Juxtapositions: The Rubik’s Hypercube

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Rubik's cube variationsMany people are familiar with the Rubik’s Cube, the 3x3x3 cube with colored faces that can be moved out of position and then, ideally, twisted back into place. This toy, originally called the Magic Cube, was invented by Ernő Rubik in 1974 while he was a lecturer at the Academy of Applied Arts and Crafts in Budapest, Hungary. In 1980 the cube made its way to other countries and spawned, among other things, a one-season cartoon series Rubik, the Amazing Cube in which some kids use a magical come-to-life Rubik’s Cube to solve mysteries, while avoiding the mandatory evil magician who wants to steal the Cube.

There is a lot more information on the official Rubik’s Cube website, including the fact that the world record for speed in solving the cube is only 16.5 seconds. The latest Rubik’s news was an announcement in May 2007 that Gene Cooperman (a professor) and Dan Kunkle (a graduate student) at Northeastern University proved that no matter how mixed up the cube, it can be solved in at most 26 moves. [This is an improvement from 27 moves, a record that stood for 10 years; the least upper bound may be in the low 20s.]

HypercubeWhat about the hypercube? The hypercube is a four-dimentional object in which each of the “faces” is actually a cube (much the same way that in a three-dimensional cube each face is a square, and in a two-dimensional square each face is a line). There are a plethora of mathematical features of the hypercube: for example, a line has 2 vertices, a square has 4, a cube has 8, and the hypercube has 16, illustrating the fact that an n-dimensional “square” has 2n vertices.

A Rubik’s Hypercube would be a four-dimensional object in which each face, instead of being a square, is a Rubik’s Cube. Does such a thing exist? Not in real life (the whole dimension thing makes a 4D object difficult to build) but on the Internet all things are possible. Yes, Don Hatch, Melinda Green, and Jay Berkenbilt created/maintain a Magic Cube 4D Applet. But it doesn’t stop there! Roice Nelson “[i]n the spirit of taking things too far” created MagicCube5D in which each face is a Rubik’s Hypercube. And if that doesn’t satisfy you, you can join the 4D Cubing Group on Yahoo (which, in the 7 days before this post, had 2 new members and 5 new messages).

The idea for this Juxtaposition came from Peter Baéz while he was researching the hypercube for a class.

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8 Responses to “Juxtapositions: The Rubik’s Hypercube”

  1. TwoPi Says:

    Back in the day, I had enough nightmares about attempting to solve an actual Rubik’s Cube in my hands. I can’t image what horrors my subconscious would have conjured up had it gotten the idea that a RC could dance and sing and talk to me as I fumbled to unscramble it.

    I sense a Clive Barker movie.

  2. Batman Says:

    Is it sad that I remember this cartoon (without nightmares)?

    Of course, that begs the question: Did you remember this cartoon, or just stumble across it in your research?

  3. Ξ Says:

    I never knew such a thing existed! Early 80s, I might not have been watching Saturday cartoons as often…although I do remember the boy band Menudo, who sang the theme song according to Wikipedia.

  4. infinity Says:

    The Rubiks Cube originated in Hungary! The national championship was in Budapest in October! One of the guys here competed in it. There were people that could do it with their feet! INSANE!

  5. Batman Says:

    The current world record for solving the cube is 9.86 seconds, set in Spain in 2007. Many more records here, including blindfolded, one-handed, and foot-ed.

  6. Golden Ratio Says:

    I’ve tried working with the 4D Applet and it is remarkably difficult. Another one of the properties of a four-dimensional hypercube is that it consists of eight three-dimensional cubes that are all interconnected. So the way the applet looks when its solves is that there are eight cubes divided into twenty seven cubes each and all twenty seven cubes in each larger cube are all the same color. There’s just one catch; you can’t see the eighth cube!
    What people see when they first open up the applet is one cube in the middle with a six additional cubes set next to it, one to a face. However, once they start turning it, they’ll notice that some of the cubes that have been rotated now have a new, eighth color that was not originally present when they started. This presents a new challenge that can make the thought of solving the puzzle quite daunting because you would have to learn how to manipulate the puzzle when you aren’t even able to see everything at once.
    The fun doesn’t end there, however. It’s rather mind-bending at first when you start to work on the hypercube. The way the applet operates is that you click on one of the one hundred and eighty nine visible cubes and the applet will perform a rotation of the hypercube that is the equivalent of one quarter turn of a three dimensional Rubik’s cube (and consequently, if you click on the same cube four times you’ll be back where you started). However, instead of just the side of a cube turning, multiple cubes will change sides with each other. Furthermore, the number of cubes changing sides is different for cube you click on. For some it’s four, others two, and still others seven. All in all, it’s remarkably hard to solve a four-dimensional Rubik’s cube and I’ve no idea where to begin.
    (And don’t get me started on the five-dimensional Rubik’s cube. I love math as much as the next math major, but it’s turned into a sickness at that point. I mean, honestly, the only thing worse is that some people have actually solved it.)

  7. Carnival of Mathematics #21: Bar-hopping at last « Secret Blogging Seminar Says:

    […] to a friendly round of Prisoner’s Dilemma, you may have missed your chance), and one on The Rubik’s Hypercube. As if the original wasn’t hard […]

  8. Lovely Exponent Says:

    the eight cubes of a hypercube are ‘interconnected’ It hurts my brain to try and comprehend such an object!

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