Hypercubes

(Rudy Rucker is a guestblogger. His latest novel, Hylozoic, describes a postsingular world in which everything is alive.)

Mark Frauenfelder mentioned that the first time he saw me I was carrying a cardboard model of an "unfolded hypercube"---so I rooted around the house and, with my wife's help, found such a model, this one dates back to 1983.

boinghypercubemodel.jpg

I think it was the hyperdimensional mathematician Tom Banchoff who told me how to make this model. You cut out 28 cardboard squares and tape them, four at a time, to make seven partial cubes. These partial cubes have no tops or bottoms, they're like square tubes. And then you tape the seven partial cubes together as shown in the photos, making a very cool shape.

[You may notice that the shape is a bit like a hyperdimensional crucifix. Indeed, if I'm not mistaken, the artist Salvador Dali actually consulted with Banchoff when Dali did his well-known painting, Crucifixion (Corpus Hypercubus).]

It's a universal joint, in that you can swivel the top part freely---which is a little surprising as its all made of straight hinges. To make this into a real hypercube, you'd fold the band of projecting cubes to match the top cube and then---this is the hard part---you "fold" the bottom cube so its faces stretch around the outsides of the other cubes. The Wikipedia Tesseract page has a little animation that helps you visualize how this might work.

The idea behind the coloring on my model is that you start with one corner of the hypercube gray, and you think of the four dimensions as adding the colors red, blue, yellow, and white, coloring the successive 15 corners accordingly. I got the coloring pattern on this model from the early mathematician and science-fictioneer Charles Howard Hinton. In 1980 I edited a book of Hinton's amazing writings, Speculations on the Fourth Dimension. You can find a lot of this book (minus my introduction) online for free.

The letters on my model have to do with the fact that this particular unfolded hypercube was a gift to my wife on our sixteenth wedding anniversery. Our family members are S, R, G, R', and I---so you can start with S and think of each of the four dimensions as adding an R, G, R', and/or I.

boingdogfeathers.jpg

One way to study a cube is to slice it into 2D cross-sections, taking the slices at various angles. By the same token, you can study hypershapes by slicing them into 3D cross-sections. Mark Newbold has written a nice Java applet called "HyperSpace Polytope Slicer" that lets you look at 3D cross-sections of four-dimensional shapes (or polytopes) like the hypercube. To use this applet, go to the link and click on the Controls button to get some interactive controls you can play with. Click on View to switch from a double view to a single view. (And better not click on Detach---at least on my machine, that often freezes up the applet.)

If you crave still more hypercube fun, I have two Windows progams written by my Master's Degree students, available for free download...including a 4D "Hyperspace Invaders" game.

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  1. OK, now this…this is just unbelievably cool. Gotta try that slicer, to see if the speculations I made with my friends about a hypercube passing corner-on through our 3D universe were accurate.

    I have a lot of trouble thinking in 4D, so I usually try to “scale up” from thinking in 3D. This works part of the time, but I don’t have enough math to recognize when it fails.

  2. [You may notice that the shape is a bit like a hyperdimensional crucifix. Indeed, if I’m not mistaken, the artist Salvador Dali actually consulted with Banchoff when Dali did his well-known painting, Crucifixion (Corpus Hypercubus).]

    I have a print of that painting hanging in my bathroom (my wife did it, and I’m not sure why, save for the fact that she’s a Christian with a major in physics and a minor in math), and this was the first thing I thought about when I saw the picture.

    Very cool.

  3. Cardboard is all well and good; just make sure you don’t build a house that looks like that.

  4. Carl Sagan did a nice little segment about dimensions and visualizing hypercubes. It helped me understand the concept a bit better.



  5. I’m just thinking: “If you gave that to your wife as an anniversary gift, and she not only got it, but loved it…BOY, did you marry the right woman for you!” :)

  6. You can build a habitrail out of these things, but sometimes your gerbils end up inside out.

  7. Rudy – Back when I was a nerdy, nerdy teenager, it was a library copy of your book on the 4th dimension that totally introduced me and my best friend to endless imaginings of new and wonderful things. I think I must have drawn and made a zillion models of hypercubes due to your inspiration. Thanks!

  8. Is that Hyperspace Polytope slicer a gas, or what?

    Got this compulsive urge to skip back to it and explore some more…

  9. Oh!

    Rudy, can you comment on how you came across the “Necker Cube” which was described in your book “The Fourth Dimension”?

    Man, I constructed one as per your directions, and my high-school mind was blown. Really enjoyed that book!

    But seriously, how did you come across the Necker Cube?

  10. It’s a universal joint, in that you can swivel the top part freely—which is a little surprising as its all made of straight hinges

    It would be very surprising if the swiveling weren’t due to the flexibility of the paper and the stretchiness of the tape. If the hypercube were made of wood or metal, it would presumably be able to rock back and forth in one dimension at a time, and that’s it.

  11. If someone got me this for our sixteenth wedding anniversary, I would marry him.

    Involves some sort of time-travel paradox, apparently.

    Am imagining wedding cake shaped like a “Crooked House.”

  12. [You may notice that the shape is a bit like a hyperdimensional crucifix. Indeed, if I’m not mistaken, the artist Salvador Dali actually consulted with Banchoff when Dali did his well-known painting, Crucifixion (Corpus Hypercubus).]

    Doubtful, since Mr. Dali painted that in 1954 (per your link to Wikipedia).

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