The Big Bang Bamma

One way to start is to take the regular polytopes. These are the regular polygons, the platonic solids and then the higher dimensional analogues. Think of the simplex, which starts as the point, line, triangle, tetrahedron, 4-simplex. Each time you add a point equidistant from the others, and join with lines. Think about it... The 2d projection (with no hidden line removal) of the n-simplex is an (n+1)-gon with all internal lines present. Look at the 4-simplex this way (pentagon), and appreciate the five tetrahedrons that make up the "surface" of the 4-simplex.Next, the cubes: point, line, square, cube, hyper-cube. Each time add a second copy a unit distance away from the first and join with lines... Appreciate the six cubes that make up the "surface" of the hypercube.Cook your brain for several months and hey-presto! You'll be seeing 4-d Seriously, you can really start visualising 4d objects this way, but it takes time. You can look at a 2d projection of a 3d object, but see it as real and rotate it in your mind. You can do the same for 4d.BTW, I fogot to mention something wonderfully fundemental...There are an infinite number of regular polygons. There are only five Platonic solids... what about the other infinitude of dimensions? How many perfect regular polytopes in each dimension? That's your homework d=2 infinite
d=3 5
d=4 ?
d=5 ?
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.This message has been edited by cavediver, 11-26-2005 08:15 AM

So you are saying if you start out with a point , and project or move the point in a perpendicular direction , you get a line . Then you project that line in a perpendicular direction and you get a square , then if you project that square in a perpendicular direction you get a cube , then if you project the cube in a perpendicular direction you get a hypercube , a 3D representation of a 4D object , is this correct ? So that you have a Larger cube (the new dimension?) surrounding a smaller cube , and all points projected are connected . So the surrounding objects of this cube are like pyramids with the top chopped off .My question is , in what direction is the cube projected ? Are all 8 points projected outwards ? I can visualise that quite easily in a 3D sense , but it still is hard to percieve this new cube as a new dimension .Okay , thats what i first thought about hypercubes , but then i saw this link http://www.well.com/~abs/SIGGRAPH96/4Dtess.html which tries to explain the formain of a hypercube from a 4D beings perspective . The hypercube is made of 8 cubes supposedly , but when looking at this picture i see only 7 hexahedrons .http://www.well.com/~abs/SIGGRAPH96/Fig6a.gifI think maybe the last one is the entire cube correct ? But then , the surrounding cubes arent actualluy cubes to us . But to a 4d being they are , so to us 3d beings , our inability to visualise the 4th dimension forces us to almost fold it in with the other 3 dimensions . Or atleast thats how i try to think of it . The 4D is the joining of the cubes to look like hexahedrons .Atleast thats the conclusion i have drawn from that links explaination .EDIT : I did a bit more searching and found this :http://members.aol.com/jmtsgibbs/draw4d.htm
Its a 3d java visualisation of a hypercube . But you can rotate the hypercube , firstly around the 3 normal dimensions , but also along the 4th dimensions . The scrollbars on the far left , near left and top control this . Now when i manipulated them along the 4th axis , i was completely stumped . And im guessing everyone else is , because i cant visualise this object rotating , atleast not yet . Maybe if i play with that thing long enough it will hit me . It also lets you do other polytopes , though i havent been game to try the others yet .This message has been edited by Darkmatic, 11-26-2005 10:53 AM

That's it If you do the above you get the 4d hypercube, not a 3d projection (a tesseract) Only becasue of perspective. They are regular cubes. Try this for a non-perspective 2d projection: hypercube 2d projectionNo supposedly about it, but you do make me think that I may have said 6 in my message above... yep, I did. Bugger That's right.There's plenty out there on this subject. Can you answer my homework problem yet?

Hi cavediver.I think I understand where you're coming from, but the thing is I already have a reasonable idea of the principles of their construction (at least I think I do)... I understand the logic of how one becomes another as we add dimensions.And I have been pondering their structure for years, yet I still can't visualize them as anything more than moving, pulsating 3D solids. No matter how hard I try, I just can't seem to make myself see all four dimensions at the same time.I'm thinking... and I may be wrong... but I'm thinking that what I want to do is visualize the fourth dimension as a perspective view, if you get me. The same way we experience the third. Technically, we don't see in three dimensions... we see a two-dimensional image focussed on our retina. We don't actually view all three dimensions in their entirety... we perceive the third dimension as depth.So, I'm guessing that what I currently experience as duration, that is, the time that elapses as I picture a hypersolid's 3D "shadow" changing shape, needs to become that perspective. If I could visualize the fourth dimension as a perspective view I would have a way of fitting all of the "shadow's" phases together into one coherent figure... just as you can use our 3D perspective to put all of those slices together to form the complete cube. Does that make sense? Well that's just it... my brain has been cooking for far longer than that and I haven't been able to do it yet. I think that perhaps you underestimate your own ability, CD. We aren't all as smart as you. Yes, but we already have an intuitive feel for that, as our eyes see that way to begin with. That's where I'm having trouble... to visualize true 4D requires you to picture something you haven't seen and have no instinctive feeling for.A 2D image can still give the illusion of 3D depth, but I've never been able to get a sense of 4D depth from any of those 4D analogues. The problem is that it's a different scenario. Trying to grasp the overall shape of a hypersolid by studying its analogue in three dimensions is like trying to figure out what a cube looks like when all you know it from is the edge-on view of its 2D shadow. Hmm... I don't know. I'm not sure how these things are calculated to begin with, but I'll give it a think.EDIT: Amended the structuralized gramaticality of a specific word string whose beginning was in the finding of thine own commencement regarding such required velocity as to necessitate a measure of pseudo-desirability.Sorry... [/Brad mode]... uh yeah... grammar.This message has been edited by Tony650, 11-26-2005 12:47 PM

Damn you spotting that! I was so looking forward to correcting you on it!

Ok, if you've played polytope generation, what about the 24-cell? Have you looked into that? It's the magic polytope in 4d. It's one of the reasons that 4d is special out of all possible dimensions.I'm looking on the net for a picture of my favourite representation but I can't find it. It's probably the construction that gave me the most insite. Esentially you take an octohedron, a truncated cube (space-station from Elite if you're old enough ), and another octahedron. They sit unit distance apart in the fourth dimension, and you join lines to construct additional octohedrons in your mind. So the top vertex of the first octo, the square face on the top of the trunc cube, and the top vertex of the other octo. That forms another octohedron "side" to the 4-solid.

Just to clarify, I haven't done any polytope generation, as such. That is to say, I haven't actually built them, myself. I've mostly just played with interactive polytopes online. Over the years, I've probably tried just about every 4D applet on the internet. Yes, but I always found the more complex ones hard to follow. I have a hard enough time keeping track of my perspective when playing with a tesseract. I find it progressively more difficult with the addition of more cells, vertices, etc. It is? Cool! What exactly is unique about four dimensions? Please elaborate! Well, if it's something you can explain relatively easily that is. If it's one of those "decades of study" things that's ok. Ok, I've been looking around and I've come across a few things. In truth, I have entirely too much to scour. I have dozens of old links to pages on the topic, and checking them all, working through all the links... ugh... it can result in a bit of a sensory overload.In any case, I found a fairly decent looking applet you might want to check. Not sure if it will help you illustrate your point about the 24-cell to me, but for what it's worth...HyperSpace Polytope SlicerIt starts with the 600-cell by default (at least on my PC) but let the applet load and then click the "Controls" button. You should be able to get a drop-down menu in the "Object" box where you can select the 24-cell (or whatever else you want).Is that any good for what you wanted to show me?As I happen to have the link handy, here's another one I found...4D 24PolytopeThough you don't appear to be able to do anything with that one.Slightly off-topic, have you ever seen the 4D version of the Rubik's Cube? There are several different versions but this was the first one I ever came across...http://www.hadron.org/%7Ehatch/MagicCube4dAppletI've had the downloadable version on my computer for years and haven't been able to solve the damn thing yet. Perhaps you'll have better luck.And, so long as I'm giving you links, take a look at these ones that I found in my travels...4DHyperbolaKleinsBottleHyperPlaneCompletely irrelevant, but I thought they were nice.

Thanks for the links! A couple of new ones there for me...re the Klein bottle. Remember that it is perfectly regular and smooth. The twisting is just an artifact of trying to represent it. 4d is maximal for regular polytopes (excluding the trivial 2d case). There are six regular polytopes in 4d, five in 3d, and only three in 5d and up. The first 5 of 4d are just pure 4d analogues of the 5 from 3d. Then there is the 24-cell, which sits on its own, unique in 4d. Just about. Notice the octahedron start and end slices, and the trunc cube in the middle (where the six square faces just touch). These are the three to which I referred.Have you thought about the spheres? Do you know how to build higher d spheres?

My pleasure, pal! I have a stack of them if you want. I appreciate the time you're putting in here with me and I'm happy to give something back. You mean the fact that it intersects itself? It's an artefact of representing its 3D cross-section? I'm not sure if I read you right. Here's hoping I understood. Ah, ok. Speaking of which, I just want you to know that I didn't ignore your homework assignment. I did try to work it out... just not with much success. The infinite number of regular 2D polygons is quite obvious and needs no explanation, but I must be honest... from three dimensions up I really wasn't sure how to proceed.I may have eventually figured out the number of Platonic solids simply by physically constructing them in my mind or on paper, but I would assume there is a way to actually calculate it. And that's just for 3D! Honestly, I don't think I have the necessary grasp of higher dimensions to have worked out any more than that... if I'd even been able to figure out that much. The Platonic solids... maybe. Polytopes in 4D and above... I seriously doubt it. Oh, well I didn't know that about it. So, basically, it has no analogue in any dimension? I couldn't figure out why I wasn't seeing them and then I realized you can change the figure's orientation, and I believe I found it under "cell first." The octahedron was obvious enough but I was having trouble finding the truncated cube. I think I'm with you now, but, just to make sure, take a look at this page...YahooIf you scroll down just a little less than half way, there's a figure there which, as far as I can tell, is exactly what I was seeing in the center of the 24-cell. It's number six on the page and listed as the "Cuboctahedron." If you scroll down a little further to number nine, the "Truncated cube"... that's what I was looking for.But the middle slice of the 24-cell matched the former, not the latter, with the squares only just meeting at their vertices before retreating again. I realize you already made the point about the square faces "just touching" and I assume that's what you meant. I don't mean to be so analytical... I just want to be sure we're on the same page here. Ugh! Sorry to be such a pain, but which "three" is that? Do you mean the three regular polytopes in 5D and above? Have I thought about them? Yep. Do I know how to build one? Um, sure... start with a point, expand it spherically to its maximum volume, and then return it to a point, all the while stretching it perpendicular to itself along the fourth spatial axis.To be honest, I find this the hardest one of all because there are no straight lines to work with. In the case of, say, a tesseract, I can somewhat comprehend its structure, even if I can't picture it. Its cells are connected by these nice flat (square) faces.Now, I may not be able to visualize how these faces can be connected while their respective cubes are perpendicular to each other, but at least I can get some general sense of the figure's structure. I can see, to a limited degree, what goes where.With a hypersphere, though, I'm kind of lost. Its curvature gives me nothing to work with. I have no problem with the concept of an "unfolded" tesseract, and can picture it folding up (though I can't picture where the folded pieces go... only their distorted "shadows" as they fold out of 3D space). But, for the life of me, I can't imagine what an unfolded hypersphere would look like.

Exactly. There is no self-intersection. It was really an internet/Wikipedia assignment rather than "work it out for yourself" !!! I would be impressed if you had reasoned it out from first principles. That said, it's not too hard in 3d. Obviously to be regular, each face must be a regular polygon. So start with the simplest polygon, look at how many you can fit around it to start building your solid. Why is not worth looking at hexagons and beyond?The magic of 3d and 4d is the existence of the dodecahedron/icosahedron and the 4d equivalent. The extra magic of 4d is the 24-cell. In 5d+ all you have are the n-simplices, the hypercubes and their duals (octahedron in 3d). Do you appreciate the concept of dual solids? Yep, it's unique to 4d. So you can see that just from a matter of mathematical symmetry, 3d and 4d are special compared to all other dimensions. Relevent to reality? I'd be surprised if it wasn't... You've got it. If you look closely, the cuboctahedron is just the special case of the trunc cube where the squares touch. It is halfway between the transformation of a cube into an octahedron. Flip back to where I was asking you to visualise the 24-cell as an octahedron either side of the cuboctahedron: The top vertex of the top octo, the top square of the cubo and the top vertex of the bottom octo make up one of the other octahedrons. Likewise with the bottom, and the four sides. So that makes 8 octahedrons forming the eight solid "faces" of the 24-cell. Ok, that will do it... but not what I was thinking:The sphere is known as the 2-sphere as it is a 2d surface, despite the fact that we usually think of it as a 3d object. The 3d filled 2-sphere is actually called the 3-ball.0-sphere is two distinct points (the poles)fill in between the points - line segment
add another line-segment and identify the boundaries (end-points)
(in other words, lay them next to each other and pinch the ends together)
Infalte. This gives a circle1-sphere is the circleFill in the circle to get the disc or 2-ball. Take a second 2-ball and place on top of each other. Identify the boundaries (pinch the edges together)
Inflate. This gives a 2-sphere2-sphere is the globeFill in the 2-sphere to get the 3-ball. Take a second 3-ball and place on top of each other. Identify the boundaries
Inflate. This gives a 3-sphereContinue ad-nauseum.

Hey CD. So is a Klein bottle an actual cross-section or a "shadow"? With a tesseract, for example, there isn't really a small cube sitting inside a larger one... the "small" cube is simply further away on the fourth axis, right?Is that the case here? Is what we see as the self-intersection of a Klein bottle in fact the "shadow" of two separate sections where one merely passes "behind" the other on the fourth axis? Oh! Uh, I mean, yeah... of course it was! I knew that!Heh, that's quite a relief, actually. I was thinking I must have been quite dumb not to be able to work out the "simple" problem you gave me. And to think, I was resisting the urge to look the answers up! I don't really know. I would guess that it's because those more complex are just different combinations of the fundamental polygons? That's just a guess, though. I can't say I'd heard of them. So I looked them up...Dual Polyhedron -- from Wolfram MathWorld...and I think I understand, more or less. I know it mentioned certain specifics which I'm missing, but, as far as I can make out, the essence of it is that where one has a face the other has a vertex, and vice versa. That's what I was thinking. In the 24-cell animation, the square faces extend until they touch vertices and then retreat. If, instead, they continued on into each other you would end up with... a truncated cube. Hmm... ok. Well, much as I wish I could, I still can't picture it. And, worse still, this doesn't even have a 3D analogue I can look at. With, say, a tesseract I can look at a square... look at a cube... and get some kind of understanding of the principles behind its construction.Is there anything at all that can give me a similar feel for the 24-cell? Or is it doomed to forever remain an abstract figure to me? Well, assuming I'm not eventually successful in picturing four dimensions, that is. Heh, I know. I was being facetious. That is, I can tell you what's required to do it, but I'll be buggered if I can actually tell you how to accomplish that. And, again, I'm with you all the way up to the final step. That last one just keeps eluding me, it seems.

Cavediver: I am still interested in this topic. I only just now found this thread and decided to read it. I have a fair understanding of the diagrams you drew. As I understand it the positron is nothing more than an electron going back in time. The illusion (or reality) of a photon creating an electron-positron pair is really an electron going backward through time absorbing a photon that changes its energy so that it travels forward in time. Of course the two (electron and positron) are seen at the same time from our perspective, but they are the same electron? I've also read where perhaps all electrons are actually the same electron travelling back in time, seen as positrons in reverse and electrons forward. It went on to say that there aren't enough positrons to balance this so it must not be true. Nevertheless, that it confusing.What I was originally wondering was, is the "shaking of hands" between electrons and photons and subsequent "scratching of the head" by electrons the only thing that produces our perception of time? I see at as though the photon instantaneously relocates itself from one electron to another. The electron then has to "decide" what information this photon is presenting, and then act accordingly. Perhaps this is crude, but is this "processing within the electron" what gives us the sensation of passage of time. It is responsible for the apparent "decrease" in light's speed through matter also, right?There must be something wrong with this idea because photons traverse "empty" space through time without electron intervention. That is unless virtual particles (electron-positron pairs) popping in and out of existence are interacting with the photons again bringing rise to the sensation of passage of time. Whew, I'm confused.This message has been edited by madeofstarstuff, 11-30-2005 05:15 PMThis message has been edited by madeofstarstuff, 11-30-2005 06:05 PM

Cavediver explaining RP2: I had you up to this point, could you clarify this sentence?

In many ways, yes. You have to have change in order for there to be a perception of time, and the predominant form of change is elecron/photon interaction. There's no real processing. It's just a case of adding two functions together: e1(x) + p(x) = e2(x) True, but what this does is challanege what we mean by single electrons and time. Yes, but this is simple absorption and re-emission. Well, yes they do. This is a very important process. We have the "bare" photon propagator, but it is "dressed" by considering one electron loop in its path, two loops, two loops interacting, 3 loops, etc. The real photon is the infinite sum of all of these possibilites:~~~~ + ~~O~~ + ~~O~~O~~ + ~~<|>~~ + ...

Let's do it backwards: Take a globe made of stretchy-stuff. Punch a hole at the south pole, squeeze your fingers in, now stretch the whole thing flat. You should have a disc, with the north pole at the centre, and the circumference is the south pole seriously stretched out.Does that help?