We’re still doing production on the little walking guy. But meanwhile my collaborator Eva Schindling and I managed to build a nice little working zoetrope of a tumbling hypercube.
For those of you who don’t know, a hypercube is something like a cube, only with four dimensions instead of three. As a cube is to a square, a hypercube is to a cube. I made a java applet some years back that lets you play with them.
Of course we don’t live in four dimensions, so it’s hard to get a feeling for what happens when you rotate four dimensional things. Eva and I thought that it would be cool, rather than looking at 4D things rotating on a computer screen, to create an animated sculpture of a rotating hypercube. That way you could look around it from all directions (at least, all 3D directions) as it does its weird 4D rotation.
Because a hypercube has four dimensions instead of three, it can rotate in some pretty fancy ways. A simple rotation only requires two dimensions. Since a hypercube has four dimensions, it can rotate one way in two of its dimensions, while rotating a different way in the other two dimensions.
In our zoetrope, we made our little hypercube tumble around a circular track (a movement that uses two dimensions), while also rotating a different way in the remaining two dimensions.
Now, this is not going to look very intuitive to us poor 3D humans. In fact, it looks as strange to us as, say, a rotating cube would look to a Flatland creature that lives its entire life in a two dimensional world.
Such a 2D creature couldn’t really see a cube — but it could see the shadow a cube makes if its silhouette is projected into the two dimensional world of Flatland:
To the Flatland creature staring at the shadow, it wouldn’t look like something rotating so much as something becoming distorted in all sorts of weird ways.
And that’s pretty much what happens when you try to make sense of a rotating hypercube. As things rotate out of our little 3D world, they look like they are changing size and shape, rather than rotating.
But see for yourself:
My brain is struggling to understand why it looks as if the cubes are turning horizontally AND vertically at the same time?? I will show this to my daughter this afternoon; she will understand it 🙂 Are these cubes 3D printed? I found this on YouTube–you are doing something like this, in 4D?!
http://www.youtube.com/watch?v=N3bDOc3Jbg4&NR=1
Yes, they are 3D printed, and it is a zoetrope, like the one in that Youtube video.
The reason the hypercubes are turning both horizontally and vertically is that they are four dimensional — so there are two dimensions for vertical turning, and another two dimensions for horizontal turning.
Of course we can only see this 4D rotation in 3D (since our world is 3D), but having the rotating 4D hypercube as a physical sculpture in 3D is much cooler than only having it as an image on a 2D screen.
what does your zoetrope setup look like? It looks like you don’t have the classic cylinder with slits spinning around with the turntable. How do you get the frames lined up otherwise?
We have several set-ups. For running it live, we use a flashing LED light. Metal strips along the bottom of the zoetrope create short intermittent electrical contacts, so that the LED will flash at the proper time.
For the set-up in the video, I manually positioned the piece into 13 differently rotated positions, took a photo at each position, and assembled those frames into a movie.
Love your NYU applet! Will have Léa try today.