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?
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.
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?
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.
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)
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.
Thanks, though I'm still trying to understand "extending a point spherically".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.
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.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.
Thanks, though I'm still trying to understand "extending a point spherically".
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).
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?
I think its symbol in this notation should be ONLY (21)1, definitely not 31, which is spherinder.
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.
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.
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.
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