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See 4-D Space With 3-D Glasses 232

purpleant writes: "A hyperplane is a 3-dimensional space that slices through the 4-dimensional space, the same way a 2-dimensional plane can slice through our 3-dimensional space. The bounding hyperplanes can be extended infinitely so that they criss-cross through each other, chopping up hyperspace into many 4-dimensional 'chunks.' Again the inner chunks are finite, and they are distributed in shells around the core polytope. The HyperStar applet displays those finite chunks, one shell at a time. The inner shells are complete -- each shell completely encases the previous shell. The outermost shells have holes in them."
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See 4-D Space With 3-D Glasses

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  • As if trying to visualize the 4th dimension wasn't enough to give you a headache it comes with a 'cross-eyed' setting.
  • All this talk about envisioning 4-d space reminds me of an interview Forced Exposure did many years ago with SF writer/mathematician Rudy Rucker, who has written extensively about dimensional spaces beyond the third, both in fiction and nonfiction. I couldn't find the interview online anywhere, but the exchange went something like:

    FE:So, what's the closest you ever came to experiencing the fourth dimension?

    Rucker: Well, there was this one time, we'd been partying all night, and I wanted to get a little higher. We were almost out of drugs, but I did have some acid. So I took it, but then I fell asleep. And when I woke up...

    FE: Oh, I've been there. That's the fourth dimension?

    • If youw ant to read an interesting description of "seeing" in a 5 dimensional plane (4 dimensions of space and 1 of time) checkout the excellent novel by Greg Egan [amazon.com] called Diaspora [amazon.com].

      While the narrative occasionally grinds to a halt for lectures on advanced mathmatics and quantum physics it is a very enjoyable "hard science fiction" read.

      Greg Egan [amazon.com], Brian Stableford [amazon.com], Greg Bear [amazon.com] and Robert Charles Wilson [amazon.com] are my personal picks for the best hard science fiction writers out there today.

      And for you movie fans that wanted to know what James Cameron's "Avatar" was going to be like, BIOS [amazon.com] by Robert Charles Wilson [amazon.com] contains a large number of very similar elements.
      • Hell yeah. Greg Egan does my head in each and every time. Admittedly he does occassionally go off the rails with the maths and physics, but I kinda like that. The thing that amazes me with Egan is that for an author he really does understand his stuff. He publishes on physics, including all the mindboggling mega dimensional super flipout brainfuck stuff. His website (sorry no URL, google it) has all sortsa homespun java apps to illustrate the math in his books.
        For a real bender, try Permutation city where he posits a cellular automata simulation that kinda busts off into it's own reality due to the internal self sufficiency of it's math (yes I'm sure he's aware of philosophical-mathematical objections to this notion). It's a brainwarp of a book.
  • so it's 5 am monday morning on the east coast and i decide to check out slashdot before going to bed for 4 hours of shuteye before work tomorrow (don't ask, crazy weekend). i am bleary eyed, brain dead, exhausted.

    and i read "The bounding hyperplanes can be extended infinitely so that they criss-cross through each other, chopping up hyperspace into many 4-dimensional 'chunks.' Again the inner chunks are finite, and they are distributed in shells around the core polytope."

    dudes! my functional iq right now is about 50! if you are going to post these kind of stories on slashdot, could you PLEASE post them around, say 3pm on a thursday? thanks ;-P

    i should be awake by then, and i promise i will come back and try to wrap my mind around this story at that time... grumble, grumble
    • Re:hold up... (Score:3, Informative)

      by Elbereth ( 58257 )
      That's nothing. I just took two Benedryl. I think I'll go to sleep now.
    • Hey that depends on what time zone your in, ehh! For me it was posted at about 10pm, so I'm full up with caffeine [thinkgeek.com] ready to do about 5 hours of University assignments. So I think I'm triping out at the moment, ooh look at the pretty colours. Geez my hand is twitching so much I can hardly type. Getting back on topic (sorry), it is certainly an interesting bit of coding. It would be pretty cool as a screen saver (preferably using OpenGL or god forbid DX) for a bit more performance. Do you think they'll GPL the source?
  • by RenHoek ( 101570 ) on Monday August 05, 2002 @04:09AM (#4010860) Homepage
    It's one of those OpenGL Windows 95 screensavers! :) Why didn't you just say so?
  • by Anonymous Coward
    "Again the inner chunks are finite, and they are distributed in shells around the core polytope."

    uhh yeah, but it all depends on the capacitive current getting to the space modulator

    come on, lets do a thorough reporting job. You didnt even mention the fucking space modulator.
  • Looking directly at four-dimensional stereograms can *really* screw up your eyes. Caution or strong hallucinogenics advised.
  • by AaronBaker2000 ( 480581 ) on Monday August 05, 2002 @04:21AM (#4010878) Homepage
    If you set the Stereo mode to "Cross-eyed," you can view the picture in 3-D using the Magic Eye [vision3d.com] technique.
  • I don't have a pair of 3d glasses, is there any kind of computer screen filter that has the same effect? Or do one's actual eyes need to be cross-eyed?
    • I found the Parallel (sometimes called "Wall-Eyed") mode to be the best, especially at this early hour. If you get your head in just the right spot and relax your eyes just right, you get a decent 3D effect without the odd color effects of the old 3D glasses method (which also works, but to me it looks odd).

      If you've viewed random dot autosteroegrams using the wall-eyed technique (or the cross-eyed technique) this is the same thing, except the movement will be a distraction.

      As always, YMMV

    • I don't have a pair of 3d glasses, is there any kind of computer screen filter that has the same effect?

      Yeah, they call it "Alcohol"

  • This sounds alot like Gene Ray, of Time Cube. [timecube.com] Creepy, man.
  • This isn't new. There have been real applications and Java applets, too that have been able to do this sort of thing. I've got an old app for the classic Mac OS called "HyperSpace" that does exactly this (draw multiple 3D cross-sections of 4-D hypercubes). It's a really, really old app.
  • Hypercube (Score:3, Informative)

    by imevil ( 260579 ) on Monday August 05, 2002 @05:00AM (#4010936)
    Here [dmawww.epfl.ch]

    Less pretty but more understandable
  • by SpatialJ ( 411269 ) on Monday August 05, 2002 @05:00AM (#4010937) Homepage Journal
    It's a pity E. Abott, author of "Flatland: A Romance of Many Dimensions" [amazon.com] has died some 80 years ago. Maybe he even would have sold the film-rights of his n-dimensional love story "soon showing in java-applet near you".

    BTW: things like the famous Stereoscopic Animated Hypercube [scorpius-farscape.tv] have been around for quite some while. There even is a game [www1.tip.nl] around to be played.

  • by tanveer1979 ( 530624 ) on Monday August 05, 2002 @05:07AM (#4010946) Homepage Journal
    the thing about opening our minds is right.
    We have always lived in three dimensions, so visualizing 4 dimensions Per Se is almost impossible coz our nuerons have been hardwired for 3 dimensions. So we can observe 4 dimensions in transit. For example if youwere a 2 dimensional being(thats not possible coz 3 is the minumum number of dimensions to sustain life) and a 3D sphere passed through your space, you will see a point, growing into a circle and then again into a point.
    So if a 4D object came it would look like a morphing 3D object.
    If mankind were able to create and use 4D's travel would be a whole new frontier. Esp since space-time is curved, Just imagine traveling a million miles instantaniosly
    Confused! Go through stephen hawkings works! you will be even more so :-)
    • Actually, a sphere passing through a 2D world would appear to be a point, then a line of increasing length, then decreasing length, then a point again.

      Or maybe that is a 3D'er view of things? I used a compact disc and a pencil to demonstrate this recently, and if my 2D example beings lived in the plane of the CD, then the pencil passing through the center of the CD would look similar to a sphere passing through a similar plane, if viewed along the plane (as a resident of said plane).

      A long time ago, I read this silly book called Planiverse about a fictitious 2D world. The author presented it in a fantasy-like format about encountering 2D beings via a computer, but the concepts were presented such that a young person (as I was then) could see what was going on. For what it's worth...

      By the way, I just looked, and it Planiverse is still around [amazon.com]. I had never read Flatland, but that is linked there on that Amazon page as well.

      • Actually, a sphere passing through a 2D world would appear to be a point, then a line of increasing length, then decreasing length, then a point again.
        If 2D lifeforms did exist (Planiverse suggests they would need zipper-like 2D organ structures) it's quite likely they would have some form of 'depth' perception - along a plane, of course. A sphere intersecting with their world could indeed be recognized as a circle, much like our brains can recognize the depth difference between a ball and a flat disc. The interesting thing about a book like Planiverse is that it inspires you to think about higher dimensions, using the simple 2D to 3D examples in the book as analogies. In my opinion, this serves as a better starting point for learning about the fourth dimension than any visualization applet.
        • (* If 2D lifeforms did exist (Planiverse suggests they would need zipper-like 2D organ structures) it's quite likely they would have some form of 'depth' perception - along a plane, of course. A sphere intersecting with their world could indeed be recognized as a circle, much like our brains can recognize the depth difference between a ball and a flat disc. *)

          If a 4th dimensional biological entity intersected our dimension, it would probably look really really funky, like a bunch of morphing blobs. Imagine sitting there in a chair watching Bay Watch, and suddenly in the middle of the room small blobs of flesh appear, grow in size, but change shape in really really odd, unnatural ways, then disappear.

          If that wouldn't make your skin crawl off, I don't know what would. It would even be odder than watching obese porn in reverse (not that I recommend it).

          Note that there are some UFO reports of odd blobs appearing, morphing funny, and then just dissappearing. Long shot, I know, but you just never know what may have came our way before.
    • For example if youwere a 2 dimensional being..., growing into a circle and then again into a point

      Wrong. If you were a 2D being and a sphere assed through your plane you'd see a dot grow into a long "wall" or "line", disappearing into a dot again.

      The book Flatland [alcyone.com] (available online in its entirity) covers all the nuances of the 2D lifestyle.

    • We have always lived in three dimensions...
      So if a 4D object came it would look like a morphing 3D object.

      Actually, if you define time as the 4th dimension, we *are* in fact 4-dimensional beings (and perceive our world in 3-space, although we are capable of perceiving motion over time, which could be argued as a 4th-dimensional perception...). We indeed do appear in 3-space as a "morphing" 3D object -- think of how a human is born, grows, matures, and dies as the "motion".

      Of course, I'm no mathemetician or physicist, so I have no idea as to whether the "time as the 4th dimension" thing is even valid, but at least it's something fun to ponder on the long commuter rail trip =)
    • We have always lived in three dimensions, so visualizing 4 dimensions Per Se is almost impossible coz our nuerons have been hardwired for 3 dimensions.


      Breaking up these limitations is not as hard as it might seem. The traditional length X width X depth is just an example of a 3d room. I understood multi-dimensionality with this simple analogy:

      Imagine the "room of cookies"

      1st dim: color (red, green, blue,...)
      2nd dim: shape (round, square, triangular,...)
      3rd dim: consistency (very hard, hard, soft,...)
      4th dim: size (from very small to very large)

      There you have it. A 4dim room that can be used to express any kind of cookie in a mathematical vector. For adding more dimensions all you have to make sure is that the new dimension os orthogonal, which means that the new component/unit has to be linear independent of all the other components/unit (which could for instance be the 5th dimension of texture (like smooth, rugged, etc.)

      (Not an native english speaker, so please excuse me for using incorrect/half correct words.)
      • (* There you have it. A 4dim room that can be used to express any kind of cookie in a mathematical vector. For adding more dimensions all you have to make sure is that the new dimension os orthogonal......*)

        OOP fans don't seem to get this. Their single-demensioned taxonomy that they use for polymorphism assumes that a single dimension is sufficient. (I know, some OO fans realize that simple-minded textbook polymorphism is not sufficient, but their "fixes" are often worse than the desease in many cases.)

        Polymorphism is pretty much one-demensional, while the real world is multi-demension (WRT number of orthogonal or semi-orthogonal factors which influence behavior and grouping.)

        Stepanov (of STL fame) has recognized this also when complaining about OOP. OO authors like Meyer gloss over it or pretend like it is a non-issue using wiggle words.

        OOP has dimension problems. Your cute shape and animal taxonomy examples can't scale beyond the dimension of the demonstration. (They pick examples where one dimension just happens to reign supream).
    • This guy doesn't know what he's talking about, he's just trying to sound smart.

      Please ignore him.
    • For example if youwere a 2 dimensional being(thats not possible coz 3 is the minumum number of dimensions to sustain life) and a 3D sphere passed through your space, you will see a point, growing into a circle and then again into a point.

      Actually, all you would see was a line that got bigger and then smaller. That assumes that you are a 2D being living in a 2D world. In that world, you would only be able to see things from along the plane that you live in. However, if the sphere passed through a plane that was perpendicular to yours, you would see what you describe. For more info on this, read the book Flatland [barnesandnoble.com]. An interesting read for stretching your mind and pretty funny in parts. At least for a /. crowd.
    • 3 is the minumum number of dimensions to sustain life

      Ok, I'll bite.
      Why is 3 is the minumum number of dimensions to sustain life?

      hehe, this ought to be amusing.

      -

      • Why is 3 is the minumum number of dimensions to sustain life?


        I can't answer this completely, but trying to draw a 2D animal with a digestive tract will give you an idea of what is meant by that statement.
        • Ah yes, the good old "mouth/digestive tract/anus would cut it in half" argument.

          Lets see how many ways I can shred that one off the top of my head...

          1: Photosynthisis does not require a digestive tract.
          2: Many bacteria re-seal the "mouth" opening and digest in a completely enclosed vacuole. The vaculoe merges with the outer surface creating a temporary "anus" to expel the waste.
          3: Many bacteria (and to a certain extent some flies) preform digestion OUTSIDE their bodies. They excrete digestive juices and adsorb/swallow the pre-digested food.
          4: Hydra and some other animals combine mouth and anus as a single opening. They are bag shaped. They swallow food, digest it, and "vomit" the waste back out.
          5: Cutting the creature in half may not be prohibitive if the two halves are held together by adheasion, suction, or some attactive force.
          6: You are assuming life has to resemble us.
          7: You are assuming the usual laws of physics apply. I'd say that's a bad assumption since we are already talking about some sort of 2D universe or enviornment.

          -
          • Ah yes, the good old "mouth/digestive tract/anus would cut it in half" argument.

            I agree - it's a very simplistic argument which makes a lot of assumptions about the definition of "life". I don't really agree with the assertion, since I have a very broad definition of what I would consider to be alive. But it's an interesting idea anyway.
    • > For example if youwere a 2 dimensional being(thats not possible coz 3 is the minumum number of dimensions to sustain life)

      My ex girlfriend could prove you wrong :)

      (sorry couldn't resist :) )

    • To grasp the 4D space idea, I always like the example of the unfolded cube.

      If you were to be a two dimentional creature, you could not see a cube as anything more than a standard 2d square. But if I were to unfold that cube into two dimentions, you would see a cross like 2D shape like remember when in primary school you made those cubes from paper? With four squares up and to the left and right of the second square down another square on each side. (giving a t shape?)

      Now just imaging in a 3D world, take a 4D cube and unfold it, now you have a simiar cross shape, but with 4 CUBES high, with a cube connected to each side of the second cube down!

      The only other thing you have to accept is that it is not possible for you to percieve 4 dimentions, so only try to percieve it in 3 dimentions..
    • It's amazing how high you can get modded up around here when pretend you know what you're talking about...

      "the thing about opening our minds is right."

      Opening one's mind is one thing. Recreational drug use is something else entirely.

      "We have always lived in three dimensions,"

      I don't know about you, but I was born and raised in 4-D space-time. If you live in three dimensions, then your life would be (by definition) quite short. Just because our perception of time differs from length, bredth and height doesn't mean that it's not a dimension just the same.

      Don't think of 3 * 10^8 m/s as a speed limit, think of it more like "300,000 km = 1 s."

      "For example if youwere a 2 dimensional being(thats not possible coz 3 is the minumum number of dimensions to sustain life)"

      Look, string theorists are having a hard enough time trying to figure out if we're living in 11 or 12 dimensions (let alone whether we're talking about space-like or time-like dimensions), the quantum mechanics folks are still scratching their heads trying to figure out what happens to all those other states that get resolved away, so could you please hold off on the xenobiological conclusions at least until those two juries come back?

      "and a 3D sphere passed through your space, you will see a point, growing into a circle and then again into a point."

      ... so you'd see a shape that changes over (wait for it...) TIME! Hey, look folks! This kid just proved 1=1! Give him a medal!

      And what's this "time" I speak of? Why, it's the fourth dimension, it's that thing you have to add to space before you can start talking about constants in this relativistic universe of ours. It's the dimension along which our universe is expanding.

      "So if a 4D object came it would look like a morphing 3D object."

      Morphing? As in "change in time?" As in "a change in space-time coordinates?" That's like saying "You wouldn't see a cone, you'd see a circle whose radius changes with height." What's "height" you ask? Why, it's the third dimension! Dur!

      "If mankind were able to create and use 4D's travel would be a whole new frontier."

      I use 4-D travel every time I get up and walk to the damn bathroom. Same thing with that morning communte. Hell, I sit on my ass and do nothing and I'm whipping about through space-time like a bat out of hell. Moving from point (x1, y1, z1, t1) to (x2, y2, z2, t2) is by definition moving through all four dimensions of space-time.

      "Esp since space-time is curved"

      You came so close to realizing how silly your whole post sounded... so close yet so far...

      "Just imagine traveling a million miles instantaniosly
      Confused! Go through stephen hawkings works! you will be even more so :-)"


      You just used the word "instantaneously" in a sentence. You obviously need to graduate from Einstein before trying to figure out Hawking.

      Burn, karma, burn!

  • I would assume that if multiple dimensions exist, then we actually live in them already, it's just that we only have the ability to percieve the 4 obvious dimensions to us. While the other Nth dimensions are still existing just not eaqsily percieved by us. (AS in that quarks may in fact exist in the "4th dimension" (or properly termed by us as the 5th dimension.... don't start with the classic rock jokes!!) which would easily explain how some of them can exist in 2 physical "3d" locations at once.. existing in the "4d" space there will be multiple intersections from the "3d" space.... espically if the particle was elongated.

    Is this a correct assumption?
  • If you think about it, it doesn't really make sense to be able to visualize a four-dimensional object using sight. Why? Because sight is a two-dimensional sense that on a daily basis we use to visualize three-dimensional space. Visualize trying to understand a 3D shape looking at a pinhole view of it (1D to 3D). Can't really be done, certainly not without a ton of staring at it.

    Seeing as how it's so difficult to jump two dimensions in visualization, how do we do it? We use the only 3D sense that we've got... touch. Since we can "see" all sides of an object at once using touch, then we are truly experiencing that object in 3D.

    So create some kind of device that you hold in your hands and moves around to simulate a hypercube. Simple answer :).

  • by Thalia ( 42305 ) on Monday August 05, 2002 @05:35AM (#4010978)
    We used to argue this in the computer science lab at college. Can the human mind gain visualization skills in four dimensional geometry? We came up with the following interesting answers:

    1. It's hard. We never see four diminsions. The brain would keep wanting to make one dimension some known continuim such as time, a color sequence, tone, or intensity. Only after this intermediate step would you get a true four dimensional geometry in your head.

    2. You would need to have a true 3D display. Current rendering of three dimensional pictures flattened onto simple two dimensional screens would never work. Imagine using a laser pointer as a point source, and imagine that you had never seen a three dimensional object; now draw a three dimensional picture of a pick-up truck using the laser pointer. At the time, we were trying to get a simple three dimensional output, like <a href="http://www.stereographics.com/frames/frame-p rod.html">Crystal Eyes</a>. Now there are liquid crystal on silicon solutions that are much cleaner, if not cheaper.

    We were students once, and poor.
    • 2. You would need to have a true 3D display.

      This is a requirement that fascinates me, how are you supposed to perceive something in 3D, when your receptors (eyes, retina) are 2D by their nature.
      As far sa my brains go, 2D surface emulator will do that just fine, so 3D screen is just a fancy toy that eases manipulation by having better interface to Real World(tm) than 2D screen.
      • Each eye is a 2D receptor, but the use of two 2D receptors allows the extrapolation of 3D data. Although the eye cannot do 3D, your brain can. And we (as humans) do not receive "raw data" from our eyes, but rather our brains' interpretation of that data.
    • Can the human mind gain visualization skills in four dimensional geometry? ... We never see four diminsions. The brain would keep wanting to make one dimension some known continuim such as time, a color sequence, tone, or intensity. Only after this intermediate step would you get a true four dimensional geometry in your head.

      Your actually wrong in the second part. While our senses can't pick up more than 3D Sensory input our brain can very well imagine (and sense) more dimenions. It's simply a matter of training.
      Indian Yogis would call that 'meditation'.

      The stuff those kind of people talk about like "when time becomes irrelevant" and such isn't some mystical BS (at least not with the honorable ones) - it's actually what you expierience when your brain is trained apropriately. Or forced into such condition by (ab)use of drugs.
      You can see "everything happen at once" like one would say. It's interesting that people reaching this kind of 'sense' have a syncronized activity of both halfs of the brain.
      Normaly we don't have that. But Yogis and people who have trained meditation can actually achieve such 'brainsyncing' at will. (a tranquil enviroment given)

      Tibetian meditation 'training' is known to train the same as modern biofeedback 'brainsyncing', often with a nearly identical setup like: "look at those 2 spots and see them as one".
  • After stareing at this for about 30 mins it made me wonder if God is a 4D being..
    • Re:God (Score:1, Funny)

      by GigsVT ( 208848 )
      If by 4D you mean totally fictional and only created to serve social leader's agendas and scare ignorant masses into submission through threats of eternal punishment, then yes, yes he is.
    • Re:God (Score:2, Interesting)

      by xscottyx ( 598497 )
      IF there is a god, I imagine that he would have to be 5D or 6D. I say that because I think that we ourselves are 4D beings, since we experience time. I imagine that God would be 5D, since I don't think that he would be limited to any particular time like us. I think that God could also be 6D, since I can't see God as being limited to one universe, but that depends on whether or not there is a multiverse. Is that weird?
      • I was just in an argument with a very smart mathmetician and long after he lost me in proofs he deigned to lower himself and explain to me what he actually meant. From what I understand it boils down to this...

        Time is not a dimension, as it is merely an expresion of the relative speed of movement.

        Now I am not sure if that makes any sense in reality but my heads hurts to much to keep thinking about it.
  • by guttentag ( 313541 ) on Monday August 05, 2002 @05:46AM (#4010993) Journal
    Each one takes a 3D image and progresses along a fourth dimensional plane I like to call:

    <fingerquotes>TIME<fingerquotes>
    This combination is stored in a media format I call a:
    <fingerquotes>MO-VIE<fingerquotes>
    4D is very subjective, so long as you are allowed to define the four dimensions involved.
    • It's talking about a shape in 4 spatial dimesions, not 3 spacial dimensions and one time dimension. Also, movies aren't 3d - not even those red-cyan ones. If movies were truly 3d, you'
      • d be able to see the entire scene in the movie from all angles at the same time. Not really of course, since you can't see in 3d, only two 2d images, but you get the idea.

        Sorry about the split post - I guess I'm not very coordinated at 4am :P.
  • Hyperplane (Score:3, Interesting)

    by t00tie ( 518552 ) on Monday August 05, 2002 @05:52AM (#4011000) Homepage
    "A hyperplane is a 3-dimensional space that slices through the 4-dimensional space".

    Thats a bit limited, and not exact enough - I expect better research from a nerd site! A hyperplane is firstly a plane - ie. linear, and it's dimension is one less than the space it lives in. The coolest hyperplanes exist over the more esoteric fields (hyperplanes in P^n come to mind). {complaint: why isn't <sup> allowed on slashdot?!}

    But see http://everything2.com/index.pl?node=hyperplane for an even better explanation.

    But then again - why listen to me - I only have an MSc in algebra ;-)

  • No really it is...

    Of course I have 20/20 vision :)
  • A few days ago I got a new 3d card and found a driver for it that would cause it to render all output as pseudo-3d anaglyph images. Naturally, I ran down to my local print shop and got a half red, half cyan color tranparency printed, cut out squares, and taped them to a strip of cardboard from an old eggo box.

    I must recommend that all slashdotters avoid doing this at all costs. The images are very blurry unless you stick your nose 2" away from the monitor. However, once you reach this point, the 3d will just be good enough that you'll want to stare at your game (or hyperspace star polytope slicer) for _WAY_ longer than is healthy. Save your eyes. Go out and buy a decent $3.00 pair of anaglyph glasses.
  • by cswiii ( 11061 )
    The abstract posted on Slashdot confused me enough, from the beginning.

    I must be too stupid to comprehend the Time Cube [timecube.com]. :)
  • Another fun site (Score:3, Informative)

    by teslatug ( 543527 ) on Monday August 05, 2002 @06:59AM (#4011082)
    To practice your non-eucledian basics, try NonEuclid [unm.edu]
  • After a bout with insomnia last night, I dove into Rudy Rucker's new novel "Spaceland: A novel of the fourth dimension" [amazon.com] and thoroughly enjoyed it. This tribute to Abbott's novel Flatland tells the story of "Joe Cube" (get it?) and his difficulties dealing with and being manipulated by fourth dimensional beings and their strange buisness plans for poor shmoes in the three-d world (which is their flatland). Pick it up if you find stuff like this interesting and you see it at the library.
  • It wouldn't suprise me that many slashdot readers have already read this book, but for those that haven't, its quite interesting...

    Try Flatland by Edwin A. Abbot. He was a headmaster at a British school in the late 1800s and he wrote a short book that explained dimensions to his students. The story he tells of a 2-D land filled with geometrically shaped characters is still very interesting and informative, and will help you understand dimensions a bit futher.

    A recently written sequel that has also received great reviews is Flatterland by Ian Stewart. Though I have not read it, I've heard only great things about it.

  • Upward, not Northward
  • by metlin ( 258108 ) on Monday August 05, 2002 @08:05AM (#4011327) Journal
    There was something similar done by a guy called Luke Dahl, only it was for audio.

    It was a Markhov chain extrapolation of the 3d sound as perceived by a 3d being and using a series of normalizations and transforms.

    It was called Frobenius Norm, and was a composition of how a 4-d sound would sound to a 5-d being, I think. I just remember it being "spiffy" and very addictive! :-)

    It was also featured in Woodstockhausen 2000 [trailwork.com].
  • It's full of stars!
  • ... Until someone makes a videogame using these shapes. Maybe something funky like Frequency Cubed, Dance Dance Dance Revolution, or Duke Nukem 4D?

    I'm serious though.
  • Tic tac toe (Score:4, Interesting)

    by tuxedo-steve ( 33545 ) on Monday August 05, 2002 @10:40AM (#4012218)
    A fun four-dimensional exercise is 4d tic-tac-toe. It's easiest if you build yourself up to it:
    1. Start with your typical 3x3 tic-tac-toe, on a piece of paper.
    2. Now add two more grids. Visualise each grid on top of the one before. It's not difficult to see how this is played. You can get three in a row on a single grid, just like normal. Or you can get three in a row by getting the middle square of each grid (3 in a row, vertically). And so on. This is basically tic-tac-toe in 3D. 3 sets of 3x3 grids. 3x3x3.
    3. Now, add another two sets of three grids. So now you've got 3x3 3x3 grids (still with me?). You can still win just like in the 3x3x3 version. But you've got another 3x3 ways in which to do it. The tricky part is, to visualise each possible `3 in a row', you've got to mentally `rotate in' any one (and only one) 3x3x3 cubic plane.
    Sorry if that's difficult to follow. If you work it through on paper, you'll see what I mean. This is what being bored in math class will lead you to think of, when plain old tic-tac-toe just doesn't seem challenging anymore. :)
  • Any n-1 dimensional plane cutting through an n dimensional space is a hyperplane. So, the correct wording of the story would have been "A hyper plane is a 3-dimensional space that cuts through a 4-dimensional space, just as a 2-dimensional hyperplane cuts through 3-dimensional space."

Two is not equal to three, even for large values of two.

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