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Gaming in the 4th Dimension 303

Posted by CmdrTaco
from the still-haven't-solved-3d dept.
Wolf pointed me to a video clip demonstrating this game: "Miegakure is a platform game where you explore the fourth dimension to solve puzzles. There is no trick; the game is entirely designed and programmed in 4D." Nothing to download yet.

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Gaming in the 4th Dimension

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  • So Many Questions (Score:5, Interesting)

    by eldavojohn (898314) * <eldavojohn@gmFREEBSDail.com minus bsd> on Wednesday March 31, 2010 @10:11AM (#31687116) Journal
    So I've traditionally known "the fourth dimension" to be something like time. Although you can call it space-time or the relationship that our three dimensional world has with our concept of time. And in games like Braid (which is like an interesting two dimensional scrolling platform with four dimensional control), you get to have fun manipulating this time so that you can predict where your little character is when you slide back in time. It's where you were before.

    In Miegakure, it appears that the player is controlling a fourth dimension except it's not too clear what fourth dimension actually represents to me. If Miegakure's fourth dimension was time, we would see some indication of natural decay of the environment to give us visual cues that it's aging. For example, if one ring were made of steel and the other of wood, the wood one would decay as we go to the future and then we would make some action that is "special" (meaning that it is not subjected to our time control) and then move the steel ring into the wood ring and blast back to when the wood ring existed. Our special action could not be undone otherwise you wouldn't get anywhere with being able to control time.

    Miegakure seemed to invent non-natural transposed states of the environment that I, for the life of me, could not understand. How did I know which blocks would appear and disappear leaving only shadows? How do I know how far to go in a fourth dimensional direction? Must the player explore the available transposed states before planning their movements along all four dimensions? So that they can construct an interleaved solution?

    And what happens with a now block exists in a shadow space and you try to transposition yourself to the point when the shadow space is occupied by another block? Does the game block you from making that transposition? What if you want to transpose to a point beyond that when it is a shadow space again? Is this a blocking mechanism that will add to the difficulty of the puzzle?

    As someone ravaged by the Adventures of Lolo series on the NES, I could see a potentially high level of addiction here.
    • Re:So Many Questions (Score:5, Interesting)

      by pushing-robot (1037830) on Wednesday March 31, 2010 @10:18AM (#31687214)

      Today's XKCD [xkcd.com] might help a bit. It's a world that has four spatial dimensions, like a hypercube. [wikipedia.org]

      We haven't been able to find any evidence of "real" higher spatial dimensions (though theories abound), but thinking in an extra dimension is an interesting mental exercise nonetheless.

      • Re: (Score:3, Insightful)

        by slim (1652)

        We haven't been able to find any evidence of "real" higher spatial dimensions

        Though superstring theory requires 10 or 11 dimensions of space (from what little I understand), so serious physicists really believe those dimensions might exist.

        • Re:So Many Questions (Score:5, Informative)

          by Per Wigren (5315) on Wednesday March 31, 2010 @10:50AM (#31687736) Homepage
          10 dimensions. There is a pretty easy to follow explanation on Youtube:
          Imagining the Tenth Dimension, Part 1 [youtube.com]
          Imagining the Tenth Dimension, Part 2 [youtube.com]
          • The video doesn't describe what the 10 dimensions in string theory really means - it simply describes what a mathematical model of 10 dimensions can probably do if we use the first 3 to model our familiar 3D world.
        • string theory (Score:3, Insightful)

          by malp (108885)

          providing no unique, testable predictions for over 20 years...

        • Re: (Score:2, Insightful)

          by losfromla (1294594)

          serious physicists really believe those dimensions might exist.

          Serious physicists work on testable, proveable concepts. String theory is little more than an exercise in imagination and really hairy mathematics. It requires far too many variables to close and new ones are added as they realize that they can't get anything that resembles reality despite the wonderfulness of the math. I've (as an informed layman) have lost interest in this black hole for research funding. Serious physicists have too. The ones still "working" on string theory are doing so because dropping it would invalidate their entire careers.

      • Re:So Many Questions (Score:5, Interesting)

        by somersault (912633) on Wednesday March 31, 2010 @10:40AM (#31687568) Homepage Journal

        This doesn't seem so much like a "fourth dimension" as a form of "subspace" or an alternate 3D reality (then again I haven't played the game and maybe am picking things up wrong from the video).

        I don't see how adding another dimension can magically allow two objects to become linked when they were unable to be linked in a lower dimension. Two circles on a piece of paper cannot physically merge with each other if you assume their boundaries are solid and cannot pass through each other. Neither can 2 rings lain on a table, or two cylinders or two spheres be overlapped without breaking them somewhere. So how would adding another dimension allow you to join two 3D objects with a hole in the middle, even if you only moved one of them into this higher dimension?

        • Re:So Many Questions (Score:5, Interesting)

          by Garble Snarky (715674) on Wednesday March 31, 2010 @10:47AM (#31687696)
          Here's one way to think about it: You have two concentric circles in a plane, they can't pass through each other in two dimensions. In three dimensions, the concept of "passing through each other" is no longer necessary for getting them "unlinked".
          • Re:So Many Questions (Score:4, Interesting)

            by somersault (912633) on Wednesday March 31, 2010 @11:06AM (#31688016) Homepage Journal

            Hmm... well that would similarly work for a sphere containing another sphere.. but a torus or any other object with a hole is surely a different class of object.. I'm not sure what the 2D representation of a torus would be..?

            • a 2d representation of a torus would be a circle.

            • Re: (Score:3, Informative)

              by SoVeryTired (967875)

              Topologically, the torus can be identified with something called S1xS1 (the cartesian product of two "one-spheres", aka circles).
              Likewise, the n-dimensional torus is the cartesian product of n copies of the circle.

              This means that the one-dimenstional torus is just the plain old circle. In one dimension, the torus and sphere are the same thing.

        • by cecille (583022)
          For rings, if you lifed one ring in the third dimension, and then moved it over then projected that back into only 2 dimensions, then they would appear linked. In the demo, they say they are shifting back into the 3rd dimension, which I'm interpreting as a projection, in which case, the links would seem linked in the third, but not necessarily in the fourth. On the other hand, what they show is more like a star-trek style phase shift. Not that I can blame them really - they're simulating a 4D world, draw
          • Yeah I was wondering about moving something into an extra dimension and combining with something from the other dimension.. I suppose the fact is that here they were combining toruses which can actually be linked in 3D.. but I'm still dubious as to whether adding a dimension makes that any easier.. seems moving them into 2D and then back to 3D would be the simple way to do it, and that moving the objects into 4D would just make it even more difficult to manipulate the two objects in such a way that they wil

            • Re: (Score:3, Informative)

              I don't really understand what you meant.. but there are no 4D toruses in the video - it's totally ok to have 3D, 2D, 1D and point objects in a mathematical 4D space. The point of allowing movements in a fourth dimension is to allow the toruses to be joined without breaking them.
              • Still having trouble visualising moving a torus into 2D, let alone 4D.. you might think that getting an orthographic perspective of a 3D object can give you a 2D representation - it works for a circle.. but it doesn't really work for any asymmetric objects, including toruses.. I suppose if you don't actually add an extra dimension to a 3D object when transferring it into 4D space then from a 4D perspective it might be possible to make it look like the two objects have become joined without breaking any boun

            • As for how the linking works for the torus, perhaps you can think about it from a 3D perspective.

              The moment the torus is lifted in the 4th dimension, you'd see its disappear because it's 4th coordinate is different from yours. It's just like a Flatlander would see a ring disappear from their world if a 3D person lifts it along the 3rd dimension.

              Then, fixed to the modified coordinate in the 4th dimension, you move the ring along 3D space such that its projection intersects the other torus. Then, you put
        • Neither can 2 rings lain on a table

          Yes you can - you simply aren't thinking in 2D. The operation required is to make 2 2D circles intersect in 2D space, but you have access to 3D space. So what you do, is to take one 2D circle up, move it in 3D space such that it intersects the other 2D circle. And then you put that 2D circle down. Now the two circles perfectly intersect each other in 2D space.

          • Have been thinking about it, and I guess this analogy just isn't working because the 2D representation of a 3D object with a hole in it probably isn't even possible.. so I can't quite imagine with the 4D representation would be, apart from to think that it still would not be possible to intersect them any more than you can cause two circles to merge by changing one of them into a sphere (which would be the real 3D form of a circle, rather than a ring or a cylinder which again are both toruses..). In 2D I'd

            • Re: (Score:3, Informative)

              You don't need to imagine the shape of the object in 4D space. It's really simple - a purely 3D object in 4D space occupies no "volume" in the 4D space because its height along the 4th dimension is zero. So when you lift a torus, or any 3D object up in the 4th dimension, the object effectively disappears from the original 3D space. Then, as long as you don't move the object back to its old coordinate along the 4th dimension, you can move it in any 3D way you want.
        • Re:So Many Questions (Score:5, Informative)

          by Monkeedude1212 (1560403) on Wednesday March 31, 2010 @11:27AM (#31688360) Journal

          Think about it this way:

          You put a box inside a safe. That safe has no doors. How do you get the box outside the safe? You slide it through the fourth dimension - so that the walls of the safe are no longer in the way. You change its XYZ co-ordinates, slide it back through the fourth dimension so its about where it began. The box is now outside the safe.

          If thats still a little tricky to understand, we'll explain it flatland style.

          You draw a circle inside of a square on a piece of paper. How do you get the circle outside of the square (assuming you can't move the lines through each other). Well, if you had the ability to take the circle off the paper, move it a few inches, and place it back on the paper, you would have moved it outside of the square with no intersection taking place.

          The same thing is happening here, you are taking two rings, sliding them among a dimension that they do not occupy (thus removing any chance for collision) and then putting them back. Its tough to wrap your mind around, I know.

        • Re:So Many Questions (Score:5, Informative)

          by Ephemeriis (315124) on Wednesday March 31, 2010 @11:29AM (#31688392) Homepage

          I don't see how adding another dimension can magically allow two objects to become linked when they were unable to be linked in a lower dimension. Two circles on a piece of paper cannot physically merge with each other if you assume their boundaries are solid and cannot pass through each other.

          Say we've got two circles drawn on a 2D plane - a sheet of paper. Assume their edges are physical boundaries - if you push them together they'll bump into each-other, not merge or join.

          Now, pick one of those 2D circles up off of the page. It no longer occupies the same 2D space that the other circle does. You can move it back and forth without it bumping into anything, because the other circle is stuck down on the piece of paper.

          If you move the two circles so that they're overlapping a bit, like a Venn diagram... And then drop that circle back onto the 2D plane of the paper, they're now overlapping or linked. Even though that would have been impossible to do in just two dimensions.

        • by Hatta (162192)

          I don't see how adding another dimension can magically allow two objects to become linked when they were unable to be linked in a lower dimension. Two circles on a piece of paper cannot physically merge with each other if you assume their boundaries are solid and cannot pass through each other.

          Two circles on a piece of paper might be the 2d projection of a 3d object like a torus. Imagine a cross section through a donut. Two circles in 2d, but one continuous solid in 3d.

        • by s-gen (890660)
          a line:
          ______________________

          a "ring" called "A":
          __A_____A_____________

          and another "ring" called "B":
          __A_____A___B_____B__

          lift "B" into the second dimension:
          _____________B_____B__
          __A_____A_____________

          slide "B" across:
          _____B_____B___________
          __A_____A_____________

          drop "B" back onto the line:
          __A__B__A__B_________

          "A" and "B" are now "linked" in the 1D universe.
    • Re:So Many Questions (Score:5, Informative)

      by Anonymous Coward on Wednesday March 31, 2010 @10:22AM (#31687274)

      Time is not "the fourth dimension." It is very much like a spacial dimension, speaking as a physicist; however, it is also very different. This is clear both from experience (ever try to move back and forth in time?) and mathematically (via the signature of the metric of spacetime).

      In this game, the fourth dimension is simply an extra spacial dimension. Consider the analog of "linking two rings" in a 2-D world: put one circle inside another. Well, if you're stuck in a plane, it cannot be done -- simply move outside of that plane into 3-D, and it's simple. In Miegakure there is a 4th spacial dimension. You can move in this fourth dimension without moving in any of the other three.

      Yeah, it's weird. I'm not entriely clear as to what the shadows represent (except, maybe, for a helpful reminder as to what is "next" to you.)

      • Re: (Score:2, Funny)

        by Anonymous Coward

        (ever try to move back and forth in time?)

        I move forth in time every day, you insensitive clod!

      • Re:So Many Questions (Score:5, Interesting)

        by pushing-robot (1037830) on Wednesday March 31, 2010 @10:41AM (#31687594)

        Yeah, it's weird. I'm not entriely clear as to what the shadows represent (except, maybe, for a helpful reminder as to what is "next" to you.)

        I think that's the idea. It's hard to tell from the short video, but the blocky nature of the world implies to me that the game limits you to arbitrary "jumps" in each dimension. Just like the world could be divided into fixed-width planes in the X, Y, and Z dimensions, it looks like the W dimension is composed of distinct layers. Which would explain the shadows; they represent what would appear if you jumped to the next adjacent "slice" of 4d-space.

      • by einhverfr (238914)

        I have often joked that if Time is the fourth dimension identical to space, than the fifth dimension must be the Dow Jones Industrial Average since that is very frequently plotted as perpendicular to time ;-)

      • Re: (Score:3, Interesting)

        by leonardluen (211265)

        It is very much like a spacial dimension, speaking as a physicist; however, it is also very different.

        How is it different? why not just consider it indeed being the same as any other spacial dimension? one in which we have a constant velocity that we currently don't, and maybe never will, know how to change. even if 2 objects in our universe have the same coordinates in 3d space they will still miss each other if their 4th dimension of time is different...ie many cars make it through an intersection because they go through at a different times...when their time is the same is when you have a crash...

        that

        • Re: (Score:3, Informative)

          by Late Adopter (1492849)

          How is it different? why not just consider it indeed being the same as any other spacial dimension?

          You can consider it however you like, if it's helpful. You have to be very careful in conversations like these to restrict your hypotheses to ones which have real observable consequences. Otherwise you wander away from science into philosophy, which is a fine conversation to have, but not one that scientists would enjoy having with you =P

          General Relativity, being the most accurate model to-date of time and space themselves, treats time as a dimension, but one with slightly different mathematical proper

    • Re:So Many Questions (Score:5, Informative)

      by Monkeedude1212 (1560403) on Wednesday March 31, 2010 @10:23AM (#31687286) Journal

      Miegakure suggests that there is a fourth spatial dimention, just like the three you are used to seeing.

      Take a read through Flatland, its a short story based on a square who lives on a 2 dimentional plane. Basically how he can only see things in 1 Dimension (a line) because him and his world are on a single plane. Now, imagine his world lives within our 3d Realm. His life doesn't change much, until we choose to interfere. Imagine if you slid a ball through his 2d plane. He would at first see nothing, then a dot, then that dot grow into a line, then it shrink, into a dot, and disappear.

      Basically someone took this idea, and imagined what it would be like if there were a 4th spatial dimension we were unaware of (physics has however shown us that there isn't one). If someone pushed a 4d Cube (or hypercube) through our 3d plane, what would we see? Nothing at first, then a cube show up, then it grows into its full size, then shrink back down, and disappear.

      Now someone has taken that idea and put it in a game. The programming is actually simpler than it seems. Instead of testing XYZ co-ordinates you are testing WXYZ co-ordinates.

      • Re:So Many Questions (Score:5, Informative)

        by ZXDunny (1376265) on Wednesday March 31, 2010 @10:48AM (#31687708) Homepage

        Take a read through Flatland, its a short story based on a square who lives on a 2 dimentional plane. Basically how he can only see things in 1 Dimension (a line) because him and his world are on a single plane.

        The XKCD alt-text contains a nice in-joke about flatland (IIRC) - all women are straight lines, and the more important a member of society, the more sides he has - a priest would be almost a circle, as he has so many sides he looks circular. The alt-text goes: "Also, I apologize for the time I climbed down into your world and everyone freaked out about the lesbian orgy overseen by a priest." Which is what the flatlanders would see when a stick-man enters their world :)

        • Take a read through Flatland, its a short story based on a square who lives on a 2 dimentional plane. Basically how he can only see things in 1 Dimension (a line) because him and his world are on a single plane.

          The XKCD alt-text contains a nice in-joke about flatland (IIRC) - all women are straight lines, and the more important a member of society, the more sides he has - a priest would be almost a circle, as he has so many sides he looks circular. The alt-text goes:

          "Also, I apologize for the time I climbed down into your world and everyone freaked out about the lesbian orgy overseen by a priest."

          Which is what the flatlanders would see when a stick-man enters their world :)

          Wonderful explanation for those of us who haven't read Flatland. Thanks!

      • If you slid a ball through his 2d plane, they'd see nothing, then a dot, then a widening circle, then a decreasing circle, then a dot, then nothing. If you were pushing a circle through, they'd see a nothing, a dot, a line (depending on the width of the circle) perhaps curved (depending on the angle of the circle), two lines moving away from each other, two dots, then two lines moving towards each other, a line, a dot and then nothing. If the circle was completely parallel, then they would see a circle.

        • by vlm (69642)

          I was unaware that physics had shown that there wasn't a 4th dimension. I'm not sure how physics or physicists could prove this.

          There's a lot of DoF thermodynamics calculations that prove, that at least at some scales of distance and energy, any extra dimensions must either not exist or interact a remarkably small amount with the other three. For instance, a rough definition of temperature is wiggling/flying in three dimensions, you can predict exactly how fast gas atoms should move at a certain temp, and interestingly enough when you resolve their movement into 3-d vectors the graph pretty much matches up. So either they don't ex

        • by ShakaUVM (157947)

          >>I was unaware that physics had shown that there wasn't a 4th dimension. I'm not sure how physics or physicists could prove this. Perhaps what you meant to say was that the math currently used by most physicists does not need a 4th dimension.

          Actually, there's a good amount of work done trying to figure out if physics would work with more than 3+1 dimensions (i.e. 3 spatial, 1 time), and a lot of people are convinced only 3+1 would work.

          http://en.wikipedia.org/wiki/Spacetime#Privileged_character_of_3. [wikipedia.org]

      • What is to say that time isn't spatial?

        • The fact that I'm not slowly moving sideways?

        • What is to say that time isn't spatial?

          You either don't understand time or you don't understand spatial.

          Suppose you are a 2 by 2 by 2 cube. One corner reaches 0,0,0 coordinates and the other corner reaches 2,2,2. I am also a cube of the same size. I could not occupy 1,1,1 simply because you are in the way.

          Now, lets suppose time is another dimension. Same scenario, except you are essentially INFINITE in your last dimension (time) because you never disappear, your matter is always physically present* (One of Newtons laws I think, energy and matter

      • by Fnord666 (889225)

        Basically someone took this idea, and imagined what it would be like if there were a 4th spatial dimension we were unaware of (physics has however shown us that there isn't one). If someone pushed a 4d Cube (or hypercube) through our 3d plane, what would we see? Nothing at first, then a cube show up, then it grows into its full size, then shrink back down, and disappear.

        See also Spaceland [amazon.com] by Rudy Rucker.

    • by kalirion (728907)

      How do I know how far to go in a fourth dimensional direction?

      Actually my question is if you can actually choose how far to go in the fourth dimensional direction, or if it's just a two-state affair.

    • by VShael (62735)

      As others have said, he's dealing in 4 spatial dimensions, not Time.

      If it helps, to have a real world analogy, imagine a maze. Even with bushes, it's basically a 2D problem. Now, leave the bushes there, but imagine you could float/hover and move over them. You're moving in a separate spatial dimension than the maze.

      Now, if there's floating bushes too, you're kind of stuck.

      What this game does, is allow you to choose which 3 spatial dimensions you see on your screen, and move around in. But at any time, you c

    • by e2d2 (115622)

      Dimensions are user defined so the 4th dimension could technically be any dimension you wanted to quantify. Think of it as an attribute of the space you want to define.

      If we're using time as the 4th dimension, which seems fairly standard, then games like Braid or Forza 3 (and probably mnany others) have already conquered this dimension, allowing one to "back up" in time or "move forward".

    • So I've traditionally known "the fourth dimension" to be something like time.

      I don't consider time to be the fourth space dimension. In a space dimension, you can move back and forth, and the spatial relationship is not restricted to a relationship of cause and effect. We don't know if it's possible to move back and forth in time yet, but we do know that the time dimension has a cause and effect relationship, therefore it cannot be a space dimension.

      Miegakure seemed to invent non-natural transposed states o

  • Or is there more to this game than the video shows? I always thought Time was the 4th dimension...
    • by slim (1652)

      It looks like this game has 4 dimensions of space.

      If time is the 4th dimension, then older games like Pac Man are 3D, conventional modern games are 4D, and this game is 5D.

      • No, because everything in a lower dimension exists potentially in a higher dimension already.

        Rather, 2D/3D/4D refers to the dimensions that one can control and manipulate.

        For example, we exist in a 3D world because, while there is time, we can only move forward through it in this dimension and cannot move freely through it.

        In the same way, someone stuck in a 2D world (like pac-man) could be spinning through a 3D world and never have control over that motion.

        In a 4D (3 spatial+spacetime), we would expect to
    • by Thanshin (1188877)

      I always thought Time was the 4th dimension...

      I always thought the 4th dimension was ROCK!

      But I guess time is ok too.

      Or PASTA!

      Ok, ok. Time's fine.

    • by Hozza (1073224)

      Yes, Lagacy of Kain's shifts between the real and 'spectral' worlds does seem to be pretty much identical, although the developers never gave it such a high concept PR treatment.

    • by sznupi (719324)

      It seems quite a bit more complex (at least potentially; might be limited in practise by level design)

      In Legacy of Kain you had only two states and nice looking, fluid, but unstoppable, shift between them. Here you can have another (fourth) set of coordinates except XYZ; a spectrum on which you can be anywhere you want.

      I guess it looks similar to Legacy of Kain because there's really no other way to project it onto 3D space...and then project it onto 2D monitor.

  • Flatland (Score:5, Funny)

    by Kagura (843695) on Wednesday March 31, 2010 @10:16AM (#31687182)
    This article was written by the most hideous of triangles.
  • There's nothing here but some fancy 3D design.
  • by jimicus (737525) on Wednesday March 31, 2010 @10:32AM (#31687428)

    Didn't they get to number six with "Baby I Want your Love Thing"?

  • The description sounds as if you choose which of the four dimensions to project onto your three, but in the video it looks like you keep the standard 3 and represent the fourth by phasing objects a fixed distance into the 4th, represented by translucency. Haven't had a chance to get hands on this yet, though.

  • I was going to make a comment about how if the 4th dimension is time then every board game ever conceived is a 4D game. I was reminded of how 3D ultrasounds are called 4D.

    However, this actually looks quite creative, although I'm having a bit of trouble determining the goal of the game. The immediate problem I see is being able to make things out with so many objects obstructing the view. In some ways this reminds me of Echochrome which I think plays with the notion of multiple dimensions in an even more dra

  • So... is this just Super Paper Mario but with an additional dimension?
  • So we're inferring a 4th spatial dimension in a game which uses our perception of a 3 dimensional construct on a 2 dimensional screen.

    I'm still waiting for the "wow" moment, and I suspect it's not going to happen here.

    • So we're inferring a 4th spatial dimension in a game,
      which uses our perception of a 3 dimensional construct,
      displayed on a 2 dimensional screen,
      stored on a 1 dimensional memory space,
      played by people with 0 life.
  • I haven't wtfv (watched the video), but 4-D can be represented in 3- and 2-D using projections, just like we regularly watch 3-D images projected into 2 dimensions (TV, video games).

    Think of a cone, a 3 dimension shape. In the 3-to-2 dimension projection, that cone can look like a triangle, a circle, an ellipse, or an ellipse with a point, all depending on how you rotate it.

    Now imagine that there's a 4-D shape whose projection changes appearance as the shape is rotated about its fourth-dimensional axis

  • It looks like this is treating the fourth dimension as more of a quick transport (can't watch the video here at work, just looked at the still shots). It would be interesting to see a game wher you could actually move through the fourth dimension incrementally and continually instead of jumping.
  • There have been a lot of games written over the last 15-20 years, certainly, that attempted to make a usable game doing 4D-2D projections.

    Some were fun, most I played with just gave me headaches. I remember one (can't recall its name) that ran on one of SGI's high end imaging systems using active shutter LCDs so it was 4D->2.5D. That one *really* gave me headaches.

    On an unrelated note, Quake ran really nicely on that box :)

  • Another 4d game (Score:3, Interesting)

    by danhaas (891773) on Wednesday March 31, 2010 @11:31AM (#31688426)
    If you want to try another 4d game while Miegakure doesn't release, check http://harmen.vanderwal.eu/hypercube/ [vanderwal.eu] The objective of the game is to push the big ball towards the blue cross, then move the cursor to the square. You will then be outside the box and have to reach the green square again, while you avoid the small ball. Try it in 2d and 3d before going to 4d.

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