Directions

So picture a space with seven labelled directions. (As shall be clear, I am not quite sure how many dimensions it really has.) We shall use the names north, south, east, west, aust, up, and down. (I am told the direction name aust comes from Rats and Gargoyles, by Mary Gentle (though I haven't read it myself, and don't know what properties it has therein.)

We shall refer to up and down as the vertical directions, north and south as the polar directions, and east, west, and aust as the parapolar directions. Each polar direction is perpendicular to each parapolar direction, and the vertical directions are also perpendicular to all the rest.

Let `n`, `s`, `e`, `w`, `a`, `u`, and `d` be operators, meaning to move one unit in the corresponding direction. The space and directions are such that these operators have the following properties:

• Associativity and commutativity. (They are generators for an abelian group.)
• `ns = ewa = ud = 1`.

So far, so banal. There's nothing terribly interesting here yet, as this space has a simple model that can be reasonably embedded in `RR^4`: a lattice of unit equilateral triangles in the `xy` plane, duplicated along a lattice of unit squares in the `zw` plane. (It may be easier to picture it without the vertical directions; that will fit in `RR^3`, since it's just a stack of triangle lattices.) An east-west-aust trip runs around a triangle in the lattice, returning you to your starting point.

Now for the fun part:

• When a human stands up and faces aust, and raises her hands up from her sides to point away from her, one hand will point north, and one hand will point south.
• When a human stands up and faces north, and raises her hands up from her sides to point away from her, one hand will point east, one hand will point west, and one hand will point aust.