rotatopes as partitions

[Marek14]

So I propose to only call "toratopes" those shapes which are smooth, i.e. which don't have any lower elements than their surfaces. Torinder doesn't belong here, since it has two 2D faces where its three separate faces meet.

However, there should be, indeed, a higher category combining rotatopes and toratopes and containing their various products. Perhaps "rotopes" might be a good name?

Ok, in that case what I called a torus is a toratope, and what I called a toratope is a rotope. The formula above is actually a formula for the number of rotopes in n dimensions. What do you think of it, by the way?

I haven't had the time to look at the formula completely, but if it omits the beasts, it will fail to count the vast majority of interesting shapes.

Ah, yes - the thing is that if we limit the prismatic product to SURFACES and ignore the inner volumes, then we get rotatopes which are smooth, but can have much lower dimension than the one they are embedded in. The simplest case here is product of two circles - the duocylinder margin. The tiger should be then gotten by "inflating" this margin much as you get torus by inflating the circle.

I think I can see what you're getting at here. You spherate the surface of the duocylinder by a circle, but because the surface of the duocylinder is 2D, it's possible to do this without changing the dimension.

It's not the "surface" that is 2D (surface of duocylinder are 2 3D "sides") - it's the margin, the planar figure where these two "sides" meet. It's relationship to the duocylinder is much as relationship of wireframe cube model to the actual cube.

But the surface of the duocylinder is topologically the same as a torus, so does this mean duocylinder*circle is topologically equivalent to torus*circle = circle^3?

It's probable, yes.

This is all reminding me of the linear algebra stuff I learned this year. Basis vectors, subspaces and stuff would be really useful here, if only this was actually linear.

The parametric equations seem to stack pretty linearly.

[PWrong]

I haven't had the time to look at the formula completely, but if it omits the beasts, it will fail to count the vast majority of interesting shapes.

I wouldn't call it a vast majority. There's infinitely many rotopes including beasts, and infinitely many not including beasts. My formula counts the number of shapes that can be made by sums and products of n-spheres. It's possible the tiger is something like (circle + circle)* circle, or (2+2)*2. If multiplication isn't distributive, then this would be different from 2*2 + 2*2. Since we're only talking about the 2D form of the duocylinder, the dimension of this object could be only 4.

It's not the "surface" that is 2D (surface of duocylinder are 2 3D "sides") - it's the margin, the planar figure where these two "sides" meet. It's relationship to the duocylinder is much as relationship of wireframe cube model to the actual cube.

It's still an important shape though. I don't think margin is a good choice of word. Most objects can only be hollow or solid, but the duocylinder comes in three different forms. Some higher rotatopes will come in many different forms, like the (3+2+2). It all depends on which parameters you vary and which ones are constant.

But the surface of the duocylinder is topologically the same as a torus, so does this mean duocylinder*circle is topologically equivalent to torus*circle = circle^3?

It's probable, yes.

I guess it's even possible they're the same shape, just like the two different equations for the glome. We need a way to find out whether two objects are equivalent. If we could prove formally that the two different sets of equations for the glome are equivalent, that would be a good start.

The parametric equations seem to stack pretty linearly.

In a way they do, but they're all based on circles, which aren't linear.
I'm wondering whether we could use the concept of basis vectors here. A set of vectors are a basis for a space if they're independent and they describe every point in the space. So (2,1,0) (1,2,0) and (0,0,5) are a basis for R^3 (that is, 3D space), but (2,1,0), (4, 2, 0) and (0,0,1) aren't a basis, because the second vector is just a multiple of the first.

The set of unit vectors in spherical coordinates (r, theta and phi) is also a basis for R^3, but it's non-linear. If we don't count the radius vector, then we have a basis for the surface of the sphere.

Clearly, the unit vectors for the torus aren't a basis for R^3, because there are four parameters. But if we put the four unit vectors into a matrix, we might get some useful information from the properties of the matrix, like the null space or something.

[Marek14]

I haven't had the time to look at the formula completely, but if it omits the beasts, it will fail to count the vast majority of interesting shapes.

I wouldn't call it a vast majority. There's infinitely many rotopes including beasts, and infinitely many not including beasts. My formula counts the number of shapes that can be made by sums and products of n-spheres. It's possible the tiger is something like (circle + circle)* circle, or (2+2)*2. If multiplication isn't distributive, then this would be different from 2*2 + 2*2. Since we're only talking about the 2D form of the duocylinder, the dimension of this object could be only 4.

Every toratope can be written as sum a1+a2+a3+...+an where sum over a's gives its dimension, and each a is itself a toratope which can be further expressed as a sum (for example every 3 can stay as 3 (sphere) or split in (2+1) - torus). The sums only generate very small portion of these figures (those that contain only a single member not equal to 1).

It's not the "surface" that is 2D (surface of duocylinder are 2 3D "sides") - it's the margin, the planar figure where these two "sides" meet. It's relationship to the duocylinder is much as relationship of wireframe cube model to the actual cube.

It's still an important shape though. I don't think margin is a good choice of word.

I copied that term from Wendy.

Most objects can only be hollow or solid, but the duocylinder comes in three different forms. Some higher rotatopes will come in many different forms, like the (3+2+2). It all depends on which parameters you vary and which ones are constant.

Not sure I understand here. Duocylinder is a product of two solid circles, and all elements it contains can be derived from this:

solid circle x solid circle - "solid" duocylinger, dimension 4
solid circle x circle - "cell" of duocylinder, dimension 3. Since there are two ways to choose a solid circle, there are 2 cells.
circle x circle - "margin" of duocylinder, dimension 2.

But the surface of the duocylinder is topologically the same as a torus, so does this mean duocylinder*circle is topologically equivalent to torus*circle = circle^3?

It's probable, yes.

I guess it's even possible they're the same shape, just like the two different equations for the glome. We need a way to find out whether two objects are equivalent. If we could prove formally that the two different sets of equations for the glome are equivalent, that would be a good start.

It doesn't look like the same shape. One reason is that the dependencies are different. If you have circle^3, the three radii must be ordered, A > B > C to avoid self-intersection. It's different with tiger - Tiger only requires C to be the smallest from the three, but has no condition A > B or B > A (as you can switch those two parameters and still have the same figure).

As for the proof of glome equivalence, you can simply square the equations and add them together:

x = r cos a cos b cos c
y = r sin a cos b cos c
z = r sin b cos c
w = r sin c

If we put r=1 (for simplicity), you can see that x^2+y^2 = cos^2 b cos^2 c. This + z^2 = cos^2 c. This + w^2 = 1, so we have x^2+y^2+z^2+w^2 = 1.

For the alternate equations

x = r cos a cos c
y = r sin a cos c
z = r cos b sin c
w = r sin b sin c

x^2+y^2 = cos^2 c, z^2+w^2 = sin^2 c so once again
x^2+y^2+z^2+w^2 = 1. Easy!

[PWrong]

Every toratope can be written as sum a1+a2+a3+...+an where sum over a's gives its dimension, and each a is itself a toratope which can be further expressed as a sum (for example every 3 can stay as 3 (sphere) or split in (2+1) - torus). The sums only generate very small portion of these figures (those that contain only a single member not equal to 1).

I think that notation is different to the one I used. I used (2+1) for a cylinder, and 1*1 for a torus. Under my former definition, every toratope is a sum of torii, where each torii is either a sphere or a product of smaller spheres. The toratopes included all rotatopes, torii, and combinations like the torinder, but not beasts. Still, it's an interesting formula, and the property of not being beastlike is interesting in itself.

Not sure I understand here. Duocylinder is a product of two solid circles, and all elements it contains can be derived from this:

solid circle x solid circle - "solid" duocylinger, dimension 4
solid circle x circle - "cell" of duocylinder, dimension 3. Since there are two ways to choose a solid circle, there are 2 cells.
circle x circle - "margin" of duocylinder, dimension 2.

Let's rewrite this using Sn for an nD sphere (geometer's notation), and Bn for an nD ball, or solid sphere.

S2*S2 - dim 2, one cell
B2*S2, S2*B2 - dim 3, two cells
B2*B2 - dim 4, one cell

The shape I mentioned, (3+2+2), comes in 4 forms, ranging from 4 to 7 dimensions. Note the number of cells in each form.
S3*S2*S2 - dim 4, 1 cell
B3*S2*S2, S3*B2*S2, S3*S2*B2 - dim 5, 3 cells
B3*B2*S2, B3*S2*B3, S3*B2*B2 - dim 6, 3 cells
B3*B2*B2 - dim 4, 1 cell

In general, consider the rotatope (a1 + a2 + ... + ak), and let N = sum(a_i). This object has N parameters. k of these are the radii of spheres, and N-k are angles. We expect that angles always vary, so the object has at least (N-k) dimensions, but equal to or less than N. So this rotatope has (k+1) forms.

Now, we have k spheres, and we get an extra dimension for each solid one. Let r be the number of solid spheres. The total dimension of the object is then k + r. Now, for each cell, we have k spheres, r of which are spheres. So the number of cells is (k choose r) = k! / ((k-r)! r!). This explains why we saw the third row of pascals triangle earlier on. The actual object is the union of all of these cells.

Ok, now let's look at the space our object is "embedded" in. A hollow torus is the 2D form of the duocylinder embedded in 3D space. The solid torus consists of one cell of the 3D form.

From now on, I'll refer to the "nD form" of a rotatope with a dash, for instance, the "3-duocylinder", or the 3-(2+2).

Let's see what we can get from a 5D rotatope, like (2+3). This has three forms, just like the duocylinder, but it ranges from 3D to 5D.

I'm not sure if this is right, but it seems like we can embed the 3-(2+3)into 4D space as either a (2*3) torus or a (3*2) torus, both hollow.
We have to be careful with solid torii. Only the last sphere in a torus can be solid. (Try to picture the ordinary 3D torus with the larger radius filled in, and the smaller radius fixed).

The 4-(2+3) has two different cells; B2 + S3 and S2 + B3. The (B2+S3) can be embedded only as a solid (3*2) torus, because only the circle is solid, and the (S2+B3) can only be embedded as a solid (2*3) torus.

I'll work on this some more in a few days.

[Marek14]

The discrepancy in notation is because you are trying to describe all the rotopes, i.e. using both prism product and torus operators in the same figure. My + notation is derived solely for the toratopes.

Toratope is, under this notation, defined recursively as:

T = T1+T2+T3+...+Tn where T1 to Tn (n>=1) are toratopes of lower dimensions (whose dimensions add to n).

There is only a single toratope of dimension 1, of course. In dimension 2, the two options are:

2
(1+1)

However, there is one more rule for toratope derivations, and that is this:

n is equivalent to (1+1+1+1...) with n 1's.

This is because the TRUE normal form for toratopes involves ONLY 1's, which are then parenthesed in various ways. Numbers greater than 1 only occur in the formula to replace uniform sums of 1's, making it more readable and making the spheres that occur more apparent.

Your method excludes the beasts, and so it finds only what we could call INANIMATE toratopes:

Inanimate toratope in n dimensions can be defined as a set of all points that have a given distance from a given inanimate toratope of lower dimension (the spheres, of course, MUST be included for this definition).
For beasts, you also allow sets of all points that have a given distance from certain contorted surface like the duocylinder margin - in general, these surfaces have the same topology as some torus, but they are "straightened out", the cost being that they now require higher dimension to live in. Duocylinder margin is a 2D surface that requires 4 dimensions of space to exist.

Beasts won't arise naturally if you restrict yourself to the "distance from a lower-dimensional figure" approach. But once you find the parametric equations, they arise, as you saw for yourself.

By the way, I think your analyze of (3+2+2) is a bit incorrect. It has not 4, but 6 different forms, since B3*S2*S2 is different from S3*B2*S2, and likewise, B3*B2*S2 is not the same thing as S3*B2*B2. To see the difference, you just consider that B3*S2*S2 has 4 "closed" dimensions (where you return to the same place after a time), and 3 "bound" ones (where you hit the boundary sooner or later and can't continue further). S3*B2*S2 has 5 closed and 2 bound dimensions.

[PWrong]

The discrepancy in notation is because you are trying to describe all the rotopes, i.e. using both prism product and torus operators in the same figure. My + notation is derived solely for the toratopes.

I'll rewrite my formula here along with new definitions. A torus is an object constructed only by torus constructions, no prism products.
The number of nD torii is T(n) = 2^(n-2) - 1.
A rotope is constructed by taking a rotatope, and replacing each sphere with a torus. The total number of rotopes in n dimensions =

Code: Select all

        n      (T(i) + k(i))!
Sum ( Product (--------------))
k     i=1      T(i)! k(i)!


where T(n) = 2^(n-2) -1, and the sum is over all partitions of n.

This is because the TRUE normal form for toratopes involves ONLY 1's, which are then parenthesed in various ways. Numbers greater than 1 only occur in the formula to replace uniform sums of 1's, making it more readable and making the spheres that occur more apparent.

I think I see what you mean now. So you could write the 4D objects as 1+1+1+1, (1+1)+1+1, (1+1)+(1+1) and so on.

For beasts, you also allow sets of all points that have a given distance from certain contorted surface like the duocylinder margin - in general, these surfaces have the same topology as some torus, but they are "straightened out", the cost being that they now require higher dimension to live in. Duocylinder margin is a 2D surface that requires 4 dimensions of space to exist.

So a tiger is the 2D form of the duocylinder, spherated by a circle. So I might write it as (2+2)*2. How many rotopes, including beasts, are there in, say, 12 dimensions? It's hard to answer that with parametric equations.

By the way, I think your analyze of (3+2+2) is a bit incorrect. It has not 4, but 6 different forms, since B3*S2*S2 is different from S3*B2*S2, and likewise, B3*B2*S2 is not the same thing as S3*B2*B2.

I don't think you understood me. B3*S2*S2 isn't a form, it's a cell. A form is the union of the cells with the same number of solid spheres. So the dim 5 form of (3+2+2) is B3*S2*S2 U S3*B2*S2 U S3*S2*B2, where U is the union symbol. Note that even though this shape sits in 7D dimensions, it has 4 kinds of content, 4D, 5D, 6D and 7D. It's content is the sum of the contents of each cell.

Lets include wireframe-like objects, so that I can use a much simpler example. The cylinder (2+1)={1,1,0}, comes in 3 forms: solid, hollow and wireframe.

The wireframe, S2*S1, is simply a pair of circles. This counts as one cell.
The hollow cylinder is the union of two different cells: a pair of solid disks B2*S1, and a cylindrical "wall" S2*B1. These count as two different cells.
The solid cylinder just has a single cell, B2*B1.

Note that because a cylinder has 3 forms, it has 3 kinds of content: Volume, surface area, and the perimeter of the edges.
The area is the sum of the areas of each cell.

Generally, the number of forms is one plus the number of non-zero k_i 's in the partition. So (5+3+3+3+1+1) = (1*5 + 0*4 + 3*3 + 0*2 + 2*1) corresponds to the set k = {1, 0, 3, 0, 2}, which has 4 non-zero components, so this object has 1+3=4 forms.

It seems to me that every toratope, and in fact every rotope, is actually a cell from a rotatope. The solid and hollow torinders (2*2+1) are both cells of the (2+2+1). Both forms of the 2*3 and 3*2 torii are cells of the (3+2)
And finally, the 2*2*2 is a cell of the 2+2+2.
I'm sure the tiger is also related to the 2+2+2 somehow.

[Marek14]

I did say that tiger is the margin of duocylinder spherated by a circle.

So a beast is the 2D form of the duocylinder, spherated by a circle. I'd gathered that much.

And this means basically the same, except that "beast" is a more general category. So is "tiger", but the name also means the most basic member of the set, the 4D case.

It seems to me that every toratope, and in fact every rotope, is actually a cell from a rotatope. The solid and hollow torinders (2*2+1) are both cells of the (2+2+1).

What exactly is a (2+2+1) in this context?

How many rotopes, including beasts, are there in, say, 12 dimensions?

Not sure if we use the same definition - according to me, rotopes include both prism product and torus operations. Let's look just at number of toratopes for now:

2D - 1
3D - 3, 2+1 - 2
4D - 4, 3+1(2), 2+2, 2+1+1 - 5
5D - 5, 4+1(5), 3+2(2), 3+1+1(2), 2+2+1,2+1+1+1 - 12
6D - 6, 5+1(12), 4+2(5), 4+1+1(5), 3+3(3), 3+2+1(2), 3+1+1+1(2), 2+2+2, 2+2+1+1, 2+1+1+1+1 - 33
7D - 7, 6+1(33), 5+2(12), 5+1+1(12), 4+3(10), 4+2+1(5), 4+1+1+1(5), 3+3+1(3), 3+2+2(2), 3+2+1+1(2), 3+1+1+1+1(2), 2+2+2+1, 2+2+1+1+1, 2+1+1+1+1+1 - 89

It gets even more complicated in higher dimensions, have to think of it more.

[PWrong]

And this means basically the same, except that "beast" is a more general category. So is "tiger", but the name also means the most basic member of the set, the 4D case.

Sorry, I meant to say tiger. My post was a bit mixed up, I'll edit it now.

What exactly is a (2+2+1) in this context?

It's a rotatope. It's the linear extension of a duocylinder. I've been using + for the prism product, and * for the torus product. From your list of toratopes, it looks like you've got it the other way round. For instance, 2+1 is a cylinder, and 2*2 is a torus (2-sphere spherated by 2-sphere).
Each * subtracts a dimension, so dim(a*b) = a+b-1, and dim(a*b*c) = a+b+c - 2.

I think it's more sensible to use + for the prism product, because it's a much simpler operation. It also reminds us that each rotatope is a partition. The torus product is more complicated because it's not commutative.

Not sure if we use the same definition - according to me, rotopes include both prism product and torus operations. Let's look just at number of toratopes for now:

2D - 1
3D - 3, 2+1 - 2
4D - 4, 3+1(2), 2+2, 2+1+1 - 5
5D - 5, 4+1(5), 3+2(2), 3+1+1(2), 2+2+1,2+1+1+1 - 12

What do the numbers in brackets indicate?

[Marek14]

What exactly is a (2+2+1) in this context?

It's a rotatope. It's the linear extension of a duocylinder. I've been using + for the prism product, and * for the torus product. From your list of toratopes, it looks like you've got it the other way round. For instance, 2+1 is a cylinder, and 2*2 is a torus (2-sphere spherated by 2-sphere).
Each * subtracts a dimension, so dim(a*b) = a+b-1, and dim(a*b*c) = a+b+c - 2.

I think it's more sensible to use + for the prism product, because it's a much simpler operation. It also reminds us that each rotatope is a partition. The torus product is more complicated because it's not commutative.

For me, it's more sensible to use + in torii, since I define them with help of partitions, where addition is used. And prism product is easier to understand with multiplication sign, as it's, in fact, a Cartesian product of two polytopes - a well-known operation.

Not sure if we use the same definition - according to me, rotopes include both prism product and torus operations. Let's look just at number of toratopes for now:

2D - 1
3D - 3, 2+1 - 2
4D - 4, 3+1(2), 2+2, 2+1+1 - 5
5D - 5, 4+1(5), 3+2(2), 3+1+1(2), 2+2+1,2+1+1+1 - 12

What do the numbers in brackets indicate?

[/quote]
Number of toratopes in that particular category. For example, in 4D, there are 2 (3+1) toratopes, as the "3" can be further divided in 2 distinct ways.

[PWrong]

For me, it's more sensible to use + in torii, since I define them with help of partitions, where addition is used. And prism product is easier to understand with multiplication sign, as it's, in fact, a Cartesian product of two polytopes - a well-known operation.

Toratopes don't have anything to do with partitions, because the torus product doesn't commute, so order is important. How can you use an algebra with commutative multiplication, and non-commutative addition?

I do understand how to translate between our notations now, at least for toratopes. If you wrote 5+3+4+6, I would write 5*4*5*7. Under my notation, each * reduces the total dimension by one, so the shape is 18-dimensional. The advantage of my notation is that all spheres are treated the same.

Number of toratopes in that particular category. For example, in 4D, there are 2 (3+1) toratopes, as the "3" can be further divided in 2 distinct ways.

I still don't see what you mean. What can you divide the 3 into? A cylinder, a torus? :? Are you including the tiger as a toratope? I also still don't see exactly how you describe other objects, like rotatopes with this notation. If you just showed me all 10 of the 4D objects in your notation, and possibly the 25 5D objects, that might clear things up for me.

I'll list all of the objects we know up to 5D, in my notation, along with their names. I can list all the non-beast shapes up to any dimension, and I think I've developed a methed to count all the beasts as well. For the toratopes, I'll include your notation in brackets, and for the beasts, I'll use my old vector notation. This notation isn't really adequate for this kind of thing.

3D:
cube = 1+1+1
cylinder = 2+1
sphere = 3
torus = 2*2

4D:
5 rotatopes:
tetracube = 1+1+1+1
cubinder = 2+1+1
duocylinder = 2+2
spherinder = 3+1
glome = 4

3 toratopes/torii :
circle*sphere = 2*3 = (2+2)
sphere*circle = 3*2 = (3+1)
circle^3 = 2*2*2 = (2+1+1)

other rotopes:
torinder = 2*2 + 1

5+3+1 = 9 rotopes, consistent with my formula.

beasts:
tiger = (2+2)*2 = a(x,y) b(z,w) c(a,b)

10 objects in 4D in all.

5D:
7 rotatopes:
1+1+1+1+1
2+1+1+1
2+2+1
3+1+1
3+2
4+1
5

7 toratopes/torii
2*4 = (2+3)
3*3 = (3+2
4*2 = (4+1)
3*2*2 = (3+1+1)
2*3*2 = (2+2+1)
2*2*3 = (2+1+2)
2*2*2*2 = (2+1+1+1)

5 other rotopes:
2*2 + 1 + 1
2*2 + 2
2*3 + 1
3*2 + 1
2*2*2 + 1

7 + 7 + 5 = 19 rotopes, consistent with my formula

6 beasts:

(2+2)*2 + 1 = a(x,y) b(z,w) c(a,b) d(v)
(3+2)*2 = a(x,y,z) b(w,v) c(a,b)
(2*2+2)*2 = a(x,y) b(a,z) c(w,v) d(a+b, c)
(2+2)*3 = a(x,y) b(z,w) c(a,b,v)
(2+2)*2*2 = a(x,y) b(z,w) c(a,b) d(c,v)
(2+2)*(2+2) = a(x,y) b(z,w) c(a,b) d(c,v)

Clearly there are some obvious problems with the vector notation, so you should probably ignore the beasts for now. I've had a significant idea that should fix these problems, but it depends heavily on my notation, so I'd rather wait until you understand that before I explain my idea.

[Marek14]

For me, it's more sensible to use + in torii, since I define them with help of partitions, where addition is used. And prism product is easier to understand with multiplication sign, as it's, in fact, a Cartesian product of two polytopes - a well-known operation.

Toratopes don't have anything to do with partitions, because the torus product doesn't commute, so order is important. How can you use an algebra with commutative multiplication, and non-commutative addition?

Oh, but the addition IS commutative! Circle*sphere is, in my notation, expressed as 2+1+1, but it could as easily be 1+2+1 or 1+1+2 - all it would affect is the ordering of coordinates in the parametric equation. Sphere*circle, on the other hand, is 3+1 - or 1+3. They don't, in fact, differ in order, but rather in quality. I changed noncommutative torus multiplication in commutative addition.

I do understand how to translate between our notations now, at least for toratopes. If you wrote 5+3+4+6, I would write 5*4*5*7.

Wrong! 5+3+4+6 is a beast (better ordered as 6+5+4+3), which your notation can't well handle. 5*4*5*7 would be written as ((5+1+1+1)+1+1+1+1)+1+1+1+1+1+1 in my notation - it is a bit more cumbersome here.

Under my notation, each * reduces the total dimension by one, so the shape is 18-dimensional. The advantage of my notation is that all spheres are treated the same.

The advantage of my notation is that the precise parametric equation can be derived more easily. For example, in the 5*4*5*7 case, there is a 5D sphere, then you add 3 more dimensions in second term, which is an 8D sphere, then 4 more dimensions in 3rd term, 12D sphere, then 6 more in the fourth, 18D sphere.

Number of toratopes in that particular category. For example, in 4D, there are 2 (3+1) toratopes, as the "3" can be further divided in 2 distinct ways.

I still don't see what you mean. What can you divide the 3 into? A cylinder, a torus? :?

Sphere or torus. Those are two toratopes of dimension 3, so whenever we have 3 later, it can mean either of these two

Are you including the tiger as a toratope?

Yes.

I also still don't see exactly how you describe other objects, like rotatopes with this notation.

This notation is developed only for toratopes But for rotatopes, prismatic products can be also used. Torinder would be (2+1)*1. Torus*tiger duoprism would be (2+1)*(2+2). Duocylinder would be 2*2.

If you just showed me all 10 of the 4D objects in your notation, and possibly the 25 5D objects, that might clear things up for me.

Let's see...
4D: Tesseract (1*1*1*1), Cubinder (2*1*1), Duocylinder (2*2), Spherinder (3*1), Torinder ((2+1)*1), Glome (4), Sphere*circle (3+1), Circle^3 ((2+1)+1), Tiger (2+2), Circle*sphere (2+1+1)

5D: Penteract (1*1*1*1*1), Cubicircle (2*1*1*1), Dual cylinder (2*2*1), Spherisquare (3*1*1), Torisquare ((2+1)*1*1), Sphericircle (3*2), Toricircle ((2+1)*2), Glominder (4*1), Sphere*cylinder ((3+1)*1), Cylinder^3 (((2+1)+1)*1), Tigerinder ((2+2)*1), Circle*spherinder ((2+1+1)*1), Petaglome (5), Glome*circle (4+1), Sphere*circle*circle ((3+1)+1), Circle^4 (((2+1)+1)+1), Tiger*circle ((2+2)+1), Circle*sphere*circle ((2+1+1)+1), Sphere tiger (3+2), Torus tiger ((2+1)+2), Sphere^2 (3+1+1), Torus*sphere ((2+1)+1+1), Circtiger (2+2+1), Circle*glome (2+1+1+1)... I only count 24?

I'll list all of the objects we know up to 5D, in my notation, along with their names. I can list all the non-beast shapes up to any dimension, and I think I've developed a methed to count all the beasts as well. For the toratopes, I'll include your notation in brackets, and for the beasts, I'll use my old vector notation. This notation isn't really adequate for this kind of thing.

3D:
cube = 1+1+1
cylinder = 2+1
sphere = 3
torus = 2*2

4D:
5 rotatopes:
tetracube = 1+1+1+1
cubinder = 2+1+1
duocylinder = 2+2
spherinder = 3+1
glome = 4

3 toratopes/torii :
circle*sphere = 2*3 = (2+2)
sphere*circle = 3*2 = (3+1)
circle^3 = 2*2*2 = (2+1+1)

other rotopes:
torinder = 2*2 + 1

5+3+1 = 9 rotopes, consistent with my formula.

beasts:
tiger = (2+2)*2 = a(x,y) b(z,w) c(a,b)

10 objects in 4D in all.

5D:
7 rotatopes:
1+1+1+1+1
2+1+1+1
2+2+1
3+1+1
3+2
4+1
5

7 toratopes/torii
2*4 = (2+3)
3*3 = (3+2
4*2 = (4+1)
3*2*2 = (3+1+1)
2*3*2 = (2+2+1)
2*2*3 = (2+1+2)
2*2*2*2 = (2+1+1+1)

5 other rotopes:
2*2 + 1 + 1
2*2 + 2
2*3 + 1
3*2 + 1
2*2*2 + 1

7 + 7 + 5 = 19 rotopes, consistent with my formula

6 beasts:

(2+2)*2 + 1 = a(x,y) b(z,w) c(a,b) d(v)
(3+2)*2 = a(x,y,z) b(w,v) c(a,b)
(2*2+2)*2 = a(x,y) b(a,z) c(w,v) d(a+b, c)
(2+2)*3 = a(x,y) b(z,w) c(a,b,v)
(2+2)*2*2 = a(x,y) b(z,w) c(a,b) d(c,v)
(2+2)*(2+2) = a(x,y) b(z,w) c(a,b) d(c,w)

Clearly there are some obvious problems with the vector notation, so you should probably ignore the beasts for now.

What problems?
When I look at the beasts - They seem a bit weird, I don't understand the *2 there.
1 is tigerinder, 2 is sphere tiger, 3 is torus tiger, 4 should be probably circtiger, 5 tiger*circle, but what is 6?

a(x,y) b(z,w) c(a,b) d(c,w) - it uses the w coordinate twice. This is not a valid toratope.

I've had a significant idea that should fix these problems, but it depends heavily on my notation, so I'd rather wait until you understand that before I explain my idea.

[Marek14]

To make things clearer: the original reason why I decided to include tigers and other beasts as toratopes was the behaviour of the toratope cuts.

When you cut a toratope with a coordinate hyperplane, you can get either a single toratope of lower dimension, or two toratopes of lower dimension, which are of the same kind. In this second case, the toratopes will differ EITHER in one coordinate of their centre, OR in one of their radii.

Let's look at an example: Sphere*circle has parameters

a(x,y,z),b(a,w)

We derive the cuts by simply eliminating one of the coordinates:

x-cut - a(y,z),b(a,w) - torus
y-cut - a(x,z),b(a,w) - torus
z-cut - a(z,y),b(a,w) - torus
w-cut - a(x,y,z),b(a) - two spheres.
Now, here's the rule for treating things like b(a): this means take a and replace it by two figures, displaced by +b and -b. In this case, the two spheres are concentric.

From this we can see that eliminating a coordinate will lead to SINGLE toratope if and only if the parameter containing that coordinate has at least two parameters left, and to two toratopes otherwise.

Let's ask a different question: What options do we have if we want a cut in shape of torus, or two torii?

A single torus looks like this: a(x,y),b(a,z). We can add the fourth coordinate to either a or b, leading to:

a(x,y,w),b(a,z)
or
a(x,y),b(a,z,w)

The first is sphere*circle, the second is circle*sphere. Both of these, therefore, can be cut to form a single torus.

However, what if I want to get two torii? I have five letters in the equations, and the torus can be duplicated through any of them. We only have four cases, since x and y are symmetrical, though. These all require another parameter, c:

c(x),a(c,y),b(a,z) are two torii displaced in the x dimension. It's a cut of c(x,w),a(c,y),b(a,z), or circle^3

a(x,y),c(a),b(a,z) are two torii with different external diameters. It's a cut of a(x,y),c(a,w),b(a,z), which is, again, circle^3.

a(x,y),b(a,z),c(b) are two torii with different internal diameters. It's a cut of a(x,y),b(a,z),c(b,w), once again - circle^3.

But there is one additional possibility, a(x,y),c(z),b(a,c)! This is two torii displaced in the z direction ("above" each other). And this is a cut of a(x,y),c(z,w),b(a,c), or tiger! This is the reason why tigers are neccessary to obtain the full theory - otherwise, some combinations of parameters would give nonexistent hyperplane cuts.

Observe the cuts in 5D:

A single GLOME - a(x,y,z,w) - is a cut of petaglome a(x,y,z,w,v)
Two displaced glomes - b(x),a(b,y,z,w) - are a cut of circle*glome b(x,v),a(b,y,z,w)
Two concentric glomes - a(x,y,z,w),b(a) - are a cut of glome*circle a(x,y,z,w),b(a,v)

A single SPHERE*CIRCLE - a(x,y,z),b(a,w) - is a cut of either a glome*circle a(x,y,z,v),b(a,w) or of a sphere^2 a(x,y,z),b(a,w,v)
Two sphere*circles displaced in x,y, or z coordinates - c(x),a(c,y,z),b(a,w) - are a cut of circle*sphere*circle c(x,v),a(c,y,z),b(a,w)
Two sphere*circles displaced in w coordinate - a(x,y,z),c(w),b(a,c) - are a cut of sphere tiger a(x,y,z),c(w,v),b(a,c)
Two concentric sphere*circles differing in diameter a - a(x,y,z),c(a),b(c,w) - are a cut of sphere*circle*circle a(x,y,z),c(a,v),b(c,w)
Two concentric sphere*circles differing in diameter b - a(x,y,z),b(a,w),c(b) - are a cut of sphere*circle*circle a(x,y,z),b(a,w),c(b,v)

A single CIRCLE^3 - a(x,y),b(a,z),c(b,w) - is a cut of either a sphere*circle*circle a(x,y,v),b(a,z),c(b,w), or of a circle*sphere*circle a(x,y),b(a,z,v),c(b,w), or of a circle*circle*sphere a(x,y),b(a,z),c(b,w,v).
Two circles^3 displaced in x or y dimension - d(x),a(d,y),b(a,z),c(b,w) - are a cut of circle^4 - d(x,v),a(d,y),b(a,z),c(b,w)
Two circles^3 displaced in z dimension - a(x,y),d(z),b(a,d),c(b,w) - are a cut of tiger*circle a(x,y),d(z,v),b(a,d),c(b,w)
Two circles^3 displaced in w dimension - a(x,y),b(a,z),d(w),c(b,d) - are a cut of torus tiger a(x,y),b(a,z),d(w,v),c(b,d)
Two concentric circles^3 differing in diameter a - a(x,y),d(a),b(d,z),c(b,w) - are a cut of circle^4 a(x,y),d(a,v),b(d,z),c(b,w)
Two concentric circles^3 differing in diameter b - a(x,y),b(a,z),d(b),c(d,w) - are a cut of circle^4 a(x,y),b(a,z),d(b,v),c(d,w)
Two concentric circles^3 differing in diameter c - a(x,y),b(a,z),c(b,w),d(c) - are a cut of circle^4 a(x,y),b(a,z),c(b,w),d(c,v)

A single TIGER - a(x,y),b(z,w),c(a,b) - is a cut of either a sphere tiger a(x,y,v),b(z,w),c(a,b), or of a circtiger a(x,y),b(z,w),c(a,b,v)
Two tigers displaced in any one dimension - d(x),a(d,y),b(z,w),c(a,b) - are a cut of a torus tiger d(x,v),a(d,y),b(z,w),c(a,b)
Two concentric tigers differing in diameter a or b - a(x,y),d(a),b(z,w),c(d,b) - are a cut of torus tiger a(x,y),d(a,v),b(z,w),c(d,b)
Two concentric tigers differing in diameter c - a(x,y),b(z,w),c(a,b),d(c) - are a cut of a tiger*circle a(x,y),b(z,w),c(a,b),d(c,v)

A single CIRCLE*SPHERE - a(x,y),b(a,z,w) - is a cut of either a sphere^2 a(x,y,v),b(a,z,w), or of a circle*glome a(x,y),b(a,z,w,v)
Two circle*spheres displaced in x or y dimension - c(x),a(c,y),b(a,z,w) - are a cut of circle*circle*sphere c(x,v),a(c,y),b(a,z,w)
Two circle*spheres displaced in z or w dimension - a(x,y),c(z),b(a,c,w) - are a cut of circtiger a(x,y),c(z,v),b(a,c,w)
Two concentric circle*spheres differing in diameter a - a(x,y),c(a),b(c,z,w) - are a cut of circle*circle*sphere a(x,y),c(a,v),b(c,z,w)
Two concentric circle*spheres differing in diameter b - a(x,y),b(a,z,w),c(b) - are a cut of circle*circle*sphere a(x,y),b(a,z,w),c(b,v)

You can see that thanks to inclusion of the beasts all possible cuts are real cuts of higher-dimensional toratopes.

Going in reverse, we find 1 cut for petaglome, 2 for glome*circle, 3 for sphere*circle*circle, 4 for circle^4, 2 for tiger*circle, 3 for circle*sphere*circle, 2 for sphere tiger, 3 for torus tiger, 2 for sphere^2, 3 for circle*circle*sphere, 2 for circtiger, and 2 for circle*glome. All 29 are accounted for.

[PWrong]

Oh, but the addition IS commutative! Circle*sphere is, in my notation, expressed as 2+1+1, but it could as easily be 1+2+1 or 1+1+2 - all it would affect is the ordering of coordinates in the parametric equation. Sphere*circle, on the other hand, is 3+1 - or 1+3. They don't, in fact, differ in order, but rather in quality. I changed noncommutative torus multiplication in commutative addition.

Ah, I think I see what you mean now, but I'm not convinced. What about these two shapes?
2*3*2 = (2+2+1) = (2+1+1)+1
2*2*3 = (2+1+2) = (2+1)+1+1
Have I translated between notations correctly? I'm not sure which of these is correct, but either addition is noncommutative, or non associative. I don't like either option.

I only count 24?

Maybe you've missed the duocylinder*duocylinder. I'm not sure what that would be in your notation.

a(x,y) b(z,w) c(a,b) d(c,w) - it uses the w coordinate twice. This is not a valid toratope.

Sorry, that second w should be a v. I've fixed it now. That shape is the result of spherating a duocylinder by another duocylinder.

When I look at the beasts - They seem a bit weird, I don't understand the *2 there.
1 is tigerinder, 2 is sphere tiger, 3 is torus tiger, 4 should be probably circtiger, 5 tiger*circle, but what is 6?

The last one is duocylinder*duocylinder. To be honest, I'm not sure if this should be a 5D shape or a 6D shape.

I decided that the tiger is (2+2)*2 in my notation, because it's the duocylinder (2+2) spherated by a circle, 2. The other beasts I found are just generalisations of this idea. Unfortunately, with my notation it's slightly harder to work out the dimension of a given object.

What problems?

Well, the torus product is define in terms of the vector notation. I think it's this definition that's the problem. For instance, a circle spherated by a circle only works if the first circle is centred on the origin. Otherwise you get some ugly shape. The other problem is it doesn't handle the duocylinder very well. I'll start a new thread to redefine the torus product.

[Marek14]

I only count 24?

Maybe you've the duocylinder*duocylinder. I'm not sure what that would be in your notation.

a(x,y) b(z,w) c(a,b) d(c,w) - it uses the w coordinate twice. This is not a valid toratope.

Sorry, that second w should be a v. I've fixed it now. That shape is the result of spherating a duocylinder by another duocylinder.

You mean this?
(2+2)*2*2 = a(x,y) b(z,w) c(a,b) d(c,v)
(2+2)*(2+2) = a(x,y) b(z,w) c(a,b) d(c,v)
Now you have two shapes with the exact same equations

When I look at the beasts - They seem a bit weird, I don't understand the *2 there.
1 is tigerinder, 2 is sphere tiger, 3 is torus tiger, 4 should be probably circtiger, 5 tiger*circle, but what is 6?

I decided the tiger would be (2+2)*2 in my notation, because it's the duocylinder (2+2) spherated by a circle, 2. The other beasts I found are just generalisations of this idea. Unfortunately, with my notation it's harder to work out the dimension of a given object.

What problems?

Well, the torus product is define in terms of the vector notation. I think it's this definition that's the problem. For instance, a circle spherated by a circle only works if the first circle is centred on the origin. Otherwise you get some ugly shape. The other problem is it doesn't handle the duocylinder very well. I'll start a new thread to redefine the torus product.

[PWrong]

You mean this?
(2+2)*2*2 = a(x,y) b(z,w) c(a,b) d(c,v)
(2+2)*(2+2) = a(x,y) b(z,w) c(a,b) d(c,v)
Now you have two shapes with the exact same equations

Damn, you're right. The duocylinder*duocylinder must not exist in 5D.
It must be either {a(x,y) b(z,w) c(a,b) d(v,u)} or {a(x,y) b(z,w) c(a,v) d(b,u)}

How come you quoted me without saying anything? :?

[Marek14]

You mean this?
(2+2)*2*2 = a(x,y) b(z,w) c(a,b) d(c,v)
(2+2)*(2+2) = a(x,y) b(z,w) c(a,b) d(c,v)
Now you have two shapes with the exact same equations

Damn, you're right. The duocylinder*duocylinder must not exist in 5D.
It must be either {a(x,y) b(z,w) c(a,b) d(v,u)} or {a(x,y) b(z,w) c(a,v) d(b,u)}

How come you quoted me without saying anything? :?

The first shape would be (2+2)*2, i.e. tiger/circle duoprism. The second would be (2+1)*(2+1), i.e. torus duoprism, not a beast at all.

[PWrong]

I've counted the number of rotopes, including beasts, in 5 and 6 dimensions. It's 24 in 5D, and 66 in 6D. I used a different notation that's easy to draw on paper, but harder on here. You start with n lines, and join them up to make a sphere.

cube = | | |
cylinder = \/ |
sphere = \|/
torus =
\/ |
. \/

Rotatopes only take up one row, while toratopes may have several.
The number of rotopes in n dimensions is now 1, 2, 4, 10, 24, 66.

Look what I found on mathworld.
http://mathworld.wolfram.com/Series-ParallelNetwork.html
Not is the sequence a_n exactly the same as ours, (and I'm sure I've seen b_n somewhere before), but equation (3) looks amazingly similar to my formula for the number of non-beast rotopes .

The only difference is that the equation is a reccurence relation. Instead of using T(n), the number of non-beast toratopes, they simply use the total number of toratopes.

Code: Select all


b(n) =
        n      (b(i) + k(i))!
Sum ( Product (--------------))
k     i=1      b(i)! k(i)!


This apparently only gives half of the rotopes. The actual number of rotopes in nD is 2*b_n.

Every rotope also corresponds to a cograph. Interestingly, a cograph is basically defined as a union of two other cographs, just like a rotope is defined as the torus product of two other rotopes. I'm still trying to work out how to tell what the corresponding cograph is for each rotope.

[PWrong]

Here's something even more fascinating. The number of toratopes- torii and beasts, is exactly half of the number of rotopes, and the rotatopes and torinder-like shapes make up the other half. It's just like the two kinds of network in that mathworld link- essentially series, and essentially parallel.

That also explains why our notations contrast so much. Yours was better at handling beasts, while mine was designed more for objects like the torinder. Our notations aren't just similar but different, they're equal and opposite.

[Marek14]

That's good to hear This might be the final proofs that the beasts are neccessary part of the whole rotope menagerie (and really, doesn't each good menagerie need beasts?)

[wendy]

Is this count of rotatopes counting the surtegmic rototopes?

Consider the rhombic tricontahedron. One can write on its rhombic faces red and blue diagonals. Then take the red diagonals, and make these run along the surface of a sphere. One gets a rototope figure.

The closest that comes in the standard list is the crossing of three cylinders, which is a rhombic dodecahedron, where the octahedron-edges have been reduced to circles.

One should be able to produce a surtegmic rototope for a wide class of figures?

or am i off topic?

W

[PWrong]

or am i off topic?

Yes. :wink:

[PWrong]

It looks like the formula isn't as similar as I thought. In fact I don't see how it works at all. Mathworlds formula is:

Code: Select all

b(n) =
        n      (b(i) + k(i) - 1)!
Sum ( Product (--------------))
k     i=1      (b(i) + k(i) - i - 1)! i!


I tried working out the first few terms using this, but I keep having to calculate the factorial of a negative number. :?

[PWrong]

Since I keep linking to this page, maybe I should point out that the formula on mathworld is correct after all. I'll try deriving the formula when I have time.