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How To See In Four Dimensions

Posted by timothy on from the wrinkle-in-time-time dept.

An anonymous reader writes "Think it's impossible to see four-dimensional objects? These videos will show you otherwise. Some mathematicians work with four-dimensional objects all the time, and they've developed some clever tricks to get a feeling for what they're like. The techniques begin by imagining how two-dimensional creatures, like those in Edwin Abbot's 'Flatland,' could get a feeling for three-dimensional objects. When those techniques are transferred up a dimension, the results are gorgeous."

34 of 227 comments (clear)

  1. Easy to see in four dimensions by religious+freak · · Score: 4, Funny

    I'm looking at my monitor in three dimensions ... wait one second ... okay, I just saw it in four :)

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    If you can read this... 01110101 01110010 00100000 01100001 00100000 01100111 01100101 01100101 01101011
    1. Re:Easy to see in four dimensions by MrNaz · · Score: 4, Interesting

      I "visualize" four dimensions and more often, when programming and setting up multi-dimensional arrays of more than three dimensions.

      All one has to do is acknowledge that adding a dimension simply adds a range of points that map to every single point in the (n-1) dimensional range. So, the easiest way to visualize a four dimensional cube is to simply imagine multiple identical cubes, side by side, for as many as the range has been specified. Five dimensions is a flat square arrangement, six is a cube arranged array of cubes, and so on. This way, an infinite number of dimensions can be visualized. Perhaps the term "mental addressing" is more appropriate a name for this mental method.

      The limit is, of course, this only works directly for finite and discrete arrays. I find it can be extrapolated to use non-discrete spectra, but describing the way that works in my head will not be possible using this clumsy tool we call "language".

      --
      I hate printers.
    2. Re:Easy to see in four dimensions by MrNaz · · Score: 5, Insightful

      After thinking about this some more, I find that the animations in the article are not at all four dimensional, as the so called "fourth" dimension they are representing exists in the same physical space as the third.

      This breaks the dimensional relationship. Imagine, if you will, a single point with no dimensions. Then extrapolate that into a line to get one dimension, imagine that line them extrapolating perpendicular to the line to form a square, and then imagine that square extruding into a cube. So far, no physical overlap has occurred. The fourth dimention as represented in these videos, does nothing but add more "balls and sticks", which is not adding another dimension, it's simply adding detail to the existing dimension.

      Likewise, those 2D imaginings of a 3D object are not visualizations of a 3D object in 2d, they are the visualization of a changing 2D object, with the simulated third dimension being time.

      In other words, the method that they have used does not actually visualize a fourth dimension in any mathematical or logical sense, they are really just optical illusions. Personally, my method of visualization that I described in my previous post is far superior, and more accurate from a logical and mathematical point of view, as it truly does represent a 1:M maping of every dimensional unit in the (n-1) dimensional space.

      P.S., I've always wanted to start a sentence with "Imagine, if you will...".

      --
      I hate printers.
    3. Re:Easy to see in four dimensions by cheater512 · · Score: 4, Interesting

      Yeah I've had arrays with double digit dimensions.
      I think my record is 16 or so.

      I dont know why but I work with them incredibly easily.
      Without them its like programming with a hand tied behind your back.

      Cant visualize them at all, I can work with them though.

    4. Re:Easy to see in four dimensions by alexj33 · · Score: 5, Informative

      I find that the animations in the article are not at all four dimensional

      Duh. That's because our screens are two dimensional, and you and I are three dimensional. Certainly you can't fault them for this? (Please tell me that I'm somehow misunderstanding this objection..)

      In other words, the method that they have used does not actually visualize a fourth dimension in any mathematical or logical sense

      That's nonsense. Their videos show the edges of the object (although distorted) as well as the interconnections of each of the vertices. What would qualify to you as a "real" mathematical or logical way of viewing these objects in a 3-D world?

      As for your previous post:

      So, the easiest way to visualize a four dimensional cube is to simply imagine multiple identical cubes, side by side, for as many as the range has been specified. Five dimensions is a flat square arrangement, six is a cube arranged array of cubes, and so on. This way, an infinite number of dimensions can be visualized. Perhaps the term "mental addressing" is more appropriate a name for this mental method.

      Okay, when you get down to it, this is stuff that any programmer knows when working with arrays. (ie- int[][][][][], etc.) Now your task is to *draw* your example for us in 3-D space.

    5. Re:Easy to see in four dimensions by Anonymous Coward · · Score: 5, Funny

      Cool story bro

    6. Re:Easy to see in four dimensions by Dun+Malg · · Score: 4, Insightful

      Imagine, if you will, that you're ignorant. That shouldn't be too hard. Do you complain that your 3D graphics card just adds more 2D pixels, where it should instead show hundreds of 2D pictures next to each other in order to represent 3D space?

      Imagine, if you will, that you're also ignorant (or perhaps a member of congress). That shouldn't be too hard...

      Do you think that humans actually see in three dimensions? We don't. We see in two dimensions. The retina is a plane. By using two planar sensory arrays, our brains use parallax to calculate depth. This is 2D vision with depth cues. Actual 3D vision would have us able to see the back side of the TV while watching a show on the front. When we talk about "visualizing" dimensions beyond the third, we're not talking about actually seeing things with our eyes. We're talking about mental pictures. We can "visualize" the back of the TV because our sensory system is accustomed to using a series of depth-cued 2D images to construct a model of the 3D world. Pushing that up to four dimensions isn't even remotely the same as drawing a ray traced 2D picture on a fucking computer monitor.

      --
      If a job's not worth doing, it's not worth doing right.
  2. it see all time by extirpater · · Score: 5, Funny

    Take LSD and sure you'll see 4th dimension.

    1. Re:it see all time by Eli+Gottlieb · · Score: 4, Funny

      I prefer melange.

  3. Scientology? by hansraj · · Score: 4, Interesting

    Why is the story tagged scientology?

    1. Re:Scientology? by Draconix · · Score: 4, Funny

      Obviously, Lord Xenu has a Slashdot account.

      --
      By reading this you acknowledge that you have read it.
    2. Re:Scientology? by Ilgaz · · Score: 4, Interesting

      I see a flash scientology , rather very big ad at bottom of article and in "videos", there are Google Adsense ads mentioning scientology youtube channel.

      It could be related to people who sees those ads (must be scientific terms used triggering them) and think the site is Scientology supported. It could be possible but it could be the adsense only too.

      BTW Google Adsense advertising Scientology Youtube channel is not really a good, pretty sight. What next? Doubleclick ads too?

    3. Re:Scientology? by elronxenu · · Score: 5, Funny

      Damn straight.

  4. Simply imagine a space defined on R^N.... by Anonymous Coward · · Score: 5, Funny

    then set N = 4....

  5. not by Holi · · Score: 5, Insightful

    Sorry it's on my screen, so it's a 2 dimensional representation of a 4 dimensional idea in 3 dimensional space.

    --
    Sorry, teleporters just kill you and then make a copy. A perfect, soul-less copy.
    1. Re:not by node+3 · · Score: 4, Funny

      Close one eye.

  6. did this years ago... by TheSHAD0W · · Score: 5, Interesting

    http://shambala.net/3d/tess3d1.gif

    1. Re:did this years ago... by doombringerltx · · Score: 4, Interesting

      The article linked to in TFS fairly crummy, but following through leads to the full videos which are really good. I even sent it to a non-math nerd friend. Its worth a look for anyone who had little trouble imagining geometric shapes in Rn. God knows that was me when I had classes that delt with that. Eventually I was like "Fuck it. It doesn't have to make sense, just get to where you can pull it off on the final." Plus its doing a good job of showing multiple methods to represent it, past what your gif shows. Right now I'm only a few chapters in, so I hope it keeps up the quality.

  7. Seeing in 4 Dimensions? by EdIII · · Score: 4, Funny

    Just go to any Burning Man concert and eat the multi colored Brownies.

  8. Just so we are clear... by Anonymous Coward · · Score: 5, Insightful

    A 4D object is mathematically projected to a 3D representation, that is then projected into a 2D representation for display on the monitor, that is then transformed by my brain back into a 3D representation, and then further needs to be transformed into a 4D object... /looks for his linear algebra textbook //begins drinking

    1. Re:Just so we are clear... by Drinking+Bleach · · Score: 5, Insightful

      In a way, it's also projected into a 1-dimensional stream of bits.

    2. Re:Just so we are clear... by smellotron · · Score: 4, Interesting

      I'd argue that it's encoded onto a 1-dimensional stream of bits. The main difference being that an encoding can be used to reconstruct the original, whereas a projection by definition loses information.

  9. Re:Carl Sagan by mgabrys_sf · · Score: 5, Informative

    Here you go. It was Cosmo's take on "flatland":

    http://www.youtube.com/watch?v=KIadtFJYWhw

  10. Interacting is the easiest way to learn by Eighty7 · · Score: 5, Interesting

    I played around with this applet a few months ago. After some practice, getting out & hitting the ball becomes easy. Getting back in is only slightly harder & I still can't hit the point reliably.

  11. I can visualize 11 dimensions by harlows_monkeys · · Score: 4, Funny

    Four? Trivial! I can visualize 11 dimensions...but 8 of them are very very small.

    1. Re:I can visualize 11 dimensions by Ibag · · Score: 4, Funny

      That reminds me of a joke:

      An engineer, physicist, and a mathematician are sitting at a bar, and the bartender asks, "Can any of you guys think about four dimensions?"

      "Sorry, not me," the engineer replies.

      The physicist chimes in, "I suppose I can, if the fourth dimension is time."

      The mathematician starts laughing. "Oh, you guys, this is easy! Picture n-dimensional space. Now, let n be equal to four..."

  12. Old maths joke... by SoVeryTired · · Score: 4, Funny

    Pff. Real mathematicians just picture N dimensions, then set N = 4.

    --
    Slashdot: news for Apple. Stuff that Apple.
  13. Seeing four dimensions. by os2fan · · Score: 4, Funny
    You can teach yourself to see in four dimensions, by using analogy and other things.

    To begin, consider that a 2d picture can either be a picture (things can fall), or a map (things don't fall). Since the corresponding 3d thing is a picture/map of four dimensions, we can build objects like houses, furniture, etc from plan and views.

    Not all seems to be aimple. A knife cuts: literally, it makes a surface by motion, and is therefore tipped by a space of N-2 dimensions. Rivers can be either "latrous" (1d) or "hedrous" (2d). A fault lake is 2d (since faults are a break of surface).

    Holes come in two types, although these are topologically the same. One can have a "bridge" or "tunnel" kind of hole: in 3d, these are the same, in 4d they are different.

    The planet rotates on clifford motion. This makes every point of the 4-sphere go around the centre. One sees this by equality of energy in modes of rotation.

    None the same, there can be seasons. If the sun does not follow in the year-circle any of the circles of the earth rotating, then there will be seasons. You don't just have hemispheres in summer vs winter, but season-zones to match the time-zones. That is, for example, Christmas (normally in summer), can fall in early spring, or late winter.

    The poles are replaced by circles of extreme climate. One has a "equator circle", and a "polar" circle. At the tropics (a singular torus-shape thing), the sun becomes to the zenith once a year. At the artic torus, the sun hugs the horizon for the equate of the shortest day.

    Because the sun is relatively still in the sky, there is no variation in the number of hours. What makes the seasons is that the the sun is lower in the horizon, even at midday.

    See, eg my site http://www.geocities.com/os2fan2/gloss/index.html

    --
    OS/2 - because choice is a terrible thing to waste.
    1. Re:Seeing four dimensions. by ColaMan · · Score: 4, Funny

      Careful.

      You read just like the timecube guy did, before he took that last hit of bad acid.

      --

      You are in a twisty maze of processor lines, all alike.
      There is a lot of hype here.
  14. Try Salvia by Nick+Ives · · Score: 4, Interesting

    One of the most common sensations (along with the sense of absolute terror at being ripped into a void in space/time) is the feeling of moving through between more than 3 dimensions of space. In my travels I usually feel like I'm spinning and being folded in about 7 different dimensions before my visions start to settle.

    To anyone who decides to take me seriously, make sure you have a sober sitter :)

    --
    Nick
  15. OT but another mathematics joke by Sycraft-fu · · Score: 5, Funny

    A physicist, and engineer, and a mathematician are sleeping in a hotel when fires break out in all their rooms. The physicist get up, does some quick calculations, and then gets the exact amount of water required to put the fire out, not a drop wasted. The engineer also does some calculations to work out the amount needed, then proceeds to flood most of the floor, to ensure that there is a sufficient tolerance for error. The mathematician wakes up, and does some extremely complex calculations but does them much quicker than the other two. He then exclaims "I have proven I can put the fire out!" and goes back to bed.

    1. Re:OT but another mathematics joke by the+phantom · · Score: 4, Funny

      Meanwhile the statistician is running from room to room lighting fires -- he needed a larger sample.

      Also, given an empty ice-bucket on the dresser, the sink in the hotel bathroom, and a burning trashcan, how does a mathematician put out the fire? Like any ordinary person, he grabs the ice-bucket, runs to bathroom, fills the bucket with water from the sink, and pours the water into the trashcan, thus putting out the fire. Now suppose that the ice-bucket is already full -- how does the mathematician put out the fire? He grabs the ice-bucket, runs to the sink, empties it, and returns it to the dresser. The problem has now been reduced to one that has been previously solved.

      --
      Rhapsody in Numbers
  16. Why parent is not insightful by doug141 · · Score: 4, Informative

    Your retinas are, even together, a 2 dimensional array. You never "saw" anything but what your brain constructed from 2 dimensional arrays. Turns out your brain is very, very good at visualizing a 3d object based on this input. Would you say you can't visualize an actor's physical body because the screen is 2 dimensional?

  17. Re:Tagit by TheSHAD0W · · Score: 4, Informative

    http://bittornado.com/torrents/Dimensions-English.torrent

    BitTorrent download for all the (English) movie files on the source website.