I agree if you're saying that there are 4-dimensional fractals... (regardless of whether you are speaking of time as 4th dimension, a 4-dimensional space, or time as a spatial dimension (upon which subject there has been some recent theorizing that I haven't caught up to yet)).
But if we're going to go that route, why not just say that any n-dimensional space will be fractable?
In the other direction, if you're saying that fractals with which we are familiar- most of which are 2-d (Sierpinski's (dp?) Triangle, the Mandelbrot and Julia sets)- must be expressable in 4-d also...
Granting that (for example) one can posit a 3-d version of Sierpinski's Triangle thingie using equilateral pyramids, there are nevertheless some problems incurred in trying to provide similar transitions for other fractal entities-
Reminds me that I was rehashing the other day my disgust with people who think that projecting a Mandelbrot onto the surface of a sphere makes it "3-d"- idiots- a 3-d Mandelbrot (if the math holds) would be a quasispherical object with smaller spheroids jutting out from it here and there in what would turn out to be a regular pattern, and similarly-located spheroids projecting from those- and that would actually be (as is also the case in 2-d) the "empty" or "null" part of the set; "from" which variously-colored fields would appear to emanate- it wouldn' be something that you could take in at a glance, but something that you would have to walk through, or cross-section.
A friend of mine did raytracings of a 3D Sierpinski tree, and a full-size framed version is on his wall. I can't find it online any more, but there IS this, which is also a 3D fractal raytracing.
Fractals are shapes of "fractional" dimensions. That's how they're defined. There are a whole lot of really good books on the subject. I've seen fractals that occupy 1, 2, and 3 dimensions, and it's very likely that they could occupy N dimensions just as easily.
...all that bangin' on about the Mandelbrot remembered to me one time when a friend and I were discussing the nature of magick, an' I said, "Well, if we ever develop a math for it, it'll likely be fractal..."
I still get an ego-boost thinking about how his eyes lit up- "Yeah!- an'- an'..."
...and that some time back in my own lj, I said that the person who comes up with a math for handling i (square root of -1) as a real number is gonna make some major money off of applications of same... I'm smart enough to be a ble to think of the possibility of such a thing, but not (I think) smart enough to be able to figure out how to make it work...
Most likely there's somebody out there who does it as a matter of course who has no idea how unusual or 'impossible' it is...
According to some theorists- sorry, don't happen to have any links just handy at the moment- time is ("merely") a(nother) spatial dimension... which still doesn't explain why our consciousness/perception of it is apparently only able to move through it in one direction (bring on the spin/polarity theorists!)... as I at least tried to point to above, in a 4-d continuum, the 4th dimension might be time, or it might be another dimension which would be more easily understood as being 'spatial'...
Hey! What about 3-d continuums in which 2 of the 3 dimensions are 2 of the 3 we're accustomed to using (height/width/depth), and the 3rd is time or another spatial dimension of the type I was just talking about? height/depth/time, or height/depth/X, with no width, for example... How does that grab you? I bet somebody else already thought of it and has gotten a Master's based on their thesis and I'm just a slowboat.
...Trying to talk about this puts us in a situation analagous to that of the poor saps who believe that projecting a Mandelbrot pattern onto the surface of a sphere, or cutting it up and wrapping it around a corner of a pyramid, or a cube, makes the Mandelbrot 3-dimensional; it does lend a certain amount of 3-dimensionality to the pattern, but it does not make it 3-dimensional, that is, it is not a 3-dimensional translation of what a Mandelbrot pattern does in 2 dimensions...
We hardly even know what the rules are for what is trying to be described here, and that does not mean that we can just make them up willy-nilly- So I feel compelled to point out that I'm not at all sure just what 'dynamic' would mean in the context in which you use it here; that the fractal moves through time? Why would it?- none of our 2-d fractals move through 2-dimensional space (unless there's something new I haven't heard of yet? -Talking fractals here, not cellular automata...)- they are all static and can be statically represented.
Even so, if we take as a given that our 4th dimension is time, there is still the possibility that some 4-d fractals might be time-static and others time-dynamic; I can't think just off the top of my head what either might look like, or how we would experience them, but I suppose that just my saying so obligates me to a certain amount of time spent thinking about it...
![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
branchandroot![[identity profile]](https://www.dreamwidth.org/img/silk/identity/openid.png){kind=small}
(no subject)
Date: 2005-09-28 05:49 am (UTC)~Zephyr~
(no subject)
Date: 2005-09-28 07:00 am (UTC)Notice the tags.
(no subject)
Date: 2005-09-28 08:45 am (UTC)(no subject)
Date: 2005-09-28 11:50 am (UTC)I'm spatial... so spatial...
Date: 2005-09-28 12:37 pm (UTC)I agree if you're saying that there are 4-dimensional fractals... (regardless of whether you are speaking of time as 4th dimension, a 4-dimensional space, or time as a spatial dimension (upon which subject there has been some recent theorizing that I haven't caught up to yet)).
But if we're going to go that route, why not just say that any n-dimensional space will be fractable?
In the other direction, if you're saying that fractals with which we are familiar- most of which are 2-d (Sierpinski's (dp?) Triangle, the Mandelbrot and Julia sets)- must be expressable in 4-d also...
Granting that (for example) one can posit a 3-d version of Sierpinski's Triangle thingie using equilateral pyramids, there are nevertheless some problems incurred in trying to provide similar transitions for other fractal entities-
Reminds me that I was rehashing the other day my disgust with people who think that projecting a Mandelbrot onto the surface of a sphere makes it "3-d"- idiots- a 3-d Mandelbrot (if the math holds) would be a quasispherical object with smaller spheroids jutting out from it here and there in what would turn out to be a regular pattern, and similarly-located spheroids projecting from those- and that would actually be (as is also the case in 2-d) the "empty" or "null" part of the set; "from" which variously-colored fields would appear to emanate- it wouldn' be something that you could take in at a glance, but something that you would have to walk through, or cross-section.
Re: I'm spatial... so spatial...
Date: 2005-09-30 02:49 am (UTC)Fractals are shapes of "fractional" dimensions. That's how they're defined. There are a whole lot of really good books on the subject. I've seen fractals that occupy 1, 2, and 3 dimensions, and it's very likely that they could occupy N dimensions just as easily.
Re: I'm spatial... so spatial...
Date: 2005-09-28 01:02 pm (UTC)I still get an ego-boost thinking about how his eyes lit up- "Yeah!- an'- an'..."
...and that some time back in my own lj, I said that the person who comes up with a math for handling i (square root of -1) as a real number is gonna make some major money off of applications of same... I'm smart enough to be a ble to think of the possibility of such a thing, but not (I think) smart enough to be able to figure out how to make it work...
Most likely there's somebody out there who does it as a matter of course who has no idea how unusual or 'impossible' it is...
(no subject)
Date: 2005-09-28 01:18 pm (UTC)http://www.upscale.utoronto.ca/GeneralInterest/Harrison/Chaos/Dimension.html
But if you mean that there should be 'dynamic' fractals that have a time component, i agree.
(no subject)
Date: 2005-09-29 03:29 am (UTC)Hey! What about 3-d continuums in which 2 of the 3 dimensions are 2 of the 3 we're accustomed to using (height/width/depth), and the 3rd is time or another spatial dimension of the type I was just talking about? height/depth/time, or height/depth/X, with no width, for example... How does that grab you? I bet somebody else already thought of it and has gotten a Master's based on their thesis and I'm just a slowboat.
...Trying to talk about this puts us in a situation analagous to that of the poor saps who believe that projecting a Mandelbrot pattern onto the surface of a sphere, or cutting it up and wrapping it around a corner of a pyramid, or a cube, makes the Mandelbrot 3-dimensional; it does lend a certain amount of 3-dimensionality to the pattern, but it does not make it 3-dimensional, that is, it is not a 3-dimensional translation of what a Mandelbrot pattern does in 2 dimensions...
We hardly even know what the rules are for what is trying to be described here, and that does not mean that we can just make them up willy-nilly- So I feel compelled to point out that I'm not at all sure just what 'dynamic' would mean in the context in which you use it here; that the fractal moves through time? Why would it?- none of our 2-d fractals move through 2-dimensional space (unless there's something new I haven't heard of yet? -Talking fractals here, not cellular automata...)- they are all static and can be statically represented.
Even so, if we take as a given that our 4th dimension is time, there is still the possibility that some 4-d fractals might be time-static and others time-dynamic; I can't think just off the top of my head what either might look like, or how we would experience them, but I suppose that just my saying so obligates me to a certain amount of time spent thinking about it...