## Disagreement with the Duocylinder

Discussion of shapes with curves and holes in various dimensions.

### Disagreement with the Duocylinder

I was thinking about the rotachora, trying to get a picture in my head about how they're formed when I noticed that I found out how to get every single rotachoron except the duocylinder. Hmm...
I finally reached this conclusion after some thought:

On the rotatopes page they say that a rotatope is an n-dimensional figure formed by extending or rotating a figure of the dimension before it. And then you find out that the five rotachora are the Hypercube, Cubinder, Duocylinder, Spherinder, and the Glome. And then these processes went through my head - you get the hypercube by extending the cube, you get the cubinder by extending the cylinder or rotating the cube, you get the spherinder by extending the sphere or rotating the cylinder, and you get the glome by rotating the sphere. Where does the duocylinder come into play? We already rotated and extended everything we could. I believe the duocylinder isn't a rotachoron at all. My depiction of a duocylinder (which I also call the cubispherinder) is a rotatetron (5D rotatope) and results when you extend the spherinder or rotate the cubinder. Furthermore, I believe those cross-sections that you see are in fact cross-sections of the cubinder tilted at a 45-degree angle.
Neues Kinder
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The duocylinder is [(w,x),(y,z)], which means that you can get it from lesser elements in this way:

[(w,x),(y,z)] = rotation of cylinder [x,(y,z)] in the wx plane.

In every case, you need to make it out so that the top and bottom circles of a cylinder arise from the same thing, by rotating it in the hedrix made from the height and the added dimension.
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Your enumeration is wrong because of one little fact:

Every rotatope can be extended in one way, BUT they can be rotated in several non-identical ways. When you rotate a 3D body through 4D, you rotate it around a plane, and in this case, the plane is one of the three coordinate planes.

Cube and sphere can be rotated in only one way (to cubinder, resp. glome), as all three coordinate planes are symmetrical. However, the cylinder has two kinds of coordinate planes, which cut it in either a square or a circle. Rotating around these planes will give you different results. (If I see it correctly, rotation around "square" plane should give spherinder, and rotation around "circle" should give duocylinder)

A way to see the duocylinder is that it's Cartesian product of two circles.

BTW - the rotation of cubinder to 5D is, once again, non-unique: rotating it around a cubic cross-section will indeed lead to the shape you describe (called spherisquare in my system), but if you rotate it around its cylindrical cross-section, you will get a different rotatope called "dual cylinder", which can be also gotten by extending duocylinger.
Marek14
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So what you're saying is that if you rotate the two connected circles of the cylinder the "ordinary" way in the same direction, and have them remain connected, you get the spherinder, and if you rotate them the "other" way (like drawing a circle on a piece of paper and turning the piece of paper around on the table), then you get the duocylinder. I only have one question. What 3D rotatope do you get when you rotate a single circle the same way?
Neues Kinder
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Neues Kinder wrote:So what you're saying is that if you rotate the two connected circles of the cylinder the "ordinary" way in the same direction, and have them remain connected, you get the spherinder, and if you rotate them the "other" way (like drawing a circle on a piece of paper and turning the piece of paper around on the table), then you get the duocylinder. I only have one question. What 3D rotatope do you get when you rotate a single circle the same way?

This sounds a bit more complex that I would put it.

If you rotate a single circle around any of its axes, you get a sphere, of course. Both coordinate axes of a circle are symmetrical. You only get two different results for a cylinder because a cylinder can be cut by coordinate planes in two very different ways.

You do realize that in 4D, the rotation happens around a plane, not around an axis as in 3D case, right?
Marek14
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The simplest way of constructing product-rototopes, is to use the notion that any axis can devolve into a circle or a square. What happens is that one replaces the simple axis x with either [w,x] for prism, and (w,x) for circle.

So we could construct the duo-cylinder as:
x line
[x,y] square
[x,(y,z)] cylinder
[(w,x),(y,z)] duocylinder

You see that regardless of whether we add w, x, y, z last, the previous 3d state is a cylinder.

Many of these figures have the symmetry construction in the group r.

There are other figures in other symmetries, such as h, hr, hh, f.

For example, one could take a {3,3,5}, and a {5,3,3}. If one draws say the {5,3,3} on a glome, and then sets the {3,3,5} and {5,3,3} to cross at the edges of the {5,3,3}, one gets a rototope with 1200 faces, each a triangle tegum (bipyramid), where the axis is bent around the curve of the glome.
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Marek14 wrote:If you rotate a single circle around any of its axes, you get a sphere, of course. Both coordinate axes of a circle are symmetrical. You only get two different results for a cylinder because a cylinder can be cut by coordinate planes in two very different ways.

Yes, you can rotate a circle around the x and y axes to get a sphere. And you can rotate a cylinder around the xy, xz, and yz planes to get - as you say - either a spherinder or a duocylinder. But, as wendy points out (inderectly), you can also rotate a cylinder around the wx, wy, and wz planes. If that is true you should also be able to rotate a circle around the z axis. And what I'm asking is what rotahedron do you get when you rotate a circle around the z axis?
Neues Kinder
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The method that i use to increase dimension, is to replace one dimension with two. That is, you can replace x with (x,y) or [x,y]. What arises out of this is a nested prismic / spheric product.

You can remove any set of brackets that are identical to the enclosing brackets.

eg as (3+(2+2)) = (3+2+2)

[x,[y,z]] = [x,y,z] cube = square prism

(x,(y,z)) = (x,y,z) sphere = circular spheric

Note also that you can remove "non-adjacent" letters, eg

cylinder = [x,(y,z)]

When one wants to look down any given set of axies, one simply removes what ever letters are not needed, and simplifies the brackets.

duocylinder = [(w,x),(y,z)]

in the wy axis = [(w),(y)] = [w,y] = square.

Marek 14 further discovers that you can for a set of w,x,y,z freely decide what the wx, wy, wz, xy, xz, yz axies ought hold. That is, you can freely populate these with square or circle sections. Of course, the 64 possibilities here reduce to 11, after orientations are taken to account, but one of the 11 (longdome, wx=xy=yz = (), and yw=wz=zx=[]), can not be expressed in a direct product of [] and () around w,x,y,z.
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Neues Kinder wrote:
Marek14 wrote:If you rotate a single circle around any of its axes, you get a sphere, of course. Both coordinate axes of a circle are symmetrical. You only get two different results for a cylinder because a cylinder can be cut by coordinate planes in two very different ways.

Yes, you can rotate a circle around the x and y axes to get a sphere. And you can rotate a cylinder around the xy, xz, and yz planes to get - as you say - either a spherinder or a duocylinder. But, as wendy points out (inderectly), you can also rotate a cylinder around the wx, wy, and wz planes. If that is true you should also be able to rotate a circle around the z axis. And what I'm asking is what rotahedron do you get when you rotate a circle around the z axis?

When you rotate the circle around the z axis, you get the same figure as if you rotated it around its center in 2D - in other words, since the axis of rotation doesn't lie within the plane of the figure, it will stay 2D figure, as none of its points will ever leave the plane while it rotates. In this particular case, the circle will stay unchanged, but this is, of course, not the general case.
Marek14
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Ahh, I get it. Getting the duocylinder is like rotating all the lines in the cylinder and not the circles. So the circles, when revolving around the center and not rotating, will form a 4D torus. And another 4D torus will fill in the gap in the surface, like a 3D tube forms the outside of the cylinder, and you need two circles to fill in the gaps in the surface. You can visualize getting a cylinder as taking a line and extending it around in a circle. As such, you can also get the duocylinder by taking a circle and extending it around in a circle, hence the (2,2) identifier. I call it the torinder, because it is made up of two tori. So there are 5 rotatopes in tetraspace, and there are 7 rotatopes in pentaspace: Pentacube (1,1,1,1,1), tetracubinder (1,1,1,2), cubispherinder - my duocylinder - (1,1,3), cubitorinder (1,2,2), spheritorinder (2,3), glominder (1,4), and the pentome (5).

And I came up with the 9 rotahexxa (6D rotatopes) just two minutes ago:
(1,1,1,1,1,1) Hexacube
(1,1,1,1,2) Pentacubinder
(1,1,1,3) Tetraspherinder
(1,1,2,2) Tetracubitorinder
(1,2,3) Cylitorinder
(2,4) Glomitorinder
(1,1,4) Tetracubiglominder
(1,5) Cubipentominder
(6) Hexome

I named the (1,2,3) the Cylitorinder because just like you extend the line and then rotate it, you extend the Torinder and then rotate it to get the Cylitorinder.
Neues Kinder
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Neues Kinder wrote:Ahh, I get it. Getting the duocylinder is like rotating all the lines in the cylinder and not the circles. So the circles, when revolving around the center and not rotating, will form a 4D torus. And another 4D torus will fill in the gap in the surface, like a 3D tube forms the outside of the cylinder, and you need two circles to fill in the gaps in the surface. You can visualize getting a cylinder as taking a line and extending it around in a circle. As such, you can also get the duocylinder by taking a circle and extending it around in a circle, hence the (2,2) identifier. I call it the torinder, because it is made up of two tori. So there are 5 rotatopes in tetraspace, and there are 7 rotatopes in pentaspace: Pentacube (1,1,1,1,1), tetracubinder (1,1,1,2), cubispherinder - my duocylinder - (1,1,3), cubitorinder (1,2,2), spheritorinder (2,3), glominder (1,4), and the pentome (5).

And I came up with the 9 rotahexxa (6D rotatopes) just two minutes ago:
(1,1,1,1,1,1) Hexacube
(1,1,1,1,2) Pentacubinder
(1,1,1,3) Tetraspherinder
(1,1,2,2) Tetracubitorinder
(1,2,3) Cylitorinder
(2,4) Glomitorinder
(1,1,4) Tetracubiglominder
(1,5) Cubipentominder
(6) Hexome

I named the (1,2,3) the Cylitorinder because just like you extend the line and then rotate it, you extend the Torinder and then rotate it to get the Cylitorinder.

You are pretty much correct, although your labels don't match with mine.

I have, in 5 and 6D (I did it even further)

(1,1,1,1,1) - penteract
(1,1,1,2) - cubicircle
(1,2,2) - dual cylinder
(1,1,3) - spherisquare
(2,3) - sphericircle
(1,4) - glominder
(5) - petaglome

(1,1,1,1,1,1) - hexeract
(1,1,1,1,2,) - tesseracticircle
(1,1,2,2) - duocubinder
(2,2,2) - tricilynder (you missed this one)
(1,1,1,3) - sphericube
(1,2,3) - sphericylinder
(3,3) - duosphere (you missed this one)
(1,1,4) - glomosquare
(2,4) - glomocircle
(1,5) - petaglominder
(6) - exaglome

I also looked through all the "extended" rotatopes in 3 to 5 dimensions. Those are shapes which are given by arbitrarily dividing the set of coordinate planes in those that cut them in circles, and those that cut them in squares. In 3D, you have cube (three squares), cylinder (two squares, one circle), and sphere (three circles). There is also a fourth, extended rotatope which has two circles and one square as its cross-sections. Can you find it?
Marek14
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Marek14 wrote:There is also a fourth, extended rotatope which has two circles and one square as its cross-sections. Can you find it?

The crind. Intersection of two perpendicular cylinders.  Keiji

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Yeah, I did think about the (3,3) and the (2,2,2) earlier today. I call the (3,3) the Duospheritorinder and the (2,2,2) the Duotorinder (I originally was going to call it the Sphericubitorinder, but it can easily be misinterpreted as the Cylitorinder)

I also came up with the rotahexxa (7D rotatopes - I meant to say rotapenta when I was doing the 6D ones) by extending or rotating the ones before them the "original" way:
(1,1,1,1,1,1,1) Heptacube
(1,1,1,1,1,2) Hexacubinder
(1,1,1,1,3) Pentacubispherinder
(1,1,1,4) Tetracubiglominder
(1,1,5) Pentomicubinder
(1,6) Hexominder
(7) Heptome

And by rotating the "other" way:
(1,1,1,2,2) Pentacubitorinder
(1,1,2,3) Cubicylitorinder
(1,2,2,2) Cubiduotorinder
(1,2,4) Sphericylitorinder
(1,3,3) Cubiduospheritorinder
(2,5) Pentomitorinder
(3,4) Glomispheritorinder (Spheriduospheritorinder)

And to better organize them and make sure you didn't miss any, you can sort them like this:

(1,1,1,1,1,1,1)
(1,1,1,1,1,2)
(1,1,1,1,3)
(1,1,1,2,2)
(1,1,1,4)
(1,1,2,3)
(1,2,2,2)
(1,1,5)
(1,2,4)
(1,3,3)
(2,2,3)
(1,6)
(2,5)
(3,4)
(7)

So that you can do even higher ones like this:

(1,1,1,1,1,1,1,1,1)
(1,1,1,1,1,1,1,2)
(1,1,1,1,1,1,3)
(1,1,1,1,1,2,2)
(1,1,1,1,1,4)
(1,1,1,1,3,2)
(1,1,1,2,2,2)
(1,1,1,1,5)
(1,1,1,2,4)
(1,1,1,3,3)
(1,1,2,2,3)
(1,2,2,2,2)
(1,1,1,6)
(1,1,2,5)
(1,1,3,4)
(1,2,2,4)
(1,2,3,3)
(2,2,2,3)
(1,1,7)
(1,2,6)
(1,3,5)
(2,2,5)
(1,4,4)
(2,3,4)
(3,3,3)
(1,8} (regular ")" after the eight makes a smiley)
(2,7)
(3,6)
(4,5)
(9)

There's a total of 30 rotaocta (9D rotatopes)

CHALLENGE: Who can name them all? (must be logical names - names like tetrahedronicubicone or Bob aren't acceptable)
Neues Kinder
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Unfortunately, one can't just use partitions to find rototopes.

For example, the same partition 2,2 gives distinct figures ([w,x],[y,z]) and [(w,x),(y,z)]. Even so, one can have something like [(1,[2,3]),(4,5)], which isn't a partition at all. It ceartianly isn't [1, (2,3), (4,5)] or anything.

In any case, the rototopes on group r correspond to the free marking of simplex edges (N vertices for N dimensions), such that they are either square or circular.

One might also note that the intersection of three cylinders gives a circle in the xy, yz and zx direction. This is the cyclotegmated octahedron, there are examples of this for every regular figure, and in every dimension. For example, the o3m3o5o yields two different rototopes, co3m3o5o and o3m3o5oc. The example listed here is co3m4o, which is a o3m4o rhombic dodecahedron, cyclated on the octahedron end. There is also a different rototope o3m4oc.

Wendy
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Neues Kinder wrote:There's a total of 30 rotaocta (9D rotatopes)

CHALLENGE: Who can name them all? (must be logical names - names like tetrahedronicubicone or Bob aren't acceptable)

Well, I can, for one.

I ended with 6D, didn't I?

I think you will be able to deduce my system from these:

7D:
(1,1,1,1,1,1,1) - hepteract
(1,1,1,1,1,2) - penteracticircle
(1,1,1,2,2) - cubiduocylinder
(1,2,2,2) - triple cylinder
(1,1,1,1,3) - tesseractisphere
(1,1,2,3) - sphericubinder
(2,2,3) - spheriduocylinder
(1,1,1,4) - glomocube
(1,2,4) - glomocylinder
(3,4) - glomosphere
(1,1,5) - petaglomosquare
(2,5) - petaglomocircle
(1,6) - exaglominder
(7) - zettaglome

8D
(1,1,1,1,1,1,1,1) - octaract
(1,1,1,1,1,1,2) - hexeracticircle
(1,1,1,1,2,2) - tesseractiduocylinder
(1,1,2,2,2) - tricubinder
(2,2,2,2) - tetracylinder
(1,1,1,1,1,3) - penteractisphere
(1,1,1,2,3) - sphericubicircle
(1,2,2,3) - dual sphericylinder
(1,1,3,3) - duospherisquare
(2,3,3) - duosphericircle
(1,1,1,1,4) - glomotesseract
(1,1,2,4) - glomocubinder
(2,2,4) - glomoduocylinder
(1,3,4) - glomospherinder
(4,4) - duoglome
(1,1,1,5) - petaglomocube
(1,2,5) - petaglomocylinder
(3,5) - petaglomosphere
(1,1,6) - exaglomosquare
(2,6) - exaglomocircle
(1,7) - zettaglominder
(8) - yottaglome

9D:

(1,1,1,1,1,1,1,1,1) - ennearact
(1,1,1,1,1,1,1,2) - hepteracticircle
(1,1,1,1,1,2,2) - penteractiduocylinder
(1,1,1,2,2,2) - spheritricylinder
(1,2,2,2,2) - quadruple cylinder
(1,1,1,1,1,1,3) - hexeractisphere
(1,1,1,1,2,3) - tesseractisphericircle
(1,1,2,2,3) - spheriduocubinder
(2,2,2,3) - spheritricylinder
(1,1,1,3,3) - duosphericube
(1,2,3,3) - duosphericylinder
(3,3,3) - trisphere
(1,1,1,1,1,4) - penteractiglome
(1,1,1,2,4) - glomocubicircle
(1,2,2,4) - dual glomocylinder
(1,1,3,4) - glomospherisquare
(2,3,4) - glomosphericircle
(1,4,4) - duoglominder
(1,1,1,1,5) - petaglomotesseract
(1,1,2,5) - petaglomocubinder
(2,2,5) - petaglomoduocylinder
(1,3,5) - petaglomospherinder
(4,5) - petaglomoglome
(1,1,1,6) - exaglomocube
(1,2,6) - exaglomocylinder
(3,6) - exaglomosphere
(1,1,7) - zettaglomosquare
(2,7) - zettaglomocircle
(1,8) - yottaglominder
(9) - xennaglome

And here are 10D, for good measure:

(1,1,1,1,1,1,1,1,1,1) - decaract
(1,1,1,1,1,1,1,1,2) - octaracticircle
(1,1,1,1,1,1,2,2) - hexeractiduocylinder
(1,1,1,1,2,2,2) - tesseractitricylinder
(1,1,2,2,2,2) - tetracubinder
(2,2,2,2,2) - pentacylinder
(1,1,1,1,1,1,1,3) - hepteractisphere
(1,1,1,1,1,2,3) - penteractisphericircle
(1,1,1,2,2,3) - cubispheriduocylinder
(1,2,2,2,3) - triple sphericylinder
(1,1,1,1,3,3) - tesseractiduosphere
(1,1,2,3,3) - duosphericubinder
(2,2,3,3) - duospheriduocylinder
(1,3,3,3) - trispherinder
(1,1,1,1,1,1,4) - hexeractiglome
(1,1,1,1,2,4) - tesseractiglomocircle
(1,1,2,2,4) - glomoduocubinder
(2,2,2,4) - glomotricylinder
(1,1,1,3,4) - glomosphericube
(1,2,3,4) - glomosphericylinder
(3,3,4) - glomoduosphere
(1,1,4,4) - duoglomosquare
(2,4,4) - duoglomocircle
(1,1,1,1,1,5) - petaglomopenteract
(1,1,1,2,5) - petaglomocubicircle
(1,2,2,5) - dual petaglomocylinder
(1,1,3,5) - petaglomospherisquare
(2,3,5) - petaglomosphericircle
(1,4,5) - petaglomoglominder
(5,5) - duopetaglome
(1,1,1,1,6) - exaglomotesseract
(1,1,2,6) - exaglomocubinder
(2,2,6) - exaglomoduocylinder
(1,3,6) - exaglomospherinder
(4,6) - exaglomoglome
(1,1,1,7) - zettaglomocube
(1,2,7) - zettaglomocylinder
(3,7) - zettaglomosphere
(1,1,8) - yottaglomosquare
(2,8) - yottaglomocircle
(1,9) - xennaglominder
(10) - dakaglome
Marek14
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By the way, here are the general rules for extending and rotating:

Extending - just add "1" to the list.
Rotating - n-dimensional figure is rotated around (n-1)-dimensional hyperplane into (n+1) dimensions. If you have the list, (n-1) dimensions mean that there is exactly one dimension of the figure it will be rotated in (not around). This dimension falls in exactly one number of the list. To produce the rotation, increase this number by 1.

Example: Let's take the glomosphericylinder (1,2,3,4). It can be extended to (1,1,2,3,4) - glomosphericubinder, or rotated in four different ways:

1. Around (2,3,4) - glomosphericircle, to (2,2,3,4) - glomospheriduocylinder
2. Around (1,1,3,4) - glomospherisquare, to (1,3,3,4) - glomoduospherinder
3. Around (1,2,2,4) - dual glomocylinder, to (1,2,4,4) - duoglomocylinder
4. Around (1,2,3,3) - duosphericylinder, to (1,2,3,5) - petaglomosphericylinder
Marek14
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And, still, somewhere along the way, we forget that the spheric product is a coherent, radial product, and that you can treat it akin to the prism or tegum products, viz p-gon () q-gon spherial. Wherever one has a prism one can have a spherion, eg pentagon-enneagon spherion.

By the time one hits nine dimensions, one can have such ungamely things as a dodecahedron-icosahedron-prism by rhombododecahedron spherion.

( x5o3o #* x3o5o ) ø (o3m4o)

Maybe i'm amazed

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the dream we dream together is reality.

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OK, I sort of understand what you're saying, but I sort of don't. Probably because I'm still in High School and I haven't been exposed to those types of equations and terms yet...

...Anyway...

Here is what I named all 30 rotaocta:

(1,1,1,1,1,1,1,1,1) Nentacube
(1,1,1,1,1,1,1,2) Octacubinder
(1,1,1,1,1,1,3) Heptacubispherinder
(1,1,1,1,1,2,2) Heptacubitorinder
(1,1,1,1,1,4) Hexacubiglominder
(1,1,1,1,2,3) Hexacubispheritorinder
(1,1,1,2,2,2) Pentacubiduotorinder
(1,1,1,1,5) Pentacubipentominder
(1,1,1,2,4) Pentacubiglomitorinder
(1,1,1,3,3) Pentacubiduospheritorinder
(1,1,2,2,3) Tetracubispheriduotorinder
(1,2,2,2,2) Cubitritorinder
(1,1,1,6) Hexomitetracubinder
(1,1,2,5) Tetracubipentomitorinder
(1,1,3,4) Tetracubiglomispheritorinder
(1,2,2,4) Cubiglomiduotorinder
(1,2,3,3) Cubiduospheriduotorinder
(2,2,2,3) Spheritritorinder
(1,1,7) Heptomicubinder
(1,2,6) Cubihexomitorinder
(1,3,5) Cubipentomispheritorinder
(2,2,5) Pentomiduotorinder
(1,4,4) Duoglomicubitorinder
(2,3,4) Glomispheriduotorinder
(3,3,3) Trispheriduotorinder
(1,8} Octominder
(2,7) Heptomitorinder
(3,6) Hexomispheritorinder
(4,5) Pentomiglomitorinder
(9) Nentome

Here are my rules for naming the rotatopes:

1. When doing the cube part of the name, always name it first if it's the highest dimension part not characterizing the torinder part, and last if it's the lowest dimension part not characterizing the torinder part. For example, (1,1,4) would be named Glomicubinder, while (1,1,1,3) would be named Tetracubispherinder, and the (1,2,5) would be named Cubipentomitorinder.

2. Also when doing the cube part of the name, always name the cube part according to the number of numbers in the identifier. If the rotatope is a torinder of some sort, then according to the number of numbers up to the second number greater than one, like (1,2,2,2) would be named the Cubitritorinder, since the third number is the second number greater than one.

3. When doing the sphere part, name every sphere part in the rotatope as an individual sphere part and according to all the dimension parts greater than 2, like the (1,1,1,3) would be named the Tetracubispherinder, the (3,4,5) would be named the Pentomiglomispheriduotorinder, despite how long the name is, and the (1,1,1,1,2) would just be named the Pentacubinder. Always place a sphere part after the cube part if it's dimension is higher and it doesn't characterize the torinder part, and always place the cube part right before the sphere part if the sphere characterizes the torinder.

4. When doing the torinder part, put the word "torinder" at the end of the name always and only if there is more than one number greater than one in the identifier, according to how many of such numbers there are minus one. Like (2,2) is named the torinder, the (2,2,2) is named the duotorinder, the (2,2,2,2) is named the tritorinder, the (2,2,2,2,2) is named the tetratorinder, etc. Also put all sphere parts characterizing the torinder part directly before n-torinder, and placed in order from highest dimension to lowest dimension. (1,1,2,3) is named Tetracubispheritorinder, and (1,1,3,6,7) would be named Tetracubiheptomihexomispheriduotorinder (please don't get into explanations about the 18th dimension and up, this is just for demonstrational purposes). To make it so that you don't have to think, just name the spheres backwards (if you label the rotatope from lowest to highest dimensional parts).

5. For an n-dimensional rotatope, if there are n ones in the identifier, or there is just the number n, then you leave "inder" out of the name, for all else put "inder" at the end.
Neues Kinder
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OK, I just need to brush up on these numbers. I see that an n-dimensional object that is all ones is that dimension's orthotope (hypercube), and an object just denoted by n in itself is the hypersphere (or apeirotope/achanetope in my nomenclature).
So I take it "1" represents linear extension into the next highest dimension. 2 is supposed to be rotations, right? 3 and above represents the number of axes of rotation? A circle is one rotation of a line, and to rotate a circle into a sphere would be two rotations. Or is the number of rotations just the total dimensions of the object formed by the rotation?

It is interesting that you all have named all of these higher dimensional things, because now we can know the name of the shape of the universe in string theory. In addition to the three extended dimensions we are familiar with, there are supposed to be six tiny dimensions (forming what Michio Kaku calls "a twisted 6D torus"). That would be (1, 1, 1, 6) exaglomocube

The way I had named these objects was simply by their bases or number of "curved" versus "straight" dimensions of the surface, so in 4D you could have a spherical hypercylinder, (2 curved, one straight) or a cylindrical hypercylinder (2 straight, one curved). Of course, in higher dimensions, it would get messy, as sperical hypercylindrical hyper-hypercylinder, etc. (I never even bothered with the 1,1,1,6 system). Then, I found you all simpler naming system.
Eric B
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Actually, the digits in the numbers represent the n-dimensional spherical parts. Like 1 represents a 1D sphere, or a line, 2 represents a 2D sphere - circle - 3 represents a sphere, 4 a glome, 5 pentome, etc. The numbers are like a set of instructions on how to get the rotatope starting with a point. Take the cylinder for instance. Its number is (1,2), meaning to take a point, extend it linearly, then extend it circularly. Or you could take a point, extend it circularly, then extend it linearly - you can do it in any order. The spherinder (1,3) tells you to take a point, extend it linearly, then extend it spherically - or take a point, extend it spherically, then extend it linearly. The duocylinder (2,2) tells you to take a point, extend it circularly twice.

Speaking of duocylinders, I found something out. When you take a circle and extend it around in a circle, you don't get a 4D torus, but you get what I call a near four-dimensional cylindrical tube. A near-dimensional object is an n-dimensional object curved or folded (n+1)-dimensionally. Say you have a rectangle and you roll it up into a tube. The resultant shape isn't 2D, because it curves 3D, and it isn't 3D either, because, being a rectangle curved in 3D space, it has no depth, so it's neither 2D nor 3D, so it's an N3D rectangular tube. To make it a cylinder, you need to attach circles at the two ends and fill in the empty space inside. If you don't fill in the empty space, then it's just an N3D cylindrical surface. it still doesn't have any volume, because it's just made up of a curved rectangle and two circles. When you talk about the volume of a hollow 3D object, you're actually talking about the volume of space inside it.

Here's another thing - how are we going to use numbers to identify prisms? I came up with a system that might work. Rotatopes could be said to be round prisms, so when naming the other prisms, let's keep the ( , ) format. To show a polytope part, surround it in brackets []. So a triangular prisminder's identifier would look like this: (, 2). You can put more than one number in the brackets, so if you're talking about a Tripentagonal Prisminder (cartesian product of a triangle, pentagon, and circle), then you write ([3, 5], 2). A Triangular Spheriprisminder would look like (, 3), and a Tripentagonal Glomiprisminder would look like ([3, 5], 4). When you're talking about what Marek14 calls duoprisms, what I call polyprisms, both parts are polytopes, so you put both parts in brackets. A duotriangular prism would be written as (, ). A tripentagonal prism would be written as (, ). A Trirectangular Pentagonal Prism could either be written as ([3, 4], ) or (, [4, 5]). But what about prisms with Archimedean parts? Like the octahedric triangular prism or the hexacosichoral tetrahedric prism? Instead of putting those parts in brackets, put their Schläfli symbols in curly braces. The tetrahedric prism would be ({3,3}, 1). The octahedric pentagonal prism would look like ({3,4}, ). The dodecahedric heptagonal spheriprisminder would look like ([{3,5}, 7], 3). And you can have shapes with 4D+ Archimedean parts, like the Hexidecachoral Tripentagonal Pentomiprisminder, which is written as ([{3,3,3}, [3, 5]], 5). What about a duotetrahedric pentaheptagonal duopentagonal duospheriprisminder? That could be written out as ([[{3,3}, {3,3}], [[5, 7], [5, 5]]], (3, 3)), but that's a lot of brackets and braces to keep up with, and you could get easily confused. There is another format that you can use instead of the double-part format. You can use the multi-part format, so now the duotetrahedric pentaheptagonal duopentagonal duospheriprisminder can be written out as ({3,3}, {3,3}, [5, 7], [5, 5], 3, 3). You can also group them all together in brackets, like this: [{3,3}, {3,3}, 5, 7, 5, 5, (3, 3)].
Neues Kinder
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Say you have a rectangle and you roll it up into a tube. The resultant shape isn't 2D, because it curves 3D, and it isn't 3D either, because, being a rectangle curved in 3D space, it has no depth, so it's neither 2D nor 3D, so it's an N3D rectangular tube. To make it a cylinder, you need to attach circles at the two ends and fill in the empty space inside.

In a notation I came up with in another thread, the object you're describing is one "cell" of the 2D form of the cylinder. The 1D form is a pair of hollow circles. The 2D form has two cells: a curved rectangle, and a pair of solid circles. The 3D form is simply a solid cylinder. PWrong
Pentonian

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The general notation that i use for the various cylinders and prism products are as follows.

1. Mirror-edge (ie most of the uniform figures) figures are given by the lining notation, eg truncated cube = x4x3o.

2. Circles and cylinders are treated as the polytope group {O}, {O,O}, with the letter O. This allows xOo = circle, xOoOo = sphere, etc.

3. When multiple x appear in the circle/sphere group, this gives rise to an ellipsoid, the further x increases the size of the axies, ie

xOo = circle xOx = ellipse

xOoOo = sphere, xOxOo = oblate ellipsoid, xOoOx = prolate ellipsoid.

ie you can write aObOcOd, and x means <, o as =, and start from zero, so

xOoOxOo means 0 < a = b < c = d.

4. & generates the prism product, so

xOo&xOo = duocylinder

xOo&x3o5x = circle . rhombododecaicosahedron

&c, &c.

W
The dream you dream alone is only a dream
the dream we dream together is reality.

\(Latex\) at https://greasyfork.org/en/users/188714-wendy-krieger wendy
Pentonian

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Actually, the digits in the numbers represent the n-dimensional spherical parts. Like 1 represents a 1D sphere, or a line, 2 represents a 2D sphere - circle - 3 represents a sphere, 4 a glome, 5 pentome, etc. The numbers are like a set of instructions on how to get the rotatope starting with a point. Take the cylinder for instance. Its number is (1,2), meaning to take a point, extend it linearly, then extend it circularly. Or you could take a point, extend it circularly, then extend it linearly - you can do it in any order. The spherinder (1,3) tells you to take a point, extend it linearly, then extend it spherically - or take a point, extend it spherically, then extend it linearly. The duocylinder (2,2) tells you to take a point, extend it circularly twice.
Thanks, though I'm still trying to understand "extending a point spherically".
Speaking of duocylinders, I found something out. When you take a circle and extend it around in a circle, you don't get a 4D torus, but you get what I call a near four-dimensional cylindrical tube. A near-dimensional object is an n-dimensional object curved or folded (n+1)-dimensionally. Say you have a rectangle and you roll it up into a tube. The resultant shape isn't 2D, because it curves 3D, and it isn't 3D either, because, being a rectangle curved in 3D space, it has no depth, so it's neither 2D nor 3D, so it's an N3D rectangular tube. To make it a cylinder, you need to attach circles at the two ends and fill in the empty space inside. If you don't fill in the empty space, then it's just an N3D cylindrical surface. it still doesn't have any volume, because it's just made up of a curved rectangle and two circles. When you talk about the volume of a hollow 3D object, you're actually talking about the volume of space inside it.
Well, as I was discussing a while ago, if you take the two ends of that hollow tube, and join them together in 4-space, that is the true shape of the Asteroids screen, and all you have to do is fill the ends in with torii, and that is the duocylinder.

But when dealing with curved spaces, the dimensionality does not change into a "near n+1". Our space may be curved, but it is not consideed "near 4", it is normal 3-space. Dealing with the surface of the brane is not the same as dealing with the overall shape (n+1) it curves into.
Eric B
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Thanks, though I'm still trying to understand "extending a point spherically".

That means to replace the point with a sphere. But it's important to consider what dimensions the sphere is in. Sometimes it's possible to "spherate" without changing the dimension at all. For instance, the duocylinder is a 2D object in 4D space. This means it has two leftover dimensions, so you can replace each point with a circle and still get a 4D object (the tiger).

I've found that the bracket notation is probably the simplest way to describe rotatopes and toratopes. You can either use xyz, or simply 111 (i.e. the same as Marek14's notation without the plusses).

The 4D rotatopes are
xyzw = 1111
(xy)zw = (11)11 = 211
(xyz)w = (111)1 = 31
(xy)(zw) = 22
(xyzw) = 4

and the other 4D shapes are:

torinder
((xy)z)w = ((11)1)1 = (21)1

circle*sphere
((xy)zw) = ((11)11) = (211)

sphere*circle
((xyz)w) = ((111)1) = (31)

circle^3
((xy)z)w) = (((11)1)1) = ((21)1)

tiger
((xy)(zw)) = ((11)(11)) = (22)

If you want to include polygons and stuff, you need to define several kinds of products. It's no good to just make up names for them, you need to work out exactly what they mean. That way we can take the product of a "duotetrahedric pentaheptagonal duopentagonal duospheriprisminder" and a catenoid, or a klein bottle and a cantor set.
Last edited by PWrong on Wed Jan 11, 2006 3:23 pm, edited 1 time in total. PWrong
Pentonian

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OK, I think I get it. Though that word "extend", while easy to understans when speaking "linearly" (a point extended once linearly is a line) or even circularly (a point rotated once circularly is a circle), when you talk of a sphere (which is not nearly as simple to be swept out by a single point rotating), and then say "replace", it throws you off.

I understand it all better in terms of numbers of "straight dimensions" versus "curved" (circular) dimensions. So I take it in the bracket notations, the numbers in parentheses are the "curved" dimensions and the ones outside are the straight ones.

Now, I've heard of the torinder. (But for some reason have missed a full description on what exactly it is. I imagine it is a cylinder rotated in 4D somehow).

But what are these other things you mention? A "tiger"? "circles phere, sphere circle, and circle3"? I see you have brackets within brackets, there. 22 vs. (22), etc. I know, for instance, the duocylinder is "circular" in two perpendicular (i.e. "straight") dimensions. So is this (22) denoting something perpendicular in circular dimensions, or something like that? Then you have two that come out as (31). Are they the same thing, but arrived at different ways?
Eric B
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Eric B wrote:Now, I've heard of the torinder. (But for some reason have missed a full description on what exactly it is. I imagine it is a cylinder rotated in 4D somehow).

Torinder is a torus/line prism. I.e. what you get when you take a torus and drag it along a line into 4th dimension. I think its symbol in this notation should be ONLY (21)1, definitely not 31, which is spherinder.

But what are these other things you mention? A "tiger"?

These require some background. We discussed these shapes thoroughly before on previous threads - you might want to read them so we wouldjn't need to repost everything in here. Basically, tiger started its existence as a 4D surface with anomalous parametric equations:

x = r1*cos a + r3*cos a*cos c
y = r1*sin a + r3*sin a*cos c
z = r2*cos b + r3*cos b*sin c
w = r3*sin b + r3*sin b*sin c

This is what we got from pondering about parametric equations of various kinds of torii.

"circles phere, sphere circle, and circle3"? I see you have brackets within brackets, there. 22 vs. (22), etc. I know, for instance, the duocylinder is "circular" in two perpendicular (i.e. "straight") dimensions. So is this (22) denoting something perpendicular in circular dimensions, or something like that? Then you have two that come out as (31). Are they the same thing, but arrived at different ways?

Every toratope is either a number or something enclosed in parenthesis. If multiple toratopes are linked together without being enclosed in parenthesis, they are combined in prism product.
(22) is a tiger, and you will probably have to read the other threads to find out what it really is. (31) is sphere*circle which can be imagined in this way:

1. have a 3D sphere and put it into 4-space.
2. replace each point of this sphere with a circle whose one dimension is the radial dimension (line from the point to the center of the sphere) and whose second dimension is the 4th dimension, perpendicular to the 3-space where the sphere lies. In effect, it's a set of all points which have a specific distance from the sphere in 4D. This is analogical of one way how a torus is constructed.

Analogically, circle*sphere is a set of points in 4D with specific distance from a circle and circle^3 is a set of points in 4D with specific distance from a torus.
Marek14
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I think its symbol in this notation should be ONLY (21)1, definitely not 31, which is spherinder.

You're right, that was misleading. I'll edit it.

However, I'm working on a efficient way to count all of these shapes without writing them all down, since I can't find a formula. Under this system, (21)1 and 31 would be equivalent, because they can both be "bracketed" in only one way i.e. (31). PWrong
Pentonian

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Torinder is a torus/line prism. I.e. what you get when you take a torus and drag it along a line into 4th dimension.

I should also mention that I sometimes use "torinder" (for want of a better name) to refer to anything that isn't a rotope or a toratope, i.e. a prism product of toratopes.

It turns out the formula I found on mathworld (in another thread) was right after all! The rotopes can easily be divided into two categories: those completely enclosed by brackets (toratopes), and those that aren't (rotatopes and torinders).

Each rototope/torinder can be turned into a toratope. Just put brackets around it.

1111 -> 4
211 -> (211)
22 -> (22)
31 -> (31)

In the old thread, I described a method for counting rotopes. You take a rotatope, and replace each n-sphere with any nD toratope (toratopes also include spheres). The only problem was I didn't allow you to replace a sphere with a beast. I've just fixed that problem, and even written a program in mathematica to list the rotopes in any dimension (see the programming forum). PWrong
Pentonian

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Marek14 wrote:Torinder is a torus/line prism. I.e. what you get when you take a torus and drag it along a line into 4th dimension. I think its symbol in this notation should be ONLY (21)1, definitely not 31, which is spherinder.

Actually, when I said "torinder" I meant "duocylinder". It was a name I came up with for that object to make the names for classifying higher-dimensional rotatopes easier. But the figure was incorrectly named. I called it the "torinder" because when you extend a circle in a circular path you get a figure which I thought was a 4D torus. But it's actually - what I call - a near-4D cylindrical tube (like a hollow cylinder is a rectangular tube).

Eric B wrote:But when dealing with curved spaces, the dimensionality does not change into a "near n+1". Our space may be curved, but it is not consideed "near 4", it is normal 3-space. Dealing with the surface of the brane is not the same as dealing with the overall shape (n+1) it curves into.

That is kinda true. If our 3D space is curved, it still is 3D space, but it is curved in 4D space to form the surface of a 4D figure. It is still 3D in the sense that there are 3 perpendicular dimensions in it, but it is also 4D in the sense that it is contained in a 4D hyperplane. It is both at the same time, but it isn't entirely 3D, because it isn't contained in a 3D hyperplane, and it isn't entirely 4D, because it has no size in 4D space.
Neues Kinder
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Now I remember that during a boring history class, I wrote the cartesian for the duocylinder, as I think it is (x**2+y**2<=1, z**2+w**2<=1) and, by projecting it on the four mutually perpendicular hyperplanes Oxyz, Oxyw, Oxzw and Oyzw, I got only cylinders, which seemed a little odd to me. Is it correct?
moonlord
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