15 August 2007

visualizing higher dimensions

Jon Shock writes at SciTalks about visualizing things in four dimensions, although it's actually less about visualization than about how we don't need to visualize so long as we can calculate -- although he does provide a link to a youtube video showing how one can obtain the projection of a hypercube onto a plane.

I've never been good at visualizing things in more than three dimensions (actually, to tell the truth, I have trouble visualizing things in my head in more than two dimensions; to visualize anything successfully in three dimensions I have to use my hands and point at things in the empty space in front of me). But a guy I knew in college claimed to be able to visualize things in five dimensions fairly easily. The fourth dimension was time -- so a four-dimensional sphere, for example, can be thought of as a three-dimensional sphere which grows from a point to a large ball and then shrinks back again. The fifth dimension was color, although I never really understood this (he wasn't very good at explaining it); I think the idea was that, say, red would correspond to a large coordinate in the "color" dimension and blue would correspond to a small coordinate, similar to weather maps showing temperature. So like weather maps allow us to visualize a surface in three-dimensional space with a two-dimensional picture, you can get an extra dimension this way (or even more than one! -- the space of colors is usually thought of as three-dimensional -- although I can't imagine really taking advantage of that).

I suppose a lot of this visualization ability comes with practice, though, and I rarely have any reason to think in more than three dimensions. I can definitely see it coming with practice when I teach calculus students about solids of revolution; at first they just don't seem to get it but eventually they seem to figure it out. The difference there is that I can tell them to draw a picture of the solid in question, and they draw a two-dimensional picture of a three-dimensional thing, which is something we're all used to; drawing a two-dimensional picture of a four-dimensional thing that doesn't throw away all the useful information is a lot more difficult, to the point where perhaps you really have to understand what the thing looks like before you can draw the picture. So they're useful for explaining results to other people, but not so much for coming up with new results.

Max said...

for the fifth dimension (or fourth, if you want to dispose of time), instead of colour, you could start with luminance (or brightness), which might be slightly easier to visualize. It's at least not too hard to visualize a shape moving through the 'brightness' dimension. On the other hand... it's much harder to visualize something that isn't a point in the brightness dimension... you have to imagine something is light and dark at the same time, or some continuum. (For example, it's easy to visualize a line that goes through the brightness dimension, from dark to light, but much harder to visualize something that's parallel to the brightness dimension, since it's dark-to-light.

*rambles*

Isabel said...

I agree with using brightness. (The other two dimensions of color -- hue (which is standard in, say, weather maps) and saturation -- are a lot harder to work with visually.) Brightness has the same problems as hue, though, in that you can't visualize a non-point. Being red and green at the same time, or light and dark at the same time, is a problem.

This is okay with weather maps because what we're visualizing is the graph of a function, whose domain is the points on the ground and whose range is possible temperatures; if temperature were some sort of multivalued function weather maps would be tricky. (This makes me wonder -- how do the people who make topographic maps handle places where a cliff actually overhangs the land under it, so there are two elevations at the same 2-D point?)

Jonathan Shock said...

Hi Isabel,

If you want to use time as the fourth dimensions then it's really very easy. A ball living in four dimensions is just a ball sitting there in front of you for however long you want (of course it could be moving about to).

Sometimes it's good to try and draw time as a direction on a graph, but in terms of visualising directions, I'd say we already have a pretty good faculty for understanding time, just sit there and wait!

In fact the idea of the ball growing and shrinking really comes from having a four dimensional ball moving in the fourth spatial dimension (so in fact it's living in five dimensions already, if you include time). As the ball intersects our three dimensional surface we will see it increasing in size and then shrinking.

We can visualise this more easily by imagining how people living on a two dimensional piece of paper would see a three dimensional ball pass through their world. They would see cross sections of the object growing and then shrinking.

On the subject of using colour or luminance as a visualisation tool, there are some shortcomings. It's quite possible to have a direction which is compact, like a circle, and if you try and do this with luminance you will find that at some point in your extra compact dimension, your luminance is double valued.

when you start using colour or brightness, you are really trying to attach something you can see to a number (the distance in the direction), I would consider that as long as you can understand the shortcomings of your imagination by seeing how we would be stumped by the third direction if we lived in two, then the language of mathematics is as elegant a construction as you will ever need.

One may be able to attach metaphors for extra dimensions in the form of visualisable labels, but I'd claim that this isn't a true visualisation of higher dimensional spaces - I don't claim to be able to really see extra dimensions either.

I hope that's of some use.

J

Scott Carter said...

Isabel,

I am in the 4-d visualization business. Sometimes I am not good at it, but I want to point you to a link:
http://tonyrobbin.home.att.net/
of an artist who explicitly uses 4-d in his work.

Also, here is a puzzle for you. If
$\int_0^1 x^3 dx = 1/4$, then what is it 1/4 of?
The answer is that it is 1/4 of the volume of a 4-d cube since the integral represents the 4-d volume of the cone on a cube. In general, $\int_0^1 x^n dx = 1/(n+1)$ is the $n+1$-dim'l volume of the cone on an n-cube. See
http://arxiv.org/abs/math/0608722
for further details.

My own higher dim'l explorations occured as a child when my mom would have me use a protractor to draw the complete graph on n-vertices --- that will shut a kid up for a while!
Little did I know that I was drawing the 1-skeleton of the (n-1)-simplex.

Similarly, hypercubes are combinatorial objects, and computations with the binomial theorem
are expressions of this.

Exercise. Consider the drawing of 3-space that sends the x-axis to
(-1,-2), the y-axis to (2,0), and the z-axis to (0,2) --- standard blackboard drawing of a coordinate system. Compute the kernel of the matrix:
[-1,2,0]
[-2,0,2]
Then trace the vector (1,1/2,1) in your drawing. You are back at the origin.

A nice projection of
4-space onto the plane is given by the matrix
[0,7,2,3]
[6,0,2,-4]
Try explicitly with a ruler drawing this out. Notebook paper helps with the underlying coordinates. Compute the kernel, and observe the 2-dimensions that land at the origin.