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Math Graphics Software Science

How To See In Four Dimensions 227

Posted by timothy
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."
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How To See In Four Dimensions

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  • by religious freak (1005821) on Sunday August 24, 2008 @04:39AM (#24724837)
    I'm looking at my monitor in three dimensions ... wait one second ... okay, I just saw it in four :)
    • by MrNaz (730548) on Sunday August 24, 2008 @06:13AM (#24725115) Homepage

      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".

      • by MrNaz (730548) on Sunday August 24, 2008 @07:13AM (#24725263) Homepage

        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...".

        • by alexj33 (968322) on Sunday August 24, 2008 @08:57AM (#24725637)

          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.

          • 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

            Wow. That makes a lot of sense. You sorta remind me of this other guy [ubertragen.com]...

      • by cheater512 (783349) <nick@nickstallman.net> on Sunday August 24, 2008 @07:36AM (#24725339) Homepage

        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.

        • by Anonymous Coward on Sunday August 24, 2008 @09:25AM (#24725737)

          Cool story bro

        • by entrylevel (559061) <jaundoh@yahoo.com> on Sunday August 24, 2008 @01:10PM (#24727129)

          I am interested in what problem space you are working with.

          In some very extreme cases, I can see it being a requirement to work the way you are, but in most real-world code, what you suggest would be far simpler to maintain (for you AND others) if you would just take a few extra minutes to think about what your data structures need to be.

          Just because you CAN, doesn't mean you SHOULD. If it is a one-off script to solve a complex problem, then you have my respect. If anyone else EVER has to grok your code, for any reason, then you are just incompetent :)

          BTW, this is probably an incredibly stupid question, but I just want to clarify. "The fourth dimension" is such an incredibly loaded term. In the context of this article, it is referring to time, correct?

          Assuming I am correct, I have always had a very simple theory I use to wrap my mind around it. Bear in mind I am a high-level programmer, not a quantum physicist. I think that we (humans) exist within the first three dimensions while we travel along the fourth. Hence we are aware of, and can, to some extent, measure the fourth, but it is very difficult to perceive it in any concrete manner.

        • Re: (Score:3, Insightful)

          by wealthychef (584778) *
          That's because while programming, your thinking about each segment of the array can be distinct in time from thinking about each of the other arrays. I.e., you do not think of other parts of the arrays (or only think about immediate neighbors or extremes) while thinking about one part of the array. To visualize something means to create it as an intuitive experience that can provide access to a higher level of thinking about it, by noticing patterns and properties that exist only in relation to the other p
      • by Yvanhoe (564877)
        Yes, you can visualize a 4D arrangement of objects but I think this article is more about visualizing a shape in 4D and making deductions on it like : is it closed ? can it be projected as a cube in a given angle ? How many edges does it have ? etc...
      • Re: (Score:3, Interesting)

        by QuoteMstr (55051)

        Personally, I visualize the fourth dimension in a figure as a kind of color, and the fifth as a variation (say, blinking) of that color; but I rarely need to go that high. This is weird stuff.

    • Re: (Score:2, Funny)

      by Hal_Porter (817932)

      "The third dimension is a theoretical realm of space and time in which the particles of dark matter of this parallel alternate reality bends light to collide with the electrical charges of the subconscious mind to create the illusion of movement where what is dark becomes light, what is light becomes dark. Some look at the third dimension and see nothingness. Others believe they see the very face of God."

      http://uk.youtube.com/watch?v=6T0UQfKTcQw [youtube.com]

  • by extirpater (132500) on Sunday August 24, 2008 @04:39AM (#24724841)

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

    • Try Salvia (Score:4, Interesting)

      by Nick Ives (317) on Sunday August 24, 2008 @07:41AM (#24725357)

      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 :)

    • Re: (Score:2, Funny)

      by Anonymous Coward

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

      I once heard the 5th dimension... it was back in the 70s, I think.

    • by rubycodez (864176)

      *lick* *lick*......I see dimensions one through three.......and five.....but no fourth dimension

      liar.

    • What really opened up my mind to really thinking about different dimensions and infinity was nitrous oxide, preferably mixed with other substances like marijuana or opiates. The fourth dimension? Many nitrous users report experiencing an infinite number of dimensions, which you then feel as being equivalent to zero dimensional space, or a space that defines itself over and over. It's almost impossible to explain with words, but if you can get to that point, it's quite beautiful and freaky. After my extensiv

  • Scientology? (Score:5, Interesting)

    by hansraj (458504) * on Sunday August 24, 2008 @04:39AM (#24724843)

    Why is the story tagged scientology?

  • by Anonymous Coward on Sunday August 24, 2008 @04:39AM (#24724845)

    then set N = 4....

    • Re: (Score:2, Offtopic)

      by siwelwerd (869956)
      I know you're being facetious, but it doesn't quite work that way. Not all 4-manifolds can be embedded in Euclidean 4-space. In fact, the best we can say is they can be embedded in 8-dimensional Euclidean space.
  • not (Score:5, Insightful)

    by Holi (250190) on Sunday August 24, 2008 @04:40AM (#24724849)

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

  • The techniques begin by imagining how two-dimensional creatures, [...] could get a feeling for three-dimensional objects.

    I guess that's already way past my abilities.

  • by TheSHAD0W (258774) on Sunday August 24, 2008 @04:44AM (#24724865) Homepage
  • Dupe! (Score:3, Funny)

    by Xfacter (1075973) on Sunday August 24, 2008 @04:46AM (#24724875)
    Learned to do this on Tralfamadore.
  • Just imagine (Score:4, Insightful)

    by Joe Jordan (453607) on Sunday August 24, 2008 @04:49AM (#24724885) Journal
    Just imagine how incredible the true nature of the universe must be if current theories hold true that 10, 11, or even possibly 26 dimensions exist in our universe.
    To think about it is mind bending, awe-inspiring, and dream provoking.
  • by EdIII (1114411) * on Sunday August 24, 2008 @04:52AM (#24724895)

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

  • Carl Sagan (Score:4, Interesting)

    by Ilgaz (86384) on Sunday August 24, 2008 @04:52AM (#24724899) Homepage

    Does anyone remember in how a good way Carl Sagan explained the problem if there are more or less than 3 dimensions exist?

    I remember he was explaining the imaginary 2d creatures not being able to see 3d creatures and so on. It was on a TV documentary. Sorry if I remember it all wrong. I was like 13 ;)

    It must be an episode of "Cosmos" http://www.imdb.com/name/nm0755981/filmoseries#tt0081846 [imdb.com]

  • by Anonymous Coward on Sunday August 24, 2008 @04:55AM (#24724907)

    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

    • by Drinking Bleach (975757) on Sunday August 24, 2008 @06:02AM (#24725079)

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

    • ...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...

      Except that it's a 2D representation that is interpreted by your brain as a 3D representation, which is then put into motion in movie form, making it essentially a 4 dimensional representation already, which you're then being asked to imagine extending into yet another (5th) dimension.

  • by Eighty7 (1130057) on Sunday August 24, 2008 @05:04AM (#24724933)
    I played around with this [vanderwal.eu] 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.
    • Seeing a forth dimension doesn't have to mean actually seeing it visually or pictorially. Since this is /. how about imagining a multi-dimensional array, with 3 indexes (x, y, & z) and then adding another (4th), Presto. Another way physical space could be defined is by relationships. When playing blind chess I don't actually see a board in my head, I just remember how all the pieces relate to each other.
      • by tenco (773732)

        Since this is /. how about imagining a multi-dimensional array, with 3 indexes (x, y, & z) and then adding another (4th), Presto.

        I'm a science nerd, you insensitive clod!

    • I feel stupid, I can't figure this out. All I can get it to do is spin.
  • Awesome. (Score:2, Interesting)

    Awesome. However, mathematicians and physicist usually don't try to "see" or "get a feeling" of higher (or infinite) dimensional objects.
    They familiarize themselves with mathematic properties of two and three-dimensional objects and space and what they mean, and then just use these properties in higher dimensional spaces.

    Trying to see these spaces or getting a feeling on how these objects would look like most likely confuses for calculations (our brain wasn't really made for this).

    Nice and interesting video

    • That completely depends on the mathematicians, and the kind of mathematics they do. For proofs that rely only on calculations, you do not need even to understand the low dimension case, just do the computations right.

      But proofs with computations are rarely elegant. Some mathematicians prefer a more geometric approach, and for that, they need to see, un to a certain level, the objects in higher dimensions.

      Furthermore, the 2D or 3D spaces we have direct access to are really limited. There are lots of phenomen

    • However, mathematicians and physicist usually don't try to "see" or "get a feeling" of higher (or infinite) dimensional objects. They familiarize themselves with mathematic properties of two and three-dimensional objects and space and what they mean, and then just use these properties in higher dimensional spaces.

      Is that you Mr. Spock? I'm sure that Albert Einstein (not a mathematician for sure), Richard Feynman, and Stephen Hawking, would beg to differ.

      Trying to see these spaces or getting a feeling on ho

  • by harlows_monkeys (106428) on Sunday August 24, 2008 @05:05AM (#24724941) Homepage
    Four? Trivial! I can visualize 11 dimensions...but 8 of them are very very small.
    • I see someone knows their string theory and quantum physics, lol. Great comment though, gave me a good laugh.
    • by Ibag (101144) on Sunday August 24, 2008 @06:37AM (#24725183)

      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..."

      • by Sycraft-fu (314770) on Sunday August 24, 2008 @08:17AM (#24725479)

        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.

        • by the phantom (107624) on Sunday August 24, 2008 @11:11AM (#24726281) Homepage
          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.
          • by melikamp (631205)

            [...] 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.

            This is the software engineer way too.

  • Buddhabrot (Score:4, Interesting)

    by Xelios (822510) on Sunday August 24, 2008 @05:24AM (#24724983)
    Buddhabrot in 4D (in 3D, in 2D). [youtube.com] The Mandelbrot fractal never looked so good.
  • by BlueParrot (965239) on Sunday August 24, 2008 @05:36AM (#24725017)

    Here is a one dimensional projection of a 5 billion dimensional sphere: _

  • rotating tesseracts (Score:3, Interesting)

    by xPsi (851544) * on Sunday August 24, 2008 @06:00AM (#24725071)
    Definitely enjoyable stuff. Of course, you could just play Portal. Oh, sorry, that's just an ordinary 3D space which happens to be multiply disconnected and topologically unsettling. For more (Euclidian!) 4D visualization tools, here [uiuc.edu] are a couple nice (but old) clips of rotating cubes and tesseracts through higher dimensions. For example, it gives you the (x,y,z) view of a cube then a simultaneous projection of that object in the (w,x) plane where w is a 4th orthogonal direction. It then proceeds to rotate the (w,x) projection in a circle to see what the 3D "shadow" in (x,y,z) space is doing. Rather than getting bigger and smaller (simulating perspective) as it moves back and forth in the 4th direction, the faces are color coded (I personally think this makes it easier to visualize). Run the simulation back and forth slowly a couple times and your brain locks in pretty well.
  • by SoVeryTired (967875) on Sunday August 24, 2008 @06:09AM (#24725097)
    Pff. Real mathematicians just picture N dimensions, then set N = 4.
  • by os2fan (254461) on Sunday August 24, 2008 @06:16AM (#24725127) Homepage
    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 [geocities.com]

  • by Anonymous Coward on Sunday August 24, 2008 @06:31AM (#24725157)

    For the same reasons you can't visualize a 3D object on a 1D space you can't visualize a 4D object on a 2D space.

    You cannot go up 2 dimensions.

    Just as we can visualize a 3D object on a 2D space we can visualize a 4D object on a 3D space.

    Thus we need something like this:
    http://dogfeathers.com/java/hyprcube.html

    *Click the Stereo button 2 times to switch it to cross-eyed view for no glasses. Simply cross your eyes to bring both shapes together in the center and it should become clear.

  • The 2nd explanation for projection of 3D objects onto a plane so as to allow the 2d lizards to perceive the objects is simply ridiculous as it requires them to have an external view point defeating the purpose altogether. The 1st example was ok, but its nearly 100 years old, something new/unique/novel would have been more interesting to watch, also the presentation drags on for too long, it should be sped up.

  • by Twinbee (767046) on Sunday August 24, 2008 @07:03AM (#24725239) Homepage

    Of course, we can't really see in 3 dimensions, otherwise, we'd be able to see through stuff. The image projected onto our eyes is a 2D image, and we have 2 eyes, so it's (x*y)+(x*y), not (x*y*z). The third dimension is a cheat and is represented as 'stuff getting smaller'.

    If we really could see in 3D, we can use the 'getting smaller' trick to visualize 4 dimensions much more easily.

    Anyone know of some images or videos on the net using reverse perspective, where things behind get bigger instead of smaller?

    • We may see in 2D, but we can perceive in 3D, due to a combination of depth perception and our brain's habit of assuming the world is made of straight lines.

      The same works for audio... A stereo recording only has one dimension spatially (a line going through the speakers). But it's quite easy to make something sound nearby (full frequency range, mostly direct sound) or far away (muted high frequencies, mostly indirect sound / reverb). Thus, you really get an extra dimension for free, same as for our eye

  • by Saint Stephen (19450) on Sunday August 24, 2008 @07:18AM (#24725273) Homepage Journal

    But I can guess how it works. A sphere passing through a plane would look at first like a dot, then a gradually wider line, then a dot. I remember flatland saying something about brightness at ends of the line.

    So, a hyperball passing through a 3-space would look like a dot, gradually expanding to a sphere, and gradually shrinking to a dot.

  • Falling WAY short (Score:3, Insightful)

    by DynaSoar (714234) on Sunday August 24, 2008 @07:19AM (#24725283) Journal

    These 2D videos show 2D diagrams of what a 4D projection into 3D would look like if it were flat. Entirely unsatisfying.

    Want a 4D-in-3D demo? Take a small balloon, blow it up then let it go flat. That's what a 4D sphere projecting into 3D would look like.

    You can imagine in 4D fairly easily if you decide to ignore your senses and decide that the smaller faces on the internal cube in a tesseract are indeed the same size (an in fact coincide with) the larger, outer faces, and so the outer pseudo-cubes are in fact cubes with all 90 degree corners. You see perpective with fake apparent angles, you can use the same trick your mind uses to see more.

    By the way, we do not see in 3 dimensions. We see in 2.5. We can't see the backs of things. We can feel in 3 dimensions if we can get our hands all the way around it.

    We do NOT see in 2 dimensions (as a previous comment stated) unless we have no depth perception. Stereoscopic vision gives us much more than flat projection, and stereointegration in the visual cortex gives us even more. In fact, a one-eyed being with stereointegration need only moves its head around and collect visual images from different angles in order to create a successfully adequate 3D concept.

    And ask the previous commenter asked, yes we do have examples of reverse perspective where things behind get bigger. Gravitational lensing of galaxies passing behind smaller, intense gravity fields (theoretically black holes or neutron stars). Can't point to any I've seen on the web offhand, but I've seen them there as well as on some astronomy shows on TV.

    • by Twinbee (767046)

      Speaking as the previous commenter, the extra information (such as stereo depth perception or blurring of non-focused areas), helps to get a good idea of the real 3D object.

      However, the jump from 2D to actual 3D where you're experiencing all widths, depths and heights at once is far greater in terms of raw information than the jump from 2D to our 'pseudo'-3D vision. Needless to say, the qualitative experience of actual 3D vision would also be vastly different (as would being able to see in 1D), and probably

  • by trifish (826353) on Sunday August 24, 2008 @07:31AM (#24725321)

    To download any of the videos directly, go here:
    http://www.sciencenews.org/pictures/mathtrek/082208/ [sciencenews.org]

  • "Think it's impossible to see four-dimensional objects? These videos will show you otherwise."

    No they won't. Submitter needs to learn math.

  • This has nothing to do with string theory, so let's all do a !stringtheory up on the tagline. There is nothing about cosmology in this; it's about methods used for years by mathematicians to visualize 4D abstract objects as they move through 3D space.

    It's not even new.

  • by cybrpnk2 (579066) on Sunday August 24, 2008 @08:54AM (#24725617) Homepage
    Anybody interested in visualizing hyperspace should learn about Alicia Boole Stott [agnesscott.edu] and her amazing story [ub.rug.nl]. She was the daughter of George Boole (of boolean algebra fame) who developed a mind-boggling series of paper cutout models of four dimensional objects that won her an honorary math doctorate in 1914. Check out these extensive photos of her work [math.rug.nl].
  • Please help, Slashdot seems to be totaly fucked since yesterday. New comments format is not working, I can't change the comments level, dropdown box is gone, just a number is there.

    Also, I can't go to my preferences, it all comes up as a big empy box, so no way to change, or even see, anything. I hope somebody is aware of this.

    This is on IE7. Also, there will be no way for me to see any replies to this. geez.

    • by mestar (121800)

      Oh, those things that look like sliders actually are really stupid sliders (you have to move them one fifth of the screen for them to actually move).

      Still, "preferences" comes as an empty box, just the words "Discussion 2" on the top, and hint of some text on the bottom, you can only see the top of the letters. ftw?

  • Torrents! (Score:3, Informative)

    by Hatta (162192) on Sunday August 24, 2008 @10:58AM (#24726213) Journal

    If you click through the sciencenews.org article, you can get to the actual website of the people who made these videos. From there you'll find that these videos are Creative Commons licensed, and available for download [dimensions-math.org] as high res MOVs. I tried the torrents, but they all stalled, so I just used wget to grab them from the US mirror [povray.org].

  • ...not Edwin Abbot.

  • Wrong name (Score:2, Informative)

    by Anonymous Coward

    Edwin Abbot Abbot wrote Flatland, not Edwin Abbot, who was Edwin Abbot Abbot's father.

  • objects all the time. Functional analysis, anyone?

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