> 4D Maze Game

  The Idea
  In the Maze
> Notes

  The Fourth Dimension
  Can You See It?
  The Tesseract
  The Tesseract, Part Two
  The Hexadecachoron
  How Much Space Is There?
  How to Point
> How to Orient Yourself
  Walls Are Opaque
  Some Mathematics

How to Orient Yourself

By now you're surely familiar with the eight directions: forward and backward, left and right, up and down, and inward and outward. I've used these directions to describe many things, and yet somehow I've managed never to mention the fact that they are relative to the observer.

The maze world has its own eight directions, which are absolute: north and south, east and west, zenith and nadir, and two that shall here remain nameless. It is these absolute directions that determine the wall colors when the color mode is set to by-direction.

An orientation is a correspondence between the relative and absolute directions.

Now, although we live in a three-dimensional world, in many ways we think like two-dimensional people. In particular, I suspect we all have the idea that an orientation is nothing more than a single direction. Wrong! Imagine, if you will, that you're standing up straight, with your arms out to your sides, and imagine that you're lifted into the air by an invisible force and can choose to turn any which way. If you choose to face north, your orientation still isn't set—you can turn to point your feet east or west, or toward zenith or nadir.

The same argument can actually be used to count orientations. You can face in any of six directions, and can then turn to point your feet toward any of the remaining four, so there are 6 × 4 = 24 possible orientations. Similarly, in the four-dimensional maze world, there are 8 × 6 × 4 = 192 possible orientations!

Clearly, we are going to need some way to deal with all those orientations.

One possibility is to continue to think like two-dimensional people. In the real world, when we have to move up or down, we almost always do one of two things: move without changing orientation, as in an elevator (i.e., slide); or change orientation, move, and immediately change back, as on a ladder. In the game, the exact same methods can be used … and they can be used not only to move up and down but also to move in and out.

By the way, you can use the color options to help keep you oriented. If you set the number of same-color dimensions to two, and the color mode to interior, and then orient yourself properly at the start of the game, the maze will be like a building with the floors painted different colors … except that the floors will form a two-dimensional array instead of a linear sequence.

Another possibility is to think like three-dimensional people and apply the methods above to only one pair of directions. If we use the methods to move in and out, the maze will be like a sequence of three-dimensional spaces; if we use them to move up and down, the maze will be like a building with three-dimensional floors. (Of course the two amount to the same thing.)

The possibility I'm most interested in, of course, is to think like four-dimensional people. I can't tell you how to do that, because I don't know, but I do have a few hints I can give you.

The first hint is that four-dimensional people living on a four-dimensional planet would pretty much be stuck on the three-dimensional surface (a big 3-sphere), and would in many ways think like three-dimensional people. So, if you can think like a three-dimensional person and get your head around the idea of a building with three-dimensional floors, that is probably a good start.

The second hint is related to the first. I'm pretty good at thinking like a three-dimensional person, or so I imagine, but I do have one serious two-dimensional limitation that I've never been able to get past: if I turn upside down, I get completely disoriented. I can turn until up (relative) becomes horizontal (absolute), and maybe even slightly past that, but if I turn too far, my mental image of the three-dimensional world around me just shuts down.

So what's the point? The point is, inside my mind, zenith is the preferred upward direction; and while having a preferred direction may not be ideal, it is apparently a useful simplification that we could consider making intentional use of. In four dimensions, we would probably want two preferred directions, one for upward and one for inward.

By the way, if you want to practice thinking like a three-dimensional person, you can use the three-dimensional mode of the maze game. I myself have “practiced” mostly in the game Descent and its sequels. I think scuba diving would work, too, but I've never tried it.

The third hint is in fact the whole reason this page exists, so wake up! Remember when we were counting orientations a minute ago? Did you notice that I cleverly neglected to spell out the four-dimensional argument? Here's how it goes.

You can face in any of eight directions, can then turn to point your feet toward any of the remaining six, and finally … hmm.

The problem is, you're not done orienting yourself, but because your body is more or less two-dimensional, you've run out of things to turn. So, clearly we need to do something about that.

First, though, we need to change terminology, because in a four-dimensional world, a two-dimensional object doesn't face in any direction—what radiates orthogonally is not a line but a plane. So, imagine orienting yourself starting not with the forward and down directions but with, say, the up and right directions, that is, with your head and right arm.

Then, to get one more thing to orient with, imagine turning your left arm (at the shoulder) so that it points inward. If you can do that, you can then hold yourself rigid and use your ability to turn any which way to point your left arm toward any of the remaining four absolute directions, while your head and right arm rotate in place. Quite a trick, eh?

For myself, since my left and right are no longer symmetrical, I want to think of my left hand as being something halfway between a hand and a head … so I imagine I'm wearing a sock puppet. I decided it ought to be a dragon, dark green with yellow eyes, and then of course it had to be named Ollie.

Thinking of yourself as having a head, a hand, and an ollie is good for more than just orienting yourself, it also allows you to visualize any rotation in terms of how your body moves. If you turn so your hand moves forward, that's a left turn; and if you turn so your ollie moves forward, that's an outward turn.

In particular, you can visualize any spin as a physical motion. If you turn so your head moves toward or away from your hand, you get the two familiar three-dimensional spins; and if you turn so your ollie moves toward or away from your hand or head, you get the four not-so-familiar spins. The default spin keys were chosen with this image in mind. The I key, for example, indicates an upward direction, so, as you'd expect, it turns your ollie toward your head.

The default change keys for the tilt angles were chosen with a different image in mind. The up arrow, for example, moves the outer face of the retina upward.

Now, turning your ollie toward your head and moving the outer face of the retina upward are really two completely different things. One is a physical motion of your four-dimensional body, the other just a change of convention regarding your three-dimensional retina. However, the two do produce similar effects on screen. This is not entirely a coincidence, and provides another way of thinking about the not-so-familiar spins.



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