MagicCube5D

September 12, 2011 6:48 PM

In the spirit of taking things too far, here is a fully functional 5-dimensional analogue of Rubik's cube.
posted by Trurl (50 comments total)

*cries*
posted by maudlin at 6:49 PM on September 12, 2011 [8 favorites]

They've gone and done it, the fools . . . they have made RUBIK'S TIMECUBE!
posted by KingEdRa at 6:52 PM on September 12, 2011 [16 favorites]

When one of my kids was big into Rubik's, we came across that page. We basically knew how a hypercube (a mere 4D) was put together, so we tried to solve one "side". We couldn't even figure out how any of the basic moves produced the changes we saw. So crazy.
posted by DU at 7:00 PM on September 12, 2011

Can you peel off the stickers? If not, I give up.
posted by Knappster at 7:04 PM on September 12, 2011 [9 favorites]

I do not understand this. Can someone explain it to me like I'm a six year old?
posted by elmer benson at 7:05 PM on September 12, 2011 [2 favorites]

*cries*
posted by maudlin at 9:49 PM

Eponysterical?

(That's my first time claiming that on mefi! I think I ding now)
posted by X-Himy at 7:07 PM on September 12, 2011 [1 favorite]

It's moving! Why is it moving?
posted by Orange Pamplemousse at 7:07 PM on September 12, 2011

Solved it.
posted by yerfatma at 7:07 PM on September 12, 2011 [3 favorites]

I do not understand this. Can someone explain it to me like I'm a six year old?

Applying an alternation operation, deleting alternating vertices of the 5-cube, creates another uniform 5-polytope, called a 5-demicube, which is also part of an infinite family called the demihypercubes.

Hope this helps.
posted by Trurl at 7:10 PM on September 12, 2011 [18 favorites]

I see.....a funky picture. Is it supposed to do anything (but I'm just missing a plugin), or is it just a funky picture?
posted by EmpressCallipygos at 7:12 PM on September 12, 2011

It reminds me of the plunge into each successive level of Tempest. So dangerously beautiful.
posted by datawrangler at 7:17 PM on September 12, 2011 [3 favorites]

I see.....a funky picture. Is it supposed to do anything (but I'm just missing a plugin), or is it just a funky picture?

I thought that too. Turns out it's software you have to install and run separately.

(I think it's a sign of the times that I'm disappointed by this.)
posted by vogon_poet at 7:21 PM on September 12, 2011

Can you peel off the stickers? If not, I give up.

Yes, you can peel them off. Of course, each sticker is a four-dimensional hypercube.
posted by benito.strauss at 7:27 PM on September 12, 2011 [34 favorites]

Yeah, it would be really cool if there were at least a youTube video of someone running the program, so we didn't have to install the software, etc., just to see it.

And also, a pony.
posted by darkstar at 7:27 PM on September 12, 2011

ahem. a five dimensional meta-pony, you mean.
posted by spindle at 7:33 PM on September 12, 2011

Yeah, it would be really cool if there were at least a youTube video of someone running the program, so we didn't have to install the software, etc., just to see it.

The search feature, it does nothing!
posted by T.D. Strange at 7:37 PM on September 12, 2011

Oh no no no, I'm not touching it! That's how the Cenobites get you.
posted by treepour at 7:38 PM on September 12, 2011 [7 favorites]

I dare you to link the links.
posted by Flashman at 7:38 PM on September 12, 2011

A five-dimensional pony which is also square. And maybe French, to be fancy. Let's call it a Pony Carré.

Hm. I wonder if such a pony is really so bizarre. Maybe it has to be homeomorphic to a more familiar manifold...
posted by stebulus at 7:39 PM on September 12, 2011 [3 favorites]

Turns out it's software you have to install and run separately.

Here is a screenshot.

Left-click-and-drag rotates the display. Right-click-and-drag zooms in and out.

The buttons in the lower-right rotate particular hyperfaces. You can filter down to the choices that apply to a particular hyperface. The Faces tab lets you toggle particular hyperfaces on and off.

The Macros and Progress tabs are fairly self-explanatory.

Or you can just go to the Scramble menu and select "Full". Then go to the Options menu and select "Solve".

Pretty!
posted by Trurl at 7:40 PM on September 12, 2011

Here's another neat Youtube link.
posted by box at 7:41 PM on September 12, 2011

Is it unsurprising to anyone else that someone writing a program like this would be completely oblivious to the fact that there are people who don't run Windows?
posted by sensate at 7:44 PM on September 12, 2011 [2 favorites]

Man, I still don't get 3-d tic-tac-toe, and you expect me to wrap my brain around this shit! There isn't enough acid in the world.
posted by not_on_display at 7:44 PM on September 12, 2011 [1 favorite]

As I opened the page, Dark Side of the Moon queued up. Rather appropriate.
posted by Slackermagee at 7:48 PM on September 12, 2011

Oh no no no, I'm not touching it! That's how the Cenobites get you.

The cube. You solved it. We came.
posted by Blazecock Pileon at 7:52 PM on September 12, 2011 [1 favorite]

Missing the timecube and batshitinsane tags.
posted by hellojed at 7:53 PM on September 12, 2011

So when you solve this, do you get a five dimensional pin-head who corrupts the world into a mirror of the Empyrean to satisfy the Ruinous Powers?
posted by Slackermagee at 7:58 PM on September 12, 2011

I can solve the 3 dimensional cube, but only by moving into the fifth dimension and back.
posted by Obscure Reference at 7:58 PM on September 12, 2011 [3 favorites]

a five dimensional meta-pony, you mean...

"Mice Ponies are not, as is commonly assumed on Earth, small white squeaking animals who spend a lot of time being experimented on requested as gifts. In fact, they are the protrusions into our dimension of hyper-intellegent pan-dimensional beings".
posted by 445supermag at 8:20 PM on September 12, 2011 [1 favorite]

I would like that as a screen saver or music visualizer. I do not, however, understand what it is I am looking at.
posted by arcticwoman at 8:23 PM on September 12, 2011

As I opened the page, Dark Side of the Moon queued up. Rather appropriate.

When I was about fifteen I smoked some horribly sticky red bud, curled up in a corner of the squat we were living in, and lapsed into a profound fugue. At some point someone put on dark side of the moon, and that first youtube video is almost exactly what I saw. Except the cubes were octahedrons, and they were pulsing like xenia coral. I then spent about a year trying to repeat the experience, with no success at all.

Until now.
posted by Ahab at 8:35 PM on September 12, 2011

sensate: "Is it unsurprising to anyone else that someone writing a program like this would be completely oblivious to the fact that there are people who don't run Windows"

Truth be told I'd have expected the writer to prefer *NIX. Or perhaps BeOS.
posted by DoctorFedora at 9:18 PM on September 12, 2011 [2 favorites]

Truth be told I'd have expected the writer to prefer *NIX. Or perhaps BeOS.

Both of the rubik's cube fans I know carry Powerbooks and write Linux code. (I helped them build a giant LED-based cube, but I have no idea how to solve it.)
posted by Mars Saxman at 9:21 PM on September 12, 2011 [4 favorites]

Yeah, one of the main reasons I've been so big into Mac OS these last several years was the discovery that it was UNIX that wouldn't shit the bed if left unsupervised. Kind of surprising that someone that nerdy is using Windows, especially if it's by choice.
posted by DoctorFedora at 9:34 PM on September 12, 2011

Be great than if you complete it (sure yerfatma - we believe ya) you year some solid stone scraping across stone and some portal or ancient tomb is opened.
posted by the noob at 9:34 PM on September 12, 2011

It reminds me of the plunge into each successive level of Tempest. So dangerously beautiful.

AVOID SPIKES
posted by Ironmouth at 9:46 PM on September 12, 2011 [4 favorites]

Somewhere, Buckaroo Banzai is smiling.
posted by cortex at 9:49 PM on September 12, 2011

(I helped them build a giant LED-based cube, but I have no idea how to solve it.)

That deserves its own FPP.
posted by Blazecock Pileon at 10:42 PM on September 12, 2011

I came here to make a Buckaroo Banzai joke. Like so many others, I have been beaten to the punch by cortex.
posted by chavenet at 11:38 PM on September 12, 2011

oh man I can almost see what's going on in the video of the puzzle being solved. I've spent enough time fiddling with 4D projection toys that I could manipulate a simulation of a 4D Rubik's Cube pretty confidently - though I couldn't solve it. On the other hand I also couldn't solve a 3D Rubik's any more as I forgot the bottom-slice moves I used to know.

But this... I don't think I could even predict what each of the possible rotations would do to the cubes. Not without a few hours of dedicated fiddling.

* * * * *

elmer benson > I do not understand this. Can someone explain it to me like I'm a six year old?

Let me try. I was devouring the collections of Martin Gardner's "Mathematical Recreations" columns when I was a six-year-old, hopefully this will make some sense...

Alright, imagine a square. Let's say it's one unit wide, and one unit tall.

+--+
|  |
+--+

Now take that square and extrude it out along an axis that's at right angles to the width and height, as if you were pulling it out from the monitor screen and leaving a trail behind. Pull it out for one unit and you have a cube. If you held a light up to one face, you might see a shadow like this.

+------+
|\    /|
| +--+ |
| |  | |
| +--+ |
|/    \|
+------+

And if you turned the cube around a little, maybe moved the light a bit, you might see a shadow like this.

+---+
|\  |\
| +-+-+
+-+-+ |
 \|  \|
  +---+

It's the same cube, but its shadow can look pretty different depending on how you turn it. Right? Maybe you wanna take a moment to get some toothpicks and peas or whatever, make a cube, and turn it around in front of a light to look at its shadow. You'll find some other interesting shapes that I don't want to try and draw in ASCII. Your brain should be able to pretty readily parse these shadows to imagine the cube that's casting them, even if you arrange the light, cube, and the surface it's casting a shadow onto such that you can only see the shadow.

Now this is where it gets interesting. Take the cube and extrude it out along an axis that's at right angles to the width, height, AND the depth. You'll have an object that is to a cube as a cube is to a square, often called a "hypercube". Yes, I know, can't be done, can it? But what you can do is create a model of the three-dimensional shadow cast by a four-dimensional object, just as my crude ASCII art is a drawing of the 2D shadow cast by a 3D object.

One way to build that model of a hypercube's shadow is to build a cube out of toothpicks and peas or whatever, then stick another toothpick into each pea at a diagonal (all these diagonal toothpicks should be at the same angle), and build another cube that connects the free ends of those diagonal toothpicks. I am not going to try to draw one in ASCII art! If you looked at it side-on it'd look pretty much like the second shadow of the cube I drew that looks like two squares overlapping each other, with diagonal lines between their corners.

Another way to build a model of a hypercube's shadow? Build another toothpick-and-peas cube, and put a toothpick in each pea. This time, they should all be sticking out from the cube like it's a little mine or something - each toothpick should be on a line that passes through the center of the cube, and through the corner it's on. Then you take a couple of skewers and cut them to the right length to make another, bigger cube whose corners are peas put on the outer ends of those diagonal toothpick. Look at it side-on and you pretty much see the first cube shadow I drew, with one cube inside a bigger one.

The kicker is that, while YOU had to build a long outer edge, if you could turn those toothpicks into a fourth spatial dimension, you would not have to get any skewers out; all the edges are the SAME LENGTH. It's just that some of them get longer when you cast the shadow. Same as the first shadow I drew - looked at as lines on paper, some of those lines are a lot longer than others, but the cube in your hand is made of twelve toothpicks, all the same length. Each of the toothpicks stuck into a pea is ninety degrees away from every other toothpick stuck into the same pea.

Now imagine taking that hypercube made of thirty-two toothpicks and sixteen peas and turning it. Just like you turned the cube made of twelve toothpicks and eight peas. I lied, the bit above wasn't where things get interesting - this is where things get interesting. Hold that hypercube up to a hyperlight, so that it casts its hypershadow onto your 3D reality. Don't look at the hypercube, just look at its hypershadow. Depending on how you hold it, it might look like either one of those two models you built earlier. Or like analogs to the other shapes the shadow of your cube took. And as you rotate the hypercube, its 3D hypershadow will twist and turn in these clearly impossible ways - it'll seem to expand and contract the same way the cube's shadow does, if you shut off the parts of your brain that want to see the shadow as a projection of a 3D object.

While you're doing that, also take a look at the hypershadow's shadow. That's a 2D projection of a 3D projection of a 4D object.

I lied. That wasn't where it gets interesting, back there. Now is where it really starts to get interesting. Get some more toothpicks and peas. You're gonna need them. And your hypercube. Stick a toothpick in each of the corners of your existing hypercube model... at right angles to width, depth, height, and the axis you stuck a toothpick in when you extruded your cube into a hypercube. Then put a pea on the end of all sixteen of the toothpicks you just added, and build another perfectly ordinary hypercube between them. Go on. I'll wait here.

Okay. Now you're going to take that hyper-hypercube and hold it in front of a hyper-hyperlight so that it casts a 4D shadow in front of the hyper-light you were illuminating your hypercube with. Which will cast a 3D shadow of the 4D shadow of the 5D object you're holding up. And that 3D shadow of the 4D shadow of the 5D object is, itself, in front of an ordinary light, and casting a shadow onto a piece of paper or something.

We are not there yet. I hope you have a lot of toothpicks and peas. You're gonna need them. And some paint.

OKAY. Take a Rubik's cube. Keep the stickers where they are and remove the plastic and metal bits that make up the mechanism. You'll be left with 54 stickers floating in space. (We are, of course, doing this in a zero-gee vacuum, so the stickers stay where we put 'em.) Got that?

Now extrude that cube of stickers into a hypercube. Easy peasy, you've done it before, right? Except instead of building the edges of the hypercube, you're gonna build a bunch of little cubes, paint them different colors, and arrange them to make up the outside of a hypercube just like those stickers made up the outside of a cube. Twenty-seven each of eight different colors, if you're making a 3x3x3x3 Rubik's hypercube. I think. It's 1AM and I don't feel like double-checking my math. Arrange them each in a cube, then put those cubes together to make a hypercube. (The hyperfaces of a hypercube are cubes, just as the faces of a cube are squares. A Flatlander who spoke English would probably call a cube a "hypersquare".)

(Incidentally, if you'll take a moment to play with your Rubik's hypercube in front of your hyperlight, you'll notice that the toothpicks you made its sub-faces out of seem to slide right through each other if you're just looking at its 3D hypershadow.)

We're almost done. You still have a lot of toothpicks and peas, right? I'm not going to try to calculate how many, if you don't have enough just go hang out at your favorite all-night diner and swipe theirs. (Protip: Waitresses will not appreciate you leaving your hyperdimensional toothpick-and-pea constructions behind in lieu of a tip.) Anyway. Build a bunch of hypercubes in however many colors you need to color each hyperhyperface (a hypercube) of your hyperhypercube. You'll need, ummm, 81 of each color I think, if you're building a 3x3x3x3x3 Rubik's hyperhypercube. Yeah, you're gonna need a few more boxes of toothpicks and another order of peas. Get some coffee while you're at it, this is gonna take some time to put together. Anyway! Once you've done THAT, hang each set of 81 hypercubes in space a hypercubic array - just like you hung each set of 27 cubes in a cubic array, and just like each set of 9 squares was hung in a square array. Then, just as before, you take all those hypercubic arrays of hypercubes and put their hyperedges next to each other to make up a hollow shell of a hyperhypercube.

We are sooooo close. Now put this 3x3x3x3x3 Rubik's hyperhypercube in front of your hyperhyperlight, which casts a 4D shadow in front of your four-dimensional hyperlight. The hyperlight shines through the hyperhypershadow of the Rubik's hyperhypercube, and casts a 3D hypershadow in front of your ordinary 3D light. Which casts a 2D shadow onto a piece of paper.

That 2D shadow is what you see on the screen as the output of the program linked by the original FPP: a 2D projection of a 3D projection of a 4D projection of a 5D object.

So you have the shell of a Rubik's hyperhypercube. Just the hyperhyperstickers floating in hyperhyperspace, rendered as a bunch of wireframe lines on the screen.

Now try turning it around for a bit. You've probably got a massive headache by now, if this is your first time trying to visualize higher-dimensional space. Get a lot of asprin, because it's only gonna get worse if you mess up your brand new Rubik's h'h'cube and try to solve it.

tl;dr: Welcome to Flatland, here's your room in the asylum next to A. Square.
posted by egypturnash at 1:05 AM on September 13, 2011 [41 favorites]

I need to go to bed now.
posted by egypturnash at 1:13 AM on September 13, 2011 [3 favorites]

Also I am not entirely sure that I precisely described the procedure to make a model of the object the program at hand simulates, a quick double-check of the linked videos does appear to be showing a bunch of arrays of hypercubes, presumably arranged along the outside h'h'faces of a h'h'cube, but I don't feel like reading all the supplementary material on the page to find out for sure. I did, however, skim far enough to discover this page, which informs me that you would have to make a total of 810 hypercube "stickers" to fill out each 3x3x3x3 hypercube outer face of the 3x3x3x3x3 Rubik's hyperhypercube. Like I said. You're gonna need a lot of toothpicks.
posted by egypturnash at 1:21 AM on September 13, 2011

also oh geeze I wish there was a Mac version of this other program the author of this hyperhypercube program wrote because I would totally get stoned and masturbate to it rotating in four dimensions for a while um maybe i'm oversharing here. BEDTIIIIIIMMMMME
posted by egypturnash at 1:25 AM on September 13, 2011 [2 favorites]

sensate i do not understand what you mean perhaps you would care to unpack/elaborate?
posted by This, of course, alludes to you at 2:06 AM on September 13, 2011

I've spent enough time fiddling with 4D projection toys that I could manipulate a simulation of a 4D Rubik's Cube pretty confidently - though I couldn't solve it.

Ahh... Fond memories of playing with MagicCube4D. If you're pretty good with a cube in 3D, a lot of the rotations start making sense and feeling predictable after a while. I remember when the 4D version started making sense and I felt like I really got it. It was exhilarating!

I finished a large portion of it as well, but I never did get around to solving it completely. I put it down late one night after realizing that I needed to figure out how to do something like a 4D version of a Sune, and I never got back around to doing it.

Anyway, for anyone who really feels they need to master this, I would advise working your way up. Become familiar and comfortable with solving it in 3D, then work in 4D until you are comfortable with what is going on. Then (if you're still sane) move on to this.
posted by Avelwood at 2:30 AM on September 13, 2011

I do not understand this. Can someone explain it to me like I'm a six year old?

The Wiki page for a 4D cube is an excellent primer.

It includes animated gifs such as a 3D projection of an 8-cell performing a simple rotation about a plane which bisects the figure from front-left to back-right and top to bottom.

But you probably already knew that.
posted by uncanny hengeman at 4:33 AM on September 13, 2011

egypturnash: you actually answered a couple questions that were bothering me about the app's rendering (for ex., why the cubes are discrete and not connected in the 2D "shadow"). Thanks!
posted by aught at 5:50 AM on September 13, 2011

My only questions is what sort of dice are required to make a SAN check against this?
posted by Godspeed.You!Black.Emperor.Penguin at 8:12 AM on September 13, 2011 [1 favorite]

AAAND.

If I've whetted your appetite, there's at least one chapter about hypergeometry in one of Martin Gardner's books. I believe it's "Mathematical Carnival" after some googling - I tossed my physical copies long ago, then misplaced my copy of the collected DVD so I can't check. Plus of course a ton of potentially fascinating info about OTHER crazy math concepts. You can also find a few interactive 4D toys on the net; play with them, you'll be able to start to get a sense of how a 2D projection of a 4D object rotates. Some of them even work with red/green 3D glasses!

Also I have always found the video for Blockhead's "The Music Scene" to be a supremely fourth-dimensional work of art.
posted by egypturnash at 8:56 AM on September 13, 2011

Imagine four Timecubes on the edge of a cliff. Say a direct copy of the Timecube nearest the cliff is sent to the back of the line of cubes and takes the place of the first cube. The formerly first Timecube becomes the second, the second becomes the third, and the fourth falls off the cliff.

The 5D Rubik's Cube works the same way.
posted by Spatch at 5:41 PM on September 13, 2011