Octacube Sculpture

October 25, 2005 4:29 PM

Octacube Sculpture The stainless-steel Octacube is a striking object of visual art and also a mental portal to the fourth dimension, a teaching tool, and a research object bringing together many branches of mathematics and physics connected to the structure of symmetry.
posted by thecollegefear (27 comments total)

Fourth Dimension
posted by thecollegefear at 4:30 PM on October 25, 2005

It would be great if everyone who views the Octacube walks away with the feeling that being kind to others is a good way to live.

Yeah right a bit of a stretch maybe but why not...
posted by elpapacito at 4:33 PM on October 25, 2005

trippy.
posted by Kifer85 at 4:45 PM on October 25, 2005

Apple plans to make a pocket-size iOctacube.
posted by brain_drain at 4:53 PM on October 25, 2005

It would be great if everyone who views MetaFilter walks away with the feeling that being kind to others is a good way to live.
posted by cleardawn at 5:07 PM on October 25, 2005

YOU ARE EDUCATED STUPID! I HAVE OCTACUBED TIME AND DEFIED THE IGNORANT WORD GOD!
posted by klangklangston at 5:11 PM on October 25, 2005

Somebody really needs to ask the TimeCube guy what he thinks of the Octacube.
posted by nmiell at 5:13 PM on October 25, 2005

I don't know, Octacube is just kind of a stupid name unless its the new Bond villain. Cubes have six sides, an eight sided cube isn't a cube, its something else.
posted by fenriq at 5:17 PM on October 25, 2005

[booming voice] Behold! The GLORY of the OCTACUBE [insert music here]
posted by blue_beetle at 5:21 PM on October 25, 2005

4D Polytope Viewer.

Something peculiar happens with polyhedra in 4 dimensions. "Everybody" knows the five regular polyhedra in 3D - the platonic solids, the tetrahedron, cube, octohedron, dodecahedron, and icosahedron. Certainly anybody that played D&D knows them, anyway. It's high school math (but very interesting high school math) to deduce that these are the only possible regular polyhedra in 3D.

There is, among polyhedra, a "duality" so that every polyhedron has a dual in life, and they pair up this way:
tetrahedron - tetrahedron (it's its own dual)
cube - octohedron (dual pair)
dodeca - icosahedron (dual pair).

The dual pairs all have central symmetry, but the tetrahedron does not (opposite a vertex of the tetrahedron is not another vertex).

In each dimension 5 and higher, there are only three regular polyhedra - the higher dimensional analogue of the tetrahedron, which is self-dual; and the higher dimensional analogues of the cube and octohedron, which are a dual pair. The tetrahedron is not centrally symmetric; the cube and octo analogues are.

However, in dimension four, you have the following wealth of regular polyhedra:
4d tetrahedron (self dual, not centrally symmetric)
4d cube & octo (dual pair, centrally symmetric)
120-cell and 600-cell (dual pair, centrally symmetric; sort of like a 4d analogue of the dodeca- and icosahedrons), and...
the 24-cell - centrally symmetric and self-dual and as such a completely singular object among all dimensions. Fascinating. And that's what this guy is calling the "octacube." But everyone else calls it the 24-cell. It's built out of 24 (3-D) octohedra, in the same way that a normal cube is built out of 6 (2-D) squares as faces.

4 dimensions also marks a breaking point for the complexity of (not necessarily regular) polytopes. It is an old fact that the realization space of any 3d polytope is very simple - topologically, it's just a ball, it's contractible, the simplest kind of space possible. A more recent result is that the realization space of a 4D polytope can have the homotopy type of any finite simplicial complex. In other words, a jump from no complexity at all to as much complexity as you want. Weird.

And of course, 4D Euclidean space is unique among all Euclidean spaces in admitting more than one differentiable structure. And that's the hardest to explain, but truly bizarre.

So what I'm trying to sayis that the 4D - it's weird, and it's wonderful. And that sculpture is beautiful and shiny.
posted by Wolfdog at 5:28 PM on October 25, 2005

Personally, I'd be more impressed if the octocube was made out of butter and/or the octocube was made out of toast and placed next to a buttery darth vader.
posted by gambit at 5:47 PM on October 25, 2005

Wow. Here I was, just content to watch the animation with an old Can album playing, and you went all supra-math.

How is 4d space conceptualized enough to give data on symetries? I assume that at this point, the "shape" becomes simply a formula administered in the same way as forumulas which represent shapes that we can conceptualize.
(Thanks, Wolfdog, that is interesting).
posted by klangklangston at 5:50 PM on October 25, 2005

Well, the cartesian coordinate system lets us think about points in 3D space as ordered triples of real numbers - any point has an (x,y,z) address, and any triple of numbers corresponds to a point in space.

The usual data for describing a cube consists of a list of vertices: (1,1,1), (1,1,-1), (1,-1,1), (1,-1,-1), etc. (Together with a little data about which vertices are connected by edges.) And you're right; once an object is situated concretely in space like that, symmetries are captured by equations - but a better word is transformations. For example, that cube has "reflection across the plane z=0" as a symmetry. You can think of that as a transformation, or a mapping - each (x,y,z) point maps to a new point (x',y',z'), and for the reflection I'm thinking of, the new coordinates are
x'=x
y'=y and
z'=-z.

This mapping qualifies as a symmetry of the cube because every vertex gets mapped to some other vertex, every edge maps onto an edge.

Another symmetry of the cube is "rotation through 90 degrees about the z-axis." In that case, the old point (x,y,z) maps to the new point (x',y',z') determined by equations
x'=-y
y'=x and
z'=z.

There are others, but those are examples of transformations of space that carry the cube to itself, which is what we mean by symmetries of the cube.

Well, 4D is at first glance not so different. It's just that points have a 4-tuple (w,x,y,z) to describe them. If you want to describe a 4D polytope, you give a list of vertices and a little extra data to describe how they're connected to form faces and edges.

The 4D analogue of the cube has 16 vertices of the form (±1,±1,±1,±1). One of its symmetries is "reflection across the hyperplane w=0" which can be written down as
w' = -w
x' = x
y' = y
z' =z.
This is so analogous to the familiar idea of reflection that we just call it reflection even if we can't visualize the entire effect simulatneously.

Another symmetry of the 4D cube is the central symmetry I mentioned before. Its transformation is simply
w' = -w
x' = -x
y' = -y
z' = -z
(For every vertex on the cube, there's a line from that vertex through the origin that passes through another vertex; this transformation simply swaps the ends of those lines).

So, yes: we write these things as equations, and can't really draw or visualize a transformation in its entirety but in 4D many of the moves and transformations - reflection, rotation - are so analogous to 3D things that it's not hard to work by analogy and intuition. Every once in a while the analogy breaks, and that's where interesting things happen.
posted by Wolfdog at 6:32 PM on October 25, 2005

*brain explodes*
posted by TheNakedPixel at 6:35 PM on October 25, 2005

Sorry to get all mathy.
posted by Wolfdog at 6:35 PM on October 25, 2005

ok, so the 4th dimension is just time, right? I mean, we already perceive that. (note: work computer a POS, can't view shockwave file. apologies if I'm saying something dumb.)
posted by shmegegge at 7:03 PM on October 25, 2005

"The fourth dimension" is just another coordinate. So you can use four coordinates to describe the spatial position and time of an event, sure. But you can also use four coordinates to describe the age, height, weight, and girth of a black bear.

A mathematician's Platonic "four dimensional space" is just the set of all 4-tuples of real numbers. For a physicist, events in space-time have a 4-coordinate address that corresponds to some point in 4D space. For a biologist, black bears have a 4-coordinate description that corresponds to a point in 4D space. Really, it's useful precisely because it's an abstraction that is capable of holding data for many different kinds of problems. Saying "the fourth dimension is just time" to me is as strange as saying "the fourth dimension is just girth" or "the fourth dimension is just temperature."
posted by Wolfdog at 7:12 PM on October 25, 2005

I'll have to go see this, the octacube is in my town. It is also in the McAllister building, where I got my ass kicked by ordinary and partial differential equations. Great post.
posted by Fat Guy at 7:13 PM on October 25, 2005

Wolfdog: Does this also have to do with using matrixes to solver mathematical problems?
And I think that the "fourth dimension=time" comes from the idea of length/width/depth extending to something else that all things that exist "exist in."
Anyway, thanks. I have a rudimentary memory of Calculus and really like the concepts of advanced math, but can rarely parse them on my own. Possibly because I tested out of math for my college program years ago, and haven't had any real opportunities for refreshers (I used to be able to do geometry in my head, but am totally lost now in trying to remember just what a sin/cosine tells us... And forget the quadratic, man, that's totally gone).
posted by klangklangston at 7:29 PM on October 25, 2005

Does this also have to do with using matrixes to solver mathematical problems?
Yes, all the transformations I'm talking about are rigid transformations that send lines to lines without bending things. Those can be described by writing down a matrix. We could write down a 3x3 matrix for each of the 24 symmetries of the ordinary 3D cube. That's the heart of things like OpenGL, too, the idea that a matrix of numbers can represent a transformation of an object, rotating it, reflecting it, moving it around, and so on. And the arithmetic of matrices makes it easy to work out the effects of a sequence of transformations executed one after the other.
posted by Wolfdog at 7:38 PM on October 25, 2005

Thanks for all the explanation, Wolfdog. I read the article and I still don't understand how this object is a "shadow" of a 4-dimensional object. I couldn't even follow the "mapping a cube" example. But still your digressions and answers have been fascinating.

I remember building a 3-D tesseract out of straws and clay or something way back when and I was able to grasp how that was a distorted 3-D representation of a 4-D object, and could pretty much imagine such an object. But what this octacube "really" is is beyond me. It's too far out. Or I'm just getting stupider in my old age.

Oh, but as for shmegegge's question about "the fourth dimension is time," it's important not to confuse that with the coordinate stuff Wolfdog was going on about. Unless we assume that time maps perfectly isomorphically onto our spatial dimensions, the "W' coordinate in those examples would not be time. When we say the fourth dimension is time we mean that we live in four dimensions. But this is about a fourth spatial dimension.

To put it another way, if you can imagine (as in Flatland) a world where people live entirely in a 2-dimensional space, like the surface of a piece of paper, they could move around on that surface, moving through time, and thus be living in 3 dimensions. But it wouldn't be the same as living in our spatial reality.
posted by soyjoy at 8:09 PM on October 25, 2005

Rudy Rucker's Spaceland is a fun-but-a-touch-corny Flatland descendant. He does a very, very good job of visualizing a 4th dimension and what 4-dimensional spaces might be like. For example, he adds vinn and vout as new directions in which 'Joe Cube' learns to travel.

In the book, if you take a step or two vinn, you can see a 3D cross-section of our space, provided that you have a spherical retina to see it with. Good stuff.
posted by ulotrichous at 8:18 PM on October 25, 2005

It's kind of funny how that nifty sculpture is just placed down on those doilies like that. Maybe they need to move it periodically to dust under it?
posted by filchyboy at 10:01 PM on October 25, 2005

[Wolfdog's link is good]

I love love love visualization software. I think one of the reasons I started failing out of math classes after calculus was the inability to visualize.
posted by Eideteker at 10:13 PM on October 25, 2005

It looks really nice, but I can't understand a damn thing about it.
posted by paperpete at 2:36 AM on October 26, 2005

Go Wolfdog!!!
posted by warbaby at 9:10 AM on October 26, 2005

right on for the bizarro 4th dimension trip . . . but, art? not on ur life.
posted by gorgor_balabala at 10:04 AM on October 26, 2005