2 = 0
82. ((III)(II)(II)) - (sqrt(x^2 + y^2 + z^2) - R1a)^2 + (sqrt(w^2 + v^2) - R1b)^2 + (sqrt(u^2 + t^2) - R1c)^2 - R2^2 = 0
83. (((II)I)(II)(II)) - (sqrt((sqrt(x^2 + y^2) - R1a)^2 + z^2) - R2)^2 + (sqrt(w^2 + v^2) - R1b)^2 + (sqrt(u^2 + t^2) - R1c)^2 - R3^2 = 0
84. ((III)(II)II) - (sqrt(x^2 + y^2 + z^2) - R1a)^2 + (sqrt(w^2 + v^2) - R1b)^2 + u^2 + t^2 - R2^2 = 0
85. (((II)I)(II)II) - (sqrt((sqrt(x^2 + y^2) - R1a)^2 + z^2) - R2)^2 + (sqrt(w^2 + v^2) - R1b)^2 + u^2 + t^2 - R3^2 = 0
86. ((III)IIII) - (sqrt(x^2 + y^2 + z^2) - R1)^2 + w^2 + v^2 + u^2 + t^2 - R2^2 = 0
87. (((II)I)IIII) - (sqrt((sqrt(x^2 + y^2) - R1)^2 + z^2) - R2)^2 + w^2 + v^2 + u^2 + t^2 - R3^2 = 0
88. ((II)(II)(II)I) - (sqrt(x^2 + y^2) - R1a)^2 + (sqrt(z^2 + w^2) - R1b)^2 + (sqrt(v^2 + u^2) - R1c)^2 + t^2 - R2^2 = 0
89. ((II)(II)III) - (sqrt(x^2 + y^2) - R1a)^2 + (sqrt(z^2 + w^2) - R1b)^2 + v^2 + u^2 + t^2 - R2^2 = 0
90. ((II)IIIII) - (sqrt(x^2 + y^2) - R1)^2 + z^2 + w^2 + v^2 + u^2 + t^2 - R2^2 = 0
in search of combinatorial objects of finite extent
ICN5D
Pentonian
 
Posts: 1047
Joined: Mon Jul 28, 2008 4:25 am
Location: Orlando, FL

Re: Publishing an article on rotatopes and toratopes

Postby PWrong » Fri Aug 29, 2014 2:51 am

I think if we can get Mathematica to generate toratopes as regions, then it will be able to calculate the volumes and surface areas for us. Then we'll be able to find an inductive formula for any surface based on its parts. So for example we want a formula for the volume of (ABC) given the volumes of A, B and C.
User avatar
PWrong
Pentonian
 
Posts: 1599
Joined: Fri Jan 30, 2004 8:21 am
Location: Perth, Australia

Re: Publishing an article on rotatopes and toratopes

Postby ICN5D » Sun Aug 31, 2014 1:56 am

I downloaded the trial version, and I've been working with the tech support in how to graph them properly. I haven't been too aggressive with learning it. But, it seems that M10 is worth it, from what I can see. If you are adept with the script commands, then maybe you can help. How about a simple rotate function for a tiger:

(sqrt(x^2 + (y*sin(a))^2) - 2)^2 + (sqrt(z^2 + (y*cos(a))^2) - 2)^2 -0.5^2 = 0

Adjusting A will turn the tiger in 4D, slicing through oblique angles. Setting A = 0 ~ 1.57 is a 90 degree turn. How does one put this equation into M10 to graph the rotating tiger?
in search of combinatorial objects of finite extent
ICN5D
Pentonian
 
Posts: 1047
Joined: Mon Jul 28, 2008 4:25 am
Location: Orlando, FL

Re: Publishing an article on rotatopes and toratopes

Postby PWrong » Wed Sep 03, 2014 4:39 am

Just to animate it, I'd do it like this:

Code: Select all
Animate[ContourPlot3D[(Sqrt[x^2 + (y Sin[t])^2] - 2)^2 + ( Sqrt[z^2 + (y Cos[t])^2] - 2)^2 == 0.5^2, {x, -4, 4}, {y, -4, 4}, {z, -4, 4}, PlotPoints -> 20], {t, 0, \[Pi], 0.1}]
User avatar
PWrong
Pentonian
 
Posts: 1599
Joined: Fri Jan 30, 2004 8:21 am
Location: Perth, Australia

Re: Publishing an article on rotatopes and toratopes

Postby PWrong » Wed Sep 03, 2014 6:44 am

I just came up with this function that works only for closed toratopes i.e. surrounded by brackets.

Code: Select all
ToratopeFunction[x_] := If[x[[0]] === Tor,   Sqrt[Total[(Table[Toratope[tor], {tor, x[[2]]}])^2]] - x[[1]], x]


In Mathematica, a list is just a function with the head "List". You can map over any function just like a list. In this case, we're mapping over an undefined function Tor that acts like a list but with an extra number, which is a radius. So we can express a toratope like this:

Code: Select all
Interval: x
Circle: Tor[r, {x, y}]
Torus: Tor[R, {Tor[r, {x, y}], z}]
Tiger: Tor[R, {Tor[r1, {x, y}], Tor[r2, {z, w}]}]


Then ToratopeFunction converts one of these to the LHS of an implicit equation (except in the case of an interval). If the input doesn't start with the head Tor, then ToratopeFunction leaves it unchanged. If it does start with the head Tor, it takes the rss of ToratopeFunction of each element in the list, and subtracts the radius.

Code: Select all
In[285]:= ToratopeFunction[x]
Out[285]= x

In[286]:= ToratopeFunction[Tor[2, {x, y}]]
Out[286]= -2 + Sqrt[x^2 + y^2]

In[290]:= ToratopeFunction[Tor[R, {Tor[r1, {x, y}], z}]]
Out[290]= -R + Sqrt[(-r1 + Sqrt[x^2 + y^2])^2 + z^2]

In[289]:= ToratopeFunction[Tor[0.5, {Tor[2, {x, y}], Tor[2, {z, w}]}]]
Out[289]= -0.5 + Sqrt[(-2 + Sqrt[x^2 + y^2])^2 + (-2 + Sqrt[w^2 + z^2])^2]
User avatar
PWrong
Pentonian
 
Posts: 1599
Joined: Fri Jan 30, 2004 8:21 am
Location: Perth, Australia

Re: Publishing an article on rotatopes and toratopes

Postby ICN5D » Sat Sep 06, 2014 12:50 am

Awesome PWrong! Thanks for those functions. I'll have to wait for an extended use activation key for M10, as it took too long for tech support to get back to me, and provide the right script syntax. How many adjustable parameters can M10 have, in any one function? If it's more than 4, than I'm all on board with it.
in search of combinatorial objects of finite extent
ICN5D
Pentonian
 
Posts: 1047
Joined: Mon Jul 28, 2008 4:25 am
Location: Orlando, FL

PreviousNext

Return to Toratopes

Who is online

Users browsing this forum: No registered users and 1 guest