shape analogy question

Discussion of shapes with curves and holes in various dimensions.

shape analogy question

Postby batmanmg » Tue Nov 08, 2011 7:40 am

I'm looking for a corresponding shape that follows a certain pattern of stepping up dimensions, but I'm stuck on the parts in 4d becuase I don't understand the pattern. The most important part is the first blank.

2 dots on a line are a slice of a circle in 2d

2 concentric circles on a plane are a slice of a torus in 3d

2 concentric spherical shells in 3d are a slice of _____________ in 4d?


3 dots on a line are a slice of a figure 8 in 2d

3 concentric circles on a plane would be a slice of a figure 8 torus looking like 2 concentric touching torus (tori?)

3 concentric sphere shells would be a slice of ___________ in 4d?

N dots :: N looped figure
N circles :: N rippled Torus
N sphere shells :: N _______________

If i could draw this might make more sense, but then again, I'm a lousy artist. :sweatdrop:
too many people have self replicating sigs. Don't copy this.
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Re: shape analogy question

Postby Secret » Tue Nov 08, 2011 1:18 pm

batmanmg wrote:I'm looking for a corresponding shape that follows a certain pattern of stepping up dimensions, but I'm stuck on the parts in 4d becuase I don't understand the pattern. The most important part is the first blank.

2 dots on a line are a slice of a circle in 2d
2 concentric circles on a plane are a slice of a torus in 3d
2 concentric spherical shells in 3d are a slice of toraspherinder dual (place one end of a open spherindrical tubing into the other opening, then glue the loose ends inside) in 4d?


Image

batmanmg wrote:3 dots on a line are a slice of a figure 8 in 2d
3 concentric circles on a plane would be a slice of a figure 8 torus looking like 2 concentric touching torus (tori?)
3 concentric sphere shells would be a slice of ___________ in 4d?


this only works if your cut is at the middle of the figure 8, else you get 1-4 circular (or higher dimensional counterparts) bits

Correct me if I'm wrong, cause I've trouble visualising the toraspherinder dual

P.S. A spherinder is a cylindrical analogue where the bases are spheres instead of circles
P.S.2. There are at least 5 distinct types of 4D torii
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Re: shape analogy question

Postby Keiji » Thu Nov 10, 2011 12:27 am

Secret wrote:P.S.2. There are at least 5 distinct types of 4D torii


There are exactly five distinct 4D toric figures. There are the four four-dimensional torii, plus the linear extension of the 3D torus, the torinder, to make up the five.

As a side note, two concentric 3D torii comprise the cross section of a ditorus.
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Re: shape analogy question

Postby gonegahgah » Thu Nov 10, 2011 9:04 am

Just to visually explain what I think that it is that you are after for the first blank, the equivalent 4D donut (donasserat?) would be a series of concentric spheres through the 4th dimension up to length r (radius) where the outer sphere grows smaller (at the angular rate of a circle) and the inner empty sphere grows bigger at the same rate. This would occur in both 4th dimension directions; as occurs for the 3D donut through the 2D plane. I have no idea what it is called; though I labelled it a donasserat.

Does that make sense? If not I'll try to explain it better.
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Re: shape analogy question

Postby gonegahgah » Thu Nov 10, 2011 9:49 am

I should qualify. I don't understand the progression of:
1. circle of 1D dots standing halfway through a 1D line, to
2. circle of 2D circles lying halfway through a 2D plane, to
3. circle of 3D spheres lying halfway through a 3D volume.

The following would be a consistent progression:
1. circle of 1D dots standing halfway through a 1D line, to
2. circle of 2D circles standing halfway through a 2D plane, to
3. circle of 3D spheres standing halfway through a 3D volume.

The problem I have with the original is with the progression of the standing circle in the 1st step to a circle lying down in the 2nd and 3rd step.
If there is a logical progression then this would also change step 3 from what I have shown it as.
Can someone please explain how the original step 1 logically progresses to step 2?
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Re: shape analogy question

Postby Secret » Thu Nov 10, 2011 4:13 pm

gonegahgah wrote:Just to visually explain what I think that it is that you are after for the first blank, the equivalent 4D donut (donasserat?) would be a series of concentric spheres through the 4th dimension up to length r (radius) where the outer sphere grows smaller (at the angular rate of a circle) and the inner empty sphere grows bigger at the same rate. This would occur in both 4th dimension directions; as occurs for the 3D donut through the 2D plane. I have no idea what it is called; though I labelled it a donasserat.

Does that make sense? If not I'll try to explain it better.


Image
(Note that the orange and purple arcs are NOT within the spheres)

gonegahgah wrote:I should qualify. I don't understand the progression of:
1. circle of 1D dots standing halfway through a 1D line, to
2. circle of 2D circles lying halfway through a 2D plane, to
3. circle of 3D spheres lying halfway through a 3D volume.

The following would be a consistent progression:
1. circle of 1D dots standing halfway through a 1D line, to
2. circle of 2D circles standing halfway through a 2D plane, to
3. circle of 3D spheres standing halfway through a 3D volume.

The problem I have with the original is with the progression of the standing circle in the 1st step to a circle lying down in the 2nd and 3rd step.
If there is a logical progression then this would also change step 3 from what I have shown it as.
Can someone please explain how the original step 1 logically progresses to step 2?


Image
They are the same
A circle lying down on a 2D plane is because you a viewing it from persepctive/at an angle in 3D
Similarly, a sphere looks squashed when viewed at an angle from 4D
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Re: shape analogy question

Postby gonegahgah » Fri Nov 11, 2011 7:04 am

Thanks secret. Your model marvelously shows for me what I was trying to verbally explain; with the purple arrows describing the radius change of the outer sphere as we move through the 4th dimension axis and the yellow arrows describing the corresponding radius change of the inner hollow sphere at the same distance in the 4th dimension.

With the progression thing my question arises because I would have thought a more logical result progression would be 2 dots a distance apart, to 2 circles a distance apart, to 2 spheres a distance apart. I'm not sure how we logically progress from 2 dots a distance apart, to 2 concentric circles, to 2 concentric spheres.
We obviously can't have to concentric dots so I would tend to say that the example could really on start with 2 concentric circles.

Or, am I missing something?
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Re: shape analogy question

Postby Secret » Fri Nov 11, 2011 10:05 am

Since dots have 0 dimensions, and concentric means sharing the same center. Concentric dots means they coincide (i.e. locating at the same place), and result in only one dot

But there is a logical way to progress things using your description, that all the cross sections are a fixed distance apart from each other
2 dots (0D objects) x distance apart
2 circles (1D objects, as opposed to disks, which are filled in circles and has 2D bounding space (or net space as referred in the forum)) x distance apart
2 spheres (2D objects, as opposed to balls, which are solid spherical objects and has 3D bounding space) x distance apart
...and so on

P.S. There is a difference between embedded space and bounding space (not sure about the official maths term for this)
e.g. a dot has 0 dimensions, but it can be located/embedded in any number of dimensions
A rolled up piece of paper (cylinder with no caps) occupies 3D but is itself 2D
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Re: shape analogy question

Postby gonegahgah » Fri Nov 11, 2011 11:55 am

Is it also a better analogue to have 2 lines of 2r length at the distance apart than 2 dots at distance apart?
A dot is a 0D object whereas circle is a 2D object and a sphere is a 3D object.
I would guess a line is pretty much a 1D analogue for any 2D object?

So progression would be:
2 lines of length 2r at distance d apart in 1D = outer circle of radiating domino lines with a circle hole in 2D standing halfway through 1D line, to
2 circles of diameter 2r at distance d apart = outer circle of radiating domino circles with a circle hole in 3D standing halfway through a 2D plane, to
2 spheres of diameter 2r at distance d apart = outer circle of radiating domino spheres with a circle hole in 4D standing halfway through a 3D volume.

For the 1D observer they would only see the frontmost line as a dot.
For the 2D observer they could see only the frontmost circle as a shaded line or move above/below them to see two shaded lines; depending upon where they look from.
For the 3D observer they could see only the frontmost sphere as a shaded circle or move around them to see two shaded circles; depending upon where they look from.
If the spheres are small enough and close enough to them then the 3D observer with stereoscopic eyes might see the sides of the backmost sphere behind the foremost sphere when they line up.

So, is it even a better analogue to have resultant lines, instead of resultant dots, to maintain an even step up through the dimensions?
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