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Rrhain Member  Posts: 6351 From: San Diego, CA, USA Joined: |
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Tony650 Member (Idle past 4058 days) Posts: 450 From: Australia Joined: |
Hi Rrhain,
Thank you for replying! So it's more a matter of perspective than something I, for instance, could actually do to a 2D being? If so, I'm not sure I understand why Sagan brings it up in the context of dimensions. Is it directly related, in some way? Do we actually need to add a dimension to view any given universe this way? For example, couldn't a microbe (limited mental capacity aside  ) living inside my body view the interior as the "outside" of the shell (my body) which "encases" the universe "inside" it? I think I see where you're coming from but how does this actually relate to dimensions? Is it because it's based on a fractal topology which, by definition, has a fractional dimensionality? Thanks again for your time, Rrhain. And please feel free to reply to anything of interest, in this thread; I'd loved to hear any thoughts you may have.
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Rrhain Member Posts: 6351 From: San Diego, CA, USA Joined: |
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Brad McFall Member (Idle past 5058 days)  Posts: 3428 From: Ithaca,NY, USA Joined: |
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Tony650 Member (Idle past 4058 days) Posts: 450 From: Australia Joined: |
Hi Rrhain,
First of all, thank you for your ongoing tuition. There don't seem to be many who share my passion for this subject (they're crazy  ) and you can't even begin to imagine how much your help means to me. Seriously!
Rrhain writes: Well, it depends upon what you mean by "actually do." You know, I actually thought that I should have been clearer about that. To clarify, I meant simply orienting it in some way, as opposed to physically altering it. I figured that he mentioned it, in this context, because it was simply a case of taking advantage of an additional dimension, and didn't require actually distorting the object in question. I mean, I can imagine how I could take a square living in Flatland and make him match Sagan's description. However, it would involve physically twisting him inside out, and stretching him out infinitely in all (two-dimensional) directions until he "surrounds" his universe. Somehow, I don't think this was Sagan's meaning. So what I meant was, basically, is it something that I could "actually do" without otherwise twisting, stretching or contorting the square? For instance, the example of the "mirror image" requires no contortion; all you need to do is rotate the square 180 through the third dimension. I hope that explains my meaning a little better. Apologies for my lack of clarity.
Rrhain writes: It is very easy to physically turn this object inside out so that the inner surface is on the outside and the outer surface is on the inside. Initially, this is what I thought Sagan meant. I figured that a circle, let's say, being infinitely flat, would be no problem for a three-dimensional being to simply grab and twist inside out, in the same fashion as your rubber band experiment. Its outer face would then be inside, and its innards would be outside. Of course, when I read that the entire cosmos would be "on the inside" I realized I must have been on the wrong track.
Rrhain writes: Now, you just need to extend that motion to everything else. Stuff everything outside the rubberband on the inside and everything inside comes out. Note, you will have to use a mapping to scale the infinite outside to the finite inside, but that in and of itself is simple to do: For every linear unit of distance you go, you progress exponentially further and further along the projection so that by the time you get to the center, you're at infinity. Ok, this is where I start having trouble. I believe I understand the concept, but I'm still not sure if we're talking about perception or reality. I didn't phrase that very well. Let's try an example, using a circle in a two-dimensional universe. Scenario 1 A circle sits within a two-dimensional universe, with side A facing out and side B facing in. Scenario 2 A circle encloses a two-dimensional universe, with side A facing in and side B facing out. Now, I think what I'm trying to get clear in my mind is this: Are you saying that we're physically turning Scenario 1 into Scenario 2? Or simply that Scenario 1 and Scenario 2 are mathematical equivalents; that in a sense, there is no difference between them? Is it just a matter of perspective, or is the idea for not just the circle but its entire universe to be turned inside out? That is, taking the circle and the two-dimensional space it sits in, and twisting it all, as a single unit, such that the circle's universe gets folded into its interior, while its inside rotates around to become its outside?
Rrhain writes: Well, in order to do it, depending upon the object, you have to traverse the next dimension up in order to do it. For example, to turn a torus inside out you either need to cut a hole in it or move it through the fourth dimension. Other topological things can happen when you move in higher dimensions. Yes, I understand the torus example. Feel free, though, to elaborate on those "other topological things" if you wish. I am always eager to learn more about that kind of thing.
Rrhain writes: One thing it points out is that things acquire a "handedness." That is, if we have established "up" and "down," then some things face to the left and some things face to the right and they always will face that way. There is no way to turn them around...unless you go through the third dimension. Yes, this is the concept I referred to as a "mirror image." This is an accurate description, isn't it? A being that was reversed in this way would be, in essence, a "mirror image" of his previous self, yes? This is something else I find interesting. If a resident of hyperspace were to "reverse" one of us like this, (assuming, of course, that we could actually live in this "mirrored" state) would our perception itself be reversed? Our eyes having traded sides, would we now see a mirror image of the world? Or is that not actually a property of binocular vision?
Rrhain writes: Since the tiger is topologically equivalent to a torus (think about it)... Heh, you're too smart for me, Rrhain. I've been thinking about it but I'm afraid I can't see how a tiger and a torus are topologically equivalent. Unless perhaps "tiger" is mathematical code for something and not what I'm actually thinking of.
Rrhain writes: Yeah, but that's a semantic argument, not a topological one. It is based upon limited understanding of the universe. There is no reason the bacteria, kept within its own dimensions, could be brought out of the body to see what lies outside it. That's what I thought. And therein lies my confusion with the original point; I'm not clear whether Sagan's "inside out" scenario is supposed to represent something real (in principle) or just a matter of perspective. As it was raised in the context of dimensions, I assume he meant something real, and the impression I'm getting from what you've said also seems to indicate this.
Rrhain writes: One of the problems is that "dimension" is fractal geometry doesn't quite mean the same thing that it means in topological metrics. That is, a fractal "dimension" is more of a description of self-similarity. Yes, I actually meant to ask you about this. Years ago, when I first learned about fractals, I vaguely recall reading something like "...therefore, in a sense, a fractal can be thought of as having a dimensionality greater than two but less than three." I've never been clear on just how "real" that dimensionality is but I've always suspected that it's more a case of having certain properties which are not strictly two-dimensional than an actual dimension.
Rrhain writes: It turns out that this definition of "dimension" results in the common ones we think of when thinking of regular, n-dimensional objects. But when objects become self-similar so that internal parts are small copies of the larger structure, you end up with a mathematical calculation that returns a fractional number, not an integer. For example, each side of a Koch snowflake has a fractal dimension of 4/3, if I recall correctly. That's because you start with an object that is three segments in composition (each side is broken into thirds) and you replace it with an object that is four segments in composition (the first third, the triangular bump which is two segments long, and the final third). In a straight line segment, getting closer has no buminess, so its fractal dimension is integral (3/3 = 1, which is what we expect from a one-dimensional line.) So would it be accurate to say that a fractal's dimensionality is the ratio of the number of new segments to the number of divisions in the replaced segment (as in the case of Koch's snowflake, where four new segments replace a single segment with three divisions)?
Rrhain writes: In the end, a Koch snowflake is a two-dimensional object. It exists in two dimensions. It has length and width but no depth. Ergo, 2-D. That's what I suspected. Thanks for the confirmation. As I understand it, the Koch snowflake will always remain within the circle enclosing the initial triangle no matter how many times you repeat the process, correct? So does the snowflake itself actually tend towards a perfect circle as the repetitions approach infinity? Is a circle, in some sense, a fractal itself? Is it enough that it changes direction at every point, or does it actually require some combination of both "zigs" and "zags" rather than simply an infinite string of tiny "zigs" to constitute a fractal? Hmm...that wasn't phrased very well, but hopefully you get my meaning. Ack! I didn't mean to get so carried away. I hope you don't mind all these questions. Thanks again for your time, Rrhain.
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Tony650 Member (Idle past 4058 days) Posts: 450 From: Australia Joined: |
Hi Brad,
I'm afraid you managed to lose me again, but one thing in particular did catch my eye.
Brad writes: In the WhiteHouse I sat a few chairs away from Sagan duing a Science and Humanities Seminar... At first, I thought you were quoting something, but I didn't have to read far to realize that it was written in your...well..."unique" style. So let me make sure I understand what I'm reading. Are you saying that you (Brad McFall) attended a Science and Humanities Seminar at the White House which was also attended by Carl Sagan? Carl Sagan, the creator of Cosmos? Carl Sagan, the author of Contact? Argh! You've lived one of my dreams; he's one guy that I would love to have met! Unfortunately, I'll never get the chance now. Such a pity.
Brad writes: ...the intro and exit to the showsCOSMOS , traveling through a dark space... You know, I've never actually seen the show, although I've heard a bit about it. I recently bought the book, though, which I'm currently reading (and very much enjoying). It took me quite a few years to get around to buying it but better late than never, I guess.
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Rrhain Member Posts: 6351 From: San Diego, CA, USA Joined: |
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Brad McFall Member (Idle past 5058 days) Posts: 3428 From: Ithaca,NY, USA Joined: |
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Tony650 Member (Idle past 4058 days) Posts: 450 From: Australia Joined: |
Hi Rrhain,
Rrhain writes: Um, how would you accomplish it if you weren't going to do anything? If you're going to take the insides and put them on the outside and take the outsides and put them on the inside, then you have to actually do it in order to actually accomplish it. Yes, once again, I apologize for my poor phrasing. I realize that you would have to do something to accomplish Sagan's example. I think what I'm confused about is the fact that, until now, the examples that I've seen of "altering" things via an extra dimension have not actually required changing the size or structure of the object from the perspective of the extra dimension; only its orientation. The "mirror image" is an example that often comes to mind because when I first read of this, I assumed that it would require changing the object in some way. However, coming down a dimension, I could see that a square could be rotated via the third dimension such that he would appear to have been physically altered in two dimensions. Beings in his universe wouldn't see it as a mere change in orientation; it would appear to them that either side of him had been physically pulled through each other until they had traded places (i.e. his actual structure would appear to have been warped). But from our vantage point, we could see that it's still exactly the same square; nothing about its structure has been changed at all. It has simply been rotated. So when I came to visualize the "inside out" scenario, I couldn't imagine how I could do it to a two-dimensional being without altering his actual structure in some way, from my perspective. I have probably been under the misunderstanding that in scenarios of this kind, what will look to a Flatlander like a physically altered object, to a three-dimensional being, need only be the same, unaltered object, merely rotated, turned, moved, etc the right way.
Rrhain writes: You simply need to do that with a vision that the band is infinitely thick. You invert it so that the stuff that's near the center is now near the outer edge and the stuff that near the outer edge is now near the inner central point. Ok, so it does require altering not only the being (or object), but its whole universe, too. So if I understand correctly, this change would be imperceptible to those inside such a universe, as they would be altered (curved?) along with the space they exist in. We could see that a two-dimensional plane had been folded inside a circle while the circle itself was folded inside out, but to those living within such a universe, no change would be noticed; they would still perceive themselves as sitting in an infinite two-dimensional space looking at the outside of an ordinary circle, correct? If this is correct, then I think what was confusing me was that Sagan was referring to a change that would be apparent not from within the universe itself, but from the dimension doing the changing. The examples that I'm used to are the inverse. I'm used to those which involve something that appears to change from within the universe but, when viewed from the next dimension, are seen not to change (or change in a different way). A cube passing through Flatland, for example, may appear to change shape, size, colour, etc to the Flatlanders, but we in Spaceland can see that all of these things are in fact static, and the only thing that actually changes is its position, as it passes through the plane.
Rrhain writes: They're not the same thing, but Scenario 2 is just an extension of Scenario 1. The former is only paying attention to the border while the latter takes into account the border and what lies on either side. Hmm...then would I be going too far to say that in Scenario 2 the circle actually has no other side? That in scenario 1 there exists two-dimensional space, circle's side A and circle's side B, but in scenario 2 all that exists is two-dimensional space and circle's side A? Or does this not work, since in scenario 1 there is a finite space inside the circle?
Rrhain writes: Um...there is a hole that runs all the way through a tiger. Ah! Of course! I apologize for making you write the whole paragraph; as soon as I read that first sentence, it hit me like a ton of bricks! I don't know why that didn't occur to me. Anyway, thank you; I understand.
Rrhain writes: No, no, no. There is no "enclosing circle" of a Koch snowflake that it is converging toward. You can draw a circle around it, yes, and that is the proof that the snowflake has a finite area but infinite perimeter, but the snowflake, itself, does not go to a circle. Oh, I see. Sorry. I knew I'd read before that it would always remain within the circle enclosing the first triangle, and I assumed this meant that it would reach the circle at infinity. Thanks for the correction.
Rrhain writes: If it did, it would converge to a finite perimeter: 2 * pi * r. Instead, the perimeter is infinitely long. This actually answers my final question, about a circle being a fractal. Again, I should have realized this.
Rrhain writes: What makes an object "fractal" is its degree of self-similarity. Take the classic fern. It is a central line with a bunch of branches off that central line. But each branch can be considered a smaller "central line" with branches off of it. And each of those branches is a smaller "central line" with branches off of it. The closer you get, the more of the large-scale structure you see. Each piece reflects the whole. Is it fair to say that, mathematics aside, there really can't be any true fractals as, in the real world, there is a practical limit to how far it can go? A fern, for example, may have a "fractal-like" structure but, eventually, the structure of matter itself makes it impossible for its pattern to repeat infinitely. Or is infinite repetition not necessary to constitute a fractal?
Rrhain writes: A circle, however, is not self similar. A circle is not made of tinier circles. Yes, a silly mistake, on my part. I should really stop writing these replies when I'm tired. Of course, if I did that, I'd probably never reply. As always, thank you for your time, Rrhain.
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Tony650 Member (Idle past 4058 days) Posts: 450 From: Australia Joined: |
Hi Brad,
Brad writes: Yes, I guess it might have been in 1984 or 5. Argh! I'm so envious! I don't suppose you've met Stephen Hawking? He's another one of my "heroes."
Brad writes: He sat behind me, a row or two back, in the Cornell Andrew Dickson White HOUSE next to the Space Science Building and in front of the Big Red Barn on the Cornell Campus. Oh, I thought you meant the presidential White House.
Brad writes: There was a small circle of a handful of scientists at that time which included Carl as a principal which was the brains"" behind CU-earlier. Were there any other big names there? Just curious who else you've met.
Brad writes: There was quite an outpouring in the town of Ithaca after he died. I can imagine. I wish I'd had the opportunity to meet him myself. But then I imagine I'd have quite a line to stand in, so to speak.
Brad writes: SpaceSciences is to the RIGHT of the second picture below during scroll.Page not found | The College of Arts & Sciences His house/residence was on the edge of a gorge near campus, that required an elevator to get DOWN into it. The seminar room LOOKED out to the tree between the two white posts in the second image above. Cool. Thanks for the link, Brad.
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Brad McFall Member (Idle past 5058 days) Posts: 3428 From: Ithaca,NY, USA Joined: |
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Brad McFall Member (Idle past 5058 days) Posts: 3428 From: Ithaca,NY, USA Joined: |
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Tony650 Member (Idle past 4058 days) Posts: 450 From: Australia Joined: |
Hi Brad,
Brad writes: Of course Gould would accept what you said about matter... You mean about it making real-life fractals impossible? Well, it depends. As I said, I don't know if a real-life fractal's structure would need to repeat infinitely. My understanding (which is not great, I should point out) is that for a pattern to truly be a fractal, it must be infinitely recurring, such that no matter how far you magnify, you will always just see the same pattern repeating over and over. Now, again, this may be wrong. The problem is, I'm really not certain which properties must exist to constitute a fractal. I have an idea of some of the properties fractals are observed to have, but are they all required, by definition? As Rrhain pointed out to me, it is an object's degree of self-similarity that makes it a fractal. So the question is, how much self-similarity is necessary? If it can have a limit then fractals could conceivably exist in real life; if it cannot have a limit then I don't see how it would be possible given that the structure of matter will eventually make it impossible for the pattern to go any further. I suspect that a true fractal would fall into the latter category, myself. I have always been of the understanding that a fractal has, by definition, a fractional "dimensionality" because its pattern is infinitely recurring, and therefore its perimeter is infinitely long. So, even if we could replicate this all the way down through the atomic and sub-atomic levels, we would still eventually reach a barrier at the smallest unit of quantized space, beyond which, the pattern could no longer continue. However, having said this, if a repeating pattern could be continued down to Planck length itself, perhaps this would be enough. As I understand it, a true fractal (with an infinite perimeter) will literally have "a bump at every point." Now, here's an idea that just occurred to me. If Planck length is the absolute limit, and it has no meaning to discuss anything "below" it, then could it, in some sense, actually be said to have zero size? Could one unit of Planck length (that is, a single quantum of space) be viewed, for all practical purposes, as a point? If so (and the impracticality of it aside, of course), would a real-life fractal whose perimeter "bumps" at every unit of Planck length indeed be a true fractal? Would it literally have an infinitely long perimeter? Incidentally, I do understand that Planck length has a non-zero size. What I'm trying to work out is whether or not there is any practical difference between a unit of zero size, and a unit which, while non-zero in size, is the smallest unit that can have any meaning in our space; is there any real-world difference between a length, below which, there is nothing smaller (which, in fact, has no meaningful "below" to speak of), and a length of simply zero? I understand the difference, mathematically, but in practical terms, can there even be a unit of absolutely zero size in a universe where it would be smaller than the smallest meaningful unit? Or, in the end, is this really just a matter of semantics? Perhaps the problem is that I'm thinking of zero size as an actual measurement of space that has its own, independent existence, instead of merely the defining property of no space, at all. This would seem to be a problem whenever words like "nothing" and "exist" start getting used in the same sentences. Boy, I went off on quite a tangent in this post. I'm utterly exhausted, at the moment, so I hope what I've written isn't completely ludicrous. It was just a thought that occurred to me, just now. I'm sure I'll be totally mortified when I re-read it after a nice, long sleep.
Brad writes: ...he wanted to know if I was a physicist,mathematician or philosopher AFTER I, BSM< corrected him, on his denotation of actual infinity. You corrected him? Heh, Brad - our resident prodigy. There's so much I'm sure I could learn from you if only I could understand more of what you write. Well, some day, perhaps.
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Brad McFall Member (Idle past 5058 days) Posts: 3428 From: Ithaca,NY, USA Joined: |
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Tony650 Member (Idle past 4058 days) Posts: 450 From: Australia Joined: |
Hi Brad,
Brad writes: I see where you might have an issue with the word "self". You mean with regard to the term "self-similar"? Well, I may. As I said, at this point it's the amount of self-similarity that I'm wondering about. That is, whether or not anything less than infinite repetition can constitute a true fractal. Also, after re-reading my idea regarding Planck length, I now have my doubts. I knew I would...ugh...why do I write these when I'm so tired? My thinking was that, despite being a non-zero value, Planck length may, for all practical purposes, be seen as a point. Or to put it another way, I reasoned that, as our universe's lower limit, Planck length may perform the same function in the real world as zero performs in the mathematical world. Now, for all I know, this may be true. However, on further consideration, I can't escape the consequences of the math. As far as I can see, no value greater than zero (even Planck length) can be a short enough distance between "bumps" to give a perimeter infinite length. As minuscule as it is, even Planck length can only repeat a finite number of times along a perimeter enclosing a finite area. I really can't see any way around this. Now, I'm speaking purely mathematically. This may not apply in the real world. To be honest, though, moving away from pure math and into practical application, I couldn't even hazard a guess on this one. There are too many factors that I simply wouldn't know how to account for. One thing that comes to mind is that if we could create a structure, in the real world, with fractal properties down to the Planck scale, wouldn't the structure's pattern itself become subject to quantum processes? Even if we could create a fractal pattern that small, would it be able to "hold" its pattern, or would it just become a haze of quantum uncertainty, with parts tunnelling through each other, popping in and out of existence, etc? Ack! It gives me a headache just thinking about it. The theory (mathematics) of it was bad enough, but now that I consider the practical implications of it, I can see a whole new Pandora's box waiting to be opened.
Brad writes: All living shapes need not have to correlate with space across the English Channel. (That was a joke). Heh, you're too smart for me, I'm afraid. I'm sure it was funny, though.
Brad writes: I know you have responded to more from me than I have given back. That's ok, I do that with just about everyone. I tend to rant a lot, and as such, I usually end up being the most talkative participant in any given (online) discussion. I recently joked to a friend that I have a chronic typing disorder which compels me to write a paragraph reply to every sentence that I read.
Brad writes: You are the only poster here really using the technology as I see fits well. Using the technology? You mean the forum technology? I use it because I reply to you, perhaps? I'm not sure if this is what you mean, but as I've said previously, I think it's a shame that you're often ignored. I think you have a lot you could contribute, if only we could understand more of what you say.
Brad writes: I am sorry I have not given you more attention. That's ok, Brad. I appreciate the attention you have given me. If not for the contributions of you and Rrhain, there wouldn't have been much to this thread, at all. Honestly, I appreciate all contributions, however big or small.
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