severoon's Journal: Visualize the Fourth Dimension

Hey all,

I've been busy working lately for an anti-spam company (I'm fighting for the good guys!), so it's been a while since I posted. I've been reading some maths books lately and I thought I'd post an interesting insight I had lately.

Ever thought about the fourth dimension? Not time, I'm talking the fourth spatial dimension. Well, I assume so cuz you're reading slashdot. Ever thought about how to visualize the fourth dimension? I have...unsuccessfully for the most part. But no more. I now have access to the color axis. What is the color axis? Let me 'splain.

Two objects cannot share the same space at the same time, more specifically no point in space may be occupied by more than one object at a time (quantum mechanics not withstanding). Let's restrict the experimental space to a plane for easy visualization. If you take two coins and lay them on the table, you cannot push them into each other such that they occupy the same part of the table surface. How do you know that this is what you're attempting to do? Easy, you can map out the boundaries of these coins in the x-y plane.

Now let's bring the third dimension into it, but not completely. By this I mean, let's assume that your perception is limited to the table surface, and you're trying to contemplate the third dimension even though you're not capable of perceiving a direction that's perpendicular to the two with which you're already familiar. This is where the color axis comes into play. Let's allow you to move the coins into the third dimension if you like, and establish the following convention: the more below the table, the more blue the object, the more above, the more red the object. Let's position the plane at the far end of this spectrum at the blue end (just for fun--you don't have to if you have need to move things down below the plane).

So you're looking at two blue coins. You push one along the color axis in the red direction, and it gets purple. If you keep pushing, it'll get red, but you don't need to go too far, just enough to get it up off the table. At this point, you push the coins into each other and you're surprised to see that they pass seemingly through each other. But, upon reflection, you realize that in three-space, they're not in the same space--they're in different planes because one is red and one is blue.

Neat, huh? Now imagine a knotted rope in three dimensions, with a simple granny knot. To undo this knot, given a fourth spatial dimension, you do not need to pull it apart the normal way. You can simply grab an overpass (that's the part of the rope that goes the underpass, the other part of the rope that forms the knot), pull it into the fourth dimension until it grows red (the rest of the rope is blue). At this point, you simply pass it through the underpass--they're not in the same space because they're different colors, remember? Then, once you've got it underneath the underpass, you pull it back along the fourth dimension until it's blue again like the rest of the rope. Ta-da. Knot undone.

That's neat-o. How come in high school when I was in all those advanced math classes looking at 386 programs simulating a rotating hypercube, no one ever thought of using color when you run out of dimensions? Even on a 2D monitor surface, this could make things a lot clearer.

How do you make a line? You drag a point along a path, leaving a trail to its original spot. A square? Drag that line perpendicular to its length, the same distance as its length and it sweeps out a square. A cube? Grab the square and drag it perpendicular to the plane of the square, sweeping out a cube. A hypercube?

Drag the (blue) cube, the length of one of its sides, along the fourth axis, sweeping out a hypercube. Two cubes, one red, one blue, connected by edges that go from blue to purple to red. Leaves me wondering what a hypersphere looks like...