Naming system for rotopes

[PWrong]

I've finally thought up a naming system for rotopes. It's based on the fact that each toratope can be created from a non-toratope, simply by putting brackets around it. For instance, the circle#sphere is simply a spherinder, stretched and attached at the ends. The sphere#circle is like a cubinder, wrapped up like a sheet around a ball.

Thus, I propose to name each toratope after it's rotatope dual, with the prefix "tor" or "tora".

tetracube -> glome
1111 -> (1111) = 4

cubinder -> toracubinder = circle#sphere
211 -> (211)

duocylinder -> torduo-cylinder = tiger
22 -> (22)

spherinder -> toraspherinder = sphere#circle
31 -> (31)

torinder -> 3-torus
(21)1 -> ((21)1)

I didn't rename the 3-torus, because toratorinder is just silly. I'll try naming the 5D rotopes tomorrow.

EDIT: Changed toracubinder to circle#sphere and toraspherinder to sphere#circle

[moonlord]

Seems neat. The 3-torus should be a 4-torus if you asked iNVERTED.

Why don't you people suffix the dimension!? A torus-4.

Also for the sake of consistency, why not using the hyper preffix? A hypersphere-3 is a sphere, a hypertorus-3 is a doughnut and so on. On the other hand, you're not using cube-3 for a box, do you? You use a hypercube-3 name... Can't understand it any more... :?

[pat]

I'm sorry, can someone point me to a thread where I can learn what this notation means again.... I think I follow the 211 and such... but I'm at a loss on the parenthesis.

[Keiji]

Why need a thread, when the wiki explains all?

http://tetraspace.alkaline.org/wiki/ind ... le=Rotopes

Moonlord, the number is more important than the object so it comes first. Also, you don't use hyper- when you use n-, because a hyperobject is the set of a 0-object, 1-object, a 2-object, etc.

[pat]

Okay, that shows the notation in use. It does little to explain the notation.

In particular, the notation "21" for cylinder and "(21)" for torus seem at odds. The torus is not the "spheration" of the cylinder (at least not in any way that I can see).

And, it seems weird that the 3-torus is written ((21)1). That seems to belie the symmetry that circle#circle#circle has.

And, if circle = 2 and circle#circle = (21) and circle#circle#circle = ((21)1), and duocylinder = 22, then how is duocylinder#circle = (22) and not (221)?

Additionally, there's nothing in the notation that says whether you're producting with a circle or a sphere or a glome or what-have-you, is there? Is it assumed that if you start with a k-dimensional object, you spherize with a k-dimensional sphere?

Also, I thought that torus products were commutative? But, the toratope list shows circle#sphere as distinct from sphere#circle.

I'm also a bit lost on the "english" versions. How is a toraspherinder (211) related to a spherinder 31? Are the names skewed in the toratopes or is my understanding that far behind?

Bleh, more later....

[pat]

I'm thinking a better notation would be just to stick with the '#' instead of the parens.... circle#circle = 2#2. Of course, this makes it look like a four-dimensional object when, in reality, it is embeddable in three-dimensions. So, maybe [2#2]₃?

[moonlord]

@iNVERTED: Well, it makes some sense now. Therefore, a 3-cube is one of the hypercubes, and so on? Ok, it does make sense.

@pat: I can't explain the notation better than PWrong, so I'll let him do it . If you take a rubber cylinder and stick the bases together, bending the cylinder, you get a torus. As for the other spherations, I must admit I don't fullly understand them.

[Nick]

@pat: I can't explain the notation better than PWrong, so I'll let him do it . If you take a rubber cylinder and stick the bases together, bending the cylinder, you get a torus. As for the other spherations, I must admit I don't fullly understand them.

Dude, I'm really lost... wth are with all of these numbers? :?

[Marek14]

And, if circle = 2 and circle#circle = (21) and circle#circle#circle = ((21)1), and duocylinder = 22, then how is duocylinder#circle = (22) and not (221)?

The thing is that tiger is quite hard to grasp this way. It's not generated from a duocylinder (which is a 4D object), but from a duocylinder margin, which is only 2D object, albeit immersed in 4 dimensions. A duocylinder is a cartesian product of two solid circles, and duocylinder margin is a cartesian product of two circle BOUNDARIES.

Additionally, there's nothing in the notation that says whether you're producting with a circle or a sphere or a glome or what-have-you, is there? Is it assumed that if you start with a k-dimensional object, you spherize with a k-dimensional sphere?

Not quite sure what you mean here. Generally, you can spherate any object with any dimension of sphere, you just need more and more dimensions to put it in.
If you spherate a circle, say, with a 1D sphere, it's equivalent to transforming it in two concentric circles. If you spherate it with a 2D sphere, or circle, it becomes a torus. If you spherate it with a 3D sphere, it becomes circle*sphere etc.

Also, I thought that torus products were commutative? But, the toratope list shows circle#sphere as distinct from sphere#circle.

That's right, they are different. If you slice circle*sphere with a coordinate hyperplane, you get either two separate spheres, or a torus. If you slice sphere*circle, you get either two concentric spheres, or a torus.

I'm also a bit lost on the "english" versions. How is a toraspherinder (211) related to a spherinder 31? Are the names skewed in the toratopes or is my understanding that far behind?

Yes, this seems to me to be a bug.

[Marek14]

@iNVERTED: Well, it makes some sense now. Therefore, a 3-cube is one of the hypercubes, and so on? Ok, it does make sense.

@pat: I can't explain the notation better than PWrong, so I'll let him do it . If you take a rubber cylinder and stick the bases together, bending the cylinder, you get a torus. As for the other spherations, I must admit I don't fullly understand them.

I'm not sure, but maybe it was me who came with the notation, so I'll try.

The notation was derived from parametric equations of toratopes. It was noted that the equations for torus are:

x = A cos a cos b +B cos b
y = A cos a sin b + B sin b
z = A sin a

You can see that these equations define a circle in xy plane (B), which then has another circle added to it. This can be also claimed to be the set of all points in 3D with given distance from a circle.
Since the first term of the equation has 2 dimensions, and the second just adds one more, it's marked by (21).
Why (21)? Actually, the better way to show it would be ((11)1), but it seemed that shortening (11) as 2 was just natural.

In 4D, there are these toratopes.

There's sphere*circle:

x = A cos a cos b cos c + B cos b cos c
y = A cos a cos b sin c + B cos b sin c
z = A cos a sin b + B sin b
w = A sin a

This is (31)

Circle*sphere:

x = A cos a cos b cos c + B cos c
y = A cos a cos b sin c + B sin c
z = A cos a sin b
w = A sin a

This is (211)

Circle^3:

x = A cos a cos b cos c + B cos b cos c + C cos c
y = A cos a cos b sin c + B cos b sin c + C sin c
z = A cos a sin b + B sin b
w = A sin a

This is ((21)1)

And the tiger (22) looks like this:

x = A cos a cos b + B cos b
y = A cos a sin b + B sin b
z = A sin a cos c + C cos c
w = A sin a sin c + C sin c

[PWrong]

In particular, the notation "21" for cylinder and "(21)" for torus seem at odds. The torus is not the "spheration" of the cylinder (at least not in any way that I can see).

Spheration isn't related to putting brackets around an object. A torus is a "bracketed" cylinder, but it's also a circle spherated by a circle. The torus product is sometimes easier to visualise, but it's not perfectly defined and it's hard to get useful information from it.

I'm also a bit lost on the "english" versions. How is a toraspherinder (211) related to a spherinder 31? Are the names skewed in the toratopes or is my understanding that far behind?

That was a mistake in the wiki. I've fixed it now. The toraspherinder is (31), and the toracubinder is (211).

And, it seems weird that the 3-torus is written ((21)1). That seems to belie the symmetry that circle#circle#circle has.

There is no symmetry. The three circles are different, and you can put them in order. Note that each set of brackets in ((21)1) contains two objects, because a circle has two dimensions.

Also, I thought that torus products were commutative? But, the toratope list shows circle#sphere as distinct from sphere#circle.

It's not commutative. The circle#sphere (toracubinder) is essentially a thick loop of string in 4D. Sphere#circle (toraspherinder) is harder to visualise, but it's not the same.

Here's another definition.
A "beast" is an object that includes a bracket containing two or more sets of brackets. For instance, (22) = ((11)(11)) and ((21)3) are beasts, but ((31)111) is not. Generally, beasts are the hardest objects to visualise.

Now, you can easily write any non-beast rotope with torus products. Start with the innermost bracket and work outwards, counting the number of elements in each bracket.
For example, ((31)111) = 3 # 2 # 4
It's harder to convert a beast into the torus product of spheres. In fact I'm not sure if it's possible.

[moonlord]

Isn't the cartesian product of two circles a sphere? Also, I'm not familiar with the parametric equations. This is how I see them:

You start with a point "". You can only extrude it, so you get a segment "1". You can rotate the segment and get a circle "2" or you can extrude it again and get a square "11". Take the circle. You can extrude it again (cylinder "21"), or you can rotate (sphere "3"). Take the cylinder. Due to it's assymetry, you can perform multiple transformations.

1. Rotate it around a base. You get "31", which is a spherinder.
2. Rotate it around a plane formed by a radius and a generator. You get a "22", which is a duocylinder, and which I currently struggle to understand.
3. Extrude it. You get a "211", which is a cubinder. I still don't think the name is accurate.

The process goes on. Given a shape's notation, you can derive its cartesian (and parametric) equations from it:

"11": Two dimensional, only extrudes: abs(x)=1, abs(y)=1 (working with unit shapes).

"2": Two dimensional, extruded to a line and then rotated: firstly, abs(x)=1 which after rotation becomes x^2 + y^2 = 1.

"21": Three dimensional, extrude, rotate, extrude: firstly, abs(x)=1, then x^2 + y^2 = 1, and finally {x^2 + y^2 = 1 and abs(z)=1}.

You can also see it as a cartesian product. Take a circle in XY (x^2 + y^2 = 1) and a line in Z (abs(z)=1). You can make the heads and tails of this...

EDIT: Now writing, I've figured out about the duocylinder. Being a 22, it's cartesian are x^2 + y^2 = 1 and z^2 + w^2 = 1 which make a 2D figure afterall... In the begining, I've assumed a dimension is common to those two circles...

[PWrong]

Isn't the cartesian product of two circles a sphere?

Depends on which circles you choose. If both are centred on the origin, you get a sphere. If not, you get a torus. Anyway, the cartesian product is completely different to the torus product. For the torus product, all circles with the same radius are identical. Rotations and translations don't matter.

EDIT: Now writing, I've figured out about the duocylinder. Being a 22, it's cartesian are x^2 + y^2 = 1 and z^2 + w^2 = 1 which make a 2D figure afterall... In the begining, I've assumed a dimension is common to those two circles...

That's right. The duocylinder actually comes in three forms (2D, 3D and 4D), but I think only the 2D form is important in most cases. I introduced forms and cells somewhere in another thread, but I can't find it.

EDIT: Here it is. The notation we used in that thread was still in development, so ignore it.
http://tetraspace.alkaline.org/forum/viewtopic.php?t=403&start=30

[moonlord]

If both are centred on the origin, you get a sphere. If not, you get a torus.

So shouldn't the sphere be treated as a special kind of torus? Moreover, if we take the cartesian for the torus (R-sqrt(x^2+y^2))^2 + z^2 = r^2 and set R=0 (displacement is zero), we get a sphere x^2 + y^2 + z^2 = r^2.

[PWrong]

So shouldn't the sphere be treated as a special kind of torus?

In a sense, it is a special kind of torus. But for classification purposes, I think it's better to keep them separate. The sphere is also a special case of the spherinder, ellipsoid, and countless other shapes, but it's not always worth mentioning this.

[moonlord]

Right. I should've thought about that... :?

[Marek14]

Yes, the key to duocylinder is that it's cartesian product of two circles in planes that only intersect in a point.

The 2D, 3D and 4D duocylinders can be better shown on lower-D analogies. I don't actually like this notation - all these forms of duocylinder are 4D, what differs is their internal dimension. It would be like calling wireframe model of the cube 1D cube.

Think cylinder. The cylinder is a 3D object. It's a cartesian product of a circle and a line. When we divide these into elements, we find that a circle has the circle itself (2D) and one boundary (1D). The line, on the other hand, has a body (1D) and two endpoints (0D). This means that you can get various lower-dimensional elements of the cylinder by combining these two lists:

1 3D cylinder (2D x 1D)
2 flat circles (2D x 0D) and 1 cylinder mantle (1D x 1D)
2 circular "edges" of the cylinder (1D x 0D)

The duocylinder is a cartesian product of two circles. So the lowest-dimensional element it has is the margin (1D x 1D) of dimension 2. Then there are two "faces" (2D x 1D and 1D x 2D) of dimension 3, and finally the duocylinder itself of dimension 4.

Analogically, cubinder (2x1x1) is made of:

1 4D cubinder (2D x 1D x 1D)
4 3D cylinders (2D x 1D x 0D and 2D x 0D x 1D) and 1 cubinder mantle (1D x 1D x 1D)
4 circles (2D x 0D x 0D) and 4 cylinder mantles (1D x 1D x 0D and 1D x 0D x 1D)
4 circular edges (1D x 0D x 0D)

And spherinder (3x1) is made of:

1 4D spherinder (3D x 1D)
2 spheres (3D x 0D) and 1 spherinder mantle (2D x 1D)
2 spherical margins (2D x 0D)

[moonlord]

I feel like we need to clear up things a little. I consider the cylindric surface a 2D shape, because it is just a surface. Likewise, a circle is 1D, because it is just a line. The "disc" is the circle, together with its interior.

To sum up:
1. A segment (S) is 1D, and has two endpoints. A circle (C) is 1D, and has no endpoints (vertices). A disc (D) is 2D and it has a face, a line and no vertices.
2. A cylindric surface SxC is 2D. It has three faces (two circles and no vertices). A cylinder DxC is 3D, and it is a cylindric surface reunited with it's interior.

I understand that, in your post, you were reffering to cylinders, disks and segments. So, let me see if I understood it fully.

A duocylinder is 4D. It is made of a duocylindric surcell and its interior. The surcell is 3D, and it also includes 2D elements, in fact a single duocylindric wireframe.

I'm not yet sure how to derive the cartesian equations for AxB, when I know the ones for A and B. Perhaps you can help.

[Marek14]

Actually, there are two duocylindric surcells - the margin, 2D element separates them.
Imagine you would stand in such a surcell. I think it would look like an inside of a cylinder to you - except that the "height" of the cylinder would be bent into a circle. This would happen in curved space, so you would see it straight - but after walking certain distance along the axis of the cylinder, you'd end up at the same place.
Now, the mantle of this cylinder would be the duocylinder's margin. Imagine you could pass through it at any point. You'd end up in the other surcell, which would look the same as first, except that the "visible circle" and the "curved space circle" dimensions would be switched. In effect, if the first surcell looked like a cylinder 10 km in diameter, and you ended up at the same place after walking 100 km, then the other one will be a cylinder 100 km in diameter, repeating after each 10 km.

The margin itself has a torus topology. However, if you were a 2D creature living in the margin, you would see it as flat (2D creature on real 3D torus could still observe some curvature).

As for the cartesian equations, it seems simple to me.
If there is equation f(x1,x2,...xn)=0 for A, and g(y1,y2,...yn)=0 for B, then the equation for A x B is just both of them at the same time, which can be summed in a single equation f^2+g^2 = 0.

However, dimension must be taken care of. If you put the equations for two circles, you get the duocylinder margin. To get the surcell equations, you'd have to use inequality and equation, to get the whole duocylinder, two inequalities would be neccessary. Actually, good results can be obtained with the max function - the equation max(x^2+y^2,z^2+w^2) = 1 gives both surcells at once.

[moonlord]

For the duocylinder, I think I got it. As for a (not so good) analogy, it is like a tennis ball, having two surfaces and a line that separates them... A not so good analogy, I believe.

What if you have f(x1,x2,...xn)=k, with non zero k? Let's try it for the cylindric surface:

x^2 + y^2 - r^2 = 0 and z^2 - h^2/4 = 0

This gives the equation for a cylindric surface

(x^2 + y^2 - r^2)^2 + (z^2 - h^2/4)^2 = 0

Or is it not? Actually, I believe it's the equation for the two circles that are the cylinder's sides.

For the cilindric surface it should be

( x^2 + y^2 - r^2 = 0 and z^2 - h^2/4 <= 0 ) or ( x^2 + y^2 - r^2<= 0 and z^2 - h^2/4 = 0 )

Which can be rewritten as... how can it be? I can't figure out how to deal with an equation and a inequation...

For the cylinder, it is, I believe,

(x^2 + y^2 - r^2)^2 + (z^2 - h^2/4)^2 <= 0

[Marek14]

Well, an inequality can be written as equation with a simple trick:

Just change "f(x)>=0" to "f(x) - abs(f(x)) = 0"

"(x^2 + y^2 - r^2)^2 + (z^2 - h^2/4)^2 <= 0"

This is wrong, for the simple reason that the left side can NEVER be less than zero, so this is exactly the same thing as having "= 0" there.

However, an equation that works for the surface of cylinder is this:
max(x^2+y^2,z^2) = 1

If the x^2+y^2 = 1 then z can be anywhere between -1 and 1, forming the mantls. If z^2 = 1, x^2+y^2 can be smaller, and you have the two disks.

[pat]

Also, I thought that torus products were commutative? But, the toratope list shows circle#sphere as distinct from sphere#circle.

That's right, they are different. If you slice circle*sphere with a coordinate hyperplane, you get either two separate spheres, or a torus. If you slice sphere*circle, you get either two concentric spheres, or a torus.

My bad, here. I was not actually thinking of the torus product.

[pat]

Here's another definition.
A "beast" is an object that includes a bracket containing two or more sets of brackets. For instance, (22) = ((11)(11)) and ((21)3) are beasts, but ((31)111) is not. Generally, beasts are the hardest objects to visualise.

Now, you can easily write any non-beast rotope with torus products. Start with the innermost bracket and work outwards, counting the number of elements in each bracket.
For example, ((31)111) = 3 # 2 # 4
It's harder to convert a beast into the torus product of spheres. In fact I'm not sure if it's possible.

As I mentioned just now, I wasn't actually thinking of the torus product. My bad. This example also clears things up a bit for me. Thanks.

[PWrong]

For instance, the circle#sphere is simply a spherinder, stretched and attached at the ends. The sphere#circle is like a cubinder, wrapped up like a sheet around a ball.

I just realised that this is a bit misleading. Although it's true that circle#sphere is like a spherinder stretched and attached at the ends, the two objects aren't related. circle#sphere is actually a toracubinder.

Here's a better visualisation.
Instead of think of the cubinder as a square with circles at every point, think of it as a circle with a square at every point. Now curl up all the squares into a sphere while keeping their position. Now you have a toracubinder.

Similarly, think of the spherinder as a sphere with lines at every point (all in the w direction). Now curl up each line into a small circle, and you'll get a toraspherinder. This one is harder to visualise.

By the way, I'm adding pages for the toracubinder and the toraspherinder to the wiki.